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+2135 0 obj << +/Type /Pages +/Count 6 +/Parent 5908 0 R +/Kids [2130 0 R 2140 0 R 2144 0 R 2154 0 R 2161 0 R 2166 0 R] +>> endobj +2175 0 obj << +/Type /Pages +/Count 6 +/Parent 5908 0 R +/Kids [2171 0 R 2177 0 R 2181 0 R 2185 0 R 2190 0 R 2196 0 R] +>> endobj +2204 0 obj << +/Type /Pages +/Count 6 +/Parent 5908 0 R +/Kids [2200 0 R 2206 0 R 2210 0 R 2215 0 R 2221 0 R 2226 0 R] +>> endobj +2236 0 obj << +/Type /Pages +/Count 6 +/Parent 5909 0 R +/Kids [2232 0 R 2238 0 R 2244 0 R 2249 0 R 2255 0 R 2260 0 R] +>> endobj +2270 0 obj << +/Type /Pages +/Count 6 +/Parent 5909 0 R +/Kids [2265 0 R 2272 0 R 2278 0 R 2284 0 R 2288 0 R 2294 0 R] +>> endobj +2305 0 obj << +/Type /Pages +/Count 6 +/Parent 5909 0 R +/Kids [2302 0 R 2307 0 R 2314 0 R 2321 0 R 2325 0 R 2330 0 R] +>> endobj +2341 0 obj << +/Type /Pages +/Count 6 +/Parent 5909 0 R +/Kids [2334 0 R 2343 0 R 2348 0 R 2353 0 R 2359 0 R 2363 0 R] +>> endobj +2372 0 obj << +/Type /Pages +/Count 6 +/Parent 5909 0 R +/Kids [2368 0 R 2374 0 R 2381 0 R 2386 0 R 2394 0 R 2398 0 R] +>> endobj +2406 0 obj << +/Type /Pages +/Count 6 +/Parent 5909 0 R +/Kids [2403 0 R 2408 0 R 2412 0 R 2417 0 R 2422 0 R 2427 0 R] +>> endobj +2436 0 obj << +/Type /Pages +/Count 6 +/Parent 5910 0 R +/Kids [2432 0 R 2438 0 R 2442 0 R 2446 0 R 2450 0 R 2455 0 R] +>> endobj +2467 0 obj << +/Type /Pages +/Count 6 +/Parent 5910 0 R +/Kids [2460 0 R 2469 0 R 2474 0 R 2478 0 R 2483 0 R 2488 0 R] +>> endobj +2497 0 obj << +/Type /Pages +/Count 6 +/Parent 5910 0 R +/Kids [2492 0 R 2499 0 R 2503 0 R 2508 0 R 2513 0 R 2518 0 R] +>> endobj +2526 0 obj << +/Type /Pages +/Count 6 +/Parent 5910 0 R +/Kids [2522 0 R 2528 0 R 2533 0 R 2538 0 R 2543 0 R 2548 0 R] +>> endobj +2557 0 obj << +/Type /Pages +/Count 6 +/Parent 5910 0 R +/Kids [2553 0 R 2559 0 R 2565 0 R 2569 0 R 2584 0 R 2594 0 R] +>> endobj +2607 0 obj << +/Type /Pages +/Count 6 +/Parent 5910 0 R +/Kids [2601 0 R 2609 0 R 2614 0 R 2618 0 R 2622 0 R 2627 0 R] +>> endobj +2637 0 obj << +/Type /Pages 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2879 0 R] +>> endobj +2890 0 obj << +/Type /Pages +/Count 6 +/Parent 5912 0 R +/Kids [2883 0 R 2892 0 R 2898 0 R 2903 0 R 2908 0 R 2912 0 R] +>> endobj +2920 0 obj << +/Type /Pages +/Count 6 +/Parent 5912 0 R +/Kids [2917 0 R 2922 0 R 2926 0 R 2935 0 R 2940 0 R 2946 0 R] +>> endobj +2954 0 obj << +/Type /Pages +/Count 6 +/Parent 5912 0 R +/Kids [2951 0 R 2956 0 R 2961 0 R 2965 0 R 2971 0 R 2975 0 R] +>> endobj +2985 0 obj << +/Type /Pages +/Count 6 +/Parent 5912 0 R +/Kids [2981 0 R 2987 0 R 2991 0 R 2996 0 R 3002 0 R 3007 0 R] +>> endobj +3017 0 obj << +/Type /Pages +/Count 6 +/Parent 5913 0 R +/Kids [3013 0 R 3019 0 R 3024 0 R 3028 0 R 3033 0 R 3038 0 R] +>> endobj +3046 0 obj << +/Type /Pages +/Count 6 +/Parent 5913 0 R +/Kids [3042 0 R 3048 0 R 3053 0 R 3059 0 R 3063 0 R 3071 0 R] +>> endobj +3078 0 obj << +/Type /Pages +/Count 6 +/Parent 5913 0 R +/Kids [3075 0 R 3080 0 R 3084 0 R 3089 0 R 3093 0 R 3097 0 R] +>> endobj +3104 0 obj << +/Type /Pages +/Count 6 +/Parent 5913 0 R 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obj << +/Type /Pages +/Count 6 +/Parent 5914 0 R +/Kids [3343 0 R 3349 0 R 3354 0 R 3359 0 R 3364 0 R 3369 0 R] +>> endobj +3378 0 obj << +/Type /Pages +/Count 6 +/Parent 5915 0 R +/Kids [3374 0 R 3380 0 R 3385 0 R 3389 0 R 3396 0 R 3400 0 R] +>> endobj +3409 0 obj << +/Type /Pages +/Count 6 +/Parent 5915 0 R +/Kids [3405 0 R 3411 0 R 3416 0 R 3420 0 R 3425 0 R 3430 0 R] +>> endobj +3437 0 obj << +/Type /Pages +/Count 6 +/Parent 5915 0 R +/Kids [3434 0 R 3439 0 R 3446 0 R 3451 0 R 3456 0 R 3460 0 R] +>> endobj +3469 0 obj << +/Type /Pages +/Count 6 +/Parent 5915 0 R +/Kids [3465 0 R 3471 0 R 3477 0 R 3483 0 R 3488 0 R 3498 0 R] +>> endobj +3506 0 obj << +/Type /Pages +/Count 6 +/Parent 5915 0 R +/Kids [3502 0 R 3508 0 R 3513 0 R 3517 0 R 3521 0 R 3526 0 R] +>> endobj +3537 0 obj << +/Type /Pages +/Count 6 +/Parent 5915 0 R +/Kids [3533 0 R 3539 0 R 3544 0 R 3549 0 R 3553 0 R 3558 0 R] +>> endobj +3568 0 obj << +/Type /Pages +/Count 6 +/Parent 5916 0 R +/Kids [3563 0 R 3570 0 R 3575 0 R 3581 0 R 3591 0 R 3598 0 R] +>> endobj +3607 0 obj << +/Type /Pages +/Count 6 +/Parent 5916 0 R +/Kids [3604 0 R 3609 0 R 3614 0 R 3619 0 R 3626 0 R 3631 0 R] +>> endobj +3640 0 obj << +/Type /Pages +/Count 6 +/Parent 5916 0 R +/Kids [3636 0 R 3642 0 R 3647 0 R 3651 0 R 3656 0 R 3664 0 R] +>> endobj +3673 0 obj << +/Type /Pages +/Count 6 +/Parent 5916 0 R +/Kids [3669 0 R 3675 0 R 3681 0 R 3685 0 R 3690 0 R 3696 0 R] +>> endobj +3707 0 obj << +/Type /Pages +/Count 6 +/Parent 5916 0 R +/Kids [3703 0 R 3712 0 R 3716 0 R 3720 0 R 3726 0 R 3731 0 R] +>> endobj +3742 0 obj << +/Type /Pages +/Count 6 +/Parent 5916 0 R +/Kids [3738 0 R 3744 0 R 3748 0 R 3752 0 R 3757 0 R 3761 0 R] +>> endobj +3769 0 obj << +/Type /Pages +/Count 6 +/Parent 5917 0 R +/Kids [3766 0 R 3771 0 R 3775 0 R 3779 0 R 3788 0 R 3792 0 R] +>> endobj +3800 0 obj << +/Type /Pages +/Count 6 +/Parent 5917 0 R +/Kids [3797 0 R 3802 0 R 3809 0 R 3815 0 R 3821 0 R 3826 0 R] +>> endobj +3833 0 obj << +/Type /Pages +/Count 6 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R 4302 0 R 4306 0 R 4311 0 R 4316 0 R 4320 0 R] +>> endobj +4332 0 obj << +/Type /Pages +/Count 6 +/Parent 5920 0 R +/Kids [4328 0 R 4335 0 R 4340 0 R 4347 0 R 4351 0 R 4355 0 R] +>> endobj +4363 0 obj << +/Type /Pages +/Count 6 +/Parent 5920 0 R +/Kids [4360 0 R 4365 0 R 4370 0 R 4379 0 R 4385 0 R 4391 0 R] +>> endobj +4398 0 obj << +/Type /Pages +/Count 6 +/Parent 5920 0 R +/Kids [4395 0 R 4400 0 R 4406 0 R 4411 0 R 4417 0 R 4421 0 R] +>> endobj +4431 0 obj << +/Type /Pages +/Count 6 +/Parent 5920 0 R +/Kids [4428 0 R 4434 0 R 4438 0 R 4442 0 R 4447 0 R 4451 0 R] +>> endobj +4459 0 obj << +/Type /Pages +/Count 6 +/Parent 5920 0 R +/Kids [4456 0 R 4461 0 R 4466 0 R 4470 0 R 4474 0 R 4478 0 R] +>> endobj +4485 0 obj << +/Type /Pages +/Count 6 +/Parent 5920 0 R +/Kids [4482 0 R 4487 0 R 4491 0 R 4496 0 R 4500 0 R 4505 0 R] +>> endobj +4514 0 obj << +/Type /Pages +/Count 6 +/Parent 5921 0 R +/Kids [4511 0 R 4516 0 R 4521 0 R 4526 0 R 4530 0 R 4534 0 R] +>> endobj +4541 0 obj << +/Type /Pages +/Count 6 +/Parent 5921 0 R +/Kids [4538 0 R 4543 0 R 4547 0 R 4552 0 R 4556 0 R 4560 0 R] +>> endobj +4568 0 obj << +/Type /Pages +/Count 6 +/Parent 5921 0 R +/Kids [4565 0 R 4570 0 R 4575 0 R 4579 0 R 4585 0 R 4589 0 R] +>> endobj +4597 0 obj << +/Type /Pages +/Count 6 +/Parent 5921 0 R +/Kids [4594 0 R 4599 0 R 4605 0 R 4610 0 R 4614 0 R 4618 0 R] +>> endobj +4626 0 obj << +/Type /Pages +/Count 6 +/Parent 5921 0 R +/Kids [4623 0 R 4628 0 R 4632 0 R 4639 0 R 4643 0 R 4647 0 R] +>> endobj +4656 0 obj << +/Type /Pages +/Count 6 +/Parent 5921 0 R +/Kids [4651 0 R 4658 0 R 4663 0 R 4667 0 R 4672 0 R 4679 0 R] +>> endobj +4687 0 obj << +/Type /Pages +/Count 6 +/Parent 5922 0 R +/Kids [4683 0 R 4788 0 R 4893 0 R 4994 0 R 5099 0 R 5202 0 R] +>> endobj +5329 0 obj << +/Type /Pages +/Count 6 +/Parent 5922 0 R +/Kids [5326 0 R 5417 0 R 5520 0 R 5626 0 R 5731 0 R 5756 0 R] +>> endobj +5764 0 obj << +/Type /Pages +/Count 6 +/Parent 5922 0 R +/Kids [5760 0 R 5766 0 R 5770 0 R 5774 0 R 5778 0 R 5782 0 R] +>> endobj +5789 0 obj << +/Type /Pages +/Count 3 +/Parent 5922 0 R +/Kids [5786 0 R 5791 0 R 5795 0 R] +>> endobj +5907 0 obj << +/Type /Pages +/Count 36 +/Parent 5923 0 R +/Kids [1715 0 R 1758 0 R 1896 0 R 1938 0 R 1974 0 R 2003 0 R] +>> endobj +5908 0 obj << +/Type /Pages +/Count 36 +/Parent 5923 0 R +/Kids [2037 0 R 2064 0 R 2096 0 R 2135 0 R 2175 0 R 2204 0 R] +>> endobj +5909 0 obj << +/Type /Pages +/Count 36 +/Parent 5923 0 R +/Kids [2236 0 R 2270 0 R 2305 0 R 2341 0 R 2372 0 R 2406 0 R] +>> endobj +5910 0 obj << +/Type /Pages +/Count 36 +/Parent 5923 0 R +/Kids [2436 0 R 2467 0 R 2497 0 R 2526 0 R 2557 0 R 2607 0 R] +>> endobj +5911 0 obj << +/Type /Pages +/Count 36 +/Parent 5923 0 R +/Kids [2637 0 R 2672 0 R 2704 0 R 2740 0 R 2771 0 R 2800 0 R] +>> endobj +5912 0 obj << +/Type /Pages +/Count 36 +/Parent 5923 0 R +/Kids [2831 0 R 2860 0 R 2890 0 R 2920 0 R 2954 0 R 2985 0 R] +>> endobj +5913 0 obj << +/Type /Pages +/Count 36 +/Parent 5924 0 R +/Kids [3017 0 R 3046 0 R 3078 0 R 3104 0 R 3133 0 R 3159 0 R] +>> endobj +5914 0 obj << +/Type /Pages +/Count 36 +/Parent 5924 0 R +/Kids [3188 0 R 3215 0 R 3247 0 R 3285 0 R 3315 0 R 3347 0 R] +>> endobj +5915 0 obj << +/Type /Pages +/Count 36 +/Parent 5924 0 R +/Kids [3378 0 R 3409 0 R 3437 0 R 3469 0 R 3506 0 R 3537 0 R] +>> endobj +5916 0 obj << +/Type /Pages +/Count 36 +/Parent 5924 0 R +/Kids [3568 0 R 3607 0 R 3640 0 R 3673 0 R 3707 0 R 3742 0 R] +>> endobj +5917 0 obj << +/Type /Pages +/Count 36 +/Parent 5924 0 R +/Kids [3769 0 R 3800 0 R 3833 0 R 3862 0 R 3895 0 R 3929 0 R] +>> endobj +5918 0 obj << +/Type /Pages +/Count 36 +/Parent 5924 0 R +/Kids [3959 0 R 3993 0 R 4023 0 R 4051 0 R 4079 0 R 4107 0 R] +>> endobj +5919 0 obj << +/Type /Pages +/Count 36 +/Parent 5925 0 R +/Kids [4142 0 R 4174 0 R 4203 0 R 4235 0 R 4266 0 R 4300 0 R] +>> endobj +5920 0 obj << +/Type /Pages +/Count 36 +/Parent 5925 0 R +/Kids [4332 0 R 4363 0 R 4398 0 R 4431 0 R 4459 0 R 4485 0 R] +>> endobj +5921 0 obj << +/Type /Pages +/Count 36 +/Parent 5925 0 R +/Kids [4514 0 R 4541 0 R 4568 0 R 4597 0 R 4626 0 R 4656 0 R] +>> endobj +5922 0 obj << +/Type /Pages +/Count 21 +/Parent 5925 0 R +/Kids [4687 0 R 5329 0 R 5764 0 R 5789 0 R] +>> endobj +5923 0 obj << +/Type /Pages +/Count 216 +/Parent 5926 0 R +/Kids [5907 0 R 5908 0 R 5909 0 R 5910 0 R 5911 0 R 5912 0 R] +>> endobj +5924 0 obj << +/Type /Pages +/Count 216 +/Parent 5926 0 R +/Kids [5913 0 R 5914 0 R 5915 0 R 5916 0 R 5917 0 R 5918 0 R] +>> endobj +5925 0 obj << +/Type /Pages +/Count 129 +/Parent 5926 0 R +/Kids [5919 0 R 5920 0 R 5921 0 R 5922 0 R] +>> endobj +5926 0 obj << +/Type /Pages +/Count 561 +/Kids [5923 0 R 5924 0 R 5925 0 R] +/CropBox[0 15 378 566] +>> endobj +5927 0 obj << +/Type /Outlines +/First 7 0 R +/Last 1599 0 R +/Count 48 +>> endobj +1703 0 obj << +/Title 1704 0 R +/A 1701 0 R +/Parent 1599 0 R +/Prev 1699 0 R +>> endobj +1699 0 obj << +/Title 1700 0 R +/A 1697 0 R +/Parent 1599 0 R +/Prev 1603 0 R +/Next 1703 0 R +>> endobj 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+/Prev 1655 0 R +/Next 1663 0 R +>> endobj +1655 0 obj << +/Title 1656 0 R +/A 1653 0 R +/Parent 1603 0 R +/Prev 1651 0 R +/Next 1659 0 R +>> endobj +1651 0 obj << +/Title 1652 0 R +/A 1649 0 R +/Parent 1603 0 R +/Prev 1647 0 R +/Next 1655 0 R +>> endobj +1647 0 obj << +/Title 1648 0 R +/A 1645 0 R +/Parent 1603 0 R +/Prev 1643 0 R +/Next 1651 0 R +>> endobj +1643 0 obj << +/Title 1644 0 R +/A 1641 0 R +/Parent 1603 0 R +/Prev 1639 0 R +/Next 1647 0 R +>> endobj +1639 0 obj << +/Title 1640 0 R +/A 1637 0 R +/Parent 1603 0 R +/Prev 1635 0 R +/Next 1643 0 R +>> endobj +1635 0 obj << +/Title 1636 0 R +/A 1633 0 R +/Parent 1603 0 R +/Prev 1631 0 R +/Next 1639 0 R +>> endobj +1631 0 obj << +/Title 1632 0 R +/A 1629 0 R +/Parent 1603 0 R +/Prev 1627 0 R +/Next 1635 0 R +>> endobj +1627 0 obj << +/Title 1628 0 R +/A 1625 0 R +/Parent 1603 0 R +/Prev 1623 0 R +/Next 1631 0 R +>> endobj +1623 0 obj << +/Title 1624 0 R +/A 1621 0 R +/Parent 1603 0 R +/Prev 1619 0 R +/Next 1627 0 R +>> endobj 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1584 0 R +/A 1581 0 R +/Parent 1555 0 R +/Prev 1579 0 R +/Next 1587 0 R +>> endobj +1579 0 obj << +/Title 1580 0 R +/A 1577 0 R +/Parent 1555 0 R +/Prev 1575 0 R +/Next 1583 0 R +>> endobj +1575 0 obj << +/Title 1576 0 R +/A 1573 0 R +/Parent 1555 0 R +/Prev 1571 0 R +/Next 1579 0 R +>> endobj +1571 0 obj << +/Title 1572 0 R +/A 1569 0 R +/Parent 1555 0 R +/Prev 1567 0 R +/Next 1575 0 R +>> endobj +1567 0 obj << +/Title 1568 0 R +/A 1565 0 R +/Parent 1555 0 R +/Prev 1563 0 R +/Next 1571 0 R +>> endobj +1563 0 obj << +/Title 1564 0 R +/A 1561 0 R +/Parent 1555 0 R +/Prev 1559 0 R +/Next 1567 0 R +>> endobj +1559 0 obj << +/Title 1560 0 R +/A 1557 0 R +/Parent 1555 0 R +/Next 1563 0 R +>> endobj +1555 0 obj << +/Title 1556 0 R +/A 1553 0 R +/Parent 1307 0 R +/Prev 1515 0 R +/First 1559 0 R +/Last 1595 0 R +/Count -10 +>> endobj +1551 0 obj << +/Title 1552 0 R +/A 1549 0 R +/Parent 1515 0 R +/Prev 1547 0 R +>> endobj +1547 0 obj << +/Title 1548 0 R +/A 1545 0 R +/Parent 1515 0 R +/Prev 1543 0 R +/Next 1551 0 R +>> endobj +1543 0 obj << +/Title 1544 0 R +/A 1541 0 R +/Parent 1515 0 R +/Prev 1539 0 R +/Next 1547 0 R +>> endobj +1539 0 obj << +/Title 1540 0 R +/A 1537 0 R +/Parent 1515 0 R +/Prev 1535 0 R +/Next 1543 0 R +>> endobj +1535 0 obj << +/Title 1536 0 R +/A 1533 0 R +/Parent 1515 0 R +/Prev 1531 0 R +/Next 1539 0 R +>> endobj +1531 0 obj << +/Title 1532 0 R +/A 1529 0 R +/Parent 1515 0 R +/Prev 1527 0 R +/Next 1535 0 R +>> endobj +1527 0 obj << +/Title 1528 0 R +/A 1525 0 R +/Parent 1515 0 R +/Prev 1523 0 R +/Next 1531 0 R +>> endobj +1523 0 obj << +/Title 1524 0 R +/A 1521 0 R +/Parent 1515 0 R +/Prev 1519 0 R +/Next 1527 0 R +>> endobj +1519 0 obj << +/Title 1520 0 R +/A 1517 0 R +/Parent 1515 0 R +/Next 1523 0 R +>> endobj +1515 0 obj << +/Title 1516 0 R +/A 1513 0 R +/Parent 1307 0 R +/Prev 1487 0 R +/Next 1555 0 R +/First 1519 0 R +/Last 1551 0 R +/Count -9 +>> endobj +1511 0 obj << +/Title 1512 0 R +/A 1509 0 R +/Parent 1487 0 R +/Prev 1507 0 R +>> endobj +1507 0 obj << +/Title 1508 0 R +/A 1505 0 R +/Parent 1487 0 R +/Prev 1503 0 R +/Next 1511 0 R +>> endobj +1503 0 obj << +/Title 1504 0 R +/A 1501 0 R +/Parent 1487 0 R +/Prev 1499 0 R +/Next 1507 0 R +>> endobj +1499 0 obj << +/Title 1500 0 R +/A 1497 0 R +/Parent 1487 0 R +/Prev 1495 0 R +/Next 1503 0 R +>> endobj +1495 0 obj << +/Title 1496 0 R +/A 1493 0 R +/Parent 1487 0 R +/Prev 1491 0 R +/Next 1499 0 R +>> endobj +1491 0 obj << +/Title 1492 0 R +/A 1489 0 R +/Parent 1487 0 R +/Next 1495 0 R +>> endobj +1487 0 obj << +/Title 1488 0 R +/A 1485 0 R +/Parent 1307 0 R +/Prev 1427 0 R +/Next 1515 0 R +/First 1491 0 R +/Last 1511 0 R +/Count -6 +>> endobj +1483 0 obj << +/Title 1484 0 R +/A 1481 0 R +/Parent 1427 0 R +/Prev 1479 0 R +>> endobj +1479 0 obj << +/Title 1480 0 R +/A 1477 0 R +/Parent 1427 0 R +/Prev 1475 0 R +/Next 1483 0 R +>> endobj +1475 0 obj << +/Title 1476 0 R +/A 1473 0 R +/Parent 1427 0 R +/Prev 1471 0 R +/Next 1479 0 R +>> endobj +1471 0 obj << +/Title 1472 0 R +/A 1469 0 R +/Parent 1427 0 R +/Prev 1467 0 R +/Next 1475 0 R +>> endobj +1467 0 obj << +/Title 1468 0 R +/A 1465 0 R +/Parent 1427 0 R +/Prev 1463 0 R +/Next 1471 0 R +>> endobj +1463 0 obj << +/Title 1464 0 R +/A 1461 0 R +/Parent 1427 0 R +/Prev 1459 0 R +/Next 1467 0 R +>> endobj +1459 0 obj << +/Title 1460 0 R +/A 1457 0 R +/Parent 1427 0 R +/Prev 1455 0 R +/Next 1463 0 R +>> endobj +1455 0 obj << +/Title 1456 0 R +/A 1453 0 R +/Parent 1427 0 R +/Prev 1451 0 R +/Next 1459 0 R +>> endobj +1451 0 obj << +/Title 1452 0 R +/A 1449 0 R +/Parent 1427 0 R +/Prev 1447 0 R +/Next 1455 0 R +>> endobj +1447 0 obj << +/Title 1448 0 R +/A 1445 0 R +/Parent 1427 0 R +/Prev 1443 0 R +/Next 1451 0 R +>> endobj +1443 0 obj << +/Title 1444 0 R +/A 1441 0 R +/Parent 1427 0 R +/Prev 1439 0 R +/Next 1447 0 R +>> endobj +1439 0 obj << +/Title 1440 0 R +/A 1437 0 R +/Parent 1427 0 R +/Prev 1435 0 R +/Next 1443 0 R +>> endobj +1435 0 obj << +/Title 1436 0 R +/A 1433 0 R +/Parent 1427 0 R +/Prev 1431 0 R +/Next 1439 0 R +>> endobj +1431 0 obj << +/Title 1432 0 R +/A 1429 0 R +/Parent 1427 0 R +/Next 1435 0 R +>> endobj +1427 0 obj << +/Title 1428 0 R +/A 1425 0 R +/Parent 1307 0 R +/Prev 1347 0 R +/Next 1487 0 R +/First 1431 0 R +/Last 1483 0 R +/Count -14 +>> endobj +1423 0 obj << +/Title 1424 0 R +/A 1421 0 R +/Parent 1347 0 R +/Prev 1419 0 R +>> endobj +1419 0 obj << +/Title 1420 0 R +/A 1417 0 R +/Parent 1347 0 R +/Prev 1415 0 R +/Next 1423 0 R +>> endobj +1415 0 obj << +/Title 1416 0 R +/A 1413 0 R +/Parent 1347 0 R +/Prev 1411 0 R +/Next 1419 0 R +>> endobj +1411 0 obj << +/Title 1412 0 R +/A 1409 0 R +/Parent 1347 0 R +/Prev 1407 0 R +/Next 1415 0 R +>> endobj +1407 0 obj << +/Title 1408 0 R +/A 1405 0 R +/Parent 1347 0 R +/Prev 1403 0 R +/Next 1411 0 R +>> endobj +1403 0 obj << +/Title 1404 0 R +/A 1401 0 R +/Parent 1347 0 R +/Prev 1399 0 R +/Next 1407 0 R +>> endobj +1399 0 obj << +/Title 1400 0 R +/A 1397 0 R +/Parent 1347 0 R +/Prev 1395 0 R +/Next 1403 0 R +>> endobj 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+/Parent 1347 0 R +/Prev 1355 0 R +/Next 1363 0 R +>> endobj +1355 0 obj << +/Title 1356 0 R +/A 1353 0 R +/Parent 1347 0 R +/Prev 1351 0 R +/Next 1359 0 R +>> endobj +1351 0 obj << +/Title 1352 0 R +/A 1349 0 R +/Parent 1347 0 R +/Next 1355 0 R +>> endobj +1347 0 obj << +/Title 1348 0 R +/A 1345 0 R +/Parent 1307 0 R +/Prev 1335 0 R +/Next 1427 0 R +/First 1351 0 R +/Last 1423 0 R +/Count -19 +>> endobj +1343 0 obj << +/Title 1344 0 R +/A 1341 0 R +/Parent 1335 0 R +/Prev 1339 0 R +>> endobj +1339 0 obj << +/Title 1340 0 R +/A 1337 0 R +/Parent 1335 0 R +/Next 1343 0 R +>> endobj +1335 0 obj << +/Title 1336 0 R +/A 1333 0 R +/Parent 1307 0 R +/Prev 1311 0 R +/Next 1347 0 R +/First 1339 0 R +/Last 1343 0 R +/Count -2 +>> endobj +1331 0 obj << +/Title 1332 0 R +/A 1329 0 R +/Parent 1311 0 R +/Prev 1327 0 R +>> endobj +1327 0 obj << +/Title 1328 0 R +/A 1325 0 R +/Parent 1311 0 R +/Prev 1323 0 R +/Next 1331 0 R +>> endobj +1323 0 obj << +/Title 1324 0 R +/A 1321 0 R +/Parent 1311 0 R +/Prev 1319 0 R +/Next 1327 0 R +>> endobj +1319 0 obj << +/Title 1320 0 R +/A 1317 0 R +/Parent 1311 0 R +/Prev 1315 0 R +/Next 1323 0 R +>> endobj +1315 0 obj << +/Title 1316 0 R +/A 1313 0 R +/Parent 1311 0 R +/Next 1319 0 R +>> endobj +1311 0 obj << +/Title 1312 0 R +/A 1309 0 R +/Parent 1307 0 R +/Next 1335 0 R +/First 1315 0 R +/Last 1331 0 R +/Count -5 +>> endobj +1307 0 obj << +/Title 1308 0 R +/A 1305 0 R +/Parent 5927 0 R +/Prev 1163 0 R +/Next 1599 0 R +/First 1311 0 R +/Last 1555 0 R +/Count 7 +>> endobj +1303 0 obj << +/Title 1304 0 R +/A 1301 0 R +/Parent 1251 0 R +/Prev 1299 0 R +>> endobj +1299 0 obj << +/Title 1300 0 R +/A 1297 0 R +/Parent 1251 0 R +/Prev 1295 0 R +/Next 1303 0 R +>> endobj +1295 0 obj << +/Title 1296 0 R +/A 1293 0 R +/Parent 1251 0 R +/Prev 1291 0 R +/Next 1299 0 R +>> endobj +1291 0 obj << +/Title 1292 0 R +/A 1289 0 R +/Parent 1251 0 R +/Prev 1287 0 R +/Next 1295 0 R +>> endobj +1287 0 obj << +/Title 1288 0 R +/A 1285 0 R +/Parent 1251 0 R +/Prev 1283 0 R +/Next 1291 0 R +>> endobj +1283 0 obj << +/Title 1284 0 R +/A 1281 0 R +/Parent 1251 0 R +/Prev 1279 0 R +/Next 1287 0 R +>> endobj +1279 0 obj << +/Title 1280 0 R +/A 1277 0 R +/Parent 1251 0 R +/Prev 1275 0 R +/Next 1283 0 R +>> endobj +1275 0 obj << +/Title 1276 0 R +/A 1273 0 R +/Parent 1251 0 R +/Prev 1271 0 R +/Next 1279 0 R +>> endobj +1271 0 obj << +/Title 1272 0 R +/A 1269 0 R +/Parent 1251 0 R +/Prev 1267 0 R +/Next 1275 0 R +>> endobj +1267 0 obj << +/Title 1268 0 R +/A 1265 0 R +/Parent 1251 0 R +/Prev 1263 0 R +/Next 1271 0 R +>> endobj +1263 0 obj << +/Title 1264 0 R +/A 1261 0 R +/Parent 1251 0 R +/Prev 1259 0 R +/Next 1267 0 R +>> endobj +1259 0 obj << +/Title 1260 0 R +/A 1257 0 R +/Parent 1251 0 R +/Prev 1255 0 R +/Next 1263 0 R +>> endobj +1255 0 obj << +/Title 1256 0 R +/A 1253 0 R +/Parent 1251 0 R +/Next 1259 0 R +>> endobj +1251 0 obj << +/Title 1252 0 R +/A 1249 0 R +/Parent 1163 0 R +/Prev 1219 0 R +/First 1255 0 R +/Last 1303 0 R +/Count -13 +>> endobj +1247 0 obj << +/Title 1248 0 R +/A 1245 0 R +/Parent 1219 0 R +/Prev 1243 0 R +>> endobj +1243 0 obj << +/Title 1244 0 R +/A 1241 0 R +/Parent 1219 0 R +/Prev 1239 0 R +/Next 1247 0 R +>> endobj +1239 0 obj << +/Title 1240 0 R +/A 1237 0 R +/Parent 1219 0 R +/Prev 1235 0 R +/Next 1243 0 R +>> endobj +1235 0 obj << +/Title 1236 0 R +/A 1233 0 R +/Parent 1219 0 R +/Prev 1231 0 R +/Next 1239 0 R +>> endobj +1231 0 obj << +/Title 1232 0 R +/A 1229 0 R +/Parent 1219 0 R +/Prev 1227 0 R +/Next 1235 0 R +>> endobj +1227 0 obj << +/Title 1228 0 R +/A 1225 0 R +/Parent 1219 0 R +/Prev 1223 0 R +/Next 1231 0 R +>> endobj +1223 0 obj << +/Title 1224 0 R +/A 1221 0 R +/Parent 1219 0 R +/Next 1227 0 R +>> endobj +1219 0 obj << +/Title 1220 0 R +/A 1217 0 R +/Parent 1163 0 R +/Prev 1167 0 R +/Next 1251 0 R +/First 1223 0 R +/Last 1247 0 R +/Count -7 +>> endobj +1215 0 obj << +/Title 1216 0 R +/A 1213 0 R +/Parent 1167 0 R +/Prev 1211 0 R +>> endobj +1211 0 obj << +/Title 1212 0 R +/A 1209 0 R +/Parent 1167 0 R +/Prev 1207 0 R +/Next 1215 0 R +>> endobj +1207 0 obj << +/Title 1208 0 R +/A 1205 0 R +/Parent 1167 0 R +/Prev 1203 0 R +/Next 1211 0 R +>> endobj +1203 0 obj << +/Title 1204 0 R +/A 1201 0 R +/Parent 1167 0 R +/Prev 1199 0 R +/Next 1207 0 R +>> endobj +1199 0 obj << +/Title 1200 0 R +/A 1197 0 R +/Parent 1167 0 R +/Prev 1195 0 R +/Next 1203 0 R +>> endobj +1195 0 obj << +/Title 1196 0 R +/A 1193 0 R +/Parent 1167 0 R +/Prev 1191 0 R +/Next 1199 0 R +>> endobj +1191 0 obj << +/Title 1192 0 R +/A 1189 0 R +/Parent 1167 0 R +/Prev 1187 0 R +/Next 1195 0 R +>> endobj +1187 0 obj << +/Title 1188 0 R +/A 1185 0 R +/Parent 1167 0 R +/Prev 1183 0 R +/Next 1191 0 R +>> endobj +1183 0 obj << +/Title 1184 0 R +/A 1181 0 R +/Parent 1167 0 R +/Prev 1179 0 R +/Next 1187 0 R +>> endobj +1179 0 obj << +/Title 1180 0 R +/A 1177 0 R +/Parent 1167 0 R +/Prev 1175 0 R +/Next 1183 0 R +>> endobj +1175 0 obj << +/Title 1176 0 R +/A 1173 0 R +/Parent 1167 0 R +/Prev 1171 0 R +/Next 1179 0 R +>> endobj +1171 0 obj << +/Title 1172 0 R +/A 1169 0 R +/Parent 1167 0 R +/Next 1175 0 R +>> endobj +1167 0 obj << +/Title 1168 0 R +/A 1165 0 R +/Parent 1163 0 R +/Next 1219 0 R +/First 1171 0 R +/Last 1215 0 R +/Count -12 +>> endobj +1163 0 obj << +/Title 1164 0 R +/A 1161 0 R +/Parent 5927 0 R +/Prev 871 0 R +/Next 1307 0 R +/First 1167 0 R +/Last 1251 0 R +/Count 3 +>> endobj +1159 0 obj << +/Title 1160 0 R +/A 1157 0 R +/Parent 1135 0 R +/Prev 1155 0 R +>> endobj +1155 0 obj << +/Title 1156 0 R +/A 1153 0 R +/Parent 1135 0 R +/Prev 1151 0 R +/Next 1159 0 R +>> endobj +1151 0 obj << +/Title 1152 0 R +/A 1149 0 R +/Parent 1135 0 R +/Prev 1147 0 R +/Next 1155 0 R +>> endobj +1147 0 obj << +/Title 1148 0 R +/A 1145 0 R +/Parent 1135 0 R +/Prev 1143 0 R +/Next 1151 0 R +>> endobj +1143 0 obj << +/Title 1144 0 R +/A 1141 0 R +/Parent 1135 0 R +/Prev 1139 0 R +/Next 1147 0 R +>> endobj +1139 0 obj << +/Title 1140 0 R +/A 1137 0 R +/Parent 1135 0 R +/Next 1143 0 R +>> endobj +1135 0 obj << +/Title 1136 0 R +/A 1133 0 R +/Parent 871 0 R +/Prev 1087 0 R +/First 1139 0 R +/Last 1159 0 R +/Count -6 +>> endobj +1131 0 obj << +/Title 1132 0 R +/A 1129 0 R +/Parent 1087 0 R +/Prev 1127 0 R +>> endobj +1127 0 obj << +/Title 1128 0 R +/A 1125 0 R +/Parent 1087 0 R +/Prev 1123 0 R +/Next 1131 0 R +>> endobj +1123 0 obj << +/Title 1124 0 R +/A 1121 0 R +/Parent 1087 0 R +/Prev 1119 0 R +/Next 1127 0 R +>> endobj +1119 0 obj << +/Title 1120 0 R +/A 1117 0 R +/Parent 1087 0 R +/Prev 1115 0 R +/Next 1123 0 R +>> endobj +1115 0 obj << +/Title 1116 0 R +/A 1113 0 R +/Parent 1087 0 R +/Prev 1111 0 R +/Next 1119 0 R +>> endobj +1111 0 obj << +/Title 1112 0 R +/A 1109 0 R +/Parent 1087 0 R +/Prev 1107 0 R +/Next 1115 0 R +>> endobj +1107 0 obj << +/Title 1108 0 R +/A 1105 0 R +/Parent 1087 0 R +/Prev 1103 0 R +/Next 1111 0 R +>> endobj +1103 0 obj << +/Title 1104 0 R +/A 1101 0 R +/Parent 1087 0 R +/Prev 1099 0 R +/Next 1107 0 R +>> endobj +1099 0 obj << +/Title 1100 0 R +/A 1097 0 R +/Parent 1087 0 R +/Prev 1095 0 R +/Next 1103 0 R +>> endobj +1095 0 obj << +/Title 1096 0 R +/A 1093 0 R +/Parent 1087 0 R +/Prev 1091 0 R +/Next 1099 0 R +>> endobj +1091 0 obj << +/Title 1092 0 R +/A 1089 0 R +/Parent 1087 0 R +/Next 1095 0 R +>> endobj +1087 0 obj << +/Title 1088 0 R +/A 1085 0 R +/Parent 871 0 R +/Prev 1027 0 R +/Next 1135 0 R +/First 1091 0 R +/Last 1131 0 R +/Count -11 +>> endobj +1083 0 obj << +/Title 1084 0 R +/A 1081 0 R +/Parent 1027 0 R +/Prev 1079 0 R +>> endobj +1079 0 obj << +/Title 1080 0 R +/A 1077 0 R +/Parent 1027 0 R +/Prev 1075 0 R +/Next 1083 0 R +>> endobj +1075 0 obj << +/Title 1076 0 R +/A 1073 0 R +/Parent 1027 0 R +/Prev 1071 0 R +/Next 1079 0 R +>> endobj +1071 0 obj << +/Title 1072 0 R +/A 1069 0 R +/Parent 1027 0 R +/Prev 1067 0 R +/Next 1075 0 R +>> endobj +1067 0 obj << +/Title 1068 0 R +/A 1065 0 R +/Parent 1027 0 R +/Prev 1063 0 R +/Next 1071 0 R +>> endobj +1063 0 obj << +/Title 1064 0 R +/A 1061 0 R +/Parent 1027 0 R +/Prev 1059 0 R +/Next 1067 0 R +>> endobj +1059 0 obj << +/Title 1060 0 R +/A 1057 0 R +/Parent 1027 0 R +/Prev 1055 0 R +/Next 1063 0 R +>> endobj +1055 0 obj << +/Title 1056 0 R +/A 1053 0 R +/Parent 1027 0 R +/Prev 1051 0 R +/Next 1059 0 R +>> endobj +1051 0 obj << +/Title 1052 0 R +/A 1049 0 R +/Parent 1027 0 R +/Prev 1047 0 R +/Next 1055 0 R +>> endobj +1047 0 obj << +/Title 1048 0 R +/A 1045 0 R +/Parent 1027 0 R +/Prev 1043 0 R +/Next 1051 0 R +>> endobj +1043 0 obj << +/Title 1044 0 R +/A 1041 0 R +/Parent 1027 0 R +/Prev 1039 0 R +/Next 1047 0 R +>> endobj +1039 0 obj << +/Title 1040 0 R +/A 1037 0 R +/Parent 1027 0 R +/Prev 1035 0 R +/Next 1043 0 R +>> endobj +1035 0 obj << +/Title 1036 0 R +/A 1033 0 R +/Parent 1027 0 R +/Prev 1031 0 R +/Next 1039 0 R +>> endobj +1031 0 obj << +/Title 1032 0 R +/A 1029 0 R +/Parent 1027 0 R +/Next 1035 0 R +>> endobj +1027 0 obj << +/Title 1028 0 R +/A 1025 0 R +/Parent 871 0 R +/Prev 999 0 R +/Next 1087 0 R +/First 1031 0 R +/Last 1083 0 R +/Count -14 +>> endobj +1023 0 obj << +/Title 1024 0 R +/A 1021 0 R +/Parent 999 0 R +/Prev 1019 0 R +>> endobj +1019 0 obj << +/Title 1020 0 R +/A 1017 0 R +/Parent 999 0 R +/Prev 1015 0 R +/Next 1023 0 R +>> endobj +1015 0 obj << +/Title 1016 0 R +/A 1013 0 R +/Parent 999 0 R +/Prev 1011 0 R +/Next 1019 0 R +>> endobj +1011 0 obj << +/Title 1012 0 R +/A 1009 0 R +/Parent 999 0 R +/Prev 1007 0 R +/Next 1015 0 R +>> endobj +1007 0 obj << +/Title 1008 0 R +/A 1005 0 R +/Parent 999 0 R +/Prev 1003 0 R +/Next 1011 0 R +>> endobj +1003 0 obj << +/Title 1004 0 R +/A 1001 0 R +/Parent 999 0 R +/Next 1007 0 R +>> endobj +999 0 obj << +/Title 1000 0 R +/A 997 0 R +/Parent 871 0 R +/Prev 959 0 R +/Next 1027 0 R +/First 1003 0 R +/Last 1023 0 R +/Count -6 +>> endobj +995 0 obj << +/Title 996 0 R +/A 993 0 R +/Parent 959 0 R +/Prev 991 0 R +>> endobj +991 0 obj << +/Title 992 0 R +/A 989 0 R +/Parent 959 0 R +/Prev 987 0 R +/Next 995 0 R +>> endobj +987 0 obj << +/Title 988 0 R +/A 985 0 R +/Parent 959 0 R +/Prev 983 0 R +/Next 991 0 R +>> endobj +983 0 obj << +/Title 984 0 R +/A 981 0 R +/Parent 959 0 R +/Prev 979 0 R +/Next 987 0 R +>> endobj +979 0 obj << +/Title 980 0 R +/A 977 0 R +/Parent 959 0 R +/Prev 975 0 R +/Next 983 0 R +>> endobj +975 0 obj << +/Title 976 0 R +/A 973 0 R +/Parent 959 0 R +/Prev 971 0 R +/Next 979 0 R +>> endobj +971 0 obj << +/Title 972 0 R +/A 969 0 R +/Parent 959 0 R +/Prev 967 0 R +/Next 975 0 R +>> endobj +967 0 obj << +/Title 968 0 R +/A 965 0 R +/Parent 959 0 R +/Prev 963 0 R +/Next 971 0 R +>> endobj +963 0 obj << +/Title 964 0 R +/A 961 0 R +/Parent 959 0 R +/Next 967 0 R +>> endobj +959 0 obj << +/Title 960 0 R +/A 957 0 R +/Parent 871 0 R +/Prev 907 0 R +/Next 999 0 R +/First 963 0 R +/Last 995 0 R +/Count -9 +>> endobj +955 0 obj << +/Title 956 0 R +/A 953 0 R +/Parent 907 0 R +/Prev 951 0 R +>> endobj +951 0 obj << +/Title 952 0 R +/A 949 0 R +/Parent 907 0 R +/Prev 947 0 R +/Next 955 0 R +>> endobj +947 0 obj << +/Title 948 0 R +/A 945 0 R +/Parent 907 0 R +/Prev 943 0 R +/Next 951 0 R +>> endobj +943 0 obj << +/Title 944 0 R +/A 941 0 R +/Parent 907 0 R +/Prev 939 0 R +/Next 947 0 R +>> endobj +939 0 obj << +/Title 940 0 R +/A 937 0 R +/Parent 907 0 R +/Prev 935 0 R +/Next 943 0 R +>> endobj +935 0 obj << +/Title 936 0 R +/A 933 0 R +/Parent 907 0 R +/Prev 931 0 R +/Next 939 0 R +>> endobj +931 0 obj << +/Title 932 0 R +/A 929 0 R +/Parent 907 0 R +/Prev 927 0 R +/Next 935 0 R +>> endobj +927 0 obj << +/Title 928 0 R +/A 925 0 R +/Parent 907 0 R +/Prev 923 0 R +/Next 931 0 R +>> endobj +923 0 obj << +/Title 924 0 R +/A 921 0 R +/Parent 907 0 R +/Prev 919 0 R +/Next 927 0 R +>> endobj +919 0 obj << +/Title 920 0 R +/A 917 0 R +/Parent 907 0 R +/Prev 915 0 R +/Next 923 0 R +>> endobj +915 0 obj << +/Title 916 0 R +/A 913 0 R +/Parent 907 0 R +/Prev 911 0 R +/Next 919 0 R +>> endobj +911 0 obj << +/Title 912 0 R +/A 909 0 R +/Parent 907 0 R +/Next 915 0 R +>> endobj +907 0 obj << +/Title 908 0 R +/A 905 0 R +/Parent 871 0 R +/Prev 875 0 R +/Next 959 0 R +/First 911 0 R +/Last 955 0 R +/Count -12 +>> endobj +903 0 obj << +/Title 904 0 R +/A 901 0 R +/Parent 875 0 R +/Prev 899 0 R +>> endobj +899 0 obj << +/Title 900 0 R +/A 897 0 R +/Parent 875 0 R +/Prev 895 0 R +/Next 903 0 R +>> endobj +895 0 obj << +/Title 896 0 R +/A 893 0 R +/Parent 875 0 R +/Prev 891 0 R +/Next 899 0 R +>> endobj +891 0 obj << +/Title 892 0 R +/A 889 0 R +/Parent 875 0 R +/Prev 887 0 R +/Next 895 0 R +>> endobj +887 0 obj << +/Title 888 0 R +/A 885 0 R +/Parent 875 0 R +/Prev 883 0 R +/Next 891 0 R +>> endobj +883 0 obj << +/Title 884 0 R +/A 881 0 R +/Parent 875 0 R +/Prev 879 0 R +/Next 887 0 R +>> endobj +879 0 obj << +/Title 880 0 R +/A 877 0 R +/Parent 875 0 R +/Next 883 0 R +>> endobj +875 0 obj << +/Title 876 0 R +/A 873 0 R +/Parent 871 0 R +/Next 907 0 R +/First 879 0 R +/Last 903 0 R +/Count -7 +>> endobj +871 0 obj << +/Title 872 0 R +/A 869 0 R +/Parent 5927 0 R +/Prev 523 0 R +/Next 1163 0 R +/First 875 0 R +/Last 1135 0 R +/Count 7 +>> endobj +867 0 obj << +/Title 868 0 R +/A 865 0 R +/Parent 819 0 R +/Prev 863 0 R +>> endobj +863 0 obj << +/Title 864 0 R +/A 861 0 R +/Parent 819 0 R +/Prev 859 0 R +/Next 867 0 R +>> endobj +859 0 obj << +/Title 860 0 R +/A 857 0 R +/Parent 819 0 R +/Prev 855 0 R +/Next 863 0 R +>> endobj +855 0 obj << +/Title 856 0 R +/A 853 0 R +/Parent 819 0 R +/Prev 851 0 R +/Next 859 0 R +>> endobj +851 0 obj << +/Title 852 0 R +/A 849 0 R +/Parent 819 0 R +/Prev 847 0 R +/Next 855 0 R +>> endobj +847 0 obj << +/Title 848 0 R +/A 845 0 R +/Parent 819 0 R +/Prev 843 0 R +/Next 851 0 R +>> endobj +843 0 obj << +/Title 844 0 R +/A 841 0 R +/Parent 819 0 R +/Prev 839 0 R +/Next 847 0 R +>> endobj +839 0 obj << +/Title 840 0 R +/A 837 0 R +/Parent 819 0 R +/Prev 835 0 R +/Next 843 0 R +>> endobj +835 0 obj << +/Title 836 0 R +/A 833 0 R +/Parent 819 0 R +/Prev 831 0 R +/Next 839 0 R +>> endobj +831 0 obj << +/Title 832 0 R +/A 829 0 R +/Parent 819 0 R +/Prev 827 0 R +/Next 835 0 R +>> endobj +827 0 obj << +/Title 828 0 R +/A 825 0 R +/Parent 819 0 R +/Prev 823 0 R +/Next 831 0 R +>> endobj +823 0 obj << +/Title 824 0 R +/A 821 0 R +/Parent 819 0 R +/Next 827 0 R +>> endobj +819 0 obj << +/Title 820 0 R +/A 817 0 R +/Parent 523 0 R +/Prev 735 0 R +/First 823 0 R +/Last 867 0 R +/Count -12 +>> endobj +815 0 obj << +/Title 816 0 R +/A 813 0 R +/Parent 735 0 R +/Prev 811 0 R +>> endobj +811 0 obj << +/Title 812 0 R +/A 809 0 R +/Parent 735 0 R +/Prev 807 0 R +/Next 815 0 R +>> endobj +807 0 obj << +/Title 808 0 R +/A 805 0 R +/Parent 735 0 R +/Prev 803 0 R +/Next 811 0 R +>> endobj +803 0 obj << +/Title 804 0 R +/A 801 0 R +/Parent 735 0 R +/Prev 799 0 R +/Next 807 0 R +>> endobj +799 0 obj << +/Title 800 0 R +/A 797 0 R +/Parent 735 0 R +/Prev 795 0 R +/Next 803 0 R +>> endobj +795 0 obj << +/Title 796 0 R +/A 793 0 R +/Parent 735 0 R +/Prev 791 0 R +/Next 799 0 R +>> endobj +791 0 obj << +/Title 792 0 R +/A 789 0 R +/Parent 735 0 R +/Prev 787 0 R +/Next 795 0 R +>> endobj +787 0 obj << +/Title 788 0 R +/A 785 0 R +/Parent 735 0 R +/Prev 783 0 R +/Next 791 0 R +>> endobj +783 0 obj << +/Title 784 0 R +/A 781 0 R +/Parent 735 0 R +/Prev 779 0 R +/Next 787 0 R +>> endobj +779 0 obj << +/Title 780 0 R +/A 777 0 R +/Parent 735 0 R +/Prev 775 0 R +/Next 783 0 R +>> endobj +775 0 obj << +/Title 776 0 R +/A 773 0 R +/Parent 735 0 R +/Prev 771 0 R +/Next 779 0 R +>> endobj +771 0 obj << +/Title 772 0 R +/A 769 0 R +/Parent 735 0 R +/Prev 767 0 R +/Next 775 0 R +>> endobj +767 0 obj << +/Title 768 0 R +/A 765 0 R +/Parent 735 0 R +/Prev 763 0 R +/Next 771 0 R +>> endobj +763 0 obj << +/Title 764 0 R +/A 761 0 R +/Parent 735 0 R +/Prev 759 0 R +/Next 767 0 R +>> endobj +759 0 obj << +/Title 760 0 R +/A 757 0 R +/Parent 735 0 R +/Prev 755 0 R +/Next 763 0 R +>> endobj +755 0 obj << +/Title 756 0 R +/A 753 0 R +/Parent 735 0 R +/Prev 751 0 R +/Next 759 0 R +>> endobj +751 0 obj << +/Title 752 0 R +/A 749 0 R +/Parent 735 0 R +/Prev 747 0 R +/Next 755 0 R +>> endobj +747 0 obj << +/Title 748 0 R +/A 745 0 R +/Parent 735 0 R +/Prev 743 0 R +/Next 751 0 R +>> endobj +743 0 obj << +/Title 744 0 R +/A 741 0 R +/Parent 735 0 R +/Prev 739 0 R +/Next 747 0 R +>> endobj +739 0 obj << +/Title 740 0 R +/A 737 0 R +/Parent 735 0 R +/Next 743 0 R +>> endobj +735 0 obj << +/Title 736 0 R +/A 733 0 R +/Parent 523 0 R +/Prev 703 0 R +/Next 819 0 R +/First 739 0 R +/Last 815 0 R +/Count -20 +>> endobj +731 0 obj << +/Title 732 0 R +/A 729 0 R +/Parent 703 0 R +/Prev 727 0 R +>> endobj +727 0 obj << +/Title 728 0 R +/A 725 0 R +/Parent 703 0 R +/Prev 723 0 R +/Next 731 0 R +>> endobj +723 0 obj << +/Title 724 0 R +/A 721 0 R +/Parent 703 0 R +/Prev 719 0 R +/Next 727 0 R +>> endobj +719 0 obj << +/Title 720 0 R +/A 717 0 R +/Parent 703 0 R +/Prev 715 0 R +/Next 723 0 R +>> endobj +715 0 obj << +/Title 716 0 R +/A 713 0 R +/Parent 703 0 R +/Prev 711 0 R +/Next 719 0 R +>> endobj +711 0 obj << +/Title 712 0 R +/A 709 0 R +/Parent 703 0 R +/Prev 707 0 R +/Next 715 0 R +>> endobj +707 0 obj << +/Title 708 0 R +/A 705 0 R +/Parent 703 0 R +/Next 711 0 R +>> endobj +703 0 obj << +/Title 704 0 R +/A 701 0 R +/Parent 523 0 R +/Prev 667 0 R +/Next 735 0 R +/First 707 0 R +/Last 731 0 R +/Count -7 +>> endobj +699 0 obj << +/Title 700 0 R +/A 697 0 R +/Parent 667 0 R +/Prev 695 0 R +>> endobj +695 0 obj << +/Title 696 0 R +/A 693 0 R +/Parent 667 0 R +/Prev 691 0 R +/Next 699 0 R +>> endobj +691 0 obj << +/Title 692 0 R +/A 689 0 R +/Parent 667 0 R +/Prev 687 0 R +/Next 695 0 R +>> endobj +687 0 obj << +/Title 688 0 R +/A 685 0 R +/Parent 667 0 R +/Prev 683 0 R +/Next 691 0 R +>> endobj +683 0 obj << +/Title 684 0 R +/A 681 0 R +/Parent 667 0 R +/Prev 679 0 R +/Next 687 0 R +>> endobj +679 0 obj << +/Title 680 0 R +/A 677 0 R +/Parent 667 0 R +/Prev 675 0 R +/Next 683 0 R +>> endobj +675 0 obj << +/Title 676 0 R +/A 673 0 R +/Parent 667 0 R +/Prev 671 0 R +/Next 679 0 R +>> endobj +671 0 obj << +/Title 672 0 R +/A 669 0 R +/Parent 667 0 R +/Next 675 0 R +>> endobj +667 0 obj << +/Title 668 0 R +/A 665 0 R +/Parent 523 0 R +/Prev 655 0 R +/Next 703 0 R +/First 671 0 R +/Last 699 0 R +/Count -8 +>> endobj +663 0 obj << +/Title 664 0 R +/A 661 0 R +/Parent 655 0 R +/Prev 659 0 R +>> endobj +659 0 obj << +/Title 660 0 R +/A 657 0 R +/Parent 655 0 R +/Next 663 0 R +>> endobj +655 0 obj << +/Title 656 0 R +/A 653 0 R +/Parent 523 0 R +/Prev 619 0 R +/Next 667 0 R +/First 659 0 R +/Last 663 0 R +/Count -2 +>> endobj +651 0 obj << +/Title 652 0 R +/A 649 0 R +/Parent 619 0 R +/Prev 647 0 R +>> endobj +647 0 obj << +/Title 648 0 R +/A 645 0 R +/Parent 619 0 R +/Prev 643 0 R +/Next 651 0 R +>> endobj +643 0 obj << +/Title 644 0 R +/A 641 0 R +/Parent 619 0 R +/Prev 639 0 R +/Next 647 0 R +>> endobj +639 0 obj << +/Title 640 0 R +/A 637 0 R +/Parent 619 0 R +/Prev 635 0 R +/Next 643 0 R +>> endobj +635 0 obj << +/Title 636 0 R +/A 633 0 R +/Parent 619 0 R +/Prev 631 0 R +/Next 639 0 R +>> endobj +631 0 obj << +/Title 632 0 R +/A 629 0 R +/Parent 619 0 R +/Prev 627 0 R +/Next 635 0 R +>> endobj +627 0 obj << +/Title 628 0 R +/A 625 0 R +/Parent 619 0 R +/Prev 623 0 R +/Next 631 0 R +>> endobj +623 0 obj << +/Title 624 0 R +/A 621 0 R +/Parent 619 0 R +/Next 627 0 R +>> endobj +619 0 obj << +/Title 620 0 R +/A 617 0 R +/Parent 523 0 R +/Prev 559 0 R +/Next 655 0 R +/First 623 0 R +/Last 651 0 R +/Count -8 +>> endobj +615 0 obj << +/Title 616 0 R +/A 613 0 R +/Parent 559 0 R +/Prev 611 0 R +>> endobj +611 0 obj << +/Title 612 0 R +/A 609 0 R +/Parent 559 0 R +/Prev 607 0 R +/Next 615 0 R +>> endobj +607 0 obj << +/Title 608 0 R +/A 605 0 R +/Parent 559 0 R +/Prev 603 0 R +/Next 611 0 R +>> endobj +603 0 obj << +/Title 604 0 R +/A 601 0 R +/Parent 559 0 R +/Prev 599 0 R +/Next 607 0 R +>> endobj +599 0 obj << +/Title 600 0 R +/A 597 0 R +/Parent 559 0 R +/Prev 595 0 R +/Next 603 0 R +>> endobj +595 0 obj << +/Title 596 0 R +/A 593 0 R +/Parent 559 0 R +/Prev 591 0 R +/Next 599 0 R +>> endobj +591 0 obj << +/Title 592 0 R +/A 589 0 R +/Parent 559 0 R +/Prev 587 0 R +/Next 595 0 R +>> endobj +587 0 obj << +/Title 588 0 R +/A 585 0 R +/Parent 559 0 R +/Prev 583 0 R +/Next 591 0 R +>> endobj +583 0 obj << +/Title 584 0 R +/A 581 0 R +/Parent 559 0 R +/Prev 579 0 R +/Next 587 0 R +>> endobj +579 0 obj << +/Title 580 0 R +/A 577 0 R +/Parent 559 0 R +/Prev 575 0 R +/Next 583 0 R +>> endobj +575 0 obj << +/Title 576 0 R +/A 573 0 R +/Parent 559 0 R +/Prev 571 0 R +/Next 579 0 R +>> endobj +571 0 obj << +/Title 572 0 R +/A 569 0 R +/Parent 559 0 R +/Prev 567 0 R +/Next 575 0 R +>> endobj +567 0 obj << +/Title 568 0 R +/A 565 0 R +/Parent 559 0 R +/Prev 563 0 R +/Next 571 0 R +>> endobj +563 0 obj << +/Title 564 0 R +/A 561 0 R +/Parent 559 0 R +/Next 567 0 R +>> endobj +559 0 obj << +/Title 560 0 R +/A 557 0 R +/Parent 523 0 R +/Prev 527 0 R +/Next 619 0 R +/First 563 0 R +/Last 615 0 R +/Count -14 +>> endobj +555 0 obj << +/Title 556 0 R +/A 553 0 R +/Parent 527 0 R +/Prev 551 0 R +>> endobj +551 0 obj << +/Title 552 0 R +/A 549 0 R +/Parent 527 0 R +/Prev 547 0 R +/Next 555 0 R +>> endobj +547 0 obj << +/Title 548 0 R +/A 545 0 R +/Parent 527 0 R +/Prev 543 0 R +/Next 551 0 R +>> endobj +543 0 obj << +/Title 544 0 R +/A 541 0 R +/Parent 527 0 R +/Prev 539 0 R +/Next 547 0 R +>> endobj +539 0 obj << +/Title 540 0 R +/A 537 0 R +/Parent 527 0 R +/Prev 535 0 R +/Next 543 0 R +>> endobj +535 0 obj << +/Title 536 0 R +/A 533 0 R +/Parent 527 0 R +/Prev 531 0 R +/Next 539 0 R +>> endobj +531 0 obj << +/Title 532 0 R +/A 529 0 R +/Parent 527 0 R +/Next 535 0 R +>> endobj +527 0 obj << +/Title 528 0 R +/A 525 0 R +/Parent 523 0 R +/Next 559 0 R +/First 531 0 R +/Last 555 0 R +/Count -7 +>> endobj +523 0 obj << +/Title 524 0 R +/A 521 0 R +/Parent 5927 0 R +/Prev 27 0 R +/Next 871 0 R +/First 527 0 R +/Last 819 0 R +/Count 8 +>> endobj +519 0 obj << +/Title 520 0 R +/A 517 0 R +/Parent 499 0 R +/Prev 515 0 R +>> endobj +515 0 obj << +/Title 516 0 R +/A 513 0 R +/Parent 499 0 R +/Prev 511 0 R +/Next 519 0 R +>> endobj +511 0 obj << +/Title 512 0 R +/A 509 0 R +/Parent 499 0 R +/Prev 507 0 R +/Next 515 0 R +>> endobj +507 0 obj << +/Title 508 0 R +/A 505 0 R +/Parent 499 0 R +/Prev 503 0 R +/Next 511 0 R +>> endobj +503 0 obj << +/Title 504 0 R +/A 501 0 R +/Parent 499 0 R +/Next 507 0 R +>> endobj +499 0 obj << +/Title 500 0 R +/A 497 0 R +/Parent 27 0 R +/Prev 427 0 R +/First 503 0 R +/Last 519 0 R +/Count -5 +>> endobj +495 0 obj << +/Title 496 0 R +/A 493 0 R +/Parent 427 0 R +/Prev 491 0 R +>> endobj +491 0 obj << +/Title 492 0 R +/A 489 0 R +/Parent 427 0 R +/Prev 487 0 R +/Next 495 0 R +>> endobj +487 0 obj << +/Title 488 0 R +/A 485 0 R +/Parent 427 0 R +/Prev 483 0 R +/Next 491 0 R +>> endobj +483 0 obj << +/Title 484 0 R +/A 481 0 R +/Parent 427 0 R +/Prev 479 0 R +/Next 487 0 R +>> endobj +479 0 obj << +/Title 480 0 R +/A 477 0 R +/Parent 427 0 R +/Prev 475 0 R +/Next 483 0 R +>> endobj +475 0 obj << +/Title 476 0 R +/A 473 0 R +/Parent 427 0 R +/Prev 471 0 R +/Next 479 0 R +>> endobj +471 0 obj << +/Title 472 0 R +/A 469 0 R +/Parent 427 0 R +/Prev 467 0 R +/Next 475 0 R +>> endobj +467 0 obj << +/Title 468 0 R +/A 465 0 R +/Parent 427 0 R +/Prev 463 0 R +/Next 471 0 R +>> endobj +463 0 obj << +/Title 464 0 R +/A 461 0 R +/Parent 427 0 R +/Prev 459 0 R +/Next 467 0 R +>> endobj +459 0 obj << +/Title 460 0 R +/A 457 0 R +/Parent 427 0 R +/Prev 455 0 R +/Next 463 0 R +>> endobj +455 0 obj << +/Title 456 0 R +/A 453 0 R +/Parent 427 0 R +/Prev 451 0 R +/Next 459 0 R +>> endobj +451 0 obj << +/Title 452 0 R +/A 449 0 R +/Parent 427 0 R +/Prev 447 0 R +/Next 455 0 R +>> endobj +447 0 obj << +/Title 448 0 R +/A 445 0 R +/Parent 427 0 R +/Prev 443 0 R +/Next 451 0 R +>> endobj +443 0 obj << +/Title 444 0 R +/A 441 0 R +/Parent 427 0 R +/Prev 439 0 R +/Next 447 0 R +>> endobj +439 0 obj << +/Title 440 0 R +/A 437 0 R +/Parent 427 0 R +/Prev 435 0 R +/Next 443 0 R +>> endobj +435 0 obj << +/Title 436 0 R +/A 433 0 R +/Parent 427 0 R +/Prev 431 0 R +/Next 439 0 R +>> endobj +431 0 obj << +/Title 432 0 R +/A 429 0 R +/Parent 427 0 R +/Next 435 0 R +>> endobj +427 0 obj << +/Title 428 0 R +/A 425 0 R +/Parent 27 0 R +/Prev 383 0 R +/Next 499 0 R +/First 431 0 R +/Last 495 0 R +/Count -17 +>> endobj +423 0 obj << +/Title 424 0 R +/A 421 0 R +/Parent 383 0 R +/Prev 419 0 R +>> endobj +419 0 obj << +/Title 420 0 R +/A 417 0 R +/Parent 383 0 R +/Prev 415 0 R +/Next 423 0 R +>> endobj +415 0 obj << +/Title 416 0 R +/A 413 0 R +/Parent 383 0 R +/Prev 411 0 R +/Next 419 0 R +>> endobj +411 0 obj << +/Title 412 0 R +/A 409 0 R +/Parent 383 0 R +/Prev 407 0 R +/Next 415 0 R +>> endobj +407 0 obj << +/Title 408 0 R +/A 405 0 R +/Parent 383 0 R +/Prev 403 0 R +/Next 411 0 R +>> endobj +403 0 obj << +/Title 404 0 R +/A 401 0 R +/Parent 383 0 R +/Prev 399 0 R +/Next 407 0 R +>> endobj +399 0 obj << +/Title 400 0 R +/A 397 0 R +/Parent 383 0 R +/Prev 395 0 R +/Next 403 0 R +>> endobj +395 0 obj << +/Title 396 0 R +/A 393 0 R +/Parent 383 0 R +/Prev 391 0 R +/Next 399 0 R +>> endobj +391 0 obj << +/Title 392 0 R +/A 389 0 R +/Parent 383 0 R +/Prev 387 0 R +/Next 395 0 R +>> endobj +387 0 obj << +/Title 388 0 R +/A 385 0 R +/Parent 383 0 R +/Next 391 0 R +>> endobj +383 0 obj << +/Title 384 0 R +/A 381 0 R +/Parent 27 0 R +/Prev 339 0 R +/Next 427 0 R +/First 387 0 R +/Last 423 0 R +/Count -10 +>> endobj +379 0 obj << +/Title 380 0 R +/A 377 0 R +/Parent 339 0 R +/Prev 375 0 R +>> endobj +375 0 obj << +/Title 376 0 R +/A 373 0 R +/Parent 339 0 R +/Prev 371 0 R +/Next 379 0 R +>> endobj +371 0 obj << +/Title 372 0 R +/A 369 0 R +/Parent 339 0 R +/Prev 367 0 R +/Next 375 0 R +>> endobj +367 0 obj << +/Title 368 0 R +/A 365 0 R +/Parent 339 0 R +/Prev 363 0 R +/Next 371 0 R +>> endobj +363 0 obj << +/Title 364 0 R +/A 361 0 R +/Parent 339 0 R +/Prev 359 0 R +/Next 367 0 R +>> endobj +359 0 obj << +/Title 360 0 R +/A 357 0 R +/Parent 339 0 R +/Prev 355 0 R +/Next 363 0 R +>> endobj +355 0 obj << +/Title 356 0 R +/A 353 0 R +/Parent 339 0 R +/Prev 351 0 R +/Next 359 0 R +>> endobj +351 0 obj << +/Title 352 0 R +/A 349 0 R +/Parent 339 0 R +/Prev 347 0 R +/Next 355 0 R +>> endobj +347 0 obj << +/Title 348 0 R +/A 345 0 R +/Parent 339 0 R +/Prev 343 0 R +/Next 351 0 R +>> endobj +343 0 obj << +/Title 344 0 R +/A 341 0 R +/Parent 339 0 R +/Next 347 0 R +>> endobj +339 0 obj << +/Title 340 0 R +/A 337 0 R +/Parent 27 0 R +/Prev 315 0 R +/Next 383 0 R +/First 343 0 R +/Last 379 0 R +/Count -10 +>> endobj +335 0 obj << +/Title 336 0 R +/A 333 0 R +/Parent 315 0 R +/Prev 331 0 R +>> endobj +331 0 obj << +/Title 332 0 R +/A 329 0 R +/Parent 315 0 R +/Prev 327 0 R +/Next 335 0 R +>> endobj +327 0 obj << +/Title 328 0 R +/A 325 0 R +/Parent 315 0 R +/Prev 323 0 R +/Next 331 0 R +>> endobj +323 0 obj << +/Title 324 0 R +/A 321 0 R +/Parent 315 0 R +/Prev 319 0 R +/Next 327 0 R +>> endobj +319 0 obj << +/Title 320 0 R +/A 317 0 R +/Parent 315 0 R +/Next 323 0 R +>> endobj +315 0 obj << +/Title 316 0 R +/A 313 0 R +/Parent 27 0 R +/Prev 207 0 R +/Next 339 0 R +/First 319 0 R +/Last 335 0 R +/Count -5 +>> endobj +311 0 obj << +/Title 312 0 R +/A 309 0 R +/Parent 207 0 R +/Prev 307 0 R +>> endobj +307 0 obj << +/Title 308 0 R +/A 305 0 R +/Parent 207 0 R +/Prev 303 0 R +/Next 311 0 R +>> endobj +303 0 obj << +/Title 304 0 R +/A 301 0 R +/Parent 207 0 R +/Prev 299 0 R +/Next 307 0 R +>> endobj +299 0 obj << +/Title 300 0 R +/A 297 0 R +/Parent 207 0 R +/Prev 295 0 R +/Next 303 0 R +>> endobj +295 0 obj << +/Title 296 0 R +/A 293 0 R +/Parent 207 0 R +/Prev 291 0 R +/Next 299 0 R +>> endobj +291 0 obj << +/Title 292 0 R +/A 289 0 R +/Parent 207 0 R +/Prev 287 0 R +/Next 295 0 R +>> endobj +287 0 obj << +/Title 288 0 R +/A 285 0 R +/Parent 207 0 R +/Prev 283 0 R +/Next 291 0 R +>> endobj +283 0 obj << +/Title 284 0 R +/A 281 0 R +/Parent 207 0 R +/Prev 279 0 R +/Next 287 0 R +>> endobj +279 0 obj << +/Title 280 0 R +/A 277 0 R +/Parent 207 0 R +/Prev 275 0 R +/Next 283 0 R +>> endobj +275 0 obj << +/Title 276 0 R +/A 273 0 R +/Parent 207 0 R +/Prev 271 0 R +/Next 279 0 R +>> endobj +271 0 obj << +/Title 272 0 R +/A 269 0 R +/Parent 207 0 R +/Prev 267 0 R +/Next 275 0 R +>> endobj +267 0 obj << +/Title 268 0 R +/A 265 0 R +/Parent 207 0 R +/Prev 263 0 R +/Next 271 0 R +>> endobj +263 0 obj << +/Title 264 0 R +/A 261 0 R +/Parent 207 0 R +/Prev 259 0 R +/Next 267 0 R +>> endobj +259 0 obj << +/Title 260 0 R +/A 257 0 R +/Parent 207 0 R +/Prev 255 0 R +/Next 263 0 R +>> endobj +255 0 obj << +/Title 256 0 R +/A 253 0 R +/Parent 207 0 R +/Prev 251 0 R +/Next 259 0 R +>> endobj +251 0 obj << +/Title 252 0 R +/A 249 0 R +/Parent 207 0 R +/Prev 247 0 R +/Next 255 0 R +>> endobj +247 0 obj << +/Title 248 0 R +/A 245 0 R +/Parent 207 0 R +/Prev 243 0 R +/Next 251 0 R +>> endobj +243 0 obj << +/Title 244 0 R +/A 241 0 R +/Parent 207 0 R +/Prev 239 0 R +/Next 247 0 R +>> endobj +239 0 obj << +/Title 240 0 R +/A 237 0 R +/Parent 207 0 R +/Prev 235 0 R +/Next 243 0 R +>> endobj +235 0 obj << +/Title 236 0 R +/A 233 0 R +/Parent 207 0 R +/Prev 231 0 R +/Next 239 0 R +>> endobj +231 0 obj << +/Title 232 0 R +/A 229 0 R +/Parent 207 0 R +/Prev 227 0 R +/Next 235 0 R +>> endobj +227 0 obj << +/Title 228 0 R +/A 225 0 R +/Parent 207 0 R +/Prev 223 0 R +/Next 231 0 R +>> endobj +223 0 obj << +/Title 224 0 R +/A 221 0 R +/Parent 207 0 R +/Prev 219 0 R +/Next 227 0 R +>> endobj +219 0 obj << +/Title 220 0 R +/A 217 0 R +/Parent 207 0 R +/Prev 215 0 R +/Next 223 0 R +>> endobj +215 0 obj << +/Title 216 0 R +/A 213 0 R +/Parent 207 0 R +/Prev 211 0 R +/Next 219 0 R +>> endobj +211 0 obj << +/Title 212 0 R +/A 209 0 R +/Parent 207 0 R +/Next 215 0 R +>> endobj +207 0 obj << +/Title 208 0 R +/A 205 0 R +/Parent 27 0 R +/Prev 119 0 R +/Next 315 0 R +/First 211 0 R +/Last 311 0 R +/Count -26 +>> endobj +203 0 obj << +/Title 204 0 R +/A 201 0 R +/Parent 119 0 R +/Prev 199 0 R +>> endobj +199 0 obj << +/Title 200 0 R +/A 197 0 R +/Parent 119 0 R +/Prev 195 0 R +/Next 203 0 R +>> endobj +195 0 obj << +/Title 196 0 R +/A 193 0 R +/Parent 119 0 R +/Prev 191 0 R +/Next 199 0 R +>> endobj +191 0 obj << +/Title 192 0 R +/A 189 0 R +/Parent 119 0 R +/Prev 187 0 R +/Next 195 0 R +>> endobj +187 0 obj << +/Title 188 0 R +/A 185 0 R +/Parent 119 0 R +/Prev 183 0 R +/Next 191 0 R +>> endobj +183 0 obj << +/Title 184 0 R +/A 181 0 R +/Parent 119 0 R +/Prev 179 0 R +/Next 187 0 R +>> endobj +179 0 obj << +/Title 180 0 R +/A 177 0 R +/Parent 119 0 R +/Prev 175 0 R +/Next 183 0 R +>> endobj +175 0 obj << +/Title 176 0 R +/A 173 0 R +/Parent 119 0 R +/Prev 171 0 R +/Next 179 0 R +>> endobj +171 0 obj << +/Title 172 0 R +/A 169 0 R +/Parent 119 0 R +/Prev 167 0 R +/Next 175 0 R +>> endobj +167 0 obj << +/Title 168 0 R +/A 165 0 R +/Parent 119 0 R +/Prev 163 0 R +/Next 171 0 R +>> endobj +163 0 obj << +/Title 164 0 R +/A 161 0 R +/Parent 119 0 R +/Prev 159 0 R +/Next 167 0 R +>> endobj +159 0 obj << +/Title 160 0 R +/A 157 0 R +/Parent 119 0 R +/Prev 155 0 R +/Next 163 0 R +>> endobj +155 0 obj << +/Title 156 0 R +/A 153 0 R +/Parent 119 0 R +/Prev 151 0 R +/Next 159 0 R +>> endobj +151 0 obj << +/Title 152 0 R +/A 149 0 R +/Parent 119 0 R +/Prev 147 0 R +/Next 155 0 R +>> endobj +147 0 obj << +/Title 148 0 R +/A 145 0 R +/Parent 119 0 R +/Prev 143 0 R +/Next 151 0 R +>> endobj +143 0 obj << +/Title 144 0 R +/A 141 0 R +/Parent 119 0 R +/Prev 139 0 R +/Next 147 0 R +>> endobj +139 0 obj << +/Title 140 0 R +/A 137 0 R +/Parent 119 0 R +/Prev 135 0 R +/Next 143 0 R +>> endobj +135 0 obj << +/Title 136 0 R +/A 133 0 R +/Parent 119 0 R +/Prev 131 0 R +/Next 139 0 R +>> endobj +131 0 obj << +/Title 132 0 R +/A 129 0 R +/Parent 119 0 R +/Prev 127 0 R +/Next 135 0 R +>> endobj +127 0 obj << +/Title 128 0 R +/A 125 0 R +/Parent 119 0 R +/Prev 123 0 R +/Next 131 0 R +>> endobj +123 0 obj << +/Title 124 0 R +/A 121 0 R +/Parent 119 0 R +/Next 127 0 R +>> endobj +119 0 obj << +/Title 120 0 R +/A 117 0 R +/Parent 27 0 R +/Prev 71 0 R +/Next 207 0 R +/First 123 0 R +/Last 203 0 R +/Count -21 +>> endobj +115 0 obj << +/Title 116 0 R +/A 113 0 R +/Parent 71 0 R +/Prev 111 0 R +>> endobj +111 0 obj << +/Title 112 0 R +/A 109 0 R +/Parent 71 0 R +/Prev 107 0 R +/Next 115 0 R +>> endobj +107 0 obj << +/Title 108 0 R +/A 105 0 R +/Parent 71 0 R +/Prev 103 0 R +/Next 111 0 R +>> endobj +103 0 obj << +/Title 104 0 R +/A 101 0 R +/Parent 71 0 R +/Prev 99 0 R +/Next 107 0 R +>> endobj +99 0 obj << +/Title 100 0 R +/A 97 0 R +/Parent 71 0 R +/Prev 95 0 R +/Next 103 0 R +>> endobj +95 0 obj << +/Title 96 0 R +/A 93 0 R +/Parent 71 0 R +/Prev 91 0 R +/Next 99 0 R +>> endobj +91 0 obj << +/Title 92 0 R +/A 89 0 R +/Parent 71 0 R +/Prev 87 0 R +/Next 95 0 R +>> endobj +87 0 obj << +/Title 88 0 R +/A 85 0 R +/Parent 71 0 R +/Prev 83 0 R +/Next 91 0 R +>> endobj +83 0 obj << +/Title 84 0 R +/A 81 0 R +/Parent 71 0 R +/Prev 79 0 R +/Next 87 0 R +>> endobj +79 0 obj << +/Title 80 0 R +/A 77 0 R +/Parent 71 0 R +/Prev 75 0 R +/Next 83 0 R +>> endobj +75 0 obj << +/Title 76 0 R +/A 73 0 R +/Parent 71 0 R +/Next 79 0 R +>> endobj +71 0 obj << +/Title 72 0 R +/A 69 0 R +/Parent 27 0 R +/Prev 31 0 R +/Next 119 0 R +/First 75 0 R +/Last 115 0 R +/Count -11 +>> endobj +67 0 obj << +/Title 68 0 R +/A 65 0 R +/Parent 31 0 R +/Prev 63 0 R +>> endobj +63 0 obj << +/Title 64 0 R +/A 61 0 R +/Parent 31 0 R +/Prev 59 0 R +/Next 67 0 R +>> endobj +59 0 obj << +/Title 60 0 R +/A 57 0 R +/Parent 31 0 R +/Prev 55 0 R +/Next 63 0 R +>> endobj +55 0 obj << +/Title 56 0 R +/A 53 0 R +/Parent 31 0 R +/Prev 51 0 R +/Next 59 0 R +>> endobj +51 0 obj << +/Title 52 0 R +/A 49 0 R +/Parent 31 0 R +/Prev 47 0 R +/Next 55 0 R +>> endobj +47 0 obj << +/Title 48 0 R +/A 45 0 R +/Parent 31 0 R +/Prev 43 0 R +/Next 51 0 R +>> endobj +43 0 obj << +/Title 44 0 R +/A 41 0 R +/Parent 31 0 R +/Prev 39 0 R +/Next 47 0 R +>> endobj +39 0 obj << +/Title 40 0 R +/A 37 0 R +/Parent 31 0 R +/Prev 35 0 R +/Next 43 0 R +>> endobj +35 0 obj << +/Title 36 0 R +/A 33 0 R +/Parent 31 0 R +/Next 39 0 R +>> endobj +31 0 obj << +/Title 32 0 R +/A 29 0 R +/Parent 27 0 R +/Next 71 0 R +/First 35 0 R +/Last 67 0 R +/Count -9 +>> endobj +27 0 obj << +/Title 28 0 R +/A 25 0 R +/Parent 5927 0 R +/Prev 7 0 R +/Next 523 0 R +/First 31 0 R +/Last 499 0 R +/Count 9 +>> endobj +23 0 obj << +/Title 24 0 R +/A 21 0 R +/Parent 7 0 R +/Prev 19 0 R +>> endobj +19 0 obj << +/Title 20 0 R +/A 17 0 R +/Parent 7 0 R +/Prev 15 0 R +/Next 23 0 R +>> endobj +15 0 obj << +/Title 16 0 R +/A 13 0 R +/Parent 7 0 R +/Prev 11 0 R +/Next 19 0 R +>> endobj +11 0 obj << +/Title 12 0 R +/A 9 0 R +/Parent 7 0 R +/Next 15 0 R +>> endobj +7 0 obj << +/Title 8 0 R +/A 5 0 R +/Parent 5927 0 R +/Next 27 0 R +/First 11 0 R +/Last 23 0 R +/Count 4 +>> endobj +5928 0 obj << +/Names [(A.1) 1606 0 R (B.1) 1610 0 R (Back\040Matter.-1) 1598 0 R (Bibliography.0) 1602 0 R (C.1) 1614 0 R (Chapter\040I.0) 30 0 R] +/Limits [(A.1) (Chapter\040I.0)] +>> endobj +5929 0 obj << +/Names [(Chapter\040II.0) 70 0 R (Chapter\040III.0) 118 0 R (Chapter\040IV.0) 206 0 R (Chapter\040IX.0) 498 0 R (Chapter\040V.0) 314 0 R (Chapter\040VI.0) 338 0 R] +/Limits [(Chapter\040II.0) (Chapter\040VI.0)] +>> endobj +5930 0 obj << +/Names [(Chapter\040VII.0) 382 0 R (Chapter\040VIII.0) 426 0 R (Chapter\040X.0) 526 0 R (Chapter\040XI.0) 558 0 R (Chapter\040XII.0) 618 0 R (Chapter\040XIII.0) 654 0 R] +/Limits [(Chapter\040VII.0) (Chapter\040XIII.0)] +>> endobj +5931 0 obj << +/Names [(Chapter\040XIV.0) 666 0 R (Chapter\040XIX.0) 906 0 R (Chapter\040XV.0) 702 0 R (Chapter\040XVI.0) 734 0 R (Chapter\040XVII.0) 818 0 R (Chapter\040XVIII.0) 874 0 R] +/Limits [(Chapter\040XIV.0) (Chapter\040XVIII.0)] +>> endobj +5932 0 obj << +/Names [(Chapter\040XX.0) 958 0 R (Chapter\040XXI.0) 998 0 R (Chapter\040XXII.0) 1026 0 R (Chapter\040XXIII.0) 1086 0 R (Chapter\040XXIV.0) 1166 0 R (Chapter\040XXIX.0) 1346 0 R] +/Limits [(Chapter\040XX.0) (Chapter\040XXIX.0)] +>> endobj +5933 0 obj << +/Names [(Chapter\040XXV.0) 1218 0 R (Chapter\040XXVI.0) 1250 0 R (Chapter\040XXVII.0) 1310 0 R (Chapter\040XXVIII.0) 1334 0 R (Chapter\040XXX.0) 1426 0 R (Chapter\040XXXI.0) 1486 0 R] +/Limits [(Chapter\040XXV.0) (Chapter\040XXXI.0)] +>> endobj +5934 0 obj << +/Names [(Chapter\040XXXII.0) 1514 0 R (Chapter\040XXXIII.0) 1554 0 R (D.1) 1618 0 R (Doc-Start) 1710 0 R (E.1) 1622 0 R (F.1) 1626 0 R] +/Limits [(Chapter\040XXXII.0) (F.1)] +>> endobj +5935 0 obj << +/Names [(Front\040Matter.-1) 6 0 R (G.1) 1630 0 R (H.1) 1634 0 R (I.1..1) 34 0 R (I.2..1) 38 0 R (I.3..1) 42 0 R] +/Limits [(Front\040Matter.-1) (I.3..1)] +>> endobj +5936 0 obj << +/Names [(I.4..1) 46 0 R (I.5..1) 50 0 R (I.6..1) 54 0 R (I.7..1) 58 0 R (I.8..1) 62 0 R (I.9..1) 66 0 R] +/Limits [(I.4..1) (I.9..1)] +>> endobj +5937 0 obj << +/Names [(II.1..1) 74 0 R (II.10..1) 110 0 R (II.11..1) 114 0 R (II.2..1) 78 0 R (II.3..1) 82 0 R (II.4..1) 86 0 R] +/Limits [(II.1..1) (II.4..1)] +>> endobj +5938 0 obj << +/Names [(II.5..1) 90 0 R (II.6..1) 94 0 R (II.7..1) 98 0 R (II.8..1) 102 0 R (II.9..1) 106 0 R (III.1..1) 122 0 R] +/Limits [(II.5..1) (III.1..1)] +>> endobj +5939 0 obj << +/Names [(III.10..1) 158 0 R (III.11..1) 162 0 R (III.12..1) 166 0 R (III.13..1) 170 0 R (III.14..1) 174 0 R (III.15..1) 178 0 R] +/Limits [(III.10..1) (III.15..1)] +>> endobj +5940 0 obj << +/Names [(III.16..1) 182 0 R (III.17..1) 186 0 R (III.18..1) 190 0 R (III.19..1) 194 0 R (III.2..1) 126 0 R (III.20..1) 198 0 R] +/Limits [(III.16..1) (III.20..1)] +>> endobj +5941 0 obj << +/Names [(III.21..1) 202 0 R (III.3..1) 130 0 R (III.4..1) 134 0 R (III.5..1) 138 0 R (III.6..1) 142 0 R (III.7..1) 146 0 R] +/Limits [(III.21..1) (III.7..1)] +>> endobj +5942 0 obj << +/Names [(III.8..1) 150 0 R (III.9..1) 154 0 R (IV.1..1) 210 0 R (IV.10..1) 246 0 R (IV.11..1) 250 0 R (IV.12..1) 254 0 R] +/Limits [(III.8..1) (IV.12..1)] +>> endobj +5943 0 obj << +/Names [(IV.13..1) 258 0 R (IV.14..1) 262 0 R (IV.15..1) 266 0 R (IV.16..1) 270 0 R (IV.17..1) 274 0 R (IV.18..1) 278 0 R] +/Limits [(IV.13..1) (IV.18..1)] +>> endobj +5944 0 obj << +/Names [(IV.19..1) 282 0 R (IV.2..1) 214 0 R (IV.20..1) 286 0 R (IV.21..1) 290 0 R (IV.22..1) 294 0 R (IV.23..1) 298 0 R] +/Limits [(IV.19..1) (IV.23..1)] +>> endobj +5945 0 obj << +/Names [(IV.24..1) 302 0 R (IV.25..1) 306 0 R (IV.26..1) 310 0 R (IV.3..1) 218 0 R (IV.4..1) 222 0 R (IV.5..1) 226 0 R] +/Limits [(IV.24..1) (IV.5..1)] +>> endobj +5946 0 obj << +/Names [(IV.6..1) 230 0 R (IV.7..1) 234 0 R (IV.8..1) 238 0 R (IV.9..1) 242 0 R (IX.1..1) 502 0 R (IX.2..1) 506 0 R] +/Limits [(IV.6..1) (IX.2..1)] +>> endobj +5947 0 obj << +/Names [(IX.3..1) 510 0 R (IX.4..1) 514 0 R (IX.5..1) 518 0 R (Index.0) 1698 0 R (J.1) 1638 0 R (K.1) 1642 0 R] +/Limits [(IX.3..1) (K.1)] +>> endobj +5948 0 obj << +/Names [(L.1) 1646 0 R (License.0) 1702 0 R (M.1) 1650 0 R (N.1) 1654 0 R (Notes\040i.0) 1134 0 R (O.1) 1658 0 R] +/Limits [(L.1) (O.1)] +>> endobj +5949 0 obj << +/Names [(P.1) 1662 0 R (Part\040I.-1) 26 0 R (Part\040II.-1) 522 0 R (Part\040III.-1) 870 0 R (Part\040IV.-1) 1162 0 R (Part\040V.-1) 1306 0 R] +/Limits [(P.1) (Part\040V.-1)] +>> endobj +5950 0 obj << +/Names [(Preface.0) 18 0 R (Project\040Gutenberg\040Boilerplate.0) 10 0 R (Q.1) 1666 0 R (R.1) 1670 0 R (S.1) 1674 0 R (T.1) 1678 0 R] +/Limits [(Preface.0) (T.1)] +>> endobj +5951 0 obj << +/Names [(Table\040of\040Contents.0) 22 0 R (Transcriber's\040Note.0) 14 0 R (U.1) 1682 0 R (V.1) 1686 0 R (V.1..1) 318 0 R (V.2..1) 322 0 R] +/Limits [(Table\040of\040Contents.0) (V.2..1)] +>> endobj +5952 0 obj << +/Names [(V.3..1) 326 0 R (V.4..1) 330 0 R (V.5..1) 334 0 R (VI.1..1) 342 0 R (VI.10..1) 378 0 R (VI.2..1) 346 0 R] +/Limits [(V.3..1) (VI.2..1)] +>> endobj +5953 0 obj << +/Names [(VI.3..1) 350 0 R (VI.4..1) 354 0 R (VI.5..1) 358 0 R (VI.6..1) 362 0 R (VI.7..1) 366 0 R (VI.8..1) 370 0 R] +/Limits [(VI.3..1) (VI.8..1)] +>> endobj +5954 0 obj << +/Names [(VI.9..1) 374 0 R (VII.1..1) 386 0 R (VII.10..1) 422 0 R (VII.2..1) 390 0 R (VII.3..1) 394 0 R (VII.4..1) 398 0 R] +/Limits [(VI.9..1) (VII.4..1)] +>> endobj +5955 0 obj << +/Names [(VII.5..1) 402 0 R (VII.6..1) 406 0 R (VII.7..1) 410 0 R (VII.8..1) 414 0 R (VII.9..1) 418 0 R (VIII.1..1) 430 0 R] +/Limits [(VII.5..1) (VIII.1..1)] +>> endobj +5956 0 obj << +/Names [(VIII.10..1) 466 0 R (VIII.11..1) 470 0 R (VIII.12..1) 474 0 R (VIII.13..1) 478 0 R (VIII.14..1) 482 0 R (VIII.15..1) 486 0 R] +/Limits [(VIII.10..1) (VIII.15..1)] +>> endobj +5957 0 obj << +/Names [(VIII.16..1) 490 0 R (VIII.17..1) 494 0 R (VIII.2..1) 434 0 R (VIII.3..1) 438 0 R (VIII.4..1) 442 0 R (VIII.5..1) 446 0 R] +/Limits [(VIII.16..1) (VIII.5..1)] +>> endobj +5958 0 obj << +/Names [(VIII.6..1) 450 0 R (VIII.7..1) 454 0 R (VIII.8..1) 458 0 R (VIII.9..1) 462 0 R (W.1) 1690 0 R (X.1..1) 530 0 R] +/Limits [(VIII.6..1) (X.1..1)] +>> endobj +5959 0 obj << +/Names [(X.2..1) 534 0 R (X.3..1) 538 0 R (X.4..1) 542 0 R (X.5..1) 546 0 R (X.6..1) 550 0 R (X.7..1) 554 0 R] +/Limits [(X.2..1) (X.7..1)] +>> endobj +5960 0 obj << +/Names [(XI.1..1) 562 0 R (XI.10..1) 598 0 R (XI.11..1) 602 0 R (XI.12..1) 606 0 R (XI.13..1) 610 0 R (XI.14..1) 614 0 R] +/Limits [(XI.1..1) (XI.14..1)] +>> endobj +5961 0 obj << +/Names [(XI.2..1) 566 0 R (XI.3..1) 570 0 R (XI.4..1) 574 0 R (XI.5..1) 578 0 R (XI.6..1) 582 0 R (XI.7..1) 586 0 R] +/Limits [(XI.2..1) (XI.7..1)] +>> endobj +5962 0 obj << +/Names [(XI.8..1) 590 0 R (XI.9..1) 594 0 R (XII.1..1) 622 0 R (XII.2..1) 626 0 R (XII.3..1) 630 0 R (XII.4..1) 634 0 R] +/Limits [(XI.8..1) (XII.4..1)] +>> endobj +5963 0 obj << +/Names [(XII.5..1) 638 0 R (XII.6..1) 642 0 R (XII.7..1) 646 0 R (XII.8..1) 650 0 R (XIII.1..1) 658 0 R (XIII.2..1) 662 0 R] +/Limits [(XII.5..1) (XIII.2..1)] +>> endobj +5964 0 obj << +/Names [(XIV.1..1) 670 0 R (XIV.2..1) 674 0 R (XIV.3..1) 678 0 R (XIV.4..1) 682 0 R (XIV.5..1) 686 0 R (XIV.6..1) 690 0 R] +/Limits [(XIV.1..1) (XIV.6..1)] +>> endobj +5965 0 obj << +/Names [(XIV.7..1) 694 0 R (XIV.8..1) 698 0 R (XIX.1..1) 910 0 R (XIX.10..1) 946 0 R (XIX.11..1) 950 0 R (XIX.12..1) 954 0 R] +/Limits [(XIV.7..1) (XIX.12..1)] +>> endobj +5966 0 obj << +/Names [(XIX.2..1) 914 0 R (XIX.3..1) 918 0 R (XIX.4..1) 922 0 R (XIX.5..1) 926 0 R (XIX.6..1) 930 0 R (XIX.7..1) 934 0 R] +/Limits [(XIX.2..1) (XIX.7..1)] +>> endobj +5967 0 obj << +/Names [(XIX.8..1) 938 0 R (XIX.9..1) 942 0 R (XV.1..1) 706 0 R (XV.2..1) 710 0 R (XV.3..1) 714 0 R (XV.4..1) 718 0 R] +/Limits [(XIX.8..1) (XV.4..1)] +>> endobj +5968 0 obj << +/Names [(XV.5..1) 722 0 R (XV.6..1) 726 0 R (XV.7..1) 730 0 R (XVI.1..1) 738 0 R (XVI.10..1) 774 0 R (XVI.11..1) 778 0 R] +/Limits [(XV.5..1) (XVI.11..1)] +>> endobj +5969 0 obj << +/Names [(XVI.12..1) 782 0 R (XVI.13..1) 786 0 R (XVI.14..1) 790 0 R (XVI.15..1) 794 0 R (XVI.16..1) 798 0 R (XVI.17..1) 802 0 R] +/Limits [(XVI.12..1) (XVI.17..1)] +>> endobj +5970 0 obj << +/Names [(XVI.18..1) 806 0 R (XVI.19..1) 810 0 R (XVI.2..1) 742 0 R (XVI.20..1) 814 0 R (XVI.3..1) 746 0 R (XVI.4..1) 750 0 R] +/Limits [(XVI.18..1) (XVI.4..1)] +>> endobj +5971 0 obj << +/Names [(XVI.5..1) 754 0 R (XVI.6..1) 758 0 R (XVI.7..1) 762 0 R (XVI.8..1) 766 0 R (XVI.9..1) 770 0 R (XVII.1..1) 822 0 R] +/Limits [(XVI.5..1) (XVII.1..1)] +>> endobj +5972 0 obj << +/Names [(XVII.10..1) 858 0 R (XVII.11..1) 862 0 R (XVII.12..1) 866 0 R (XVII.2..1) 826 0 R (XVII.3..1) 830 0 R (XVII.4..1) 834 0 R] +/Limits [(XVII.10..1) (XVII.4..1)] +>> endobj +5973 0 obj << +/Names [(XVII.5..1) 838 0 R (XVII.6..1) 842 0 R (XVII.7..1) 846 0 R (XVII.8..1) 850 0 R (XVII.9..1) 854 0 R (XVIII.1..1) 878 0 R] +/Limits [(XVII.5..1) (XVIII.1..1)] +>> endobj +5974 0 obj << +/Names [(XVIII.2..1) 882 0 R (XVIII.3..1) 886 0 R (XVIII.4..1) 890 0 R (XVIII.5..1) 894 0 R (XVIII.6..1) 898 0 R (XVIII.7..1) 902 0 R] +/Limits [(XVIII.2..1) (XVIII.7..1)] +>> endobj +5975 0 obj << +/Names [(XX.1..1) 962 0 R (XX.2..1) 966 0 R (XX.3..1) 970 0 R (XX.4..1) 974 0 R (XX.5..1) 978 0 R (XX.6..1) 982 0 R] +/Limits [(XX.1..1) (XX.6..1)] +>> endobj +5976 0 obj << +/Names [(XX.7..1) 986 0 R (XX.8..1) 990 0 R (XX.9..1) 994 0 R (XXI.1..1) 1002 0 R (XXI.2..1) 1006 0 R (XXI.3..1) 1010 0 R] +/Limits [(XX.7..1) (XXI.3..1)] +>> endobj +5977 0 obj << +/Names [(XXI.4..1) 1014 0 R (XXI.5..1) 1018 0 R (XXI.6..1) 1022 0 R (XXII.1..1) 1030 0 R (XXII.10..1) 1066 0 R (XXII.11..1) 1070 0 R] +/Limits [(XXI.4..1) (XXII.11..1)] +>> endobj +5978 0 obj << +/Names [(XXII.12..1) 1074 0 R (XXII.13..1) 1078 0 R (XXII.14..1) 1082 0 R (XXII.2..1) 1034 0 R (XXII.3..1) 1038 0 R (XXII.4..1) 1042 0 R] +/Limits [(XXII.12..1) (XXII.4..1)] +>> endobj +5979 0 obj << +/Names [(XXII.5..1) 1046 0 R (XXII.6..1) 1050 0 R (XXII.7..1) 1054 0 R (XXII.8..1) 1058 0 R (XXII.9..1) 1062 0 R (XXIII.1..1) 1090 0 R] +/Limits [(XXII.5..1) (XXIII.1..1)] +>> endobj +5980 0 obj << +/Names [(XXIII.10..1) 1126 0 R (XXIII.11..1) 1130 0 R (XXIII.2..1) 1094 0 R (XXIII.3..1) 1098 0 R (XXIII.4..1) 1102 0 R (XXIII.5..1) 1106 0 R] +/Limits [(XXIII.10..1) (XXIII.5..1)] +>> endobj +5981 0 obj << +/Names [(XXIII.6..1) 1110 0 R (XXIII.7..1) 1114 0 R (XXIII.8..1) 1118 0 R (XXIII.9..1) 1122 0 R (XXIV.1..1) 1170 0 R (XXIV.10..1) 1206 0 R] +/Limits [(XXIII.6..1) (XXIV.10..1)] +>> endobj +5982 0 obj << +/Names [(XXIV.11..1) 1210 0 R (XXIV.12..1) 1214 0 R (XXIV.2..1) 1174 0 R (XXIV.3..1) 1178 0 R (XXIV.4..1) 1182 0 R (XXIV.5..1) 1186 0 R] +/Limits [(XXIV.11..1) (XXIV.5..1)] +>> endobj +5983 0 obj << +/Names [(XXIV.6..1) 1190 0 R (XXIV.7..1) 1194 0 R (XXIV.8..1) 1198 0 R (XXIV.9..1) 1202 0 R (XXIX.1..1) 1350 0 R (XXIX.10..1) 1386 0 R] +/Limits [(XXIV.6..1) (XXIX.10..1)] +>> endobj +5984 0 obj << +/Names [(XXIX.11..1) 1390 0 R (XXIX.12..1) 1394 0 R (XXIX.13..1) 1398 0 R (XXIX.14..1) 1402 0 R (XXIX.15..1) 1406 0 R (XXIX.16..1) 1410 0 R] +/Limits [(XXIX.11..1) (XXIX.16..1)] +>> endobj +5985 0 obj << +/Names [(XXIX.17..1) 1414 0 R (XXIX.18..1) 1418 0 R (XXIX.19..1) 1422 0 R (XXIX.2..1) 1354 0 R (XXIX.3..1) 1358 0 R (XXIX.4..1) 1362 0 R] +/Limits [(XXIX.17..1) (XXIX.4..1)] +>> endobj +5986 0 obj << +/Names [(XXIX.5..1) 1366 0 R (XXIX.6..1) 1370 0 R (XXIX.7..1) 1374 0 R (XXIX.8..1) 1378 0 R (XXIX.9..1) 1382 0 R (XXV.1..1) 1222 0 R] +/Limits [(XXIX.5..1) (XXV.1..1)] +>> endobj +5987 0 obj << +/Names [(XXV.2..1) 1226 0 R (XXV.3..1) 1230 0 R (XXV.4..1) 1234 0 R (XXV.5..1) 1238 0 R (XXV.6..1) 1242 0 R (XXV.7..1) 1246 0 R] +/Limits [(XXV.2..1) (XXV.7..1)] +>> endobj +5988 0 obj << +/Names [(XXVI.1..1) 1254 0 R (XXVI.10..1) 1290 0 R (XXVI.11..1) 1294 0 R (XXVI.12..1) 1298 0 R (XXVI.13..1) 1302 0 R (XXVI.2..1) 1258 0 R] +/Limits [(XXVI.1..1) (XXVI.2..1)] +>> endobj +5989 0 obj << +/Names [(XXVI.3..1) 1262 0 R (XXVI.4..1) 1266 0 R (XXVI.5..1) 1270 0 R (XXVI.6..1) 1274 0 R (XXVI.7..1) 1278 0 R (XXVI.8..1) 1282 0 R] +/Limits [(XXVI.3..1) (XXVI.8..1)] +>> endobj +5990 0 obj << +/Names [(XXVI.9..1) 1286 0 R (XXVII.1..1) 1314 0 R (XXVII.2..1) 1318 0 R (XXVII.3..1) 1322 0 R (XXVII.4..1) 1326 0 R (XXVII.5..1) 1330 0 R] +/Limits [(XXVI.9..1) (XXVII.5..1)] +>> endobj +5991 0 obj << +/Names [(XXVIII.1..1) 1338 0 R (XXVIII.2..1) 1342 0 R (XXX.1..1) 1430 0 R (XXX.10..1) 1466 0 R (XXX.11..1) 1470 0 R (XXX.12..1) 1474 0 R] +/Limits [(XXVIII.1..1) (XXX.12..1)] +>> endobj +5992 0 obj << +/Names [(XXX.13..1) 1478 0 R (XXX.14..1) 1482 0 R (XXX.2..1) 1434 0 R (XXX.3..1) 1438 0 R (XXX.4..1) 1442 0 R (XXX.5..1) 1446 0 R] +/Limits [(XXX.13..1) (XXX.5..1)] +>> endobj +5993 0 obj << +/Names [(XXX.6..1) 1450 0 R (XXX.7..1) 1454 0 R (XXX.8..1) 1458 0 R (XXX.9..1) 1462 0 R (XXXI.1..1) 1490 0 R (XXXI.2..1) 1494 0 R] +/Limits [(XXX.6..1) (XXXI.2..1)] +>> endobj +5994 0 obj << +/Names [(XXXI.3..1) 1498 0 R (XXXI.4..1) 1502 0 R (XXXI.5..1) 1506 0 R (XXXI.6..1) 1510 0 R (XXXII.1..1) 1518 0 R (XXXII.2..1) 1522 0 R] +/Limits [(XXXI.3..1) (XXXII.2..1)] +>> endobj +5995 0 obj << +/Names [(XXXII.3..1) 1526 0 R (XXXII.4..1) 1530 0 R (XXXII.5..1) 1534 0 R (XXXII.6..1) 1538 0 R (XXXII.7..1) 1542 0 R (XXXII.8..1) 1546 0 R] +/Limits [(XXXII.3..1) (XXXII.8..1)] +>> endobj +5996 0 obj << +/Names [(XXXII.9..1) 1550 0 R (XXXIII.1..1) 1558 0 R (XXXIII.10..1) 1594 0 R (XXXIII.2..1) 1562 0 R (XXXIII.3..1) 1566 0 R (XXXIII.4..1) 1570 0 R] +/Limits [(XXXII.9..1) (XXXIII.4..1)] +>> endobj +5997 0 obj << +/Names [(XXXIII.5..1) 1574 0 R (XXXIII.6..1) 1578 0 R (XXXIII.7..1) 1582 0 R (XXXIII.8..1) 1586 0 R (XXXIII.9..1) 1590 0 R (Y.1) 1694 0 R] +/Limits [(XXXIII.5..1) (Y.1)] +>> endobj +5998 0 obj << +/Names [(chapter*.8) 1770 0 R (note.i.1..1) 1138 0 R (note.i.2..1) 1142 0 R (note.ii.1..1) 1146 0 R (note.ii.2..1) 1150 0 R (note.ii.3..1) 1154 0 R] +/Limits [(chapter*.8) (note.ii.3..1)] +>> endobj +5999 0 obj << +/Names [(note.ii.4..1) 1158 0 R (page.1) 1902 0 R (page.10) 1961 0 R (page.100) 2452 0 R (page.101) 2457 0 R (page.102) 2462 0 R] +/Limits [(note.ii.4..1) (page.102)] +>> endobj +6000 0 obj << +/Names [(page.103) 2471 0 R (page.104) 2476 0 R (page.105) 2480 0 R (page.106) 2485 0 R (page.107) 2490 0 R (page.108) 2494 0 R] +/Limits [(page.103) (page.108)] +>> endobj +6001 0 obj << +/Names [(page.109) 2501 0 R (page.11) 1967 0 R (page.110) 2505 0 R (page.111) 2510 0 R (page.112) 2515 0 R (page.113) 2520 0 R] +/Limits [(page.109) (page.113)] +>> endobj +6002 0 obj << +/Names [(page.114) 2524 0 R (page.115) 2530 0 R (page.116) 2535 0 R (page.117) 2540 0 R (page.118) 2545 0 R (page.119) 2550 0 R] +/Limits [(page.114) (page.119)] +>> endobj +6003 0 obj << +/Names [(page.12) 1972 0 R (page.120) 2555 0 R (page.121) 2561 0 R (page.122) 2567 0 R (page.123) 2571 0 R (page.124) 2586 0 R] +/Limits [(page.12) (page.124)] +>> endobj +6004 0 obj << +/Names [(page.125) 2596 0 R (page.126) 2603 0 R (page.127) 2611 0 R (page.128) 2616 0 R (page.129) 2620 0 R (page.13) 1978 0 R] +/Limits [(page.125) (page.13)] +>> endobj +6005 0 obj << +/Names [(page.130) 2624 0 R (page.131) 2629 0 R (page.132) 2634 0 R (page.133) 2641 0 R (page.134) 2646 0 R (page.135) 2650 0 R] +/Limits [(page.130) (page.135)] +>> endobj +6006 0 obj << +/Names [(page.136) 2659 0 R (page.137) 2665 0 R (page.138) 2670 0 R (page.139) 2676 0 R (page.14) 1983 0 R (page.140) 2681 0 R] +/Limits [(page.136) (page.140)] +>> endobj +6007 0 obj << +/Names [(page.141) 2686 0 R (page.142) 2691 0 R (page.143) 2697 0 R (page.144) 2702 0 R (page.145) 2708 0 R (page.146) 2714 0 R] +/Limits [(page.141) (page.146)] +>> endobj +6008 0 obj << +/Names [(page.147) 2723 0 R (page.148) 2729 0 R (page.149) 2735 0 R (page.15) 1988 0 R (page.150) 2739 0 R (page.151) 2744 0 R] +/Limits [(page.147) (page.151)] +>> endobj +6009 0 obj << +/Names [(page.152) 2749 0 R (page.153) 2754 0 R (page.154) 2762 0 R (page.155) 2766 0 R (page.156) 2770 0 R (page.157) 2775 0 R] +/Limits [(page.152) (page.157)] +>> endobj +6010 0 obj << +/Names [(page.158) 2779 0 R (page.159) 2783 0 R (page.16) 1993 0 R (page.160) 2787 0 R (page.161) 2791 0 R (page.162) 2799 0 R] +/Limits [(page.158) (page.162)] +>> endobj +6011 0 obj << +/Names [(page.163) 2804 0 R (page.164) 2808 0 R (page.165) 2813 0 R (page.166) 2819 0 R (page.167) 2824 0 R (page.168) 2829 0 R] +/Limits [(page.163) (page.168)] +>> endobj +6012 0 obj << +/Names [(page.169) 2835 0 R (page.17) 1997 0 R (page.170) 2840 0 R (page.171) 2844 0 R (page.172) 2848 0 R (page.173) 2854 0 R] +/Limits [(page.169) (page.173)] +>> endobj +6013 0 obj << +/Names [(page.174) 2859 0 R (page.175) 2864 0 R (page.176) 2868 0 R (page.177) 2873 0 R (page.178) 2877 0 R (page.179) 2881 0 R] +/Limits [(page.174) (page.179)] +>> endobj +6014 0 obj << +/Names [(page.18) 2002 0 R (page.180) 2885 0 R (page.181) 2894 0 R (page.182) 2900 0 R (page.183) 2905 0 R (page.184) 2910 0 R] +/Limits [(page.18) (page.184)] +>> endobj +6015 0 obj << +/Names [(page.185) 2914 0 R (page.186) 2919 0 R (page.187) 2924 0 R (page.188) 2928 0 R (page.189) 2937 0 R (page.19) 2007 0 R] +/Limits [(page.185) (page.19)] +>> endobj +6016 0 obj << +/Names [(page.190) 2942 0 R (page.191) 2948 0 R (page.192) 2953 0 R (page.193) 2958 0 R (page.194) 2963 0 R (page.195) 2967 0 R] +/Limits [(page.190) (page.195)] +>> endobj +6017 0 obj << +/Names [(page.196) 2973 0 R (page.197) 2977 0 R (page.198) 2983 0 R (page.199) 2989 0 R (page.2) 1908 0 R (page.20) 2011 0 R] +/Limits [(page.196) (page.20)] +>> endobj +6018 0 obj << +/Names [(page.200) 2993 0 R (page.201) 2998 0 R (page.202) 3004 0 R (page.203) 3009 0 R (page.204) 3015 0 R (page.205) 3021 0 R] +/Limits [(page.200) (page.205)] +>> endobj +6019 0 obj << +/Names [(page.206) 3026 0 R (page.207) 3030 0 R (page.208) 3035 0 R (page.209) 3040 0 R (page.21) 2019 0 R (page.210) 3044 0 R] +/Limits [(page.206) (page.210)] +>> endobj +6020 0 obj << +/Names [(page.211) 3050 0 R (page.212) 3055 0 R (page.213) 3061 0 R (page.214) 3065 0 R (page.215) 3073 0 R (page.216) 3077 0 R] +/Limits [(page.211) (page.216)] +>> endobj +6021 0 obj << +/Names [(page.217) 3082 0 R (page.218) 3086 0 R (page.219) 3091 0 R (page.22) 2023 0 R (page.220) 3095 0 R (page.221) 3099 0 R] +/Limits [(page.217) (page.221)] +>> endobj +6022 0 obj << +/Names [(page.222) 3103 0 R (page.223) 3108 0 R (page.224) 3112 0 R (page.225) 3118 0 R (page.226) 3122 0 R (page.227) 3126 0 R] +/Limits [(page.222) (page.227)] +>> endobj +6023 0 obj << +/Names [(page.228) 3131 0 R (page.229) 3137 0 R (page.23) 2029 0 R (page.230) 3142 0 R (page.231) 3146 0 R (page.232) 3150 0 R] +/Limits [(page.228) (page.232)] +>> endobj +6024 0 obj << +/Names [(page.233) 3154 0 R (page.234) 3158 0 R (page.235) 3163 0 R (page.236) 3168 0 R (page.237) 3173 0 R (page.238) 3178 0 R] +/Limits [(page.233) (page.238)] +>> endobj +6025 0 obj << +/Names [(page.239) 3183 0 R (page.24) 2035 0 R (page.240) 3187 0 R (page.241) 3192 0 R (page.242) 3196 0 R (page.243) 3201 0 R] +/Limits [(page.239) (page.243)] +>> endobj +6026 0 obj << +/Names [(page.244) 3205 0 R (page.245) 3210 0 R (page.246) 3214 0 R (page.247) 3219 0 R (page.248) 3224 0 R (page.249) 3228 0 R] +/Limits [(page.244) (page.249)] +>> endobj +6027 0 obj << +/Names [(page.25) 2041 0 R (page.250) 3232 0 R (page.251) 3239 0 R (page.252) 3245 0 R (page.253) 3251 0 R (page.254) 3258 0 R] +/Limits [(page.25) (page.254)] +>> endobj +6028 0 obj << +/Names [(page.255) 3263 0 R (page.256) 3267 0 R (page.257) 3275 0 R (page.258) 3279 0 R (page.259) 3289 0 R (page.26) 2045 0 R] +/Limits [(page.255) (page.26)] +>> endobj +6029 0 obj << +/Names [(page.260) 3293 0 R (page.261) 3298 0 R (page.262) 3303 0 R (page.263) 3307 0 R (page.264) 3312 0 R (page.265) 3319 0 R] +/Limits [(page.260) (page.265)] +>> endobj +6030 0 obj << +/Names [(page.266) 3324 0 R (page.267) 3329 0 R (page.268) 3333 0 R (page.269) 3338 0 R (page.27) 2049 0 R (page.270) 3345 0 R] +/Limits [(page.266) (page.270)] +>> endobj +6031 0 obj << +/Names [(page.271) 3351 0 R (page.272) 3356 0 R (page.273) 3361 0 R (page.274) 3366 0 R (page.275) 3371 0 R (page.276) 3376 0 R] +/Limits [(page.271) (page.276)] +>> endobj +6032 0 obj << +/Names [(page.277) 3382 0 R (page.278) 3387 0 R (page.279) 3391 0 R (page.28) 2053 0 R (page.280) 3398 0 R (page.281) 3402 0 R] +/Limits [(page.277) (page.281)] +>> endobj +6033 0 obj << +/Names [(page.282) 3407 0 R (page.283) 3413 0 R (page.284) 3418 0 R (page.285) 3422 0 R (page.286) 3427 0 R (page.287) 3432 0 R] +/Limits [(page.282) (page.287)] +>> endobj +6034 0 obj << +/Names [(page.288) 3436 0 R (page.289) 3441 0 R (page.29) 2058 0 R (page.290) 3448 0 R (page.291) 3453 0 R (page.292) 3458 0 R] +/Limits [(page.288) (page.292)] +>> endobj +6035 0 obj << +/Names [(page.293) 3462 0 R (page.294) 3467 0 R (page.295) 3473 0 R (page.296) 3479 0 R (page.297) 3485 0 R (page.298) 3490 0 R] +/Limits [(page.293) (page.298)] +>> endobj +6036 0 obj << +/Names [(page.299) 3500 0 R (page.3) 1916 0 R (page.30) 2063 0 R (page.300) 3504 0 R (page.301) 3510 0 R (page.302) 3515 0 R] +/Limits [(page.299) (page.302)] +>> endobj +6037 0 obj << +/Names [(page.303) 3519 0 R (page.304) 3523 0 R (page.305) 3528 0 R (page.306) 3535 0 R (page.307) 3541 0 R (page.308) 3546 0 R] +/Limits [(page.303) (page.308)] +>> endobj +6038 0 obj << +/Names [(page.309) 3551 0 R (page.31) 2068 0 R (page.310) 3555 0 R (page.311) 3560 0 R (page.312) 3565 0 R (page.313) 3572 0 R] +/Limits [(page.309) (page.313)] +>> endobj +6039 0 obj << +/Names [(page.314) 3577 0 R (page.315) 3583 0 R (page.316) 3593 0 R (page.317) 3600 0 R (page.318) 3606 0 R (page.319) 3611 0 R] +/Limits [(page.314) (page.319)] +>> endobj +6040 0 obj << +/Names [(page.32) 2073 0 R (page.320) 3616 0 R (page.321) 3621 0 R (page.322) 3628 0 R (page.323) 3633 0 R (page.324) 3638 0 R] +/Limits [(page.32) (page.324)] +>> endobj +6041 0 obj << +/Names [(page.325) 3644 0 R (page.326) 3649 0 R (page.327) 3653 0 R (page.328) 3658 0 R (page.329) 3666 0 R (page.33) 2080 0 R] +/Limits [(page.325) (page.33)] +>> endobj +6042 0 obj << +/Names [(page.330) 3671 0 R (page.331) 3677 0 R (page.332) 3683 0 R (page.333) 3687 0 R (page.334) 3692 0 R (page.335) 3698 0 R] +/Limits [(page.330) (page.335)] +>> endobj +6043 0 obj << +/Names [(page.336) 3705 0 R (page.337) 3714 0 R (page.338) 3718 0 R (page.339) 3722 0 R (page.34) 2084 0 R (page.340) 3728 0 R] +/Limits [(page.336) (page.340)] +>> endobj +6044 0 obj << +/Names [(page.341) 3733 0 R (page.342) 3740 0 R (page.343) 3746 0 R (page.344) 3750 0 R (page.345) 3754 0 R (page.346) 3759 0 R] +/Limits [(page.341) (page.346)] +>> endobj +6045 0 obj << +/Names [(page.347) 3763 0 R (page.348) 3768 0 R (page.349) 3773 0 R (page.35) 2090 0 R (page.350) 3777 0 R (page.351) 3781 0 R] +/Limits [(page.347) (page.351)] +>> endobj +6046 0 obj << +/Names [(page.352) 3790 0 R (page.353) 3794 0 R (page.354) 3799 0 R (page.355) 3804 0 R (page.356) 3811 0 R (page.357) 3817 0 R] +/Limits [(page.352) (page.357)] +>> endobj +6047 0 obj << +/Names [(page.358) 3823 0 R (page.359) 3828 0 R (page.36) 2094 0 R (page.360) 3832 0 R (page.361) 3837 0 R (page.362) 3841 0 R] +/Limits [(page.358) (page.362)] +>> endobj +6048 0 obj << +/Names [(page.363) 3847 0 R (page.364) 3851 0 R (page.365) 3855 0 R (page.366) 3860 0 R (page.367) 3866 0 R (page.368) 3872 0 R] +/Limits [(page.363) (page.368)] +>> endobj +6049 0 obj << +/Names [(page.369) 3877 0 R (page.37) 2100 0 R (page.370) 3882 0 R (page.371) 3886 0 R (page.372) 3894 0 R (page.373) 3899 0 R] +/Limits [(page.369) (page.373)] +>> endobj +6050 0 obj << +/Names [(page.374) 3903 0 R (page.375) 3909 0 R (page.376) 3914 0 R (page.377) 3923 0 R (page.378) 3928 0 R (page.379) 3933 0 R] +/Limits [(page.374) (page.379)] +>> endobj +6051 0 obj << +/Names [(page.38) 2105 0 R (page.380) 3937 0 R (page.381) 3942 0 R (page.382) 3946 0 R (page.383) 3953 0 R (page.384) 3958 0 R] +/Limits [(page.38) (page.384)] +>> endobj +6052 0 obj << +/Names [(page.385) 3963 0 R (page.386) 3968 0 R (page.387) 3974 0 R (page.388) 3979 0 R (page.389) 3984 0 R (page.39) 2111 0 R] +/Limits [(page.385) (page.39)] +>> endobj +6053 0 obj << +/Names [(page.390) 3991 0 R (page.391) 3999 0 R (page.392) 4003 0 R (page.393) 4008 0 R (page.394) 4012 0 R (page.395) 4017 0 R] +/Limits [(page.390) (page.395)] +>> endobj +6054 0 obj << +/Names [(page.396) 4021 0 R (page.397) 4027 0 R (page.398) 4031 0 R (page.399) 4035 0 R (page.4) 1925 0 R (page.40) 2119 0 R] +/Limits [(page.396) (page.40)] +>> endobj +6055 0 obj << +/Names [(page.400) 4041 0 R (page.401) 4045 0 R (page.402) 4050 0 R (page.403) 4055 0 R (page.404) 4059 0 R (page.405) 4064 0 R] +/Limits [(page.400) (page.405)] +>> endobj +6056 0 obj << +/Names [(page.406) 4068 0 R (page.407) 4072 0 R (page.408) 4078 0 R (page.409) 4083 0 R (page.41) 2126 0 R (page.410) 4087 0 R] +/Limits [(page.406) (page.410)] +>> endobj +6057 0 obj << +/Names [(page.411) 4091 0 R (page.412) 4097 0 R (page.413) 4101 0 R (page.414) 4106 0 R (page.415) 4111 0 R (page.416) 4117 0 R] +/Limits [(page.411) (page.416)] +>> endobj +6058 0 obj << +/Names [(page.417) 4121 0 R (page.418) 4126 0 R (page.419) 4134 0 R (page.42) 2132 0 R (page.420) 4140 0 R (page.421) 4146 0 R] +/Limits [(page.417) (page.421)] +>> endobj +6059 0 obj << +/Names [(page.422) 4153 0 R (page.423) 4157 0 R (page.424) 4162 0 R (page.425) 4166 0 R (page.426) 4172 0 R (page.427) 4178 0 R] +/Limits [(page.422) (page.427)] +>> endobj +6060 0 obj << +/Names [(page.428) 4182 0 R (page.429) 4187 0 R (page.43) 2142 0 R (page.430) 4191 0 R (page.431) 4197 0 R (page.432) 4202 0 R] +/Limits [(page.428) (page.432)] +>> endobj +6061 0 obj << +/Names [(page.433) 4207 0 R (page.434) 4212 0 R (page.435) 4216 0 R (page.436) 4220 0 R (page.437) 4225 0 R (page.438) 4230 0 R] +/Limits [(page.433) (page.438)] +>> endobj +6062 0 obj << +/Names [(page.439) 4239 0 R (page.44) 2146 0 R (page.440) 4244 0 R (page.441) 4249 0 R (page.442) 4253 0 R (page.443) 4260 0 R] +/Limits [(page.439) (page.443)] +>> endobj +6063 0 obj << +/Names [(page.444) 4265 0 R (page.445) 4270 0 R (page.446) 4276 0 R (page.447) 4284 0 R (page.448) 4290 0 R (page.449) 4294 0 R] +/Limits [(page.444) (page.449)] +>> endobj +6064 0 obj << +/Names [(page.45) 2156 0 R (page.450) 4298 0 R (page.451) 4304 0 R (page.452) 4308 0 R (page.453) 4313 0 R (page.454) 4318 0 R] +/Limits [(page.45) (page.454)] +>> endobj +6065 0 obj << +/Names [(page.455) 4322 0 R (page.456) 4330 0 R (page.457) 4337 0 R (page.458) 4342 0 R (page.459) 4349 0 R (page.46) 2163 0 R] +/Limits [(page.455) (page.46)] +>> endobj +6066 0 obj << +/Names [(page.460) 4353 0 R (page.461) 4357 0 R (page.462) 4362 0 R (page.463) 4367 0 R (page.464) 4372 0 R (page.465) 4381 0 R] +/Limits [(page.460) (page.465)] +>> endobj +6067 0 obj << +/Names [(page.466) 4387 0 R (page.467) 4393 0 R (page.468) 4397 0 R (page.469) 4402 0 R (page.47) 2168 0 R (page.470) 4408 0 R] +/Limits [(page.466) (page.470)] +>> endobj +6068 0 obj << +/Names [(page.471) 4413 0 R (page.472) 4419 0 R (page.473) 4423 0 R (page.474) 4430 0 R (page.475) 4436 0 R (page.476) 4440 0 R] +/Limits [(page.471) (page.476)] +>> endobj +6069 0 obj << +/Names [(page.477) 4444 0 R (page.478) 4449 0 R (page.479) 4453 0 R (page.48) 2173 0 R (page.480) 4458 0 R (page.481) 4463 0 R] +/Limits [(page.477) (page.481)] +>> endobj +6070 0 obj << +/Names [(page.482) 4468 0 R (page.483) 4472 0 R (page.484) 4476 0 R (page.485) 4480 0 R (page.486) 4484 0 R (page.487) 4489 0 R] +/Limits [(page.482) (page.487)] +>> endobj +6071 0 obj << +/Names [(page.488) 4493 0 R (page.489) 4498 0 R (page.49) 2179 0 R (page.490) 4502 0 R (page.491) 4507 0 R (page.492) 4513 0 R] +/Limits [(page.488) (page.492)] +>> endobj +6072 0 obj << +/Names [(page.493) 4518 0 R (page.494) 4523 0 R (page.495) 4528 0 R (page.496) 4532 0 R (page.497) 4536 0 R (page.498) 4540 0 R] +/Limits [(page.493) (page.498)] +>> endobj +6073 0 obj << +/Names [(page.499) 4545 0 R (page.5) 1931 0 R (page.50) 2183 0 R (page.500) 4549 0 R (page.501) 4554 0 R (page.502) 4558 0 R] +/Limits [(page.499) (page.502)] +>> endobj +6074 0 obj << +/Names [(page.503) 4562 0 R (page.504) 4567 0 R (page.505) 4572 0 R (page.506) 4577 0 R (page.507) 4581 0 R (page.508) 4587 0 R] +/Limits [(page.503) (page.508)] +>> endobj +6075 0 obj << +/Names [(page.509) 4591 0 R (page.51) 2187 0 R (page.510) 4596 0 R (page.511) 4601 0 R (page.512) 4607 0 R (page.513) 4612 0 R] +/Limits [(page.509) (page.513)] +>> endobj +6076 0 obj << +/Names [(page.514) 4616 0 R (page.515) 4620 0 R (page.516) 4625 0 R (page.517) 4630 0 R (page.518) 4634 0 R (page.519) 4641 0 R] +/Limits [(page.514) (page.519)] +>> endobj +6077 0 obj << +/Names [(page.52) 2192 0 R (page.520) 4645 0 R (page.521) 4649 0 R (page.522) 4653 0 R (page.523) 4660 0 R (page.524) 4665 0 R] +/Limits [(page.52) (page.524)] +>> endobj +6078 0 obj << +/Names [(page.525) 4669 0 R (page.526) 4674 0 R (page.527) 4681 0 R (page.528) 4685 0 R (page.529) 4790 0 R (page.53) 2198 0 R] +/Limits [(page.525) (page.53)] +>> endobj +6079 0 obj << +/Names [(page.530) 4895 0 R (page.531) 4996 0 R (page.532) 5101 0 R (page.533) 5204 0 R (page.534) 5328 0 R (page.535) 5419 0 R] +/Limits [(page.530) (page.535)] +>> endobj +6080 0 obj << +/Names [(page.536) 5522 0 R (page.537) 5628 0 R (page.538) 5733 0 R (page.539) 5758 0 R (page.54) 2202 0 R (page.55) 2208 0 R] +/Limits [(page.536) (page.55)] +>> endobj +6081 0 obj << +/Names [(page.56) 2212 0 R (page.57) 2217 0 R (page.58) 2223 0 R (page.59) 2228 0 R (page.6) 1937 0 R (page.60) 2234 0 R] +/Limits [(page.56) (page.60)] +>> endobj +6082 0 obj << +/Names [(page.61) 2240 0 R (page.62) 2246 0 R (page.63) 2251 0 R (page.64) 2257 0 R (page.65) 2262 0 R (page.66) 2267 0 R] +/Limits [(page.61) (page.66)] +>> endobj +6083 0 obj << +/Names [(page.67) 2274 0 R (page.68) 2280 0 R (page.69) 2286 0 R (page.7) 1942 0 R (page.70) 2290 0 R (page.71) 2296 0 R] +/Limits [(page.67) (page.71)] +>> endobj +6084 0 obj << +/Names [(page.72) 2304 0 R (page.73) 2309 0 R (page.74) 2316 0 R (page.75) 2323 0 R (page.76) 2327 0 R (page.77) 2332 0 R] +/Limits [(page.72) (page.77)] +>> endobj +6085 0 obj << +/Names [(page.78) 2336 0 R (page.79) 2345 0 R (page.8) 1947 0 R (page.80) 2350 0 R (page.81) 2355 0 R (page.82) 2361 0 R] +/Limits [(page.78) (page.82)] +>> endobj +6086 0 obj << +/Names [(page.83) 2365 0 R (page.84) 2370 0 R (page.85) 2376 0 R (page.86) 2383 0 R (page.87) 2388 0 R (page.88) 2396 0 R] +/Limits [(page.83) (page.88)] +>> endobj +6087 0 obj << +/Names [(page.89) 2400 0 R (page.9) 1952 0 R (page.90) 2405 0 R (page.91) 2410 0 R (page.92) 2414 0 R (page.93) 2419 0 R] +/Limits [(page.89) (page.93)] +>> endobj +6088 0 obj << +/Names [(page.94) 2424 0 R (page.95) 2429 0 R (page.96) 2434 0 R (page.97) 2440 0 R (page.98) 2444 0 R (page.99) 2448 0 R] +/Limits [(page.94) (page.99)] +>> endobj +6089 0 obj << +/Names [(page.A) 5762 0 R (page.B) 5768 0 R (page.C) 5772 0 R (page.D) 5776 0 R (page.E) 5780 0 R (page.F) 5784 0 R] +/Limits [(page.A) (page.F)] +>> endobj +6090 0 obj << +/Names [(page.G) 5788 0 R (page.H) 5793 0 R (page.I) 5797 0 R (page.a) 1709 0 R (page.b) 1719 0 R (page.i) 1727 0 R] +/Limits [(page.G) (page.i)] +>> endobj +6091 0 obj << +/Names [(page.ii) 1733 0 R (page.iii) 1739 0 R (page.iv) 1748 0 R (page.ix) 1847 0 R (page.v) 1754 0 R (page.vi) 1768 0 R] +/Limits [(page.ii) (page.vi)] +>> endobj +6092 0 obj << +/Names [(page.vii) 1793 0 R (page.viii) 1820 0 R (page.x) 1873 0 R (page.xi) 1893 0 R (section*.1) 1711 0 R (section*.10) 1772 0 R] +/Limits [(page.vii) (section*.10)] +>> endobj +6093 0 obj << +/Names [(section*.100) 2120 0 R (section*.101) 2121 0 R (section*.102) 2128 0 R (section*.103) 2133 0 R (section*.104) 2147 0 R (section*.105) 2148 0 R] +/Limits [(section*.100) (section*.105)] +>> endobj +6094 0 obj << +/Names [(section*.106) 1781 0 R (section*.107) 2149 0 R (section*.108) 2150 0 R (section*.109) 2157 0 R (section*.11) 1773 0 R (section*.110) 2158 0 R] +/Limits [(section*.106) (section*.110)] +>> endobj +6095 0 obj << +/Names [(section*.111) 2169 0 R (section*.112) 2174 0 R (section*.113) 2188 0 R (section*.114) 2194 0 R (section*.115) 2203 0 R (section*.116) 2213 0 R] +/Limits [(section*.111) (section*.116)] +>> endobj +6096 0 obj << +/Names [(section*.117) 2218 0 R (section*.118) 2219 0 R (section*.119) 2224 0 R (section*.12) 1774 0 R (section*.120) 2229 0 R (section*.121) 2241 0 R] +/Limits [(section*.117) (section*.121)] +>> endobj +6097 0 obj << +/Names [(section*.122) 2242 0 R (section*.123) 2247 0 R (section*.124) 2252 0 R (section*.125) 2253 0 R (section*.126) 2258 0 R (section*.127) 2263 0 R] +/Limits [(section*.122) (section*.127)] +>> endobj +6098 0 obj << +/Names [(section*.128) 2268 0 R (section*.129) 2275 0 R (section*.13) 1775 0 R (section*.130) 2276 0 R (section*.131) 2281 0 R (section*.132) 2282 0 R] +/Limits [(section*.128) (section*.132)] +>> endobj +6099 0 obj << +/Names [(section*.133) 2291 0 R (section*.134) 2297 0 R (section*.135) 2298 0 R (section*.136) 1782 0 R (section*.137) 2299 0 R (section*.138) 2300 0 R] +/Limits [(section*.133) (section*.138)] +>> endobj +6100 0 obj << +/Names [(section*.139) 2310 0 R (section*.14) 1776 0 R (section*.140) 2317 0 R (section*.141) 2318 0 R (section*.142) 2328 0 R (section*.143) 2337 0 R] +/Limits [(section*.139) (section*.143)] +>> endobj +6101 0 obj << +/Names [(section*.144) 2338 0 R (section*.145) 1803 0 R (section*.146) 2339 0 R (section*.147) 2340 0 R (section*.148) 2346 0 R (section*.149) 2351 0 R] +/Limits [(section*.144) (section*.149)] +>> endobj +6102 0 obj << +/Names [(section*.15) 1777 0 R (section*.150) 2356 0 R (section*.151) 2357 0 R (section*.152) 2366 0 R (section*.153) 2371 0 R (section*.154) 2377 0 R] +/Limits [(section*.15) (section*.154)] +>> endobj +6103 0 obj << +/Names [(section*.155) 2378 0 R (section*.156) 2379 0 R (section*.157) 2389 0 R (section*.158) 2390 0 R (section*.159) 1804 0 R (section*.16) 1794 0 R] +/Limits [(section*.155) (section*.16)] +>> endobj +6104 0 obj << +/Names [(section*.160) 2391 0 R (section*.161) 2392 0 R (section*.162) 2401 0 R (section*.163) 2415 0 R (section*.164) 2420 0 R (section*.165) 2425 0 R] +/Limits [(section*.160) (section*.165)] +>> endobj +6105 0 obj << +/Names [(section*.166) 2430 0 R (section*.167) 2435 0 R (section*.168) 2159 0 R (section*.169) 2453 0 R (section*.17) 1795 0 R (section*.170) 2458 0 R] +/Limits [(section*.166) (section*.170)] +>> endobj +6106 0 obj << +/Names [(section*.171) 2463 0 R (section*.172) 2464 0 R (section*.173) 1805 0 R (section*.174) 2465 0 R (section*.175) 2466 0 R (section*.176) 2472 0 R] +/Limits [(section*.171) (section*.176)] +>> endobj +6107 0 obj << +/Names [(section*.177) 2481 0 R (section*.178) 2486 0 R (section*.179) 2495 0 R (section*.18) 1796 0 R (section*.180) 2496 0 R (section*.181) 2506 0 R] +/Limits [(section*.177) (section*.181)] +>> endobj +6108 0 obj << +/Names [(section*.182) 2511 0 R (section*.183) 2516 0 R (section*.184) 2525 0 R (section*.185) 2531 0 R (section*.186) 2536 0 R (section*.187) 2551 0 R] +/Limits [(section*.182) (section*.187)] +>> endobj +6109 0 obj << +/Names [(section*.188) 2556 0 R (section*.189) 2562 0 R (section*.19) 1797 0 R (section*.190) 2563 0 R (section*.191) 2572 0 R (section*.192) 2573 0 R] +/Limits [(section*.188) (section*.192)] +>> endobj +6110 0 obj << +/Names [(section*.193) 1806 0 R (section*.194) 2574 0 R (section*.195) 2575 0 R (section*.196) 2587 0 R (section*.197) 2588 0 R (section*.198) 2589 0 R] +/Limits [(section*.193) (section*.198)] +>> endobj +6111 0 obj << +/Names [(section*.199) 2115 0 R (section*.2) 1712 0 R (section*.20) 1798 0 R (section*.200) 2604 0 R (section*.201) 2605 0 R (section*.202) 1807 0 R] +/Limits [(section*.199) (section*.202)] +>> endobj +6112 0 obj << +/Names [(section*.203) 2606 0 R (section*.204) 2612 0 R (section*.205) 2625 0 R (section*.206) 2630 0 R (section*.207) 2635 0 R (section*.208) 2636 0 R] +/Limits [(section*.203) (section*.208)] +>> endobj +6113 0 obj << +/Names [(section*.209) 2642 0 R (section*.21) 1799 0 R (section*.210) 2651 0 R (section*.211) 2652 0 R (section*.212) 1808 0 R (section*.213) 2653 0 R] +/Limits [(section*.209) (section*.213)] +>> endobj +6114 0 obj << +/Names [(section*.214) 2654 0 R (section*.215) 2655 0 R (section*.216) 2660 0 R (section*.217) 2661 0 R (section*.218) 2666 0 R (section*.219) 2671 0 R] +/Limits [(section*.214) (section*.219)] +>> endobj +6115 0 obj << +/Names [(section*.22) 1800 0 R (section*.220) 2677 0 R (section*.221) 2682 0 R (section*.222) 2687 0 R (section*.223) 2692 0 R (section*.224) 2693 0 R] +/Limits [(section*.22) (section*.224)] +>> endobj +6116 0 obj << +/Names [(section*.225) 2698 0 R (section*.226) 2703 0 R (section*.227) 2715 0 R (section*.228) 2716 0 R (section*.229) 1809 0 R (section*.23) 1801 0 R] +/Limits [(section*.225) (section*.23)] +>> endobj +6117 0 obj << +/Names [(section*.230) 2717 0 R (section*.231) 2719 0 R (section*.232) 2724 0 R (section*.233) 2725 0 R (section*.234) 2730 0 R (section*.235) 2731 0 R] +/Limits [(section*.230) (section*.235)] +>> endobj +6118 0 obj << +/Names [(section*.236) 2745 0 R (section*.237) 2750 0 R (section*.238) 2755 0 R (section*.239) 2756 0 R (section*.24) 1802 0 R (section*.240) 1830 0 R] +/Limits [(section*.236) (section*.240)] +>> endobj +6119 0 obj << +/Names [(section*.241) 2757 0 R (section*.242) 2758 0 R (section*.243) 2792 0 R (section*.244) 2793 0 R (section*.245) 1831 0 R (section*.246) 2794 0 R] +/Limits [(section*.241) (section*.246)] +>> endobj +6120 0 obj << +/Names [(section*.247) 2795 0 R (section*.248) 2809 0 R (section*.249) 2814 0 R (section*.25) 1821 0 R (section*.250) 2815 0 R (section*.251) 2820 0 R] +/Limits [(section*.247) (section*.251)] +>> endobj +6121 0 obj << +/Names [(section*.252) 2825 0 R (section*.253) 2830 0 R (section*.254) 2836 0 R (section*.255) 2869 0 R (section*.256) 2886 0 R (section*.257) 2887 0 R] +/Limits [(section*.252) (section*.257)] +>> endobj +6122 0 obj << +/Names [(section*.258) 1832 0 R (section*.259) 2888 0 R (section*.26) 1822 0 R (section*.260) 2889 0 R (section*.261) 2895 0 R (section*.262) 2896 0 R] +/Limits [(section*.258) (section*.262)] +>> endobj +6123 0 obj << +/Names [(section*.263) 2901 0 R (section*.264) 2906 0 R (section*.265) 2915 0 R (section*.266) 2929 0 R (section*.267) 2930 0 R (section*.268) 1833 0 R] +/Limits [(section*.263) (section*.268)] +>> endobj +6124 0 obj << +/Names [(section*.269) 2931 0 R (section*.27) 1823 0 R (section*.270) 2932 0 R (section*.271) 2938 0 R (section*.272) 2943 0 R (section*.273) 2944 0 R] +/Limits [(section*.269) (section*.273)] +>> endobj +6125 0 obj << +/Names [(section*.274) 2949 0 R (section*.275) 2959 0 R (section*.276) 2968 0 R (section*.277) 2969 0 R (section*.278) 2978 0 R (section*.279) 2979 0 R] +/Limits [(section*.274) (section*.279)] +>> endobj +6126 0 obj << +/Names [(section*.28) 1824 0 R (section*.280) 2994 0 R (section*.281) 2999 0 R (section*.282) 3000 0 R (section*.283) 3005 0 R (section*.284) 3016 0 R] +/Limits [(section*.28) (section*.284)] +>> endobj +6127 0 obj << +/Names [(section*.285) 3022 0 R (section*.286) 3031 0 R (section*.287) 2855 0 R (section*.288) 3036 0 R (section*.289) 3045 0 R (section*.29) 1825 0 R] +/Limits [(section*.285) (section*.29)] +>> endobj +6128 0 obj << +/Names [(section*.290) 3051 0 R (section*.291) 3056 0 R (section*.292) 3066 0 R (section*.293) 3067 0 R (section*.294) 1834 0 R (section*.295) 3068 0 R] +/Limits [(section*.290) (section*.295)] +>> endobj +6129 0 obj << +/Names [(section*.296) 3069 0 R (section*.297) 3113 0 R (section*.298) 3114 0 R (section*.299) 3127 0 R (section*.3) 1713 0 R (section*.30) 1826 0 R] +/Limits [(section*.296) (section*.30)] +>> endobj +6130 0 obj << +/Names [(section*.300) 3164 0 R (section*.301) 3169 0 R (section*.302) 3174 0 R (section*.303) 3179 0 R (section*.304) 3197 0 R (section*.305) 3206 0 R] +/Limits [(section*.300) (section*.305)] +>> endobj +6131 0 obj << +/Names [(section*.306) 3220 0 R (section*.307) 2319 0 R (section*.308) 3233 0 R (section*.309) 3234 0 R (section*.31) 1827 0 R (section*.310) 1835 0 R] +/Limits [(section*.306) (section*.310)] +>> endobj +6132 0 obj << +/Names [(section*.311) 3235 0 R (section*.312) 3240 0 R (section*.313) 3241 0 R (section*.314) 3246 0 R (section*.315) 3252 0 R (section*.316) 3253 0 R] +/Limits [(section*.311) (section*.316)] +>> endobj +6133 0 obj << +/Names [(section*.317) 3259 0 R (section*.318) 3268 0 R (section*.319) 3269 0 R (section*.32) 1828 0 R (section*.320) 1836 0 R (section*.321) 3270 0 R] +/Limits [(section*.317) (section*.321)] +>> endobj +6134 0 obj << +/Names [(section*.322) 3271 0 R (section*.323) 3280 0 R (section*.324) 3281 0 R (section*.325) 3294 0 R (section*.326) 3299 0 R (section*.327) 3308 0 R] +/Limits [(section*.322) (section*.327)] +>> endobj +6135 0 obj << +/Names [(section*.328) 3313 0 R (section*.329) 3314 0 R (section*.33) 1829 0 R (section*.330) 3320 0 R (section*.331) 3325 0 R (section*.332) 3334 0 R] +/Limits [(section*.328) (section*.332)] +>> endobj +6136 0 obj << +/Names [(section*.333) 3339 0 R (section*.334) 3340 0 R (section*.335) 1857 0 R (section*.336) 3341 0 R (section*.337) 3346 0 R (section*.338) 3352 0 R] +/Limits [(section*.333) (section*.338)] +>> endobj +6137 0 obj << +/Names [(section*.339) 3357 0 R (section*.34) 1848 0 R (section*.340) 3362 0 R (section*.341) 3367 0 R (section*.342) 3372 0 R (section*.343) 3377 0 R] +/Limits [(section*.339) (section*.343)] +>> endobj +6138 0 obj << +/Names [(section*.344) 3383 0 R (section*.345) 3392 0 R (section*.346) 3393 0 R (section*.347) 1858 0 R (section*.348) 3394 0 R (section*.349) 3403 0 R] +/Limits [(section*.344) (section*.349)] +>> endobj +6139 0 obj << +/Names [(section*.35) 1849 0 R (section*.350) 3408 0 R (section*.351) 3414 0 R (section*.352) 3423 0 R (section*.353) 3428 0 R (section*.354) 3442 0 R] +/Limits [(section*.35) (section*.354)] +>> endobj +6140 0 obj << +/Names [(section*.355) 3443 0 R (section*.356) 1859 0 R (section*.357) 3444 0 R (section*.358) 3449 0 R (section*.359) 3454 0 R (section*.36) 1850 0 R] +/Limits [(section*.355) (section*.36)] +>> endobj +6141 0 obj << +/Names [(section*.360) 3463 0 R (section*.361) 3468 0 R (section*.362) 3474 0 R (section*.363) 3475 0 R (section*.364) 3480 0 R (section*.365) 3486 0 R] +/Limits [(section*.360) (section*.365)] +>> endobj +6142 0 obj << +/Names [(section*.366) 3491 0 R (section*.367) 3492 0 R (section*.368) 3505 0 R (section*.369) 3511 0 R (section*.37) 1851 0 R (section*.370) 3524 0 R] +/Limits [(section*.366) (section*.370)] +>> endobj +6143 0 obj << +/Names [(section*.371) 3529 0 R (section*.372) 3530 0 R (section*.373) 1860 0 R (section*.374) 3531 0 R (section*.375) 3536 0 R (section*.376) 3542 0 R] +/Limits [(section*.371) (section*.376)] +>> endobj +6144 0 obj << +/Names [(section*.377) 3547 0 R (section*.378) 3556 0 R (section*.379) 3561 0 R (section*.38) 1852 0 R (section*.380) 3566 0 R (section*.381) 3567 0 R] +/Limits [(section*.377) (section*.381)] +>> endobj +6145 0 obj << +/Names [(section*.382) 3573 0 R (section*.383) 3578 0 R (section*.384) 3579 0 R (section*.385) 3584 0 R (section*.386) 1861 0 R (section*.387) 3585 0 R] +/Limits [(section*.382) (section*.387)] +>> endobj +6146 0 obj << +/Names [(section*.388) 3586 0 R (section*.389) 3588 0 R (section*.39) 1853 0 R (section*.390) 3594 0 R (section*.391) 3595 0 R (section*.392) 3596 0 R] +/Limits [(section*.388) (section*.392)] +>> endobj +6147 0 obj << +/Names [(section*.393) 3601 0 R (section*.394) 3602 0 R (section*.395) 3612 0 R (section*.396) 2591 0 R (section*.397) 3622 0 R (section*.398) 3623 0 R] +/Limits [(section*.393) (section*.398)] +>> endobj +6148 0 obj << +/Names [(section*.399) 1862 0 R (section*.4) 1721 0 R (section*.40) 1854 0 R (section*.400) 3624 0 R (section*.401) 3629 0 R (section*.402) 3634 0 R] +/Limits [(section*.399) (section*.402)] +>> endobj +6149 0 obj << +/Names [(section*.403) 3639 0 R (section*.404) 3645 0 R (section*.405) 3654 0 R (section*.406) 3659 0 R (section*.407) 3667 0 R (section*.408) 3672 0 R] +/Limits [(section*.403) (section*.408)] +>> endobj +6150 0 obj << +/Names [(section*.409) 3678 0 R (section*.41) 1855 0 R (section*.410) 3679 0 R (section*.411) 3688 0 R (section*.412) 3699 0 R (section*.413) 3700 0 R] +/Limits [(section*.409) (section*.413)] +>> endobj +6151 0 obj << +/Names [(section*.414) 1863 0 R (section*.415) 3701 0 R (section*.416) 3706 0 R (section*.417) 3723 0 R (section*.418) 3729 0 R (section*.419) 3734 0 R] +/Limits [(section*.414) (section*.419)] +>> endobj +6152 0 obj << +/Names [(section*.42) 1856 0 R (section*.420) 3755 0 R (section*.421) 3764 0 R (section*.422) 3782 0 R (section*.423) 3783 0 R (section*.424) 1883 0 R] +/Limits [(section*.42) (section*.424)] +>> endobj +6153 0 obj << +/Names [(section*.425) 3784 0 R (section*.426) 3785 0 R (section*.427) 3786 0 R (section*.428) 3795 0 R (section*.429) 3805 0 R (section*.43) 1874 0 R] +/Limits [(section*.425) (section*.43)] +>> endobj +6154 0 obj << +/Names [(section*.430) 3812 0 R (section*.431) 3818 0 R (section*.432) 3824 0 R (section*.433) 3842 0 R (section*.434) 3843 0 R (section*.435) 3708 0 R] +/Limits [(section*.430) (section*.435)] +>> endobj +6155 0 obj << +/Names [(section*.436) 3856 0 R (section*.437) 3861 0 R (section*.438) 3873 0 R (section*.439) 3878 0 R (section*.44) 1875 0 R (section*.440) 2592 0 R] +/Limits [(section*.436) (section*.440)] +>> endobj +6156 0 obj << +/Names [(section*.441) 3887 0 R (section*.442) 3888 0 R (section*.443) 1884 0 R (section*.444) 3889 0 R (section*.445) 3890 0 R (section*.446) 3904 0 R] +/Limits [(section*.441) (section*.446)] +>> endobj +6157 0 obj << +/Names [(section*.447) 3905 0 R (section*.448) 3910 0 R (section*.449) 3915 0 R (section*.45) 1876 0 R (section*.450) 3916 0 R (section*.451) 1885 0 R] +/Limits [(section*.447) (section*.451)] +>> endobj +6158 0 obj << +/Names [(section*.452) 3917 0 R (section*.453) 3938 0 R (section*.454) 3947 0 R (section*.455) 3948 0 R (section*.456) 1886 0 R (section*.457) 3949 0 R] +/Limits [(section*.452) (section*.457)] +>> endobj +6159 0 obj << +/Names [(section*.458) 3954 0 R (section*.459) 3964 0 R (section*.46) 1877 0 R (section*.460) 3969 0 R (section*.461) 3970 0 R (section*.462) 3975 0 R] +/Limits [(section*.458) (section*.462)] +>> endobj +6160 0 obj << +/Names [(section*.463) 3980 0 R (section*.464) 3985 0 R (section*.465) 3992 0 R (section*.466) 3924 0 R (section*.467) 4004 0 R (section*.468) 4013 0 R] +/Limits [(section*.463) (section*.468)] +>> endobj +6161 0 obj << +/Names [(section*.469) 4022 0 R (section*.47) 1878 0 R (section*.470) 4036 0 R (section*.471) 4037 0 R (section*.472) 4046 0 R (section*.473) 4060 0 R] +/Limits [(section*.469) (section*.473)] +>> endobj +6162 0 obj << +/Names [(section*.474) 4073 0 R (section*.475) 4074 0 R (section*.476) 4092 0 R (section*.477) 4093 0 R (section*.478) 4102 0 R (section*.479) 4127 0 R] +/Limits [(section*.474) (section*.479)] +>> endobj +6163 0 obj << +/Names [(section*.48) 1879 0 R (section*.480) 4128 0 R (section*.481) 1887 0 R (section*.482) 4129 0 R (section*.483) 4135 0 R (section*.484) 4141 0 R] +/Limits [(section*.48) (section*.484)] +>> endobj +6164 0 obj << +/Names [(section*.485) 4147 0 R (section*.486) 4158 0 R (section*.487) 4167 0 R (section*.488) 4168 0 R (section*.489) 4173 0 R (section*.49) 1881 0 R] +/Limits [(section*.485) (section*.49)] +>> endobj +6165 0 obj << +/Names [(section*.490) 4183 0 R (section*.491) 4192 0 R (section*.492) 4193 0 R (section*.493) 4198 0 R (section*.494) 4208 0 R (section*.495) 4221 0 R] +/Limits [(section*.490) (section*.495)] +>> endobj +6166 0 obj << +/Names [(section*.496) 4231 0 R (section*.497) 4232 0 R (section*.498) 1888 0 R (section*.499) 4233 0 R (section*.5) 1755 0 R (section*.50) 1882 0 R] +/Limits [(section*.496) (section*.50)] +>> endobj +6167 0 obj << +/Names [(section*.500) 4234 0 R (section*.501) 4240 0 R (section*.502) 4254 0 R (section*.503) 4255 0 R (section*.504) 4261 0 R (section*.505) 4277 0 R] +/Limits [(section*.500) (section*.505)] +>> endobj +6168 0 obj << +/Names [(section*.506) 4278 0 R (section*.507) 1897 0 R (section*.508) 4279 0 R (section*.509) 4285 0 R (section*.51) 1894 0 R (section*.510) 4286 0 R] +/Limits [(section*.506) (section*.510)] +>> endobj +6169 0 obj << +/Names [(section*.511) 4299 0 R (section*.512) 4309 0 R (section*.513) 4314 0 R (section*.514) 4323 0 R (section*.515) 4331 0 R (section*.516) 4343 0 R] +/Limits [(section*.511) (section*.516)] +>> endobj +6170 0 obj << +/Names [(section*.517) 4344 0 R (section*.518) 4333 0 R (section*.519) 4345 0 R (section*.52) 1895 0 R (section*.520) 4358 0 R (section*.521) 4373 0 R] +/Limits [(section*.517) (section*.521)] +>> endobj +6171 0 obj << +/Names [(section*.522) 4374 0 R (section*.523) 1898 0 R (section*.524) 4375 0 R (section*.525) 4376 0 R (section*.526) 4382 0 R (section*.527) 4388 0 R] +/Limits [(section*.522) (section*.527)] +>> endobj +6172 0 obj << +/Names [(section*.528) 4409 0 R (section*.529) 4414 0 R (section*.53) 1903 0 R (section*.530) 4445 0 R (section*.531) 4454 0 R (section*.532) 4464 0 R] +/Limits [(section*.528) (section*.532)] +>> endobj +6173 0 obj << +/Names [(section*.533) 4494 0 R (section*.534) 4503 0 R (section*.535) 4508 0 R (section*.536) 4136 0 R (section*.537) 4509 0 R (section*.538) 4519 0 R] +/Limits [(section*.533) (section*.538)] +>> endobj +6174 0 obj << +/Names [(section*.539) 4524 0 R (section*.54) 1909 0 R (section*.540) 4550 0 R (section*.541) 4563 0 R (section*.542) 4573 0 R (section*.543) 4582 0 R] +/Limits [(section*.539) (section*.543)] +>> endobj +6175 0 obj << +/Names [(section*.544) 4583 0 R (section*.545) 4592 0 R (section*.546) 4602 0 R (section*.547) 4603 0 R (section*.548) 4608 0 R (section*.549) 4621 0 R] +/Limits [(section*.544) (section*.549)] +>> endobj +6176 0 obj << +/Names [(section*.55) 1910 0 R (section*.550) 4635 0 R (section*.551) 4636 0 R (section*.552) 4637 0 R (section*.553) 4654 0 R (section*.554) 4655 0 R] +/Limits [(section*.55) (section*.554)] +>> endobj +6177 0 obj << +/Names [(section*.555) 4661 0 R (section*.556) 4670 0 R (section*.557) 4675 0 R (section*.558) 4676 0 R (section*.559) 4677 0 R (section*.56) 1778 0 R] +/Limits [(section*.555) (section*.56)] +>> endobj +6178 0 obj << +/Names [(section*.560) 4686 0 R (section*.561) 4791 0 R (section*.562) 5763 0 R (section*.57) 1911 0 R (section*.58) 1912 0 R (section*.59) 1917 0 R] +/Limits [(section*.560) (section*.59)] +>> endobj +6179 0 obj << +/Names [(section*.6) 1756 0 R (section*.60) 1926 0 R (section*.61) 1927 0 R (section*.62) 1932 0 R (section*.63) 1933 0 R (section*.64) 1943 0 R] +/Limits [(section*.6) (section*.64)] +>> endobj +6180 0 obj << +/Names [(section*.65) 1948 0 R (section*.66) 1953 0 R (section*.67) 1954 0 R (section*.68) 1779 0 R (section*.69) 1955 0 R (section*.7) 1769 0 R] +/Limits [(section*.65) (section*.7)] +>> endobj +6181 0 obj << +/Names [(section*.70) 1956 0 R (section*.71) 1957 0 R (section*.72) 1962 0 R (section*.73) 1963 0 R (section*.74) 1968 0 R (section*.75) 1973 0 R] +/Limits [(section*.70) (section*.75)] +>> endobj +6182 0 obj << +/Names [(section*.76) 1979 0 R (section*.77) 1984 0 R (section*.78) 1989 0 R (section*.79) 1998 0 R (section*.80) 2012 0 R (section*.81) 2013 0 R] +/Limits [(section*.76) (section*.81)] +>> endobj +6183 0 obj << +/Names [(section*.82) 1780 0 R (section*.83) 2014 0 R (section*.84) 2015 0 R (section*.85) 2024 0 R (section*.86) 2025 0 R (section*.87) 2036 0 R] +/Limits [(section*.82) (section*.87)] +>> endobj +6184 0 obj << +/Names [(section*.88) 2054 0 R (section*.89) 2059 0 R (section*.9) 1771 0 R (section*.90) 2069 0 R (section*.91) 2074 0 R (section*.92) 2075 0 R] +/Limits [(section*.88) (section*.92)] +>> endobj +6185 0 obj << +/Names [(section*.93) 2076 0 R (section*.94) 2085 0 R (section*.95) 2086 0 R (section*.96) 2095 0 R (section*.97) 2101 0 R (section*.98) 2106 0 R] +/Limits [(section*.93) (section*.98)] +>> endobj +6186 0 obj << +/Names [(section*.99) 2112 0 R] +/Limits [(section*.99) (section*.99)] +>> endobj +6187 0 obj << +/Kids [5928 0 R 5929 0 R 5930 0 R 5931 0 R 5932 0 R 5933 0 R] +/Limits [(A.1) (Chapter\040XXXI.0)] +>> endobj +6188 0 obj << +/Kids [5934 0 R 5935 0 R 5936 0 R 5937 0 R 5938 0 R 5939 0 R] +/Limits [(Chapter\040XXXII.0) (III.15..1)] +>> endobj +6189 0 obj << +/Kids [5940 0 R 5941 0 R 5942 0 R 5943 0 R 5944 0 R 5945 0 R] +/Limits [(III.16..1) (IV.5..1)] +>> endobj +6190 0 obj << +/Kids [5946 0 R 5947 0 R 5948 0 R 5949 0 R 5950 0 R 5951 0 R] +/Limits [(IV.6..1) (V.2..1)] +>> endobj +6191 0 obj << +/Kids [5952 0 R 5953 0 R 5954 0 R 5955 0 R 5956 0 R 5957 0 R] +/Limits [(V.3..1) (VIII.5..1)] +>> endobj +6192 0 obj << +/Kids [5958 0 R 5959 0 R 5960 0 R 5961 0 R 5962 0 R 5963 0 R] +/Limits [(VIII.6..1) (XIII.2..1)] +>> endobj +6193 0 obj << +/Kids [5964 0 R 5965 0 R 5966 0 R 5967 0 R 5968 0 R 5969 0 R] +/Limits [(XIV.1..1) (XVI.17..1)] +>> endobj +6194 0 obj << +/Kids [5970 0 R 5971 0 R 5972 0 R 5973 0 R 5974 0 R 5975 0 R] +/Limits [(XVI.18..1) (XX.6..1)] +>> endobj +6195 0 obj << +/Kids [5976 0 R 5977 0 R 5978 0 R 5979 0 R 5980 0 R 5981 0 R] +/Limits [(XX.7..1) (XXIV.10..1)] +>> endobj +6196 0 obj << +/Kids [5982 0 R 5983 0 R 5984 0 R 5985 0 R 5986 0 R 5987 0 R] +/Limits [(XXIV.11..1) (XXV.7..1)] +>> endobj +6197 0 obj << +/Kids [5988 0 R 5989 0 R 5990 0 R 5991 0 R 5992 0 R 5993 0 R] +/Limits [(XXVI.1..1) (XXXI.2..1)] +>> endobj +6198 0 obj << +/Kids [5994 0 R 5995 0 R 5996 0 R 5997 0 R 5998 0 R 5999 0 R] +/Limits [(XXXI.3..1) (page.102)] +>> endobj +6199 0 obj << +/Kids [6000 0 R 6001 0 R 6002 0 R 6003 0 R 6004 0 R 6005 0 R] +/Limits [(page.103) (page.135)] +>> endobj +6200 0 obj << +/Kids [6006 0 R 6007 0 R 6008 0 R 6009 0 R 6010 0 R 6011 0 R] +/Limits [(page.136) (page.168)] +>> endobj +6201 0 obj << +/Kids [6012 0 R 6013 0 R 6014 0 R 6015 0 R 6016 0 R 6017 0 R] +/Limits [(page.169) (page.20)] +>> endobj +6202 0 obj << +/Kids [6018 0 R 6019 0 R 6020 0 R 6021 0 R 6022 0 R 6023 0 R] +/Limits [(page.200) (page.232)] +>> endobj +6203 0 obj << +/Kids [6024 0 R 6025 0 R 6026 0 R 6027 0 R 6028 0 R 6029 0 R] +/Limits [(page.233) (page.265)] +>> endobj +6204 0 obj << +/Kids [6030 0 R 6031 0 R 6032 0 R 6033 0 R 6034 0 R 6035 0 R] +/Limits [(page.266) (page.298)] +>> endobj +6205 0 obj << +/Kids [6036 0 R 6037 0 R 6038 0 R 6039 0 R 6040 0 R 6041 0 R] +/Limits [(page.299) (page.33)] +>> endobj +6206 0 obj << +/Kids [6042 0 R 6043 0 R 6044 0 R 6045 0 R 6046 0 R 6047 0 R] +/Limits [(page.330) (page.362)] +>> endobj +6207 0 obj << +/Kids [6048 0 R 6049 0 R 6050 0 R 6051 0 R 6052 0 R 6053 0 R] +/Limits [(page.363) (page.395)] +>> endobj +6208 0 obj << +/Kids [6054 0 R 6055 0 R 6056 0 R 6057 0 R 6058 0 R 6059 0 R] +/Limits [(page.396) (page.427)] +>> endobj +6209 0 obj << +/Kids [6060 0 R 6061 0 R 6062 0 R 6063 0 R 6064 0 R 6065 0 R] +/Limits [(page.428) (page.46)] +>> endobj +6210 0 obj << +/Kids [6066 0 R 6067 0 R 6068 0 R 6069 0 R 6070 0 R 6071 0 R] +/Limits [(page.460) (page.492)] +>> endobj +6211 0 obj << +/Kids [6072 0 R 6073 0 R 6074 0 R 6075 0 R 6076 0 R 6077 0 R] +/Limits [(page.493) (page.524)] +>> endobj +6212 0 obj << +/Kids [6078 0 R 6079 0 R 6080 0 R 6081 0 R 6082 0 R 6083 0 R] +/Limits [(page.525) (page.71)] +>> endobj +6213 0 obj << +/Kids [6084 0 R 6085 0 R 6086 0 R 6087 0 R 6088 0 R 6089 0 R] +/Limits [(page.72) (page.F)] +>> endobj +6214 0 obj << +/Kids [6090 0 R 6091 0 R 6092 0 R 6093 0 R 6094 0 R 6095 0 R] +/Limits [(page.G) (section*.116)] +>> endobj +6215 0 obj << +/Kids [6096 0 R 6097 0 R 6098 0 R 6099 0 R 6100 0 R 6101 0 R] +/Limits [(section*.117) (section*.149)] +>> endobj +6216 0 obj << +/Kids [6102 0 R 6103 0 R 6104 0 R 6105 0 R 6106 0 R 6107 0 R] +/Limits [(section*.15) (section*.181)] +>> endobj +6217 0 obj << +/Kids [6108 0 R 6109 0 R 6110 0 R 6111 0 R 6112 0 R 6113 0 R] +/Limits [(section*.182) (section*.213)] +>> endobj +6218 0 obj << +/Kids [6114 0 R 6115 0 R 6116 0 R 6117 0 R 6118 0 R 6119 0 R] +/Limits [(section*.214) (section*.246)] +>> endobj +6219 0 obj << +/Kids [6120 0 R 6121 0 R 6122 0 R 6123 0 R 6124 0 R 6125 0 R] +/Limits [(section*.247) (section*.279)] +>> endobj +6220 0 obj << +/Kids [6126 0 R 6127 0 R 6128 0 R 6129 0 R 6130 0 R 6131 0 R] +/Limits [(section*.28) (section*.310)] +>> endobj +6221 0 obj << +/Kids [6132 0 R 6133 0 R 6134 0 R 6135 0 R 6136 0 R 6137 0 R] +/Limits [(section*.311) (section*.343)] +>> endobj +6222 0 obj << +/Kids [6138 0 R 6139 0 R 6140 0 R 6141 0 R 6142 0 R 6143 0 R] +/Limits [(section*.344) (section*.376)] +>> endobj +6223 0 obj << +/Kids [6144 0 R 6145 0 R 6146 0 R 6147 0 R 6148 0 R 6149 0 R] +/Limits [(section*.377) (section*.408)] +>> endobj +6224 0 obj << +/Kids [6150 0 R 6151 0 R 6152 0 R 6153 0 R 6154 0 R 6155 0 R] +/Limits [(section*.409) (section*.440)] +>> endobj +6225 0 obj << +/Kids [6156 0 R 6157 0 R 6158 0 R 6159 0 R 6160 0 R 6161 0 R] +/Limits [(section*.441) (section*.473)] +>> endobj +6226 0 obj << +/Kids [6162 0 R 6163 0 R 6164 0 R 6165 0 R 6166 0 R 6167 0 R] +/Limits [(section*.474) (section*.505)] +>> endobj +6227 0 obj << +/Kids [6168 0 R 6169 0 R 6170 0 R 6171 0 R 6172 0 R 6173 0 R] +/Limits [(section*.506) (section*.538)] +>> endobj +6228 0 obj << +/Kids [6174 0 R 6175 0 R 6176 0 R 6177 0 R 6178 0 R 6179 0 R] +/Limits [(section*.539) (section*.64)] +>> endobj +6229 0 obj << +/Kids [6180 0 R 6181 0 R 6182 0 R 6183 0 R 6184 0 R 6185 0 R] +/Limits [(section*.65) (section*.98)] +>> endobj +6230 0 obj << +/Kids [6186 0 R] +/Limits [(section*.99) (section*.99)] +>> endobj +6231 0 obj << +/Kids [6187 0 R 6188 0 R 6189 0 R 6190 0 R 6191 0 R 6192 0 R] +/Limits [(A.1) (XIII.2..1)] +>> endobj +6232 0 obj << +/Kids [6193 0 R 6194 0 R 6195 0 R 6196 0 R 6197 0 R 6198 0 R] +/Limits [(XIV.1..1) (page.102)] +>> endobj +6233 0 obj << +/Kids [6199 0 R 6200 0 R 6201 0 R 6202 0 R 6203 0 R 6204 0 R] +/Limits [(page.103) (page.298)] +>> endobj +6234 0 obj << +/Kids [6205 0 R 6206 0 R 6207 0 R 6208 0 R 6209 0 R 6210 0 R] +/Limits [(page.299) (page.492)] +>> endobj +6235 0 obj << +/Kids [6211 0 R 6212 0 R 6213 0 R 6214 0 R 6215 0 R 6216 0 R] +/Limits [(page.493) (section*.181)] +>> endobj +6236 0 obj << +/Kids [6217 0 R 6218 0 R 6219 0 R 6220 0 R 6221 0 R 6222 0 R] +/Limits [(section*.182) (section*.376)] +>> endobj +6237 0 obj << +/Kids [6223 0 R 6224 0 R 6225 0 R 6226 0 R 6227 0 R 6228 0 R] +/Limits [(section*.377) (section*.64)] +>> endobj +6238 0 obj << +/Kids [6229 0 R 6230 0 R] +/Limits [(section*.65) (section*.99)] +>> endobj +6239 0 obj << +/Kids [6231 0 R 6232 0 R 6233 0 R 6234 0 R 6235 0 R 6236 0 R] +/Limits [(A.1) (section*.376)] +>> endobj +6240 0 obj << +/Kids [6237 0 R 6238 0 R] +/Limits [(section*.377) (section*.99)] +>> endobj +6241 0 obj << +/Kids [6239 0 R 6240 0 R] +/Limits [(A.1) (section*.99)] +>> endobj +6242 0 obj << +/Dests 6241 0 R +>> endobj +6243 0 obj << +/Type /Catalog +/Pages 5926 0 R +/Outlines 5927 0 R +/Names 6242 0 R +/PageMode/UseNone/ViewerPreferences<</DisplayDocTitle true>>/PageLayout/SinglePage/PageLabels << /Nums [0 << /S /a >> 2 << /S /r >> 13 << /S /D >> 552 << /S /A >> ] >> +/OpenAction 1705 0 R +>> endobj +6244 0 obj << +/Author(John Maynard Keynes)/Title(The Project Gutenberg eBook \04332625: A treatise on probability)/Subject()/Creator(LaTeX with hyperref 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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % +% % +% Project Gutenberg's A Treatise on Probability, by John Maynard Keynes % +% % +% This eBook is for the use of anyone anywhere at no cost and with % +% almost no restrictions whatsoever. You may copy it, give it away or % +% re-use it under the terms of the Project Gutenberg License included % +% with this eBook or online at www.gutenberg.org % +% % +% % +% Title: A Treatise on Probability % +% % +% Author: John Maynard Keynes % +% % +% Release Date: May 31, 2010 [EBook #32625] % +% % +% Language: English % +% % +% Character set encoding: ISO-8859-1 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK A TREATISE ON PROBABILITY *** % +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{32625} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Standard DP encoding. Required. %% +%% fontenc: For boldface small-caps. Required. %% +%% babel: For Greek snippets. Required. %% +%% type1ec: Coax text Greek into Type 1 fonts. Required. %% +%% %% +%% calc: Infix arithmetic. Required. %% +%% %% +%% textcomp: Better ditto marks. 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\ifx\begin#1\def\begin@stack{b} + \let\@next\eat@environment + \else\ifx\lintertext#1 + \let\@next\linter@text + \else\ifx\rintertext#1 + \let\@next\rinter@text + \else\ifx\\#1 + \addto@DPbody{\DPh@@kr&\DP@rint}\@xp\addto@DPbody\@xp{\@xp{\the\DP@lign@no}&\\}\DP@crtrue + \global\advance\DP@lign@no\@ne + \let\@next\parse@cr + \else\check@braces#1!Q!Q!Q!\ifbr@ce\addto@DPbody{{#1}}\else + \addto@DPbody{#1}\fi + \let\@next\parseb@dy + \fi\fi\fi\fi\fi\fi\fi\fi\@next} +\def\process@DPb@dy{\let\lintertext\@gobble\let\rintertext\@gobble + \@xp\start@align\@xp\tw@\@xp\st@rredtrue\@xp\m@ne\the\DP@lignb@dy} +\def\linter@text#1{\@xp\DPlint\@xp{\the\DP@lign@no}{#1}\parseb@dy} +\def\rinter@text#1{\@xp\DPrint\@xp{\the\DP@lign@no}{#1}\parseb@dy} +\def\DPlint#1#2{\@xp\def\csname DP@lint:#1\endcsname{\text{#2}}} +\def\DPrint#1#2{\@xp\def\csname DP@rint:#1\endcsname{\text{#2}}} +\def\DP@lint#1{\ifbalancedlrint\@xp\ifx\csname +DP@lint:#1\endcsname\relax\phantom + {\csname DP@rint:#1\endcsname}\else\csname DP@lint:#1\endcsname\fi + \else\csname DP@lint:#1\endcsname\fi} +\def\DP@rint#1{\ifbalancedlrint\@xp\ifx\csname +DP@rint:#1\endcsname\relax\phantom + {\csname DP@lint:#1\endcsname}\else\csname DP@rint:#1\endcsname\fi + \else\csname DP@rint:#1\endcsname\fi} +\def\eat@command#1#2{\ifcase#1\addto@DPbody{\intertext{#2}}\or + \addto@DPbody{\shortintertext{#2}}\fi\DP@crtrue + \global\advance\DP@lign@no\@ne\parseb@dy} +\def\parse@cr{\new@ifnextchar*{\parse@crst}{\parse@crst{}}} +\def\parse@crst#1{\addto@DPbody{#1}\new@ifnextchar[{\parse@crb}{\parseb@dy}} +\def\parse@crb[#1]{\addto@DPbody{[#1]}\parseb@dy} +\def\check@braces#1#2!Q!Q!Q!{\def\dp@lignt@stm@cro{#2}\ifx + \empty\dp@lignt@stm@cro\br@cefalse\else\br@cetrue\fi} +\def\eat@environment#1{\addto@DPbody{\begin{#1}}\begingroup + \def\@currenvir{#1}\let\@next\digest@env\@next} +\def\digest@env#1\end#2{% + \edef\begin@stack{\push@begins#1\begin\end \@xp\@gobble\begin@stack}% + \ifx\@empty\begin@stack + \@checkend{#2} + \endgroup\let\@next\parseb@dy\fi + \addto@DPbody{#1\end{#2}} + \@next} +\def\lintertext{lint}\def\rintertext{rint} +\newif\ifbalancedlrint +\let\DPh@@kl\empty\let\DPh@@kr\empty +\def\DPg@therl{&\omit\hfil$\displaystyle} +\def\DPg@therr{$\hfil} + +\newenvironment{DPalign*}[1][a]{% + \if m#1\balancedlrintfalse\else\balancedlrinttrue\fi + \global\DP@lign@no\z@\DP@crfalse + \DP@lignb@dy{&\DP@lint0&}\parseb@dy +}{% + \endalign +} +\newenvironment{DPgather*}[1][a]{% + \if m#1\balancedlrintfalse\else\balancedlrinttrue\fi + \global\DP@lign@no\z@\DP@crfalse + \let\DPh@@kl\DPg@therl + \let\DPh@@kr\DPg@therr + \DP@lignb@dy{&\DP@lint0&\DPh@@kl}\parseb@dy +}{% + \endalign +} +\makeatother + +%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{document} + +\pagestyle{empty} +\pagenumbering{alph} +\phantomsection +\pdfbookmark[-1]{Front Matter}{Front Matter} + +%%%% PG BOILERPLATE %%%% +\Pagelabel{PGBoilerplate} +\phantomsection +\pdfbookmark[0]{PG Boilerplate}{Project Gutenberg Boilerplate} + +\begin{center} +\begin{minipage}{\textwidth} +\small +\begin{PGtext} +Project Gutenberg's A Treatise on Probability, by John Maynard Keynes + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: A Treatise on Probability + +Author: John Maynard Keynes + +Release Date: May 31, 2010 [EBook #32625] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK A TREATISE ON PROBABILITY *** +\end{PGtext} +\end{minipage} +\end{center} + +\clearpage + + +%%%% Credits and transcriber's note %%%% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. Hwang, Ralph Janke, and the Online +Distributed Proofreading Team at http://www.pgdp.net (This +file was produced from images generously made available +by The Internet Archive) +\end{PGtext} +\end{minipage} +\end{center} +\vfill + +\begin{minipage}{0.85\textwidth} +\small +\pdfbookmark[0]{Transcriber's Note}{Transcriber's Note} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% + +\frontmatter + +\pagenumbering{roman} +\pagestyle{empty} + +\normalsize + +\null\vfill\vfill +\begin{center} +BY THE SAME AUTHOR +\medskip + +\setlength{\fboxsep}{12pt}% +\settowidth{\TmpLen}{\textbf{INDIAN CURRENCY AND FINANCE}}% +\framebox{% + \begin{minipage}{\TmpLen} + \bigskip + + \centering + \textbf{INDIAN CURRENCY AND FINANCE} + \medskip + + 8vo. Pp.~viii + 263. 1913. + \medskip + + \textbf{7s.\ 6d.}\ net. + \bigskip + + \textbf{THE ECONOMIC CONSEQUENCES + OF THE PEACE} + \medskip + + 8vo. Pp.~vii + 279. 1919. + \medskip + + \textbf{8s.\ 6d.}\ net. + \bigskip + \end{minipage} +} +\end{center} +\vfill\vfill\vfill +\cleardoublepage + +%% -----File: 001.png---Folio ii------- + +\TitlePage[noclear]{A TREATISE ON PROBABILITY} + +%% -----File: 002.png---Folio iii------- + +\null\vfill +\begin{center} +\includegraphics[width=1.5in]{./images/macmillan.pdf} +\medskip + +\footnotesize MACMILLAN AND CO., \textsc{Limited} + +\scriptsize +LONDON · BOMBAY · CALCUTTA · MADRAS \\ +MELBOURNE +\medskip + +\footnotesize THE MACMILLAN COMPANY + +\scriptsize +NEW YORK · BOSTON · CHICAGO \\ +DALLAS · SAN FRANCISCO +\medskip + +\footnotesize THE MACMILLAN CO. OF CANADA, \textsc{Ltd.} + +\scriptsize +TORONTO +\end{center} +\vfill +\newpage + +%% -----File: 003.png---Folio iv------- + +\begin{center} +\MyHuge\bfseries A TREATISE \\ +ON PROBABILITY +\vfill\vfill + +\normalfont\footnotesize BY\\[8pt] +\large JOHN\ \ MAYNARD\ \ KEYNES\\[8pt] +\tiny FELLOW OF KING'S COLLEGE, CAMBRIDGE +\vfill\vfill\vfill + +\normalsize\settowidth{\TmpLen}{ST. MARTIN'S STREET, LONDON\quad}% +\makebox[\TmpLen][s]{MACMILLAN AND CO., LIMITED} \\ +\makebox[\TmpLen][s]{ST. MARTIN'S STREET, LONDON} + +\oldstylenums{1921} +\end{center} + +%% -----File: 004.png---Folio v------- + +\stretchyspace + +\Preface + +The subject matter of this book was first broached in the brain +of Leibniz, who, in the dissertation, written in his twenty-third +year, on the mode of electing the kings of Poland, conceived +of Probability as a branch of Logic. A few years before, ``un +problème,'' in the words of Poisson, ``proposé à un austère +janséniste par un homme du monde, a été l'origine du calcul +des probabilitiés.'' In the intervening centuries the algebraical +exercises, in which the Chevalier de la Méré interested Pascal, +have so far predominated in the learned world over the profounder +enquiries of the philosopher into those processes of +human faculty which, by determining reasonable preference, +guide our choice, that Probability is oftener reckoned with Mathematics +than with Logic. There is much here, therefore, which is +novel and, being novel, unsifted, inaccurate, or deficient. I +propound my systematic conception of this subject for criticism +and enlargement at the hand of others, doubtful whether I +myself am likely to get much further, by waiting longer, +with a work, which, beginning as a Fellowship Dissertation, +and interrupted by the war, has already extended over +many years. + +It may be perceived that I have been much influenced by +W.~E. Johnson, G.~E. Moore, and Bertrand Russell, that is +to say by Cambridge, which, with great debts to the writers +of Continental Europe, yet continues in direct succession +the English tradition of Locke and Berkeley and Hume, of +Mill and Sidgwick, who, in spite of their divergences of +%% -----File: 005.png---Folio vi------- +doctrine, are united in a preference for what is matter of +fact, and have conceived their subject as a branch rather of +science than of the creative imagination, prose writers, hoping +to be understood. + +\hfill J. M. KEYNES.{\qquad} +\bigskip + +\footnotesize +\settowidth{\TmpLen}{\textsc{King's College, Cambridge},}% +\begin{minipage}{\TmpLen} +\centering +\textsc{King's College, Cambridge}, \\ +\textit{May} 1, 1920. +\end{minipage} +\normalsize + +%% -----File: 006.png---Folio vii------- + +\clearpage +\fancyhead[CE]{\CtrHeading{A Treatise on Probability}} +\fancyhead[CO]{\CtrHeading{Contents}} +\phantomsection +\pdfbookmark[0]{Contents}{Table of Contents} +\tableofcontents + +\iffalse +CONTENTS + +PART I +FUNDAMENTAL IDEAS + +CHAPTER I PAGE +The Meaning of Probability 3 + +CHAPTER II +Probability in Relation to the Theory of Knowledge 10 + +CHAPTER III +The Measurement of Probabilities 20 + +CHAPTER IV +The Principle of Indifference 41 + +CHAPTER V +Other Methods of Determining Probabilities 65 + +CHAPTER VI +The Weight of Arguments 71 +%% -----File: 007.png---Folio viii------- + +CHAPTER VII +Historical Retrospect 79 + +CHAPTER VIII +The Frequency Theory of Probability 92 + +CHAPTER IX +The Constructive Theory of Part I. summarised 111 + + +PART II +FUNDAMENTAL THEOREMS + +CHAPTER X +Introductory 115 + +CHAPTER XI + +The Theory of Groups, with special reference to Logical +Consistence, Inference and Logical Priority 123 + +CHAPTER XII +The Definitions and Axioms of Inference and Probability 133 + +CHAPTER XIII +The Fundamental Theorems of Necessary Inference 139 + +CHAPTER XIV +The Fundamental Theorems of Probable Inference 144 +%% -----File: 008.png---Folio ix------- + +CHAPTER XV PAGE +Numerical Measurement and Approximation of Probabilities 158 + +CHAPTER XVI +Observations on the Theorems of Chapter XIV., and +their Developments, including Testimony 164 + +CHAPTER XVII +Some Problems in Inverse Probability, including Averages 186 + +PART III +INDUCTION AND ANALOGY + +CHAPTER XVIII +Introduction 217 + +CHAPTER XIX +The Nature of Argument by Analogy 222 + +CHAPTER XX +The Value of Multiplication of Instances, or Pure Induction 233 + +CHAPTER XXI +The Nature of Inductive Argument continued 242 + +CHAPTER XXII +The Justification of these Methods 251 + +CHAPTER XXIII +Some Historical Notes on Induction 265 + +Notes on Part III 274 +%% -----File: 009.png---Folio x------- + +PART IV +SOME PHILOSOPHICAL APPLICATIONS OF PROBABILITY + +CHAPTER XXIV PAGE +The Meanings of Objective Chance, and of Randomness 281 + +CHAPTER XXV +Some Problems arising out of the Discussion of Chance 293 + +CHAPTER XXVI +The Application of Probability to Conduct 307 + + +PART V +THE FOUNDATIONS OF STATISTICAL INFERENCE + +CHAPTER XXVII +The Nature of Statistical Inference 327 + +CHAPTER XXVIII +The Law of Great Numbers 332 + +CHAPTER XXIX +The Use of à priori Probabilities for the Prediction of +Statistical Frequency---the Theorems of Bernoulli, +Poisson, and Tchebycheff 337 + +CHAPTER XXX +The Mathematical use of Statistical Frequencies for the +Determination of Probability à posteriori--the Methods +of Laplace 367 +%% -----File: 010.png---Folio xi------- + +CHAPTER XXXI PAGE +The Inversion of Bernoulli's Theorem 384 + +CHAPTER XXXII +The Inductive Use of Statistical Frequencies for the +Determination of Probability à posteriori--the Methods +of Lexis 391 + +CHAPTER XXXIII +Outline of a Constructive Theory 406 + +BIBLIOGRAPHY 429 + +INDEX 459 +\fi +%% -----File: 011.png---Folio xii------- +%[Blank Page] +%% -----File: 012.png---Folio 1------- +\index{Morgan, \textit{vide} De Morgan}% +\index{Succession, Law of!\textit{See} Rule of}% + +\mainmatter + +\Part{I}{Fundamental Ideas} +%% -----File: 013.png---Folio 2------- +%[Blank Page] +%% -----File: 014.png---Folio 3------- +\index{Logic, academic}% +\index{Proposition, characterisation of}% + +\Chapter{I}{The Meaning of Probability} + +\begin{Quote} +``J'ai dit plus d'une fois qu'il faudrait une nouvelle espèce de logique, qui +traiteroit des degrés de Probabilité.''---\textsc{Leibniz}. +\end{Quote} + +\Paragraph{1.} \First{Part} of our knowledge we obtain direct; and part by +\index{Knowledge!kinds of}% +argument. The Theory of Probability is concerned with that +part which we obtain by argument, and it treats of the different +degrees in which the results so obtained are conclusive or inconclusive. + +In most branches of academic logic, such as the theory of the +syllogism or the geometry of ideal space, all the arguments aim +at demonstrative certainty. They claim to be \emph{conclusive}. But +many other arguments are rational and claim some weight without +pretending to be certain. In Metaphysics, in Science, and in +Conduct, most of the arguments, upon which we habitually base +our rational beliefs, are admitted to be inconclusive in a greater +or less degree. Thus for a philosophical treatment of these +branches of knowledge, the study of probability is required. + +The course which the history of thought has led Logic to follow +has encouraged the view that doubtful arguments are not within +its scope. But in the actual exercise of reason we do not wait +on certainty, or doom it irrational to depend on a doubtful +argument. If logic investigates the general principles of valid +thought, the study of arguments, to which it is rational to attach +\emph{some} weight, is as much a part of it as the study of those which +are demonstrative. + +\Paragraph{2.} The terms \emph{certain} and \emph{probable} describe the various degrees +of rational belief about a proposition which different amounts of +knowledge authorise us to entertain. All propositions are true +or false, but the knowledge we have of them depends on our +circumstances; and while it is often convenient to speak of +%% -----File: 015.png---Folio 4------- +\index{Belief, rational|ifoll}% +\index{Probability, and relevant knowledge}% +\index{Probability relation}% +\index{Proposition, characterisation of}% +propositions as certain or probable, this expresses strictly a +relationship in which they stand to a \textit{corpus} of knowledge, actual or +\index{Knowledge!kinds of}% +hypothetical, and not a characteristic of the propositions in themselves. +A proposition is capable at the same time of varying degrees +of this relationship, depending upon the knowledge to which it is +related, so that it is without significance to call a proposition probable +unless we specify the knowledge to which we are relating it. + +To this extent, therefore, probability may be called subjective. +But in the sense important to logic, probability is not +subjective. It is not, that is to say, subject to human caprice. +A proposition is not probable because we think it so. When once +the facts are given which determine our knowledge, what is +probable or improbable in these circumstances has been fixed +objectively, and is independent of our opinion. The Theory of +Probability is logical, therefore, because it is concerned with the +degree of belief which it is \emph{rational} to entertain in given conditions, +and not merely with the actual beliefs of particular individuals, +which may or may not be rational. + +Given the body of direct knowledge which constitutes our +ultimate premisses, this theory tells us what further rational +beliefs, certain or probable, can be derived by valid argument +from our direct knowledge. This involves purely logical relations +between the propositions which embody our direct knowledge +and the propositions about which we seek indirect knowledge. +What particular propositions we select as the premisses +of \emph{our} argument naturally depends on subjective factors peculiar +to ourselves; but the relations, in which other propositions stand +to these, and which entitle us to probable beliefs, are objective +and logical. + +\Paragraph{3.} Let our premisses consist of any set of propositions~$h$, and +our conclusion consist of any set of propositions~$a$, then, if a +knowledge of~$h$ justifies a rational belief in~$a$ of degree~$\alpha$, we say +that there is a \emph{probability-relation} of degree~$\alpha$ between $a$~and~$h$.\footnote + {This will be written $a/h = \alpha$.} + +In ordinary speech we often describe the \emph{conclusion} as being +doubtful, uncertain, or only probable. But, strictly, these terms +ought to be applied, either to the degree of our \emph{rational belief} in +the conclusion, or to the relation or argument between two sets +of propositions, knowledge of which would afford grounds for a +corresponding degree of rational belief.\footnote + {See also \Chapref{II}. §\;5.} +%% -----File: 016.png---Folio 5------- +\index{Event, probability of}% +\index{Probability, and relevant knowledge!objective relation of}% + +\Paragraph{4.} With the term ``event,'' which has taken hitherto so important +a place in the phraseology of the subject, I shall dispense +altogether.\footnote + {Except in those chapters (\Chapref[Chap.]{XVII}., for example) where I am dealing + chiefly with the work of others.} +Writers on Probability have generally dealt +with what they term the ``happening'' of ``events.'' In the +problems which they first studied this did not involve much +departure from common usage. But these expressions are now +used in a way which is vague and ambiguous; and it will be +more than a verbal improvement to discuss the truth and the +probability of \emph{propositions} instead of the occurrence and the +probability of \emph{events}.\footnote + {The first writer I know of to notice this was Ancillon in \textit{Doutes sur les +\index{Ancillon|inote}% + bases du calcul des probabilités} (1794): ``Dire qu'un fait passé, présent ou à + venir est probable, c'est dire qu'une proposition est probable.'' The point was + emphasised by Boole, \textit{Laws of Thought}, pp.\ 7~and~167. See also Czuber, + \textit{Wahrscheinlichkeitsrechnung}, vol.~i.\ p.~5, and Stumpf, \textit{Über den Begriff der mathematischen + Wahrscheinlichkeit}.} + +\Paragraph{5.} These general ideas are not likely to provoke much +criticism. In the ordinary course of thought and argument, +we are constantly assuming that knowledge of one statement, +while not \emph{proving} the truth of a second, yields nevertheless +\emph{some ground} for believing it. We assert that we \emph{ought} on the +evidence to prefer such and such a belief. We claim rational +grounds for assertions which are not conclusively demonstrated. +We allow, in fact, that statements may be unproved, without, for +that reason, being unfounded. And it does not seem on reflection +that the information we convey by these expressions is wholly +subjective. When we argue that Darwin gives valid grounds +for our accepting his theory of natural selection, we do not simply +mean that we are psychologically inclined to agree with him; +it is certain that we also intend to convey our belief that +we are acting rationally in regarding his theory as probable. +We believe that there is some real objective relation +between Darwin's evidence and his conclusions, which is independent +of the mere fact of our belief, and which is just as real +and objective, though of a different degree, as that which would +exist if the argument were as demonstrative as a syllogism. +We are claiming, in fact, to cognise correctly a logical connection +between one set of propositions which we call our evidence and +which we suppose ourselves to know, and another set which we +call our conclusions, and to which we attach more or less weight +%% -----File: 017.png---Folio 6------- +\index{Probability, and relevant knowledge!mathematical}% +\index{Relation, of probability}% +according to the grounds supplied by the first. It is this type +of objective relation between sets of propositions---the type +which we claim to be correctly perceiving when we make such +assertions as these---to which the reader's attention must be +directed. + +\Paragraph{6.} It is not straining the use of words to speak of this as the +relation of probability. It is true that mathematicians have +employed the term in a narrower sense; for they have often +confined it to the limited class of instances in which the relation +is adapted to an algebraical treatment. But in common usage +the word has never received this limitation. + +Students of probability in the sense which is meant by the +authors of typical treatises on \textit{Wahrscheinlichkeitsrechnung} or +\textit{Calcul des probabilités}, will find that I do eventually reach topics +with which they are familiar. But in making a serious attempt +to deal with the fundamental difficulties with which all students +of mathematical probabilities have met and which are notoriously +unsolved, we must begin at the beginning (or almost at the +beginning) and treat our subject widely. As soon as mathematical +probability ceases to be the merest algebra or pretends +to guide our decisions, it immediately meets with problems +against which its own weapons are quite powerless. And even +if we wish later on to use probability in a narrow sense, it will +be well to know first what it means in the widest. + +\Paragraph{7.} Between two sets of propositions, therefore, there exists +a relation, in virtue of which, if we know the first, we can attach +to the latter some degree of rational belief. This relation is the +subject-matter of the logic of probability. + +A great deal of confusion and error has arisen out of a +failure to take due account of this \emph{relational} aspect of probability. +From the premisses ``$a$~implies~$b$'' and ``$a$~is true,'' we +can conclude something about~$b$---namely that $b$~is true---which +does not involve~$a$. But, if $a$ is so related to~$b$, that a knowledge +of it renders a probable belief in~$b$ rational, we cannot conclude +anything whatever about~$b$ which has not reference to~$a$; and it +is not true that every set of self-consistent premisses which +includes~$a$ has this same relation to~$b$. It is as useless, therefore, +to say ``$b$~is probable'' as it would be to say ``$b$~is equal,'' +or ``$b$~is greater than,'' and as unwarranted to conclude that, +because $a$~makes $b$~probable, therefore $a$~and~$c$ together make $b$~probable, +%% -----File: 018.png---Folio 7------- +\index{Evidence, and measurement of Probability}% +\index{Probability, and relevant knowledge!dependent on evidence}% +as to argue that because $a$~is less than~$b$, therefore $a$~and~$c$ +together are less than~$b$. + +Thus, when in ordinary speech we name some opinion as +probable without further qualification, the phrase is generally +elliptical. We mean that it is probable when certain considerations, +implicitly or explicitly present to our minds at the moment, +are taken into account. We use the word for the sake of shortness, +just as we speak of a place as being three miles distant, +when we mean three miles distant from where we are then situated, +or from some starting-point to which we tacitly refer. No +proposition is in itself either probable or improbable, just as no +place can be intrinsically distant; and the probability of the +same statement varies with the evidence presented, which is, +as it were, its origin of reference. We may fix our attention +on our own knowledge and, treating this as our origin, consider +the probabilities of all other suppositions,---according to the +usual practice which leads to the elliptical form of common +speech; or we may, equally well, fix it on a proposed conclusion +and consider what degree of probability this would derive from +various sets of assumptions, which might constitute the \textit{corpus} of +knowledge of ourselves or others, or which are merely +hypotheses. + +Reflection will show that this account harmonises with +familiar experience. There is nothing novel in the supposition +that the probability of a theory turns upon the evidence by which +it is supported; and it is common to assert that an opinion was +probable on the evidence at first to hand, but on further information +was untenable. As our knowledge or our hypothesis changes, +\index{Hypothesis}% +our conclusions have new probabilities, not in themselves, but +relatively to these new premisses. New logical relations have +now become important, namely those between the conclusions +which we are investigating and our new assumptions; but the +old relations between the conclusions and the former assumptions +still exist and are just as real as these new ones. It would be +as absurd to deny that an opinion \emph{was} probable, when at a later +stage certain objections have come to light, as to deny, when +we have reached our destination, that it was ever three miles +distant; and the opinion still \emph{is} probable in relation to the old +hypotheses, just as the destination is still three miles distant +from our starting-point. +%% -----File: 019.png---Folio 8------- +\index{Logic, academic!of probability}% +\index{Probability@{`\textit{Probability}'}}% +\index{Probability, and relevant knowledge!objective relation of}% +\index{Probability, and relevant knowledge!philosophical definition of}% +\index{Probability relation}% + +\Paragraph{8.} A \emph{definition} of probability is not possible, unless it contents +us to define degrees of the probability-relation by reference to +degrees of rational belief. We cannot analyse the probability-relation +in terms of simpler ideas. As soon as we have passed +from the logic of implication and the categories of truth and +falsehood to the logic of probability and the categories of knowledge, +ignorance, and rational belief, we are paying attention to +a new logical relation in which, although it is logical, we were +not previously interested, and which cannot be explained or +defined in terms of our previous notions. + +This opinion is, from the nature of the case, incapable of positive +proof. The presumption in its favour must arise partly +out of our failure to find a definition, and partly because the +notion presents itself to the mind as something new and independent. +If the statement that an opinion was probable on the +evidence at first to hand, but became untenable on further information, +is not solely concerned with psychological belief, I +do not know how the element of logical doubt is to be defined, +or how its substance is to be stated, in terms of the other +indefinables of formal logic. The attempts at definition, which +have been made hitherto, will be criticised in later chapters. +I do not believe that any of them accurately represent that particular +logical relation which we have in our minds when we +speak of the probability of an argument. + +In the great majority of cases the term ``probable'' seems to +be used consistently by different persons to describe the same +concept. Differences of opinion have not been due, I think, to +a radical ambiguity of language. In any case a desire to reduce +the indefinables of logic can easily be carried too far. Even if +a definition is discoverable in the end, there is no harm in postponing +it until our enquiry into the object of definition is far +advanced. In the case of ``probability'' the object before the +mind is so familiar that the danger of misdescribing its qualities +through lack of a definition is less than if it were a highly abstract +entity far removed from the normal channels of thought. + +\Paragraph{9.} This chapter has served briefly to indicate, though not +to define, the subject matter of the book. Its object has +been to emphasise the existence of \emph{a logical relation between two +sets of propositions} in cases where it is not possible to argue +demonstratively from one to the other. This is a contention +%% -----File: 020.png---Folio 9------- +of a most fundamental character. It is not entirely novel, but +has seldom received due emphasis, is often overlooked, and +sometimes denied. The view, that probability arises out of +the existence of a specific relation between premiss and conclusion, +depends for its acceptance upon a reflective judgment on the +true character of the concept. It will be our object to discuss, +under the title of Probability, the principal properties of this +relation. First, however, we must digress in order to consider +briefly what we mean by \emph{knowledge}, \emph{rational belief}, and \emph{argument}. +%% -----File: 021.png---Folio 10------- +\index{Belief, rational}% + + +\Chapter{II}{Probability in Relation to the Theory of Knowledge} +\index{Knowledge}% + +\Paragraph{1.} \First{I do} not wish to become involved in questions of epistemology +to which I do not know the answer; and I am anxious to reach +as soon as possible the particular part of philosophy or logic +which is the subject of this book. But some explanation is +necessary if the reader is to be put in a position to understand +the point of view from which the author sets out; I will, therefore, +expand some part of what has been outlined or assumed +in the first chapter. + +\Paragraph{2.} There is, first of all, the distinction between that part of +our belief which is rational and that part which is not. If a +man believes something for a reason which is preposterous or +for no reason at all, and what he believes turns out to be true for +some reason not known to him, he cannot be said to believe it +\emph{rationally}, although he believes it and it is in fact true. On the +other hand, a man may rationally believe a proposition to be +\emph{probable}, when it is in fact false. The distinction between +rational belief and mere belief, therefore, is not the same as the +distinction between true beliefs and false beliefs. The highest +degree of rational belief, which is termed \emph{certain} rational belief, +corresponds to \emph{knowledge}. We may be said to know a thing +when we have a certain rational belief in it, and \textit{vice versa}. For +reasons which will appear from our account of probable degrees +of rational belief in the following paragraph, it is preferable to +regard \emph{knowledge} as fundamental and to define \emph{rational belief} by +reference to it. + +\Paragraph{3.} We come next to the distinction between that part of our +rational belief which is certain and that part which is only +probable. Belief, whether rational or not, is capable of degree. +The highest degree of rational belief, or rational certainty of +\index{Certainty}% +%% -----File: 022.png---Folio 11------- +\index{Belief, rational!degrees of}% +\index{Johnson, W. E.!propositions@{and propositions}|inote}% +\index{Probability, and relevant knowledge!three senses of}% +\index{Proposition, characterisation of!primary and secondary}% +belief, and its relation to knowledge have been introduced above. +What, however, is the relation to knowledge of \emph{probable} degrees +of rational belief? + +The proposition (\emph{say},~$q$) that we \emph{know} in this case is not the +same as the proposition (\emph{say},~$p$) in which we have a probable +degree (\emph{say},~$\DPtypo{a}{\alpha}$) of rational belief. If the evidence upon which +we base our belief is~$h$, then what we \emph{know}, namely~$q$, is that +the proposition~$p$ bears the probability-relation of degree~$\DPtypo{a}{\alpha}$ to +the set of propositions~$h$; and this knowledge of ours justifies +us in a rational belief of degree~$\DPtypo{a}{\alpha}$ in the proposition~$p$. It will +be convenient to call propositions such as~$p$, which do not contain +assertions about probability-relations, ``primary propositions''; +and propositions such as~$q$, which assert the existence of a +probability-relation, ``secondary propositions.''\footnote + {This classification of ``primary'' and ``secondary'' propositions was + suggested to me by Mr.~W.~E. Johnson.} + +\Paragraph{4.} Thus knowledge of a proposition always corresponds to +certainty of rational belief in it and at the same time to actual +truth in the proposition itself. We cannot know a proposition +unless it is in fact true. A probable degree of rational belief +in a proposition, on the other hand, arises out of knowledge of +some corresponding secondary proposition. A man may rationally +believe a proposition to be probable when it is in fact false, +if the secondary proposition on which he depends is true and +certain; while a man cannot rationally believe a proposition +to be probable even when it is in fact true, if the secondary +proposition on which he depends is not true. Thus rational +belief of whatever degree can only arise out of knowledge, +although the knowledge may be of a proposition secondary, in +the above sense, to the proposition in which the rational degree +of belief is entertained. + +\Paragraph{5.} At this point it is desirable to colligate the three senses +in which the term \emph{probability} has been so far employed. In its +most fundamental sense, I think, it refers to the logical relation +between two sets of propositions, which in §\;4~of \Chapref{I}\@. I +have termed the probability-relation. It is with this that I shall +be mainly concerned in the greater part of this Treatise. Derivative +from this sense, we have the sense in which, as above, the +term \emph{probable} is applied to the degrees of rational belief arising +out of knowledge of secondary propositions which assert the +%% -----File: 023.png---Folio 12------- +\index{Acquaintance, direct}% +\index{Proposition, characterisation of!knowledge of}% +existence of probability-relations in the fundamental logical sense. +Further it is often convenient, and not necessarily misleading, +to apply the term \emph{probable} to the proposition which is the object +of the probable degree of rational belief, and which bears the +probability-relation in question to the propositions comprising +the evidence. + +\Paragraph{6.} I turn now to the distinction between direct and indirect +knowledge---between that part of our rational belief which we +\index{Knowledge!direct and indirect}% +know directly and that part which we know by argument. + +We start from things, of various classes, with which we have, +what I choose to call without reference to other uses of this term, +\emph{direct acquaintance}. Acquaintance with such things does not in +itself constitute knowledge, although knowledge arises out of +acquaintance with them. The most important classes of things +with which we have direct acquaintance are our own sensations, +which we may be said to \emph{experience}, the ideas or meanings, about +which we have thoughts and which we may be said to \emph{understand}, +and facts or characteristics or relations of sense-data or meanings, +which we may be said to \emph{perceive};---experience, understanding, +and perception being three forms of direct acquaintance. + +The objects of knowledge and belief---as opposed to the +objects of direct acquaintance which I term sensations, meanings, +and perceptions---I shall term \emph{propositions}. + +Now our knowledge of propositions seems to be obtained in +two ways: directly, as the result of contemplating the objects +of acquaintance; and indirectly, \emph{by argument}, through perceiving +the probability-relation of the proposition, about which we seek +knowledge, to other propositions. In the second case, at any +rate at first, what we know is not the proposition itself but a +secondary proposition involving it. When we know a secondary +proposition involving the proposition $p$ as subject, we may be +said to have indirect knowledge \emph{about}~$p$. + +Indirect knowledge about~$p$ may in suitable conditions lead +to rational belief in~$p$ of an appropriate degree. If this degree +is that of certainty, then we have not merely indirect knowledge +\emph{about}~$p$, but indirect knowledge \emph{of}~$p$. + +\Paragraph{7.} Let us take examples of direct knowledge. From acquaintance +with a sensation of yellow I can pass directly to a +knowledge of the proposition ``I have a sensation of yellow.'' +From acquaintance with a sensation of yellow and with the +%% -----File: 024.png---Folio 13------- +\index{Probability relation}% +\index{Proposition, characterisation of!primary and secondary}% +meanings of ``yellow,'' ``colour,'' ``existence,'' I may be able +to pass to a direct knowledge of the propositions ``I understand +\index{Knowledge!incomplete and proper}% +the meaning of yellow,'' ``my sensation of yellow exists,'' ``yellow +is a colour.'' Thus, by some mental process of which it is +difficult to give an account, we are able to pass from direct +acquaintance with things to a knowledge of propositions about +the things of which we have sensations or understand the +meaning. + +Next, by the contemplation of propositions of which we have +direct knowledge, we are able to pass indirectly to knowledge of or +about other propositions. The mental process by which we pass +from direct knowledge to indirect knowledge is in some cases and +in some degree capable of analysis. We pass from a knowledge +of the proposition~$a$ to a knowledge about the proposition~$b$ by perceiving +a logical relation between them. With this logical relation +we have direct acquaintance. The logic of knowledge is +mainly occupied with a study of the logical relations, direct +acquaintance with which permits direct knowledge of the +secondary proposition asserting the probability-relation, and so +to indirect knowledge about, and in some cases of, the primary +proposition. + +It is not always possible, however, to analyse the mental +process in the case of indirect knowledge, or to say by the perception +of \emph{what} logical relation we have passed from the knowledge +of one proposition to knowledge about another. But +although in some cases we \emph{seem} to pass directly from one proposition +to another, I am inclined to believe that in all legitimate +transitions of this kind some logical relation of the proper kind +must exist between the propositions, even when we are not +explicitly aware of it. In any case, whenever we pass to +knowledge about one proposition by the contemplation of it in +relation to another proposition of which we have knowledge---even +when the process is unanalysable---I call it an \emph{argument.} +\index{Argument}% +The knowledge, such as we have in ordinary thought by passing +from one proposition to another without being able to say what +logical relations, if any, we have perceived between them, may +be termed uncompleted knowledge. And knowledge, which +results from a distinct apprehension of the relevant logical +relations, may be termed knowledge proper. + +\Paragraph{8.} In this way, therefore, I distinguish between direct and +%% -----File: 025.png---Folio 14------- +indirect knowledge, between that part of our rational belief which +\index{Knowledge!of logical relations}% +is based on direct knowledge and that part which is based on +argument. About what \emph{kinds} of things we are capable of knowing +propositions directly, it is not easy to say. About our +own existence, our own sense-data, some logical ideas, and some +logical relations, it is usually agreed that we have direct knowledge. +Of the law of gravity, of the appearance of the other +side of the moon, of the cure for phthisis, of the contents of +Bradshaw, it is usually agreed that we do \emph{not} have direct knowledge. +But many questions are in doubt. Of \emph{which} logical +ideas and relations we have direct acquaintance, as to whether +we can ever know directly the existence of \emph{other people}, and as +to when we are knowing propositions about sense-data directly +and when we are interpreting them---it is not possible to give +a clear answer. Moreover, there is another and peculiar kind +of derivative knowledge---by memory. +\index{Memory}% + +At a given moment there is a great deal of our knowledge +which we know neither directly nor by argument---we remember +it. We may remember it as knowledge, but forget how we originally +knew it. What we once knew and now consciously remember, +can fairly be called knowledge. But it is not easy to +draw the line between conscious memory, unconscious memory +or habit, and pure instinct or irrational associations of ideas +(acquired or inherited)---the last of which cannot fairly be called +knowledge, for unlike the first two it did not even arise (in us at +least) out of knowledge. Especially in such a case as that of +what our eyes tell us, it is difficult to distinguish between the +different ways in which our beliefs have arisen. We cannot +always tell, therefore, what is remembered knowledge and what is +not knowledge at all; and when knowledge is remembered, we +do not always remember at the same time whether, originally, it +was direct or indirect. + +Although it is with knowledge by argument that I shall be +mainly concerned in this book there is one kind of direct knowledge, +namely of secondary propositions, with which I cannot +help but be involved. In the case of every argument, it is only +directly that we can know the secondary proposition which makes +the argument itself valid and rational. When we know something +by argument this must be through direct acquaintance +with some logical relation between the conclusion and the premiss. +%% -----File: 026.png---Folio 15------- +\index{Bernoulli, Jac.|inote}% +\index{Fries|inote}% +\index{Laplace|inote}% +In \emph{all} knowledge, therefore, there is some direct element; and +logic can never be made purely mechanical. All it can do is +so to arrange the reasoning that the logical relations, which +have to be perceived directly, are made explicit and are of a +simple kind. + +\Paragraph{9.} It must be added that the term \emph{certainty} is sometimes used +\index{Certainty!truth@{and truth}}% +in a merely psychological sense to describe a state of mind +without reference to the logical grounds of the belief. With +this sense I am not concerned. It is also used to describe the +highest degree of rational belief; and this is the sense relevant +to our present purpose. The peculiarity of certainty is that +knowledge of a secondary proposition involving certainty, +together with knowledge of what stands in this secondary +proposition in the position of evidence, leads to \emph{knowledge of}, +and not merely \emph{about}, the corresponding primary proposition. +Knowledge, on the other hand, of a secondary proposition involving +a degree of probability lower than certainty, together +with knowledge of the premiss of the secondary proposition, +leads only to a \emph{rational belief of the appropriate degree} in the +primary proposition. The knowledge present in this latter case +I have called knowledge \emph{about} the primary proposition or conclusion +of the argument, as distinct from knowledge \emph{of} it. + +Of probability we can say no more than that it is a lower degree +of rational belief than certainty; and we may say, if we like, +that it deals with degrees of certainty.\footnote + {This view has often been taken, \eg, by Bernoulli and, incidentally, by + Laplace; also by Fries (see Czuber, \textit{Entwicklung}, p.~12). The view, occasionally + held, that probability is concerned with degrees of truth, arises out of a + confusion between certainty and truth. Perhaps the Aristotelian doctrine + that future events are neither true nor false arose in this way.} +Or we may make +probability the more fundamental of the two and regard certainty +as a special case of probability, as being, in fact, the \emph{maximum +probability}. Speaking somewhat loosely we may say that, if +our premisses make the conclusion certain, then it \emph{follows} from +the premisses; and if they make it very probable, then it very +nearly follows from them. + +It is sometimes useful to use the term ``impossibility'' as +\index{Impossibility}% +the negative correlative of ``certainty,'' although the former +sometimes has a different set of associations. If $a$~is certain, +then the contradictory of~$a$ is impossible. If a knowledge of~$a$ +makes $b$~certain, then a knowledge of~$a$ makes the contradictory +%% -----File: 027.png---Folio 16------- +\index{Belief, rational}% +\index{Modality and probability|inote}% +of~$b$ impossible. Thus a proposition is impossible with respect +to a given premiss, if it is disproved by the premiss; and the +relation of impossibility is the relation of minimum probability.\footnote + {Necessity and Impossibility, in the senses in which these terms are used + in the theory of Modality, seem to correspond to the relations of Certainty and + Impossibility in the theory of probability, the other modals, which comprise + the intermediate degrees of possibility, corresponding to the intermediate + degrees of probability. Almost up to the end of the seventeenth century + the traditional treatment of modals is, in fact, a primitive attempt to bring + the relations of probability within the scope of formal logic.} + +\Paragraph{10.} We have distinguished between rational belief and irrational +belief and also between rational beliefs which are certain in degree +and those which are only probable. Knowledge has been +distinguished according as it is direct or indirect, according as it +is of primary or secondary propositions, and according as it is +\emph{of} or merely \emph{about} its object. + +In order that we may have a rational belief in a proposition~$p$ +of the degree of certainty, it is necessary that one of two conditions +should be fulfilled---(i.)~that we know $p$ directly; or (ii.)~that +we know a set of propositions~$h$, and also know some secondary +proposition~$q$ asserting a certainty-relation between $p$~and~$h$. +In the latter case $h$~may include secondary as well as primary +propositions, but it is a necessary condition that all the propositions~$h$ +should be \emph{known}. In order that we may have rational +belief in~$p$ of a lower degree of probability than certainty, it is +necessary that we know a set of propositions~$h$, and also know +some secondary proposition~$q$ asserting a probability-relation +between $p$~and~$h$. + +In the above account one possibility has been ruled out. It +is assumed that we cannot have a rational belief in~$p$ of a degree +less than certainty except through knowing a secondary proposition +of the prescribed type. Such belief can only arise, that +is to say, by means of the perception of some probability-relation. +To employ a common use of terms (though one inconsistent with +the use adopted above), I have assumed that all direct knowledge +is certain. All knowledge, that is to say, which is obtained in a +manner strictly direct by contemplation of the objects of acquaintance +and without any admixture whatever of argument and the +contemplation of the logical bearing of any other knowledge on +this, corresponds to \emph{certain} rational belief and not to a merely +probable degree of rational belief. It is true that there do \emph{seem} +to be degrees of knowledge and rational belief, when the source of +%% -----File: 028.png---Folio 17------- +\index{Proposition, characterisation of!self-evident}% +\index{Relativity, of knowledge}% +the belief is solely in acquaintance, as there are when its source +is in argument. But I think that this appearance arises partly +out of the difficulty of distinguishing direct from indirect knowledge, +\index{Knowledge!probable and vague}% +\index{Knowledge!relativity of}% +and partly out of a confusion between \emph{probable} knowledge +and \emph{vague} knowledge. I cannot attempt here to analyse +the meaning of vague knowledge. It is certainly not the same +thing as knowledge proper, whether certain or probable, and +it does not seem likely that it is susceptible of strict logical +treatment. At any rate I do not know how to deal with it, +and in spite of its importance I will not complicate a difficult +subject by endeavouring to treat adequately the theory of vague +knowledge. + +I assume then that only true propositions can be known, +that the term ``probable knowledge'' ought to be replaced by +the term ``probable degree of rational belief,'' and that a probable +degree of rational belief cannot arise directly but only as the +result of an argument, out of the knowledge, that is to say, of +a secondary proposition asserting some logical probability-relation +in which the object of the belief stands to some known +proposition. With arguments, if they exist, the \emph{ultimate} premisses +of which are known in some other manner than that +described above, such as might be called ``probable knowledge,'' +my theory is not adequate to deal without modification.\footnote + {I do not mean to imply, however, at any rate at present, that the ultimate + premisses of an argument need always be \emph{primary} propositions.} + +For the objects of certain belief which is based on direct +knowledge, as opposed to certain belief arising indirectly, there +is a well-established expression; propositions, in which our +rational belief is both certain and direct, are said to be +\emph{self-evident}. + +\Paragraph{11.} In conclusion, the relativity of knowledge to the individual +may be briefly touched on. Some part of knowledge---knowledge +of our own existence or of our own sensations---is clearly relative +to individual experience. We cannot speak of knowledge +absolutely---only of the knowledge of a particular person. Other +parts of knowledge---knowledge of the axioms of logic, for example---may +seem more objective. But we must admit, I think, +that this too is relative to the constitution of the human mind, +and that the constitution of the human mind may vary in some +degree from man to man. What is self-evident to me and what +%% -----File: 029.png---Folio 18------- +I really know, may be only a probable belief to you, or may form +no part of your rational beliefs at all. And this may be true +not only of such things as \emph{my} existence, but of some logical axioms +also. Some men---indeed it is obviously the case---may have a +greater power of logical intuition than others. Further, the +difference between some kinds of propositions over which human +intuition seems to have power, and some over which it has none, +may depend wholly upon the constitution of our minds and +have no significance for a perfectly objective logic. We can no +more assume that all true secondary propositions are or ought +to be universally known than that all true primary propositions +are known. The perceptions of some relations of probability +may be outside the powers of some or all of us. + +What we know and what probability we can attribute to our +rational beliefs is, therefore, subjective in the sense of being +relative to the individual. But given the body of premisses which +our subjective powers and circumstances supply to us, and given +the kinds of logical relations, upon which arguments can be based +and which we have the capacity to perceive, the conclusions, +which it is rational for us to draw, stand to these premisses in an +objective and wholly logical relation. Our logic is concerned +with drawing conclusions by a series of steps of certain specified +kinds from a \emph{limited} body of premisses. + +With these brief indications as to the relation of Probability, +as I understand it, to the Theory of Knowledge, I pass from +problems of ultimate analysis and definition, which are not the +primary subject matter of this book, to the logical theory and +superstructure, which occupies an intermediate position between +the ultimate problems and the applications of the theory, whether +such applications take a generalised mathematical form or a +concrete and particular one. For this purpose it would only +encumber the exposition, without adding to its clearness or its +accuracy, if I were to employ the perfectly exact terminology +and minute refinements of language, which are necessary for the +avoidance of error in very fundamental enquiries. While taking +pains, therefore, to avoid any divergence between the substance +of this chapter and of those which succeed it, and to employ only +such periphrases as could be translated, \emph{if desired}, into perfectly +exact language, I shall not cut myself off from the convenient, +but looser, expressions, which have been habitually employed +%% -----File: 030.png---Folio 19------- +\index{Moore, G. E.}% +\index{Russell, Bertrand}% +by previous writers and have the advantage of being, in a general +way at least, immediately intelligible to the reader.\footnote + {This question, which faces all contemporary writers on logical and philosophical + subjects, is in my opinion much more a question of \emph{style}---and therefore + to be settled on the same sort of considerations as other such questions---than + is generally supposed. There are occasions for very exact methods of statement, + such as are employed in Mr.~Russell's \textit{Principia Mathematica}. But there + are advantages also in writing the English of Hume. Mr.~Moore has developed + in \textit{Principia Ethica} an intermediate style which in his hands has force and + beauty. But those writers, who strain after exaggerated precision without + going the whole hog with Mr.~Russell, are sometimes merely pedantic. They + lose the reader's attention, and the repetitious complication of their phrases + eludes his comprehension, without their really attaining, to compensate, + a complete precision. Confusion of thought is not always best avoided by + technical and unaccustomed expressions, to which the mind has no immediate + reaction of understanding; it is possible, under cover of a careful formalism, + to make statements, which, if expressed in plain language, the mind would + immediately repudiate. There is much to be said, therefore, in favour of + understanding the substance of what you are saying \emph{all the time}, and of never + reducing the substantives of your argument to the mental status of an $x$~or~$y$.} +%% -----File: 031.png---Folio 20------- +\index{Bentham, measurement of Probability}% +\index{Donkin, W. F.}% +\index{Forbes, J. D.|inote}% +\index{Probability, and relevant knowledge!measurement of|ifoll}% + + +\Chapter{III}{The Measurement of Probabilities} + +\Paragraph{1.} \First{I have} spoken of probability as being concerned with \emph{degrees} +of rational belief. This phrase implies that it is in some sense +quantitative and perhaps capable of measurement. The theory +of probable arguments must be much occupied, therefore, with +\emph{comparisons} of the respective weights which attach to different +arguments. With this question we will now concern ourselves. + +It has been assumed hitherto as a matter of course that +probability is, in the full and literal sense of the word, measurable. +I shall have to limit, not extend, the popular doctrine. But, +keeping my own theories in the background for the moment, I +will begin by discussing some existing opinions on the subject. + +\Paragraph{2.} It has been sometimes supposed that a numerical comparison +between the degrees of any pair of probabilities is not only conceivable +but is actually within our power. Bentham, for instance, +in his \textit{Rationale of Judicial Evidence},\footnote + {Book~i.\ chap~vi.\ (referred to by Venn).} +proposed a scale on which +witnesses might mark the degree of their certainty; and others +have suggested seriously a `barometer of probability.'\footnote + {The reader may be reminded of Gibbon's proposal that:---``A Theological + Barometer might be formed, of which the Cardinal (Baronius) and our countryman, + Dr.~Middleton, should constitute the opposite and remote extremities, + as the former sunk to the lowest degree of credulity, which was compatible with + learning, and the latter rose to the highest pitch of scepticism, in any wise + consistent with Religion.''} + +That such comparison is \emph{theoretically possible}, whether or not +we are actually competent in every case to make the comparison, +has been the generally accepted opinion. The following quotation\footnote + {W.~F. Donkin, \textit{Phil.~Mag.}, 1851. He is replying to an article by J.~D. + Forbes (\textit{Phil.~Mag.}, Aug. 1849) which had cast doubt upon this opinion.} +puts this point of view very well: + +``I do not see on what ground it can be doubted that every +%% -----File: 032.png---Folio 21------- +\index{De Morgan}% +\index{Forbes, J. D.}% +definite state of belief concerning a proposed hypothesis is in +itself capable of being represented by a numerical expression, +however difficult or impracticable it may be to ascertain its +actual value. It would be very difficult to estimate in numbers +the \textit{vis viva} of all of the particles of a human body at any instant; +but no one doubts that it is capable of numerical expression. I +mention this because I am not sure that Professor Forbes has +distinguished the difficulty of \emph{ascertaining numbers} in certain +cases from a supposed difficulty of \emph{expression by means of numbers}. +The former difficulty is real, but merely relative to our knowledge +and skill; the latter, if real, would be absolute and inherent in +the subject-matter, which I conceive is not the case.'' + +De~Morgan held the same opinion on the ground that, wherever +we have differences of degree, numerical comparison \emph{must} be +theoretically possible.\footnote + {``Whenever the terms greater and less can be applied, there twice, thrice, + etc., can be conceived, though not perhaps measured by us.''---``Theory of Probabilities,'' + \textit{Encyclopaedia Metropolitana}, p.~395. He is a little more guarded in + his \textit{Formal Logic}, pp.\ 174,~175; but arrives at the same conclusion so far as + probability is concerned.} +He assumes, that is to say, that all +probabilities can be placed in an \emph{order} of magnitude, and argues +from this that they must be measurable. Philosophers, however, +who are mathematicians, would no longer agree that, even if the +premiss is sound, the conclusion follows from it. Objects can +be arranged in an order, which we can reasonably call one of +degree or magnitude, without its being possible to conceive a +system of measurement of the differences between the individuals. + +This opinion may also have been held by others, if not by +De~Morgan, in part because of the narrow associations which +Probability has had for them. The Calculus of Probability has +received far more attention than its logic, and mathematicians, +under no compulsion to deal with the whole of the subject, have +naturally confined their attention to those special cases, the existence +of which will be demonstrated at a later stage, where +algebraical representation is possible. Probability has become +associated, therefore, in the minds of theorists with those problems +in which we are presented with a number of exclusive and exhaustive +alternatives of equal probability; and the principles, which +are readily applicable in such circumstances, have been supposed, +without much further enquiry, to possess general validity. + +\Paragraph{3.} It is also the case that theories of probability have been +%% -----File: 033.png---Folio 22------- +\index{Insurance}% +propounded and widely accepted, according to which its numerical +character is necessarily involved in the definition. It is often +said, for instance, that probability is the ratio of the number of +``favourable cases'' to the total number of ``cases.'' If this +definition is accurate, it follows that every probability can be +properly represented by a number and in fact \emph{is} a number; for +a ratio is not a quantity at all. In the case also of definitions +based upon statistical frequency, there must be by definition a +numerical ratio corresponding to every probability. These +definitions and the theories based on them will be discussed in +\Chapref{VIII}.; they are connected with fundamental differences +of opinion with which it is not necessary to burden the present +argument. + +\Paragraph{4.} If we pass from the opinions of theorists to the experience +of practical men, it might perhaps be held that a presumption +in favour of the numerical valuation of all probabilities can be +based on the practice of underwriters and the willingness of +Lloyd's to insure against practically any risk. Underwriters are +actually willing, it might be urged, to name a numerical measure +in every case, and to back their opinion with money. But this +practice shows no more than that many probabilities are greater +or less than some numerical measure, not that they themselves +are numerically definite. It is sufficient for the underwriter if +the premium he names \emph{exceeds} the probable risk. But, apart +from this, I doubt whether in extreme cases the process of thought, +through which he goes before naming a premium, is wholly +rational and determinate; or that two equally intelligent brokers +acting on the same evidence would always arrive at the same +result. In the case, for instance, of insurances effected before +a Budget, the figures quoted must be partly arbitrary. There is +in them an element of caprice, and the broker's state of mind, +when he quotes a figure, is like a bookmaker's when he names +odds. Whilst he may be able to make sure of a profit, on the +principles of the bookmaker, yet the individual figures that make +up the book are, within certain limits, arbitrary. He may be +almost certain, that is to say, that there will not be new taxes on +more than one of the articles tea, sugar, and whisky; there +may be an opinion abroad, reasonable or unreasonable, that the +likelihood is in the order---whisky, tea, sugar; and he may, +therefore be able to effect insurances for equal amounts in each +%% -----File: 034.png---Folio 23------- +at $30$~per cent, $40$~per cent, and $45$~per cent. He has thus made +sure of a profit of $15$~per cent, however absurd and arbitrary his +quotations may be. It is not necessary for the success of underwriting +on these lines that the probabilities of these new taxes +are really measurable by the figures $\frac{3}{10}$,~$\frac{4}{10}$, and~$\frac{45}{100}$; it is sufficient +that there should be merchants willing to insure at these rates. +These merchants, moreover, may be wise to insure even if the +quotations are partly arbitrary; for they may run the risk of insolvency +unless their possible loss is thus limited. That the +transaction is in principle one of bookmaking is shown by the +fact that, if there is a specially large demand for insurance against +one of the possibilities, the rate rises;---the probability has not +changed, but the ``book'' is in danger of being upset. A Presidential +election in the United States supplies a more precise +example. On August~23, 1912, $60$~per cent was quoted at Lloyd's +to pay a total loss should Dr.~Woodrow Wilson be elected, $30$~per +cent should Mr.~Taft be elected, and $20$~per cent should Mr.~Roosevelt +be elected. A broker, who could effect insurances +in equal amounts against the election of each candidate, would be +certain at these rates of a profit of 10 per cent. Subsequent +modifications of these terms would largely depend upon the +number of applicants for each kind of policy. Is it possible to +maintain that these figures in any way represent reasoned +numerical estimates of probability? + +In some insurances the arbitrary element seems even greater. +Consider, for instance, the reinsurance rates for the \textit{Waratah}, +a vessel which disappeared in South African waters. The +lapse of time made rates rise; the departure of ships in search of +her made them fall; some nameless wreckage is found and they +rise; it is remembered that in similar circumstances thirty +years ago a vessel floated, helpless but not seriously damaged, +for two months, and they fall. Can it be pretended that the +figures which were quoted from day to day---$75$~per cent, $83$~per +cent, $78$~per cent---were rationally determinate, or that the +actual figure was not within wide limits arbitrary and due to +the caprice of individuals? In fact underwriters themselves +distinguish between risks which are properly insurable, either +because their probability can be estimated between comparatively +narrow numerical limits or because it is possible to make a ``book'' +which covers all possibilities, and other risks which cannot be +%% -----File: 035.png---Folio 24------- +\index{Leibniz|inote}% +\index{Probability, and relevant knowledge!law@{and law}}% +dealt with in this way and which cannot form the basis of a regular +business of insurance,---although an occasional gamble may be +indulged in. I believe, therefore, that the practice of underwriters +weakens rather than supports the contention that all +probabilities can be measured and estimated numerically. + +\Paragraph{5.} Another set of practical men, the lawyers, have been more +subtle in this matter than the philosophers.\footnote + {Leibniz notes the subtle distinctions made by Jurisconsults between + degrees of probability; and in the preface to a work, projected but unfinished, + which was to have been entitled \textit{Ad stateram juris de gradibus probationum et + probabilitatum} he recommends them as models of logic in contingent questions + (Couturat, \textit{Logique de~Leibniz}, p.~240).} +A distinction, +interesting for our present purpose, between probabilities, which +can be estimated within somewhat narrow limits, and those which +cannot, has arisen in a series of judicial decisions respecting +damages. The following extract\footnote + {I have considerably compressed the original report (Sapwell \textit{v.}~Bass).} +from the \textit{Times Law Reports} +seems to me to deal very clearly in a mixture of popular and legal +phraseology, with the logical point at issue: + +This was an action brought by a breeder of racehorses to +recover damages for breach of a contract. The contract was +that Cyllene, a racehorse owned by the defendant, should in the +season of the year~1909 serve one of the plaintiff's brood +mares. In the summer of~1908 the defendant, without the consent +of the plaintiff, sold Cyllene for~£30,000 to go to South +America. The plaintiff claimed a sum equal to the average +profit he had made through having a mare served by Cyllene +during the past four years. During those four years he had +had four colts which had sold at~£3300. Upon that basis his +loss came to 700~guineas. + +Mr.~Justice Jelf said that he was desirous, if he properly +could, to find some mode of legally making the defendant compensate +the plaintiff; but the question of damages presented +formidable and, to his mind, insuperable difficulties. The +damages, if any, recoverable here must be either the estimated +loss of profit or else nominal damages. The estimate could only +be based on a succession of contingencies. Thus it was assumed +that (\textit{inter alia}) Cyllene would be alive and well at the time of the +intended service; that the mare sent would be well bred and not +barren; that she would not slip her foal; and that the foal would +be born alive and healthy. In a case of this kind he could only +%% -----File: 036.png---Folio 25------- +rely on the weighing of chances; and the law generally regarded +damages which depended on the weighing of chances as too +remote, and therefore irrecoverable. It was drawing the line +between an estimate of damage based on probabilities, as in +``Simpson \textit{v.}\ L.~and~N.W. Railway Co.''\ (1,~Q.B.D.,~274), where +Cockburn,~C.J., said: ``To some extent, no doubt, the damage +must be a matter of speculation, but that is no reason for not +awarding any damages at all,'' and a claim for damages of a +totally problematical character. He (Mr.~Justice Jelf) thought +the present case was well over the line. Having referred to +``Mayne on Damages'' (8th~ed., p.~70), he pointed out that +in ``Watson \textit{v.}\ Ambergah Railway Co.''\ (15,~Jur.,~448) Patteson,~J., +seemed to think that the chance of a prize might be taken into +account in estimating the damages for breach of a contract to +send a machine for loading barges by railway too late for a show; +but Erle,~J., appeared to think such damage was too remote. +In his Lordship's view the chance of winning a prize was not of +sufficiently ascertainable value at the time the contract was made +to be within the contemplation of the parties. Further, in the +present case, the contingencies were far more numerous and +uncertain. He would enter judgment for the plaintiff for nominal +damages, which were all he was entitled to. They would be +assessed at~1s. + +One other similar case may be quoted in further elucidation +of the same point, and because it also illustrates another point---the +importance of making clear the assumptions relative to which +the probability is calculated. This case\footnote + {Chaplin \textit{v.}\ Hicks (1911).} +arose out of an offer of +a Beauty Prize\footnote + {The prize was to be a theatrical engagement and, according to the article, + \DPtypo{he}{the} probability of subsequent marriage into the peerage.} +by the \textit{Daily Express}. Out of $6000$ photographs +submitted, a number were to be selected and published in the +newspaper in the following manner: + +The United Kingdom was to be divided into districts and the +photographs of the selected candidates living in each district were +to be submitted to the readers of the paper in the district, who +were to select by their votes those whom they considered the +most beautiful, and a Mr.~Seymour Hicks was then to make an +appointment with the $50$~ladies obtaining the greatest number +of votes and himself select $12$ of them. The plaintiff, who came +%% -----File: 037.png---Folio 26------- +out head of one of the districts, submitted that she had not been +given a reasonable opportunity of keeping an appointment, that +she had thereby lost the value of her chance of one of the $12$~prizes, +and claimed damages accordingly. The jury found that +the defendant had not taken reasonable means to give the +plaintiff an opportunity of presenting herself for selection, and +assessed the damages, provided they were capable of assessment, +at~£100, the question of the possibility of assessment being postponed. +This was argued before Mr.~Justice Pickford, and subsequently +in the Court of Appeal before Lord Justices Vaughan +Williams, Fletcher Moulton, and Harwell. Two questions arose---relative +to what evidence ought the probability to be calculated, +and was it numerically measurable? Counsel for the +defendant contended that, ``if the value of the plaintiff's chance +was to be considered, it must be the value as it stood at the beginning +of the competition, not as it stood after she had been selected +as one of the~$50$. As $6000$ photographs had been sent in, and there +was also the personal taste of the defendant as final arbiter to +be considered, the value of the chance of success was really incalculable.'' +The first contention that she ought to be considered +as one of~$6000$ not as one of~$50$ was plainly preposterous and did +not hoodwink the court. But the other point, the personal +taste of the arbiter, presented more difficulty. In estimating +the chance, ought the Court to receive and take account of +evidence respecting the arbiter's preferences in types of beauty? +Mr.~Justice Pickford, without illuminating the question, held that +the damages were capable of estimation. Lord Justice Vaughan +Williams in giving judgment in the Court of Appeal argued as +follows: + +As he understood it, there were some $50$~competitors, and +there were $12$~prizes of equal value, so that the average chance +of success was about one in four. It was then said that the +questions which might arise in the minds of the persons who had +to give the decisions were so numerous that it was impossible to +apply the doctrine of averages. He did not agree. Then it +was said that if precision and certainty were impossible in any +case it would be right to describe the damages as unassessable. +He agreed that there might be damages so unassessable that the +doctrine of averages was not possible of application because the +figures necessary to be applied were not forthcoming. Several +%% -----File: 038.png---Folio 27------- +cases were to be found in the reports where it had been so held, +but he denied the proposition that because precision and certainty +had not been arrived at, the jury had no function or duty to +determine the damages\ldots. He (the Lord Justice) denied that +the mere fact that you could not assess with precision and certainty +relieved a wrongdoer from paying damages for his breach of +duty. He would not lay down that in every case it could be left +to the jury to assess the damages; there were cases where the +loss was so dependent on the mere unrestricted volition of another +person that it was impossible to arrive at any assessable loss +from the breach. It was true that there was no market here; +the right to compete was personal and could not be transferred. +He could not admit that a competitor who found herself one of +50 could have gone into the market and sold her right to compete. +At the same time the jury might reasonably have asked themselves +the question whether, if there was a right to compete, it +could have been transferred, and at what price. Under these +circumstances he thought the matter was one for the jury. + +The attitude of the Lord Justice is clear. The plaintiff had +evidently suffered damage, and justice required that she should +be compensated. But it was equally evident, that, relative to +the completest information available and account being taken of +the arbiter's personal taste, the probability could be by no means +estimated with numerical precision. Further, it was impossible +to say how much weight ought to be attached to the fact that +the plaintiff had been \emph{head} of her district (there were \emph{fewer} than +$50$~districts); yet it was plain that it made her chance \emph{better} than +the chances of those of the $50$ left in, who were not head of their +districts. Let rough justice be done, therefore. Let the case +be simplified by ignoring some part of the evidence. The +``doctrine of averages'' is then applicable, or, in other words, +the plaintiff's loss may be assessed at twelve-fiftieths of the +value of the prize.\footnote + {The jury in assessing the damages at~£100, however, cannot have argued + so subtly as this; for the average value of a prize (I have omitted the details + bearing on their value) could not have been fairly estimated so high as~£400.} + +\Paragraph{6.} How does the matter stand, then? Whether or not such +a thing is theoretically conceivable, no exercise of the practical +judgment is possible, by which a numerical value can actually +be given to the probability of every argument. So far from +%% -----File: 039.png---Folio 28------- +\index{Laplace|inote}% +\index{Probability, and relevant knowledge!similarity@{and similarity}}% +our being able to measure them, it is not even clear that we are +always able to place them in an order of magnitude. Nor has +any theoretical rule for their evaluation ever been suggested. + +The doubt, in view of these facts, whether any two probabilities +are in every case even theoretically capable of comparison +in terms of numbers, has not, however, received serious consideration. +There seems to me to be exceedingly strong reasons for +entertaining the doubt. Let us examine a few more instances. + +\Paragraph{7.} Consider an induction or a generalisation. It is usually +held that each additional instance increases the generalisation's +probability. A conclusion, which is based on three experiments +in which the unessential conditions are varied, is more trustworthy +than if it were based on two. But what reason or +principle can be adduced for attributing a numerical measure to +the increase?\footnote + {It is true that Laplace and others (even amongst contemporary writers) + have believed that the probability of an induction is measurable by means of a + formula known as the \emph{rule of succession}, according to which the probability of an + induction based on $n$~instances is~$\dfrac{n+1}{n+2}$. Those who have been convinced by + the reasoning employed to establish this rule must be asked to postpone judgment + until it has been examined in \Chapref{XXX}\@. But we may point out here + the absurdity of supposing that the odds are $2$~to~$1$ in favour of a generalisation + based on a single instance---a conclusion which this formula would seem to + justify.} + +Or, to take another class of instances, we may sometimes +have some reason for supposing that one object belongs to a +certain category if it has points of similarity to other known +members of the category (\eg\ if we are considering whether +a certain picture should be ascribed to a certain painter), and +the greater the similarity the greater the probability of our +conclusion. But we cannot in these cases \emph{measure} the increase; +we can say that the presence of certain peculiar marks in a +picture increases the probability that the artist of whom those +marks are known to be characteristic painted it, but we cannot +say that the presence of these marks makes it two or three or +any other number of times more probable than it would have +been without them. We can say that one thing is more like a +second object than it is like a third; but there will very seldom be +any meaning in saying that it is twice as like. Probability is, so +far as measurement is concerned, closely analogous to similarity.\footnote + {There are very few writers on probability who have explicitly admitted + that probabilities, though in some sense quantitative, may be incapable of + numerical comparison. Edgeworth, ``Philosophy of Chance'' (\textit{Mind}, 1884, p.~225), +\index{Edgeworth|inote}% +\index{Goldschmidt|inote}% + admitted that ``there may well be important quantitative, although not + numerical, estimates'' of probabilities. Goldschmidt (\textit{Wahrscheinlichkeitsrechnung}, + p.~43) may also be cited as holding a somewhat similar opinion. He + maintains that a lack of comparability in the grounds often stands in the way + of the measurability of the probable in ordinary usage, and that there are not + necessarily good reasons for measuring the value of one argument against + that of another. On the other hand, a numerical statement for the degree of the + probable, although generally impossible, is not in itself contradictory to the + notion; and of three statements, relating to the same circumstances, we can + well say that one is more probable than another, and that one is the most + probable of the three.} +%% -----File: 040.png---Folio 29------- + +Or consider the ordinary circumstances of life. We are out +for a walk---what is the probability that we shall reach home +alive? Has this always a numerical measure? If a thunderstorm +bursts upon us, the probability is less than it was before; +but is it changed by some definite numerical amount? There +might, of course, be data which would make these probabilities +numerically comparable; it might be argued that a knowledge +of the statistics of death by lightning would make such a comparison +possible. But if such information is not included within +the knowledge to which the probability is referred, this fact is +not relevant to the probability actually in question and cannot +affect its value. In some cases, moreover, where general statistics +are available, the numerical probability which might be derived +from them is inapplicable because of the presence of additional +knowledge with regard to the particular case. Gibbon calculated +\index{Gibbon}% +his prospects of life from the volumes of vital statistics +and the calculations of actuaries. But if a doctor had been called +to his assistance the nice precision of these calculations would +have become useless; Gibbon's prospects would have been better +or worse than before, but he would no longer have been able to +calculate to within a day or week the period for which he then +possessed an even chance of survival. + +In these instances we can, perhaps, arrange the probabilities +in an order of magnitude and assert that the new datum +strengthens or weakens the argument, although there is no +basis for an estimate \emph{how much} stronger or weaker the new +argument is than the old. But in another class of instances is +it even possible to arrange the probabilities in an \emph{order} of magnitude, +or to say that one is the greater and the other less? + +\Paragraph{8.} Consider three sets of experiments, each directed towards +establishing a generalisation. The first set is more numerous; +%% -----File: 041.png---Folio 30------- +in the second set the irrelevant conditions have been more +carefully varied; in the third case the generalisation in view +is wider in scope than in the others. Which of these generalisations +is on such evidence the most probable? There is, surely, +no answer; there is neither equality nor inequality between +them. We cannot always weigh the analogy against the induction, +or the scope of the generalisation against the bulk of the +evidence in support of it. If we have \emph{more} grounds than +before, comparison is possible; but, if the grounds in the two +cases are quite different, even a comparison of more and less, +let alone numerical measurement, may be impossible. + +This leads up to a contention, which I have heard supported, +that, although not all measurements and not all comparisons of +probability are within our power, yet we can say in the case of +every argument whether it is \emph{more} or \emph{less} likely than not. Is our +expectation of rain, when we start out for a walk, always \emph{more} +likely than not, or \emph{less} likely than not, or \emph{as} likely as not? I am +prepared to argue that on some occasions \emph{none} of these alternatives +hold, and that it will be an arbitrary matter to decide for or +against the umbrella. If the barometer is high, but the clouds are +black, it is not always rational that one should prevail over the +other in our minds, or even that we should balance them,---though +it will be rational to allow caprice to determine us and +to waste no time on the debate. + +\Paragraph{9.} Some cases, therefore, there certainly are in which no +rational basis has been discovered for numerical comparison. It +is not the case here that the method of calculation, prescribed +by theory, is beyond our powers or too laborious for actual +application. \emph{No} method of calculation, however impracticable, +has been suggested. Nor have we any \textit{prima facie} indications of +the existence of a common unit to which the magnitudes of all +probabilities are naturally referrible. A degree of probability +is not composed of some homogeneous material, and is not +apparently divisible into parts of like character with one +another. An assertion, that the magnitude of a given probability +is in a numerical ratio to the magnitude of every +other, seems, therefore, unless it is based on one of the current +\emph{definitions} of probability, with which I shall deal separately +in later chapters, to be altogether devoid of the kind of support, +which can usually be supplied in the case of quantities of which +%% -----File: 042.png---Folio 31------- +the mensurability is not open to denial. It will be worth +while, however, to pursue the argument a little further. + +\Paragraph{10.} There appear to be four alternatives. Either in some +cases there is no probability at all; or probabilities do not all +belong to a single set of magnitudes measurable in terms of a +common unit; or these measures always exist, but in many +cases are, and \emph{must remain}, unknown; or probabilities do +belong to such a set and their measures are \emph{capable} of being +determined by us, although we are not always able so to +determine them in practice. + +\Paragraph{11.} Laplace and his followers excluded the first two alternatives. +\index{Laplace}% +They argued that every conclusion has its place in +the numerical range of probabilities from $0$~to~$1$, \emph{if only we knew +it}, and they developed their theory of \emph{unknown} probabilities. + +In dealing with this contention, we must be clear as to what +we mean by saying that a probability is \emph{unknown}. Do we mean +unknown through lack of skill in arguing from given evidence, +or unknown through lack of evidence? The first is alone +admissible, for new evidence would give us a new probability, +not a fuller knowledge of the old one; we have not discovered +the probability of a statement on given evidence, by determining +its probability in relation to quite different evidence. We must +not allow the theory of unknown probabilities to gain plausibility +from the second sense. A relation of probability does not yield +us, as a rule, information of much value, unless it invests the +conclusion with a probability which lies between narrow numerical +limits. In ordinary practice, therefore, we do not always regard +ourselves as \emph{knowing} the probability of a conclusion, unless we +can estimate it numerically. We are apt, that is to say, to +restrict the use of the expression \emph{probable} to these numerical +examples, and to allege in other cases that the probability is +unknown. We might say, for example, that we do not know, +when we go on a railway journey, the probability of death in a +railway accident, unless we are told the statistics of accidents +in former years; or that we do not know our chances in a lottery, +unless we are told the number of the tickets. But it must be +clear upon reflection that if we use the term in this sense,---which +is no doubt a perfectly legitimate sense,---we ought to say that +in the case of some arguments a relation of probability does not +exist, and not that it is unknown. For it is not \emph{this} probability +%% -----File: 043.png---Folio 32------- +that we have discovered, when the accession of new evidence +makes it possible to frame a numerical estimate. + +Possibly this theory of unknown probabilities may also gain +strength from our practice of estimating arguments, which, as +I maintain, have \emph{no} numerical value, by reference to those that +have. We frame two ideal arguments, that is to say, in which +the general character of the evidence largely resembles what is +actually within our knowledge, but which is so constituted as +to yield a numerical value, and we judge that the probability of +the actual argument lies between these two. Since our standards, +therefore, are referred to numerical measures in many cases +where actual measurement is impossible, and since the probability +lies \emph{between} two numerical measures, we come to believe that it +must also, if only we knew it, possess such a measure itself. + +\Paragraph{12.} To say, then, that a probability is unknown ought to +mean that it is unknown to us through our lack of skill in arguing +from given evidence. The evidence justifies a certain degree of +knowledge, but the weakness of our reasoning power prevents our +knowing what this degree is. At the best, in such cases, we only +know \emph{vaguely} with what degree of probability the premisses invest +the conclusion. That probabilities can be unknown in this sense +or known with less distinctness than the argument justifies, +is clearly the case. We can through stupidity fail to make any +estimate of a probability at all, just as we may through the +same cause estimate a probability wrongly. As soon as we +distinguish between the degree of belief which it is rational to +entertain and the degree of belief actually entertained, we have +in effect admitted that the true probability is \emph{not} known to +everybody. + +But this admission must not be allowed to carry us too far. +Probability is, \textit{vide} \Chapref{II}. (§\;12), relative in a sense to the +principles of \emph{human} reason. The degree of probability, which +it is rational for \emph{us} to entertain, does not presume perfect logical +insight, and is relative in part to the secondary propositions +which we in fact know; and it is not dependent upon whether +more perfect logical insight is or is not conceivable. It is the +degree of probability to which those logical processes lead, of +which our minds are capable; or, in the language of \Chapref{II}., +which those secondary propositions justify, which we in fact know. +If we do not take this view of probability, if we do not limit it +%% -----File: 044.png---Folio 33------- +in this way and make it, to this extent, relative to human +powers, we are altogether adrift in the unknown; for we cannot +ever know what degree of probability would be justified by the +perception of logical relations which we are, and must always be, +incapable of comprehending. + +\Paragraph{13.} Those who have maintained that, where we cannot assign +a numerical probability, this is not because there is none, but +simply because we do not know it, have really meant, I feel +sure, that with some addition to our knowledge a numerical +value would be assignable, that is to say that our conclusions +would have a numerical probability relative to \emph{slightly different} +premisses. Unless, therefore, the reader clings to the opinion +that, in every one of the instances I have cited in the earlier +paragraphs of this chapter, it is theoretically possible on \emph{that} +evidence to assign a numerical value to the probability, we are +left with the first two of the alternatives of §\;10, which were +as follows: either in some cases there is no probability at all; +or probabilities do not all belong to a single set of magnitudes +measurable in terms of a common unit. It would be difficult to +maintain that there is \emph{no} logical relation whatever between +our premiss and our conclusion in those cases where we cannot +assign a numerical value to the probability; and if this is so, +it is really a question of whether the logical relation has characteristics, +other than mensurability, of a kind to justify us in +calling it a probability-relation. Which of the two we favour is, +therefore, partly a matter of definition. We might, that is to +say, pick out from probabilities (in the widest sense) a set, if there +is one, all of which are measurable in terms of a common unit, +and call the members of this set, and them only, probabilities (in +the narrow sense). To restrict the term `probability' in this +way would be, I think, very inconvenient. For it is possible, +as I shall show, to find \emph{several} sets, the members of each of +which are measurable in terms of a unit common to all the +members of that set; so that it would be in some degree +arbitrary\footnote + {Not altogether; for it would be natural to select the set to which the + relation of certainty belongs.} +which we chose. Further, the distinction between +probabilities, which would be thus measurable and those which +would not, is not fundamental. + +At any rate I aim here at dealing with probability in its +%% -----File: 045.png---Folio 34------- +\index{Measurement of Probability}% +\index{Probability, and relevant knowledge!comparison of}% +widest sense, and am averse to confining its scope to a limited +type of argument. If the opinion that not all probabilities can +be measured seems paradoxical, it may be due to this divergence +from a usage which the reader may expect. Common usage, +even if it involves, as a rule, a flavour of numerical measurement, +does not \emph{consistently} exclude those probabilities which are incapable +of it. The confused attempts, which have been made, +to deal with numerically indeterminate probabilities under the +title of unknown probabilities, show how difficult it is to +confine the discussion within the intended limits, if the original +definition is too narrow. + +\Paragraph{14.} I maintain, then, in what follows, that there are some pairs +of probabilities between the members of which \emph{no} comparison +of magnitude is possible; that we can say, nevertheless, of some +pairs of relations of probability that the one is greater and the +other less, although it is not possible to measure the difference +between them; and that in a very special type of case, to be +dealt with later, a meaning can be given to a \emph{numerical} comparison +of magnitude. I think that the results of observation, of which +examples have been given earlier in this chapter, are consistent +with this account. + +By saying that not all probabilities are measurable, I mean +that it is not possible to say of every pair of conclusions, about +which we have some knowledge, that the degree of our rational +belief in one bears any numerical relation to the degree of our +rational belief in the other; and by saying that not all probabilities +are comparable in respect of more and less, I mean that +it is not always possible to say that the degree of our rational +belief in one conclusion is either equal to, greater than, or less +than the degree of our belief in another. + +We must now examine a philosophical theory of the quantitative +properties of probability, which would explain and +justify the conclusions, which reflection discovers, if the preceding +discussion is correct, in the practice of ordinary argument. We +must bear in mind that our theory must apply to all probabilities +and not to a limited class only, and that, as we do not adopt a +definition of probability which presupposes its numerical mensurability, +we cannot directly argue from differences in degree +to a numerical measurement of these differences. The problem +is subtle and difficult, and the following solution is, therefore, +%% -----File: 046.png---Folio 35------- +\index{Evidence, and measurement of Probability}% +\index{Probability, and relevant knowledge!series of}% +\index{Relation, of probability!of `\textit{between}'}% +proposed with hesitation; but I am strongly convinced that +something resembling the conclusion here set forth is true. + +\Paragraph{15.} The so-called magnitudes or degrees of knowledge or +probability, in virtue of which one is greater and another less, +really arise out of an \emph{order} in which it is possible to place them. +Certainty, impossibility, and a probability, which has an intermediate +value, for example, constitute an ordered series in which +the probability lies \emph{between} certainty and impossibility. In the +same way there may exist a second probability which lies \emph{between} +certainty and the first probability. When, therefore, we say that +one probability is greater than another, this precisely means that +the degree of our rational belief in the first case lies \emph{between} +certainty and the degree of our rational belief in the second case. + +On this theory it is easy to see why comparisons of more +and less are not always possible. They exist between two probabilities, +only when they and certainty all lie on the same ordered +series. But if more than one distinct series of probabilities +\index{Series of probabilities}% +exist, then it is clear that only those, which belong to the \emph{same} +series, can be compared. If the attribute `greater' as applied +to one of two terms arises solely out of the relative order of the +terms in a series, then comparisons of greater and less must +always be possible between terms which are members of the +same series, and can never be possible between two terms which +are not members of the same series. Some probabilities are not +comparable in respect of more and less, because there exists +more than one path, so to speak, between proof and disproof, +between certainty and impossibility; and neither of two probabilities, +which lie on independent paths, bears to the other and +to certainty the relation of `between' which is necessary for +quantitative comparison. + +If we are comparing the probabilities of two arguments, +where the conclusion is the same in both and the evidence of +one exceeds the evidence of the other by the inclusion of some +fact which is favourably relevant, in such a case a relation seems +clearly to exist between the two in virtue of which one lies +\emph{nearer} to certainty than the other. Several types of argument +can be instanced in which the existence of such a relation is +equally apparent. But we cannot assume its presence in every +case or in comparing in respect of more and less the probabilities +of every pair of arguments. +%% -----File: 047.png---Folio 36------- +\index{Probability, and relevant knowledge!similarity@{and similarity}}% + +\Paragraph{16.} Analogous instances are by no means rare, in which, by a +convenient looseness, the phraseology of quantity is misapplied +in the same manner as in the case of probability. The simplest +example is that of colour. When we describe the colour of +one object as bluer than that of another, or say that it has more +green in it, we do not mean that there are quantities blue and +green of which the object's colour possesses more or less; we +mean that the colour has a certain position in an order of colours +and that it is nearer some standard colour than is the colour +with which we compare it. + +Another example is afforded by the cardinal numbers. We +say that the number three is greater than the number two, but +we do not mean that these numbers are quantities one of which +possesses a greater magnitude than the other. The one is +greater than the other by reason of its position in the order of +numbers; it is further distant from the origin zero. One number +is greater than another if the second number lies \emph{between} zero +and the first. + +But the closest analogy is that of similarity. When we say +of three objects $A$,~$B$, and~$C$ that $B$~is more like~$A$ than $C$~is, we +mean, not that there is any respect in which $B$~is in itself quantitatively +greater than~$C$, but that, if the three objects are placed +in an order of similarity, $B$~is nearer to~$A$ than $C$~is. There are +also, as in the case of probability, \emph{different} orders of similarity. +For instance, a book bound in blue morocco is more like a book +bound in red morocco than if it were bound in blue calf; and a +book bound in red calf is more like the book in red morocco than +if it were in blue calf. But there may be no comparison between +the degree of similarity which exists between books bound in +red morocco and blue morocco, and that which exists between +books bound in red morocco and red calf. This illustration +deserves special attention, as the analogy between orders of +similarity and probability is so great that its apprehension will +greatly assist that of the ideas I wish to convey. We say +that one argument is more probable than another (\ie\ nearer to +certainty) in the same kind of way as we can describe one object +as more like than another to a standard object of comparison. + +\Paragraph{17.} Nothing has been said up to this point which bears on +the question whether probabilities are ever capable of \emph{numerical} +comparison. It is true of some types of ordered series that +%% -----File: 048.png---Folio 37------- +\index{Addition, of probabilities}% +\index{Probability, and relevant knowledge!measurement of}% +there are measurable relations of distance between their members +as well as order, and that the relation of one of its members +to an `origin' can be numerically compared with the relation +of another member to the same origin. But the legitimacy of +such comparisons must be matter for special enquiry in each +case. + +It will not be possible to explain in detail how and in what +sense a meaning can sometimes be given to the numerical measurement +of probabilities until \Partref{II}\@. is reached. But this chapter +will be more complete if I indicate briefly the conclusions at +which we shall arrive later. It will be shown that a process +of compounding probabilities can be defined with such properties +that it can be conveniently called a process of \emph{addition}. It will +sometimes be the case, therefore, that we can say that one +probability~$C$ is equal to the \emph{sum} of two other probabilities $A$~and~$B$, +\ie~$C=A+B$. If in such a case $A$~and~$B$ are equal, then +we may write this $C=2A$ and say that $C$~is double~$A$\@. Similarly +if $D=C+A$, we may write $D=3A$, and so on. We can attach a +meaning, therefore, to the equation $P=n· A$, where $P$~and~$A$ are +relations of probability, and $n$~is a number. The relation of +certainty has been commonly taken as the unit of such conventional +measurements. Hence if $P$~represents certainty, +we should say, in ordinary language, that the magnitude of the +probability~$A$ is~$\frac{1}{n}$. It will be shown also that we can define a +process, applicable to probabilities, which has the properties of +arithmetical multiplication. Where numerical measurement is +possible, we can in consequence perform algebraical operations +of considerable complexity. The attention, out of proportion +to their real importance, which has been paid, on account of the +opportunities of mathematical manipulation which they afford, +to the limited class of numerical probabilities, seems to be +a part explanation of the belief, which it is the principal object +of this chapter to prove erroneous, that \emph{all} probabilities must +belong to it. + +\Paragraph{18.} We must look, then, at the quantitative characteristics of +probability in the following way. Some sets of probabilities +we can place in an ordered series, in which we can say of any +pair that one is nearer than the other to certainty,---that the +argument in one case is nearer proof than in the other, and that +there is more reason for one conclusion than for the other. But +%% -----File: 049.png---Folio 38------- +we can only build up these ordered series in special cases. If we +are given two distinct arguments, there is no general presumption +that their two probabilities and certainty can be placed +in an order. The burden of establishing the existence of such +an order lies on us in each separate case. An endeavour will +be made later to explain in a systematic way how and in +what circumstances such orders can be established. The +argument for the theory here proposed will then be strengthened. +For the present it has been shown to be agreeable to common +sense to suppose that an order exists in some cases and not in +others. + +\Paragraph{19.} Some of the principal properties of ordered series of +probabilities are as follows: +\begin{Subpars} +\item[(i.)] Every probability lies on a path between impossibility +and certainty; it is always true to say of a degree +of probability, which is not identical either with +impossibility or with certainty, that it lies \emph{between} +them. Thus certainty, impossibility and \emph{any} other +degree of probability form an ordered series. This +is the same thing as to say that every argument +amounts to proof, or disproof, or occupies an intermediate +position. + +\item[(ii.)] A path or series, composed of degrees of probability, +is not in general compact. It is not necessarily true, +that is to say, that any pair of probabilities in the +same series have a probability between them. + +\item[(iii.)] The same degree of probability can lie on more than +one path (\ie~can belong to more than one series). +Hence, if $B$~lies between $A$~and~$C$, and also lies between +$A'$~and~$C'$ it does not follow that of $A$~and~$A'$ either lies +between the other and certainty. The fact, that the +same probability can belong to more than one distinct +series, has its analogy in the case of similarity. + +\item[(iv.)] If $ABC$ forms an ordered series, $B$~lying between $A$~and~$C$, +and $BCD$~forms an ordered series, $C$~lying between +$B$~and~$D$, then $ABCD$ forms an ordered series, $B$~lying +between $A$~and~$D$. +\end{Subpars} + +\Paragraph{20.} The different series of probabilities and their mutual relations +\index{Probability, and relevant knowledge!series of}% +\index{Series of probabilities}% +can be most easily pictured by means of a diagram. Let us +represent an ordered series by points lying upon a path, all the +%% -----File: 050.png---Folio 39------- +points on a given path belonging to the same series. It follows +from~(i.)\ that the points $O$~and~$I$, representing the relations of +impossibility and certainty, lie on every path, and that all paths +lie wholly between these points. It follows from~(iii.)\ that the +same point can lie on more than one path. It is possible, therefore, +for paths to intersect and cross. It follows from~(iv.)\ that +the probability represented by a given point is greater than that +represented by any other point which can be reached by passing +along a path with a motion constantly towards the point of +impossibility, and less than that represented by any point which +can be reached by moving along a path towards the point of +certainty. As there are independent paths there will be some +pairs of points representing relations of probability such that we +cannot reach one by moving from the other along a path always +in the same direction. + +These properties are illustrated in the annexed diagram, +$O$~represents impossibility, $I$~certainty, and $A$~a numerically +measurable probability intermediate +between $O$~and~$I$; $U$,~$V$,~$W$, +$X$,~$Y$,~$Z$ are non-numerical +probabilities, of which, however, +$V$~is less than the numerical +\begin{wrapfigure}{r}{2.5in}%[illustration] + \includegraphics[width=2.5in]{./images/050.pdf} +\end{wrapfigure} +probability~$A$, and is also less +than $W$,~$X$, and~$Y$\@. $X$~and~$Y$ +are both greater than~$W$, and greater than~$V$, but are not +comparable with one another, or with~$A$\@. $V$~and~$Z$ are both +less than $W$,~$X$, and~$Y$, but are not comparable with one +another; $U$~is not quantitatively comparable with any of the +probabilities $V$,~$W$, $X$,~$Y$,~$Z$\@. Probabilities which are numerically +comparable will all belong to one series, and the path of this +series, which we may call the numerical path or strand, will be +represented by~$OAI$. + +\Paragraph{21.} The chief results which have been reached so far are +collected together below, and expressed with precision:--- +\begin{Subpars} +\item[(i.)] There are amongst degrees of probability or rational +belief various sets, each set composing an ordered +series. These series are ordered by virtue of a relation +of `between.' If $B$~is `between' $A$~and~$C$, $ABC$~form a +series. + +\item[(ii.)] There are two degrees of probability $O$ and $I$ \emph{between} +\index{Relation, of probability!of `\textit{between}'}% +%% -----File: 051.png---Folio 40------- +which \emph{all} other probabilities lie. If, that is to say, $A$~is +a probability, $OAI$~form a series. $O$~represents impossibility +and $I$~certainty. + +\item[(iii.)] If $A$~lies between $O$~and~$B$, we may write this~$\widehat{AB}$, +so that $\widehat{OA}$~and~$\widehat{AI}$ are true for all probabilities. + +\item[(iv.)] If~$\widehat{AB}$, the probability~$B$ is said to be greater than +the probability~$A$, and this can be expressed by~$B>A$. + +\item[(v.)] If the conclusion~$a$ bears the relation of probability~$P$ +to the premiss~$h$, or if, in other words, the hypothesis~$h$ +invests the conclusion~$a$ with probability~$P$, this may +be written~$aPh$. It may also be written~$a/h=P$. +\end{Subpars} + +This latter expression, which proves to be the more useful of the +two for most purposes, is of fundamental importance. If $aPh$ +and~$a'Ph'$, \ie~if the probability of $a$~relative to~$h$ is the +same as the probability of $a'$~relative to~$h'$, this may be written +$a/h=a'/h'$. The value of the symbol~$a/h$, which represents +what is called by other writers `the probability of~$a$,' lies in +the fact that it contains explicit reference to the \textit{data} to which +the probability relates the conclusion, and avoids the numerous +errors which have arisen out of the omission of this reference. +%% -----File: 052.png---Folio 41------- +\index{Bernoulli, Jac.}% + + +\Chapter{IV}{The Principle of Indifference} + +\begin{Quote} +\textsc{Absolute}. `Sure, Sir, this is not very reasonable, to summon my affection +for a lady I know nothing of.' + +\textsc{Sir Anthony}. `I am sure, Sir, 'tis more unreasonable in you to object +to a lady you know nothing of.'\footnote + {Quoted by Mr.~Bosanquet with reference to the Principle of Non-Sufficient + Reason.} +\end{Quote} + +\Paragraph{1.} \First{In} the last chapter it was assumed that in some cases the +probabilities of two arguments may be \emph{equal}. It was also argued +that there are other cases in which one probability is, in some +sense, greater than another. But so far there has been nothing +to show \emph{how} we are to know when two probabilities are equal or +unequal. The recognition of equality, when it exists, will be +dealt with in this chapter, and the recognition of inequality in +the next. An historical account of the various theories about +this problem, which have been held from time to time, will be +given in \Chapref{VII}. + +\Paragraph{2.} The determination of equality between probabilities has +received hitherto much more attention than the determination +of inequality. This has been due to the stress which has been +laid on the mathematical side of the subject. In order that +numerical measurement may be possible, we must be given a +number of \emph{equally} probable alternatives. The discovery of a +rule, by which equiprobability could be established, was, therefore, +\index{Equiprobability}% +essential. A rule, adequate to the purpose, introduced by +James Bernoulli, who was the real founder of mathematical +probability,\footnote + {See also \Chapref[Chap.]{VII}.} +has been widely adopted, generally under the +title of \emph{The Principle of Non-Sufficient Reason}, down to the +\index{Principle of Non-Sufficient Reason}% +present time. This description is clumsy and unsatisfactory, +and, if it is justifiable to break away from tradition, I prefer to +call it \emph{The Principle of Indifference}. +%% -----File: 053.png---Folio 42------- +\index{Kries, von}% + +The Principle of Indifference asserts that if there is no \emph{known} +\index{Principle of Indifference}% +reason for predicating of our subject one rather than another of +several alternatives, then relatively to such knowledge the +assertions of each of these alternatives have an \emph{equal} probability. +Thus \emph{equal} probabilities must be assigned to each of several +arguments, if there is an absence of positive ground for assigning +\emph{unequal} ones. + +This rule, as it stands, may lead to paradoxical and even +contradictory conclusions. I propose to criticise it in detail, +and then to consider whether any valid modification of it is +discoverable. For several of the criticisms which follow I am +much indebted to Von Kries's \textit{Die Principien der Wahrscheinlichkeit}.\footnote + {Published in 1886. A brief account of Von Kries's principal conclusions + will be given on \Pageref{87}. A useful summary of his book will be found in a review + by Meinong, published in the \textit{Göttingische gelehrte Anzeigen} for 1890 (pp.~56--75).} + +\Paragraph{3.} If every probability was necessarily either greater than, +equal to, or less than any other, the Principle of Indifference +would be plausible. For if the evidence affords no ground for +attributing unequal probabilities to the alternative predications, +it seems to follow that they must be equal. If, on the other hand, +there need be neither equality nor inequality between probabilities, +this method of reasoning fails. Apart, however, from +this objection, which is based on the arguments of \Chapref{III}., +the plausibility of the principle will be most easily shaken by an +exhibition of the contradictions which it involves. These fall +under three or four distinct heads. In §§\;4--9 my criticism will +be purely destructive, and I shall not attempt in these paragraphs +to indicate my own way out of the difficulties. + +\Paragraph{4.} Consider a proposition, about the subject of which we know +only the meaning, and about the truth of which, as applied to +this subject, we possess no external relevant evidence. It has +been held that there are here two exhaustive and exclusive +alternatives---the truth of the proposition and the truth of its +contradictory---while our knowledge of the subject affords no +ground for preferring one to the other. Thus if $a$~and~$\bar a$ are +contradictories, about the subject of which we have no outside +knowledge, it is inferred that the probability of each is~$\frac{1}{2}$.\footnote + {Cf.\ (\eg)\ the well-known passage in Jevons's \textit{Principles of Science}, vol.~i.\ +\index{Jevons!equiprobability@{and equiprobability}|inote}% + p.~243, in which he assigns the probability~$\frac{1}{2}$ to the proposition ``A Platythliptic + Coefficient is positive.'' Jevons points out, by way of proof, that no other + probability could reasonably be given. This, of course, involves the assumption + that every proposition must have some numerical probability. Such a contention + was first criticised, so far as I am aware, by Bishop Terrot in the \textit{Edin.\ + Phil.\ Trans.}\ for 1856. It was deliberately rejected by Boole in his last published + work on probability: ``It is a plain consequence,'' he says (\textit{Edin.\ Phil.\ + Trans.}\ vol.~xxi.\ p.~624), ``of the logical theory of probabilities, that the state + of expectation which accompanies entire ignorance of an event is properly + represented, not by the fraction~$\frac{1}{2}$, but by the indefinite form~$\frac{0}{0}$.'' Jevons's + particular example, however, is also open to the objection that we do not even + know the \emph{meaning} of the subject of the proposition. Would he maintain that + there is any sense in saying that for those who know no Arabic the probability + of every statement expressed in Arabic is even? How far has he been + influenced in the choice of his example by known characteristics of the predicate + `positive'? Would he have assigned the probability~$\frac{1}{2}$ to the proposition + `A Platythliptic Coefficient is a perfect cube'? What about the proposition + `A Platythliptic Coefficient is allogeneous'?} +In +%% -----File: 054.png---Folio 43------- +\index{Boole|inote}% +\index{Terrot, Bishop|inote}% +the same way the probabilities of two other propositions, $b$~and~$c$, +having the same subject as~$a$, may be each~$\frac{1}{2}$. But without +having any evidence bearing on the subject of these propositions +we may know that the predicates are contraries amongst themselves, +and, therefore, exclusive alternatives---a supposition which +leads by means of the same principle to values inconsistent with +those just obtained. If, for instance, having no evidence relevant +to the colour of this book, we could conclude that $\frac{1}{2}$~is the probability +of `This book is red,' we could conclude equally that the +probability of each of the propositions `This book is black' and +`This book is blue' is also~$\frac{1}{2}$. So that we are faced with the +impossible case of \emph{three} exclusive alternatives all as likely as not. +A defender of the Principle of Indifference might rejoin that we +are assuming knowledge of the proposition: `Two different +colours cannot be predicated of the same subject at the same +time'; and that, if we know this, it constitutes relevant outside +evidence. But such evidence is about the predicate, not +about the subject. Thus the defender of the Principle will be +driven on, either to confine it to cases where we know nothing +about either the subject or the predicate, which would be to +emasculate it for all practical purposes, or else to revise and +amplify it, which is what we propose to do ourselves. + +The difficulty cannot be met by saying that we must know +and take account of the \emph{number} of possible contraries. For the +number of contraries to any proposition on any evidence is always +infinite; $\bar ab$~is contrary to~$a$ for all values of~$b$. The same point +can be put in a form which does not involve contraries or +contradictories. For example, $a/h=\frac{1}{2}$ and $ab/h=\frac{1}{2}$, if $h$~is +%% -----File: 055.png---Folio 44------- +\index{Kries, von|inote}% +\index{Stumpf|inote}% +irrelevant both to~$a$ and to~$b$, in the sense required by the crude +Principle of Indifference.\footnote + {$a/h$ stands for `the probability of~$a$ on hypothesis~$h$.'} +It follows from this that, if $a$~is true, +$b$~must be true also. If it follows from the absence of positive +\textit{data} that `$A$~is a red book' has a probability of~$\frac{1}{2}$, and that the +probability of `$A$~is red' is also~$\frac{1}{2}$, then we may deduce that, if +$A$~is red, it must certainly be a book. + +We may take it, then, that the probability of a proposition, +about the subject of which we have no extraneous evidence, is +\emph{not} necessarily~$\frac{1}{2}$. Whether or not this conclusion discredits the +Principle of Indifference, it is important on its own account, and +will help later on to confute some famous conclusions of Laplace's +\index{Laplace!school of}% +school. + +\Paragraph{5.} Objection can now be made in a somewhat different shape. +Let us suppose as before that there is no positive evidence relating +to the subjects of the propositions under examination which +would lead us to discriminate in any way between certain +alternative predicates. If, to take an example, we have no +information whatever as to the area or population of the +countries of the world, a man is as likely to be an inhabitant +of Great Britain as of France, there being no reason to prefer +one alternative to the other.\footnote + {This example raises a difficulty similar to that raised by Von Kries's + example of the meteor. Stumpf has propounded an invalid solution of Von + Kries's difficulty. Against the example proposed here, Stumpf's solution has + less plausibility than against Von Kries's.} +He is also as likely to be an +inhabitant of Ireland as of France. And on the same principle +he is as likely to be an inhabitant of the British Isles as of +France. And yet these conclusions are plainly inconsistent. +For our first two propositions together yield the conclusion +that he is twice as likely to be an inhabitant of the British +Isles as of France. + +Unless we argue, as I do not think we can, that the knowledge +that the British Isles are composed of Great Britain and Ireland +is a ground for supposing that a man is more likely to inhabit +them than France, there is no way out of the contradiction. It +is not plausible to maintain, when we are considering the relative +populations of different areas, that the number of \emph{names} of subdivisions +which are within our knowledge, is, in the absence of +any evidence as to their size, a piece of relevant evidence. + +At any rate, many other similar examples could be invented, +%% -----File: 056.png---Folio 45------- +\index{Kries, von|inote}% +\index{Nitsche, A.|inote}% +which would require a special explanation in each case; for the +above is an instance of a perfectly general difficulty. The +possible alternatives may be $a$,~$b$,~$c$, and~$d$, and there may be no +means of discriminating between them; but equally there may +be no means of discriminating between ($a$~or~$b$), $c$, and~$d$. +This difficulty could be made striking in a variety of ways, but +it will be better to criticise the principle further from a somewhat +different side. + +\Paragraph{6.} Consider the specific volume of a given substance.\footnote + {This example is taken from Von Kries, \textit{op.~cit.} p.~24. Von Kries does + not seem to me to explain correctly how the contradiction arises.} +Let us +suppose that we know the specific volume to lie between $1$~and~$3$, +but that we have no information as to whereabouts in this interval +its exact value is to be found. The Principle of Indifference +would allow us to assume that it is as likely to lie between $1$~and~$2$ +as between $2$~and~$3$; for there is no reason for supposing that it +lies in one interval rather than in the other. But now consider +the specific density. The specific density is the reciprocal of +the specific volume, so that if the latter is~$v$ the former is~$\frac{1}{v}$. +Our \textit{data} remaining as before, we know that the specific density +must lie between $1$~and~$\frac{1}{3}$, and, by the same use of the Principle +of Indifference as before, that it is as likely to be between +$1$~and~$\frac{2}{3}$ as between $\frac{2}{3}$~and~$\frac{1}{3}$. But the specific volume being +a determinate function of the specific density, if the latter lies +between $1$~and~$\frac{2}{3}$, the former lies between $1$~and~$1\frac{1}{2}$, and if the +latter lies between $\frac{2}{3}$~and~$\frac{1}{3}$, the former lies between $1\frac{1}{2}$~and~$3$. +It follows, therefore, that the specific volume is as likely to lie +between $1$~and~$1\frac{1}{2}$ as between $1\frac{1}{2}$~and~$3$; +whereas we have already proved, relatively to precisely the same \textit{data}, that it is as likely +to lie between $1$~and~$2$ as between $2$~and~$3$. Moreover, any other +function of the specific volume would have suited our purpose +equally well, and by a suitable choice of this function we might +have proved in a similar manner that any division whatever +of the interval $1$~to~$3$ yields sub-intervals of equal probability. +Specific volume and specific density are simply alternative +methods of measuring the \emph{same} objective quantity; and there +are many methods which might be adopted, each yielding on the +application of the Principle of Indifference a different probability +for a given objective variation in the quantity.\footnote + {A. Nitsche (``Die Dimensionen der Wahrscheinlichkeit und die Evidenz der + Ungewissheit,'' \textit{Vierteljahrsschr.\ f.~wissensch. Philos.}\ vol.~xvi.\ p.~29, 1892), in + criticising Von~Kries, argues that the alternatives to which the principle must + be applied are the smallest \emph{physically} distinguishable intervals, and that the + probability of the specific volume's lying within a certain range of values turns + on the \emph{number} of such distinguishable intervals in the range. This procedure + might conceivably provide the correct method of computation, but it does not + therefore restore the credit of the Principle of Indifference. For it is argued, + not that the results of applying the principle are always wrong, but that it does + not lead unambiguously to the correct procedure. If we do not know the + number of distinguishable intervals we have \emph{no} reason for supposing that the + specific volume lies between $1$~and~$2$ rather than $2$~and~$3$, and the principle can + therefore be applied as it has been applied above. And even if we do know + the number and reckon intervals as equal which contain an equal number of + `physically distinguishable' parts, is it certain that this does not simply + provide us with a new system of measurement, which has the same conventional + basis as the methods of specific volume and specific density, and is no + more the one correct measure than these are?} +%% -----File: 057.png---Folio 46------- +\index{Kries, von|inote}% + +The arbitrary nature of particular methods of measurement +of this and of many other physical quantities is easily explained. +The objective quality measured may not, strictly speaking, possess +numerical quantitativeness, although it has the properties necessary +for measurement by means of correlation with numbers. +The values which it can assume may be capable of being +ranged in an order, and it will sometimes happen that the series +which is thus formed is \emph{continuous}, so that a value can always +be found whose order in the series is between any two selected +values; but it does not follow from this that there is any meaning +in the assertion that one value is \emph{twice} another value. The +relations of continuous order can exist between the terms of a +series of values, without the relations of numerical quantitativeness +necessarily existing also, and in such cases we can adopt a +largely arbitrary measure of the successive terms, which yields +results which may be satisfactory for many purposes, those, +for instance, of mathematical physics, though not for those of +probability. This method is to select some other series of +quantities or numbers, each of the terms of which corresponds +in order to one and only one of the terms of the series which +we wish to measure\DPtypo{,}{.} For instance, the series of characteristics, +differing in degree, which are measured by specific +volume, have this relation to the series of numerical ratios +between the volumes of equal masses of the substances, the +specific volumes of which are in question, and of water. They +have it also to the corresponding ratios which give rise to the +measure of specific density. But these only yield conventional +measurements, and the numbers with which we correlate the +%% -----File: 058.png---Folio 47------- +\index{Borel|inote}% +\index{Crofton|inote}% +\index{Czuber|inote}% +\index{Lammel@{Lämmel}|inote}% +\index{Probability, and relevant knowledge!geometrical@{`\textit{geometrical}'}}% +terms which we wish to measure can be selected in a variety of +ways. It follows that equal intervals between the numbers +which represent the ratios do not necessarily correspond to equal +intervals between the qualities under measurement; for these +numerical differences depend upon which convention of measurement +we have selected. + +\Paragraph{7.} A somewhat analogous difficulty arises in connection with +the problems of what is known as `geometrical' or `local' +probability.\footnote + {The best accounts of this subject are to be found in Czuber, \textit{Geometrische + Wahrscheinlichkeiten und Mittelwerte}; Czuber, \textit{Wahrscheinlichkeitsrechnung}, + vol.~i.\ pp.~75--109; Crofton, \textit{Encycl.\ Brit.}\ (9th~edit.), article `Probability'; + Borel, \textit{Eléments de la théorie des probabilités}, chaps.\ vi.--viii.; a few other + references are given in the following pages, and a number of discussions of + individual problems will be found in the mathematical volumes of the + \textit{Educational Times}. The interest of the subject is primarily mathematical, + and no discussion of its principal problems will be attempted here.} +In these problems we are concerned with the position +of a point or infinitesimal area or volume within a continuum.\footnote + {As Czuber points out (\textit{Wahrscheinlichkeiterechnung}, vol.~i.\ p.~84), all + problems, whether geometrical or arithmetical, which deal with a continuum + and with non-enumerable aggregates, are commonly discussed under the name of + `geometrical probability.' See also Lämmel, \textit{Untersuchungen}.} +\index{Geometrical probability}% +The number of cases here is indefinite, but the Principle +of Indifference has been held to justify the supposition that equal +lengths or areas or volumes of the continuum are, in the absence +of discriminating evidence, equally likely to contain the point. +It has long been known that this assumption leads in numerous +cases to contradictory conclusions. If, for instance, two points +$A$~and~$A'$ are taken at random on the surface of a sphere, and we +seek the probability that the lesser of the two arcs of the great +circle~$AA'$ is less than~$a$, we get one result by assuming that the +probability of a point's lying on a given portion of the sphere's +surface is proportional to the area of that portion, and another +result by assuming that, if a point lies on a given great circle, the +probability of its lying on a given arc of that circle is proportional +to the length of the arc, each of these assumptions being equally +justified by the Principle of Indifference. + +Or consider the following problem: if a chord in a circle is +drawn at random, what is the probability that it will be less +than the side of the inscribed equilateral triangle. One can +argue:--- + +\begin{Subpars} +\item[(\textit{a})] It is indifferent at what point one end of the chord lies. +If we suppose this end fixed, the direction is then +%% -----File: 059.png---Folio 48------- +\index{Bertrand|inote}% +\index{Poincaré, Henri}% +\index{Probability, and relevant knowledge!geometrical@{`\textit{geometrical}'}}% +chosen at random. In this case the answer is easily +shown to be~$\frac{2}{3}$. + +\item[(\textit{b})] It is indifferent in what direction we suppose the chord +to lie. Beginning with this apparently not less justifiable +assumption, we find that the answer is~$\frac{1}{2}$. + +\item[(\textit{c})] To choose a chord at random, one must choose its +middle point at random. If the chord is to be less +than the side of the inscribed equilateral triangle, the +middle point must be at a greater distance from the +centre than half the radius. But the area at a +greater distance than this is $\frac{3}{4}$~of the whole. Hence +our answer is~$\frac{3}{4}$.\footnote + {Bertrand, \textit{Calcul des probabilités}, p.~5.} +\end{Subpars} + +In general, if $x$~and~$f(x)$ are both continuous variables, varying +always in the same or in the opposite sense, and $x$~must lie +between $a$~and~$b$, then the probability that $x$~lies between $c$~and~$d$, +where $a<c<d<b$, seems to be $\dfrac{d-c}{b-a}$, and the probability +that $f(x)$~lies between $f(c)$~and~$f(d)$ to be $\dfrac{f(d)-f(c)}{f(b)-f(a)}$. These +expressions, which represent the probabilities of necessarily +concordant conclusions, are not, as they ought to be, equal.\footnote + {See (\eg)\ Borel, \textit{Éléments de la théorie des probabilités}, p.~85.} +\index{Borel}% + +\Paragraph{8.} More than one attempt has been made to separate the +cases in which the Principle of Indifference can be legitimately +applied to examples of geometrical probability from those in +which it cannot. M.~Borel argues that the mathematician can +\emph{define} the geometrical probability that a point~$M$ lies on a certain +segment~$PQ$ of~$AD$ as proportional to the length of the segment, +but that this definition is \emph{conventional} until its consequences +have been confirmed \textit{à~posteriori} by their conformity with the +results of empirical observation. He points out that in actual +cases there are generally some considerations present which +lead us to prefer one of the possible assumptions to the others. +Whether or not this is so, the proposed procedure amounts to +an abandonment of the Principle of Indifference as a valid +criterion, and leaves our choice undetermined when further +evidence is not forthcoming. + +M.~Poincaré, who also held that judgments of equiprobability +in such cases depend upon a `convention,' endeavoured to minimise +%% -----File: 060.png---Folio 49------- +the importance of the arbitrary element by showing that, +under certain conditions, the result is independent of the particular +convention which is chosen. Instead of assuming that the +point is equally likely to lie in every infinitesimal interval~$dx$ +we may represent the probability of its lying in this interval by +the function~$\phi(x)\,dx$. M.~Poincaré showed that, in the game of +\textit{rouge et noir}, for instance, where we have a number of compartments +arranged in a circle coloured alternately black and white, +if we can assume that $\phi(x)$~is a regular function, continuous and +with continuous differential coefficients, then, whatever the +particular form of the function, the probability of black is +approximately equal to that of white.\footnote + {Poincaré, \textit{Calcul des probabilités}, pp.~126 \textit{et seq}.} + +Whether or not investigations on these lines prove to have +a practical value, they have not, I think, any theoretical importance. +If, as I maintain, the probability~$\phi(x)$ is not necessarily +numerical, it is not a generally justifiable assumption to +take its continuity for granted. We have, in the particular +example quoted, a number of alternatives, half of which lead to +black and half to white; the assumption of continuity amounts +to the assumption that for every white alternative there is a +black alternative whose probability is very nearly equal to that +of the white. Naturally in such a case we can get an approximately +equal probability for the whites as a whole and for the +blacks as a whole, without assuming equal probability for each +alternative individually. But this fact has no bearing on the +theoretical difficulties which we are discussing. + +M.~Bertrand is so much impressed by the contradictions of +\index{Bertrand}% +geometrical probability that he wishes to exclude all examples +in which the number of alternatives is \emph{infinite}.\footnote + {Bertrand, \textit{Calcul des probabilités}, p.~4: ``L'infini n'est pas un nombre; + on ne doit pas, sans explication, l'introduire dans les raisonnements. La + précision illusoire des mots pourrait faire naitre des contradictions. Choisir + au hasard, entre un nombre infini de cas possibles, n'est pas une indication + suffisante.''} +It will be argued +in the sequel that something resembling this is true. The discussion +of this question will be resumed in §§\;21--25. + +\Paragraph{9.} There is yet another group of cases, distinct in character +from those considered so far, in which the principle does not +seem to provide us with unambiguous guidance. The typical +example is that of an urn containing black and white balls in an +%% -----File: 061.png---Folio 50------- +\index{Boole|inote}% +\index{Kries, von}% +\index{Nitsche, A.|inote}% +\index{Peirce|inote}% +\index{Stumpf|inote}% +unknown proportion.\footnote + {The difficulty in question was first pointed out by Boole, \textit{Laws of Thought}, + pp.~369--370. After discussing the Law of Succession, Boole proceeds to show + that ``there are other hypotheses, as strictly involving the principle of the + `equal distribution of knowledge or ignorance' which would also conduct to + conflicting results.'' See also Von Kries, \textit{op.\ cit\DPtypo{}{.}}\ pp.~31--34,~59, and Stumpf, + \textit{Über den Begriff der mathematischen Wahrscheinlichkeit}, Bavarian Academy, + 1892, pp.~64--68.} +The Principle of Indifference can be +claimed to support the most usual hypothesis, namely, that all +possible numerical \emph{ratios} of black and white are equally probable. +But we might equally well assume that all possible \emph{constitutions}\footnote + {If $A$~and~$B$ are two balls, $A$~white, $B$~black, and $A$~black, $B$~white, are + different `constitutions.' But if we consider different numerical ratios, these + two cases are indistinguishable, and count as one only.} +of the system of balls are equally probable, so that each individual +ball is assumed equally likely to be black or white. It would +follow from this that an approximately equal number of black +and white balls is more probable than a large excess of one colour. +On this hypothesis, moreover, the drawing of one ball and the +resulting knowledge of its colour leaves unaltered the probabilities +of the various possible constitutions of the rest of the bag; +whereas on the first hypothesis knowledge of the colour of one +ball, drawn and not replaced, manifestly alters the probability +of the colour of the next ball to be drawn. Either of these hypotheses +seems to satisfy the Principle of Indifference, and a believer +in the absolute validity of the principle will doubtless adopt that +one which enters his mind first.\footnote + {C.~S.\ Peirce in his \textit{Theory of Probable Inference} (Johns Hopkins \textit{Studies in + Logic}), pp.~172,~173, argues that the `constitution' hypothesis is alone valid, + on the ground that, of the two hypotheses, only this one is consistent with itself. + I agree with his conclusion, and shall give at the close of the chapter the fundamental + considerations which lead to the rejection of the `ratio' hypothesis. + Stumpf points out that the probability of drawing a white ball is, in any + case,~$\frac{1}{2}$. This is true; but the probability of a second white clearly depends + upon which of the two hypotheses has been preferred. Nitsche (\textit{loc.\ cit.}\ p.~31) + seems to miss the point of the difficulty in the same way.} + +The same point is very clearly illustrated by an example +which I take from Von Kries. Two cards, chosen from different +packs, are placed face downwards on the table; one is taken +up and found to be of a black suit: what is the chance that the +other is black also? One would naturally reply that the +chance is even. But this is based on the supposition, relatively +unpopular with writers on the subject, that every `constitution' +is equally probable, \ie~that each individual card is as likely +to be black as red. If we prefer this assumption, we must relinquish +%% -----File: 062.png---Folio 51------- +\index{Poisson|inote}% +the text-book theory that the drawing of a black ball from +an urn, containing black and white balls in unknown proportions, +affects our knowledge as to the proportion of black and white +amongst the remaining balls. + +The alternative---or text-book---theory assumes that there +are three equal possibilities---one of each colour, both black, both +red. If both cards are black, we are twice as likely to turn up +a black card than if only one is black. \emph{After} we have turned up +a black, the probability that the other is black is, therefore, twice +as great as the probability that it is red. The chance of the +second's being black is therefore~$\frac{2}{3}$.\footnote + {This is Poisson's solution, \textit{Recherches}, p.~96.} +The Principle of Indifference +has nothing to say against either solution. Until some further +criterion has been proposed we seem compelled to agree with +Poincaré that a preference for either hypothesis is wholly arbitrary. + +\Paragraph{10.} Such, then, are the kinds of result to which an unguarded +use of the Principle of Indifference may lead us. The difficulties, +to which attention has been drawn, have been noticed before; +but the discredit has not been emphatically thrown on the +original source of error. Yet the principle certainly remains as +a \emph{negative} criterion; two propositions cannot be equally probable, +so long as there \emph{is} any ground for discriminating between them. +The principle is a necessary, but not, as it seems, a sufficient +condition. + +The enunciation of some sufficient rule is certainly essential if +we are to make any progress in the subject. But the difficulty +of discovering a correct principle is considerable. This difficulty +is partly responsible, I think, for the doubts which philosophers +and many others have often felt regarding any practical application +of the Calculus. Many candid persons, when confronted +with the results of Probability, feel a strong sense of the uncertainty +of the logical basis upon which it seems to rest. It is +difficult to find an intelligible account of the meaning of `probability,' +or of how we are ever to determine the probability of any +particular proposition; and yet treatises on the subject profess +to arrive at complicated results of the greatest precision and the +most profound practical importance. + +The incautious methods and exaggerated claims of the school +of Laplace have undoubtedly contributed towards the existence +\index{Laplace!school of}% +of these sentiments. But the general scepticism, which I believe +%% -----File: 063.png---Folio 52------- +\index{Probability relation!intuition of}% +\index{Psychology and probability}% +to be much more widely spread than the literature of the subject +admits, is more fundamental. In this matter Hume need not +\index{Hume}% +have felt ``affrighted and confounded with that forelorn solitude, +in which I am placed in my philosophy,'' or have fancied himself +``some strange uncouth monster, who not being able to mingle +and unite in society, has been expell'd all human commerce, +and left utterly abandon'd and disconsolate.'' In his views on +probability, he stands for the plain man against the sophisms +and ingenuities of ``metaphysicians, logicians, mathematicians, +and even theologians.'' + +Yet such scepticism goes too far. The judgments of probability, +upon which we depend for almost all our beliefs in matters +of experience, undoubtedly depend on a strong psychological +propensity in us to consider objects in a particular light. But +this is no ground for supposing that they are nothing more than +``lively imaginations.'' The same is true of the judgments in +virtue of which we assent to other logical arguments; and yet +in such cases we believe that there may be present some element +of objective validity, transcending the psychological impulsion, +with which primarily we are presented. So also in the case of +probability, we may believe that our judgments can penetrate +into the real world, even though their credentials are subjective. + +\Paragraph{11.} We must now inquire how far it is possible to rehabilitate +the Principle of Indifference or find a substitute for it. There +are several distinct difficulties which need attention in a discussion +of the problems raised in the preceding paragraphs. +Our first object must be to make the Principle itself more precise +by disclosing how far its application is mechanical and how far +it involves an appeal to logical intuition. + +\Paragraph{12.} Without compromising the objective character of relations +of probability, we must nevertheless admit that there is little +likelihood of our discovering a method of recognising particular +probabilities, without any assistance whatever from intuition or +direct judgment. Inasmuch as it is always assumed that we can +sometimes judge directly that a conclusion \emph{follows from} a premiss, +it is no great extension of this assumption to suppose that we +can sometimes recognise that a conclusion \emph{partially follows from}, +or stands in a relation of probability to, a premiss. Moreover, +the failure to explain or define `probability' in terms of other +logical notions, creates a presumption that particular relations +%% -----File: 064.png---Folio 53------- +\index{Evidence, and measurement of Probability!relevant and irrelevant}% +\index{Principle of Indifference!analysis of}% +of probability must be, in the first instance, directly recognised +as such, and cannot be evolved by rule out of \textit{data} which themselves +contain no statements of probability. + +On the other hand, although we cannot exclude every element +of direct judgment, these judgments may be limited and controlled, +perhaps, by logical rules and principles which possess a +general application. While we may possess a faculty of direct +recognition of many relations of probability, as in the case of +many other logical relations, yet some may be much more +easily recognisable than others. The object of a logical system +of probability is to enable us to know the relations, which +cannot be easily perceived, by means of other relations which +we can recognise more distinctly---to convert, in fact, vague +\index{Knowledge!vague and distinct}% +knowledge into more distinct knowledge.\footnote + {As it is the aim of trigonometry to determine the position of an object, + which is in a sense visible, not by a direct observation of it, but by observing + some other object together with certain relations, so an indirect method of this + kind is the aim of all logical system. If the truth of \emph{some} propositions, and the + validity of \emph{some} arguments, could not be recognised directly, we could make no + progress. We may have, moreover, some power of direct recognition where it + is not necessary in our logical system that we should make use of it. In these + cases the method of logical proof increases the certainty of knowledge, which + we might be able to possess in a more doubtful manner without it. In other + cases, that, for instance, of a complicated mathematical theorem, it enables + us to know propositions to be true, which are altogether beyond the reach of + our direct insight; just as we can often obtain knowledge about the position + of a partially visible or even invisible object by starting with observations of + other objects.} + +\Paragraph{13.} Let us seek to distinguish between the element of direct +judgment and the element of mechanical rule in the Principle +of Indifference. The enunciation of this principle, as it is +ordinarily expressed, cloaks, but does not avoid, the former +element. It is in part a formula and in part an appeal to direct +inspection; but in addition to the obscurity and ambiguity of +the formula, the appeal to intuition is not as explicit as it should +be. The principle states that `there must be no known +reason for preferring one of a set of alternatives to any other.' +What does this mean? What are `reasons,' and how are +we to know whether they do or do not justify us in preferring +one alternative to another? I do not know any discussion +of Probability in which this question has been so much as +asked. If, for example, we are considering the probability +of drawing a black ball from an urn containing balls which are +%% -----File: 065.png---Folio 54------- +\index{Evidence, and measurement of Probability!relevant and irrelevant}% +\index{Relevance, judgments of}% +black and white, we assume that the difference of \emph{colour} between +the balls is not a reason for preferring either alternative. +But how do we know this, unless by a judgment that, on the +evidence in hand, our knowledge of the colours of the balls is +\emph{irrelevant} to the probability in question? We know of \emph{some} +respects in which the alternatives differ; but we judge that a +knowledge of \emph{these} differences is not relevant. If, on the other +hand, we were taking the balls out of the urn with a magnet, +and knew that the black balls were of iron and the white of tin, +we might regard the fact, that a ball was iron and not tin, as +very important in determining the probability of its being +drawn. Before, then, we can begin to apply the Principle of +Indifference, we must have made a number of direct judgments +\index{Judgments}% +to the effect that the probabilities under consideration are unaffected +by the inclusion in the evidence of certain particular +details. We have no right to say of any known difference +between the two alternatives that it is `no reason' for preferring +one of them, unless we have judged that a knowledge of this +difference is irrelevant to the probability in question. + +\Paragraph{14.} A brief digression is now necessary, in order to introduce +some new terms. There are in general two principal types of +probabilities, the magnitudes of which we seek to compare,---those +in which the evidence is the same and the conclusions +different, and those in which the evidence is different but the +conclusion the same. Other types of comparison may be required, +but these two are by far the commonest. In the first +we compare the likelihood of two conclusions on given evidence; +in the second we consider what difference a change of evidence +makes to the likelihood of a given conclusion. In symbolic +language we may wish to compare $x/h$~with~$y/h$, or $x/h$~with~$x/h_1h$. +We may call the first type judgments of \emph{preference}, or, +when there is equality between $x/h$~and~$y/h$, of \emph{indifference}; and +the second type we may call judgments of \emph{relevance}, or, when there +is equality between $x/h$~and~$x/h_1h$, of \emph{irrelevance}. In the first +\index{Irrelevance!judgments of}% +we consider whether or not $x$~is to be preferred to~$y$ on evidence~$h$; +in the second we consider whether the addition of~$h_1$ to evidence~$h$ +is relevant to~$x$. + +The Principle of Indifference endeavours to formulate a rule +which will justify judgments of \emph{indifference}. But the rule that +there must be no ground for preferring one alternative to another, +%% -----File: 066.png---Folio 55------- +\index{Evidence, and measurement of Probability!independent and complementary}% +\index{Principle of Indifference!modification of}% +involves, if it is to be a guiding rule at all, and not a \textit{petitio +principii}, an appeal to judgments of \emph{irrelevance}. +\index{Irrelevance!definition of}% + +The simplest definition of Irrelevance is as follows: $h_1$~is +irrelevant to~$x$ on evidence~$h$, if the probability of~$x$ on evidence~$hh_1$ +is the same as its probability on evidence~$h$.\footnote + {That is to say, $h_1$~is irrelevant to~$x/h$ if $x/h_1h = x/h$.} +But for a reason +which will appear in \Chapref{VI}., a stricter and more complicated +definition, as follows, is theoretically preferable: $h_1$~is irrelevant +to~$x$ on evidence~$h$, if there is no proposition, inferrible from~$h_1h$ +but not from~$h$, such that its addition to evidence~$h$ affects the +probability of~$x$.\footnote + {That is to say, $h_1$~is irrelevant to~$x/h$, if there is no propsition~$h'_1$ such that + $h'_1/h_1h = 1$, $h'_1/h \neq 1$, and $x/h'_1h \neq x/h$.} +Any proposition which is irrelevant in the +strict sense is, of course, also irrelevant in the simpler sense; +but if we were to adopt the simpler definition, it would sometimes +occur that a part of evidence would be relevant, which taken as +a whole was irrelevant. The more elaborate definition by avoiding +this proves in the sequel more convenient. If the condition +$x/h_1h = x/h$ alone is satisfied, we may say that the evidence~$h_1$ +is `irrelevant as a whole.'\footnote + {Where no misunderstanding can arise, the qualification `as a whole' will + be sometimes omitted.} + +It will be convenient to define also two other phrases. $h_1$~and~$h_2$ +are independent and complementary parts of the evidence, +if between them they make up~$h$ and neither can be inferred from +the other. If $x$~is the conclusion, and $h_1$~and~$h_2$ are independent +and complementary parts of the evidence, then $h_1$~is relevant if +the addition of it to~$h_2$ affects the probability of~$x$.\footnote + {\Ie\ (in symbolism) $h_1$~and~$h_2$ are independent and complementary parts of~$h$ + if $h_1h_2 = h$, $h_1/h_2 \neq 1$, and $h_2/h_1 \neq 1$. Also $h_1$~is relevant if $x/h \neq x/h_2$.} + +Some propositions regarding irrelevance will be proved in +\Partref{II}\@. If $\bar h_1$~is the contradictory of~$h_1$ and $x/h_1h=x/h$, then +$x/\bar h_1h = x/h$. Thus the contradictory of irrelevant evidence is +also irrelevant. Also, if $x/yh = x/h$, it follows that $y/xh = y/h$. +Hence if, on initial evidence~$h$, $y$~is irrelevant to~$x$, then, on the +same initial evidence, $x$~is irrelevant to~$y$, \ie~if in a given state +of knowledge one occurrence has no bearing on another then +equally the second has no bearing on the first. + +\Paragraph{15.} This distinction enables us to formulate the Principle \DPtypo{or}{of} +Indifference at any rate more precisely. There must be no +\emph{relevant} evidence relating to one alternative, unless there is +\emph{corresponding} evidence relating to the other; our relevant +%% -----File: 067.png---Folio 56------- +evidence, that is to say, must be symmetrical with regard to the +alternatives, and must be applicable to each in the same manner. +This is the rule at which the Principle of Indifference somewhat +obscurely aims. We must first determine what parts of our +evidence are relevant on the whole by a series of judgments of relevance, +not easily reduced to rule, of the type described above. +If this relevant evidence is \emph{of the same form} for both alternatives, +then the Principle authorises a judgment of indifference. + +\Paragraph{16.} This rule can be expressed more precisely in symbolic +\index{Propositional function}% +language. Let us assume, to begin with, that the alternative +conclusions are expressible in the forms $\phi(a)$~and~$\phi(b)$, where +$\phi(x)$~is a propositional function.\footnote + {If $\phi(a)$, $\phi(b)$, etc., are propositions, and $x$~is a variable, capable of taking + the values $a$,~$b$, etc, then $\phi(x)$~is a propositional function.} +The difference between them, +that is to say, can be represented in terms of a single variable. + +The Principle of Indifference is applicable to the alternatives +$\phi(a)$~and~$\phi(b)$, when the evidence~$h$ is so constituted that, if $f(a)$~is +an independent part of~$h$ (see §\;14) which is relevant to~$\phi(a)$, +and does not contain any independent parts which are irrelevant +to~$\phi(a)$, then $h$~includes~$f(b)$ also. + +The rule can be extended by successive steps to cases in +which we have more than one variable. We can, if the necessary +conditions are fulfilled, successively compare the probabilities +of $\phi(a_1a_2)$~and~$\phi(b_1a_2)$, and of $\phi(b_1a_2)$~and~$\phi(b_1b_2)$ and establish +equality between $\phi(a_1a_2)$~and~$\phi(b_1b_2)$. + +This elucidation is suited to most of the cases to which the +Principle of Indifference is ordinarily applied. Thus in the +favourite examples in which balls are drawn from urns, we can +infer from our evidence no relevant proposition about white balls, +such that we cannot infer a corresponding proposition about +black balls. Most of the examples, to which the mathematical +theory of chances has been applied, and which depend upon the +Principle of Indifference, can be arranged, I think, in the forms +which the rule requires as formulated above. + +\Paragraph{17.} We can now clear up the difficulties which arose over the +group of cases dealt with in §\;9, the typical example of which was +the problem of the urn containing black and white balls in an +unknown proportion. This more precise enunciation of the +Principle enables us to show that of the two solutions the equiprobability +of each `constitution' is alone legitimate, and the +%% -----File: 068.png---Folio 57------- +\index{Evidence, and measurement of Probability!external}% +equiprobability of each numerical ratio erroneous. Let us write +the alternative `The proportion of black balls is~$x$' $\equiv\phi(x)$, and +the datum `There are $n$~balls in the bag, with regard to none +of which it is known whether they are black or white'~$\equiv h$. +On the `ratio' hypothesis it is argued that the Principle of +Indifference justifies the judgment of indifference, $\phi(x)/h= +\phi(y)/h$. In order that this may be valid, it must be possible to +state the relevant evidence in the form $f(x)\,f(y)$. But this is +not the case. If $x=\frac{1}{2}$ and~$y=\frac{1}{4}$, we have relevant knowledge +about the way in which a proportion of black balls of one half +can arise, which is not identical with our knowledge of the way +in which a proportion of one quarter can arise. If there are four +balls, $A$,~$B$, $C$,~$D$, one half are black, if $A$,~$B$ or $A$,~$C$ or $A$,~$D$ or +$B$,~$C$ or $B$,~$D$ or $C$,~$D$ are black; and one quarter are black, +if $A$~or $B$ or $C$ or~$D$ are black. These propositions are not identical +in form, and only by a false judgment of irrelevance can we +ignore them. On the `constitution' hypothesis, however, +where $A$,~$B$ black and $A$,~$C$ black are treated as distinct alternatives, +this want of symmetry in our relevant evidence cannot +arise. + +\Paragraph{18.} We can also deal with the point which was illustrated by +the difficulty raised in §\;4. We considered there the probabilities +of~$a$ and its contradictory~$\bar a$ when there is no external evidence +relevant to either. What exactly do we mean by saying that +there is \emph{no} relevant evidence? Is the addition of the word +\emph{external} significant? If $a$~represents a particular proposition, +we must know something about it, namely, its \emph{meaning}. May +not the apprehension of its meaning afford us some relevant +evidence? If so, such evidence must not be excluded. If, then, +we say that there is \emph{no} relevant evidence, we must mean no +evidence beyond what arises from the mere apprehension of the +meaning of the symbol~$a$. If we attach \emph{no} meaning to the +symbol, it is useless to discuss the value of the probability; for +the probability, which belongs to a proposition as an object of +knowledge, not as a form of words, cannot in such a case exist. + +What exactly does the symbol~$a$ stand for in the above? +Does it stand for any proposition of which we know no more +than that it is a proposition? Or does it stand for a particular +proposition which we understand but of which we know no more +than is involved in understanding it? In the former case we +%% -----File: 069.png---Folio 58------- +\index{Logic, academic!of implication}% +cannot extend our result to a proposition of which we know even +the meaning; for we should then know \emph{more} than that it is a +proposition; and in the latter case we cannot say what the +probability of~$a$ is as compared with that of its contradictory, +until we know \emph{what particular proposition} it stands for; for, as +we have seen, the proposition itself may supply relevant evidence. + +This suggests that a source of much confusion may lie in the\Pagelabel{58} +use of symbols and the notion of variables in probability. In +\index{Variables in Probability}% +the logic of implication, which deals not with probability but +with truth, what is true of a variable must be equally true of all +instances of the variable. In Probability, on the other hand, +we must be on our guard wherever a variable occurs. In Implication +we may conclude that $\psi$~is true of anything of which +$\phi$~is true. In Probability we may conclude no more than that +$\psi$~is probable of anything of which we \emph{only} know that $\phi$~is true of +it. If $x$~stands for anything of which $\phi(x)$~is true, as soon as +we substitute in probability any particular value, whose meaning +we know, for~$x$, the value of the probability may be affected; +for knowledge, which was irrelevant before, may now become +relevant. Take the following example: Does $\phi(a)/\psi(a)= +\phi(b)/\psi(b)$? That is to say, is the probability of $\phi$'s~being true +of~$a$, given only that $\psi$~is true of~$a$, equal to the probability of +$\phi$'s~being true of~$b$, given only that $\psi$~is true of~$b$? If this simply +means that the probability of an object's satisfying~$\phi$ about +which nothing is known except that it satisfies~$\psi$ is equal to +ditto ditto, the equation is an identity. For in this case $\phi(a)/\psi(a)$ +\emph{means} the same as~$\phi(b)/\psi(b)$, \ie~we know \emph{nothing} about $x$~and~$y$ +except that they satisfy~$\psi$, and there is nothing whatever by +which we can distinguish $a$~from~$b$. But if $a$~and~$b$ represent +specific entities, which we can distinguish, then the equality +does not necessarily hold. If, for instance, $\phi(x)$~stands for `$x$~is +Socrates,' then it is plainly false that $\phi(a)/\psi(a)=\phi(b)/\psi(b)$, where +$a$~stands for Socrates and $b$~does not. + +\Paragraph{19.} Bearing this danger in mind, we can now give further +precision to the enunciation of the Principle of Indifference given +\index{Principle of Indifference!modification of}% +in §\;16. Our knowledge of the meaning of $a$ must be taken +account of \emph{so far as it is relevant}; and the Principle is only satisfied +if we have corresponding knowledge about the meaning of~$b$. +Thus $\phi(a)/h=\phi(b)/h$ may be true for one pair of values $a$,~$b$, and +not true for another pair of values $a'$,~$b'$. +%% -----File: 070.png---Folio 59------- + +This makes it possible to explain in part the contradiction +discussed in §\;4. Even if it were true that the probability of~$a$ is~$\frac{1}{2}$, +when we know nothing except that $a$~is a proposition, it does +not follow that the probability of `This book is red' is~$\frac{1}{2}$, when +we know the meanings of `book' and `red,' even if we know no +more than this. Knowledge arising directly out of acquaintance +with the meaning of `red' may be sufficient to enable us to infer +that `red' and `not-red' are not satisfactory alternatives to +which to apply the Principle of Indifference. How this may +come about will be discussed in §§\;20,~21. + +But the contradictions are not yet really solved; for some +of the difficulties discussed in §\;4 can arise even when we know +no more of $a$ and $b$ than that they are \emph{different} propositions. In +fact, although we have now stated more clearly than before how +the Principle should be enunciated, it is not yet possible to explain +or to avoid all the contradictions to which it led us in §§\;4 to~7. +For this purpose we must proceed to a further qualification. + +\Paragraph{20.} The examples, in which the Principle of Indifference +broke down, had a great deal in common. We broke up the +field of possibility, as we may term it, into a number of areas +by a series of disjunctive judgments. But the alternative areas +were not \emph{ultimate}. They were capable of further subdivision +into other areas \emph{similar in kind} to the former. The paradoxes +and contradictions arose, in each case, when the alternatives, +which the Principle of Indifference treated as equivalent, actually +contained or might contain a different or an indefinite number of +more elementary units. + +In the type of cases in which the Principle of Indifference +seemed to permit the assertion that, in the absence of relevant +evidence, a proposition is as likely as its contradictory, its contradictory +is not an ultimate and indivisible alternative (in the +sense to be explained in §\;21 below), even if the proposition itself +satisfies this condition. For its contradictory can be disjunctively +resolved into an indefinite number of sets of contraries to +the proposition. It was out of this that our difficulties first arose. +`This book is not red' includes amongst others the alternatives +`This book is black' and `This book is blue.' It is not, therefore, +an ultimate alternative. + +In the same way the contradiction of §\;5 arose out of the possibility +of splitting the alternatives `He inhabits the British +%% -----File: 071.png---Folio 60------- +Isles' into the sub-alternatives `He inhabits Ireland or he +inhabits Great Britain.' And in the third type of case, to +which the example of specific volume and density belongs, the +alternative `$v$~lies in the interval $1$~to~$2$' can be broken up into +the sub-alternatives `$v$~lies in the interval $1$~to~$1\frac{1}{2}$ or $1\frac{1}{2}$~to~$2$.' + +\Paragraph{21.} This, then, seems to point the way to the qualification of +which we are in search. We must enunciate some formal rule +which will exclude those cases, in which one of the alternatives +involved is itself a disjunction of sub-alternatives \emph{of the same +form}. For this purpose the following condition is proposed. + +Let the alternatives, the equiprobability of which we seek to +establish by means of the Principle of Indifference, be $\phi(a_1)$, +$\phi(a_2) \ldots \phi(a_r)$,\footnote + {The more complicated cases in which the propositional function, of which + the alternatives are instances, involves more than one variable (see §\;16), can be + dealt with in a similar manner \textit{mutatis mutandis}.} +and let the evidence be~$h$. Then it is a necessary +condition for the application of the principle, that these +should be, relatively to the evidence, indivisible alternatives of +the form~$\phi(x)$. We may define a divisible alternative in the +following manner: + +An alternative $\phi(a_r)$ is \emph{divisible} if +\begin{align*} +\text{(i.)}\quad & [\phi(a_r)\equiv\phi(a_{r'})+\phi(a_{r''})]/h=1,\\ +\text{(ii.)}\quad & \phi(a_{r'})· \phi(a_{r''})/h=0,\\ +\text{(iii.)}\quad & \phi(a_{r'})/h\neq 0 \quad \text{and} \quad \phi(a_{r''})/h \neq 0 +\end{align*} + +The condition that the sub-alternatives must be \emph{of the same +form} as the original alternatives, \ie\ expressible by means of the +same propositional function~$\phi(x)$, deserves attention. It might +be the case that the original alternatives had nothing substantial +in common; \ie\ $\phi(x)\equiv (x=x)$ is the only propositional function +common to all of them, the alternatives being $a_1$, $a_2,\ldots, a_r$. In +these circumstances the condition in question cannot be satisfied. +For the proposition~$a_r$ can always be resolved into the disjunction +$a_rb+a_r\bar b$, where $b$~is any proposition and $\bar b$~its contradictory. If, +on the other hand, the alternatives which we are comparing can +be expressed in the forms $\phi(a_1)$~and~$\phi(a_2)$, where the function~$\phi(x)$ +is distinct from~$x$, it is not necessarily the case that either +of these can be resolved into a disjunctive combination of terms +which can be expressed in their turn in the same form. + +Dispensing with symbolism, we can express these conditions +as follows: Our knowledge must not enable us to split up the +%% -----File: 072.png---Folio 61------- +alternative~$\phi(a_r)$ into a disjunction of two sub-alternatives, (i.)~which +are themselves expressible in the same form~$\phi$, (ii.)~which +are mutually exclusive, and (iii.)~which, on the evidence, are +possible. + +In short, the Principle of Indifference is not applicable to a +pair of alternatives, if we know that either of them is capable of +being further split up into a pair of possible but incompatible +alternatives of the same form as the original pair. + +\Paragraph{22.} This rule commends itself to common sense. If we +know that the two alternatives are compounded of a different +number or of an indefinite number of sub-alternatives which are +in other respects similar, so far as our evidence goes to the +original alternatives, then this is a relevant fact of which we +must take account. And as it affects the two alternatives in +differing and unsymmetrical ways, it breaks down the fundamental +condition for the valid application of the Principle of +Indifference. + +Neither this consideration nor that discussed in §§\;18 and~19 +substantially modify the Principle of Indifference as enunciated +in §\;16. They have only served to make explicit what was +always implicit in the Principle, by explaining the manner in +which our knowledge of the \emph{form and meaning} of the alternatives +may be a relevant part of the evidence. The apparent contradictions +arose from paying attention to what we may term +the \emph{extraneous} evidence only, to the neglect of such part of the +evidence as bore upon the form and meaning of the alternatives. + +\Paragraph{23.} The application of this result to the examples cited in §\;18 +is not difficult. It excludes the class of cases in which a proposition +and its contradictory constitute the alternatives. For +if $b$~is the proposition and $\bar b$~its contradictory, we cannot find +a propositional function~$\phi(x)$ which will satisfy the necessary +conditions. It deals also with the type of contradiction which +arose in considering the probability that an individual taken at +random was an inhabitant of a given region. If, on the other +hand, the term `country' is so defined that one country cannot +include two countries, then an individual is, relatively to suitable +hypotheses, as likely to be an inhabitant of one as of another. +For the function~$\phi(x)$, where $\phi(x)\equiv$ `the individual is an inhabitant +of country~$x$,' satisfies the conditions. And it deals +with the example of ranges of specific volume and specific density, +%% -----File: 073.png---Folio 62------- +\index{Probability, and relevant knowledge!geometrical@{`\textit{geometrical}'}}% +because there is no range which does not contain within itself two +similar ranges. As there are in this case no definite units by +which we can define \emph{equal} ranges, the device, which will be referred +to in §\;25 for dealing with geometrical probabilities, is not available. + +\Paragraph{24.} It is worth while to add that the qualification of §\;21 is +fatal to the practical utility of the Principle of Indifference in +those cases only in which it is possible to find \emph{no} ultimate alternatives +which satisfy the conditions. For if the original alternatives +each comprise a definite number of indivisible and indifferent +sub-alternatives, we can compute their probabilities. It is often +the case, however, that we cannot by any process of finite subdivision +arrive at indivisible sub-alternatives, or that, if we can, +they are not on the evidence indifferent. In the examples given +above, for instance, where $\phi(x)\equiv x$, or where $x$ is a part of unspecified +magnitude in a continuum, there are \emph{no} indivisible +sub-alternatives. The first type comprises all cases, amongst +others, in which we weigh the probabilities of a proposition and +its contradictory; and the second includes a great number of +cases in which physical or geometrical quantities are involved. + +\Paragraph{25.} We can now return to the numerous paradoxes which +arise in the study of geometrical probability (see §§\;7,~8). The +\index{Geometrical probability}% +qualification of §\;21 enables us, I think, to discover the source +of the confusion. Our alternatives in these problems relate to +certain areas or segments or arcs, and however small the elements +are which we adopt as our alternatives, they are made up of yet +smaller elements which would also serve as alternatives. Our +rule, therefore, is not satisfied, and, as long as we enunciate them +in this shape, we cannot employ the Principle of Indifference. +But it is easy in most cases to discover another set of alternatives +which do satisfy the condition, and which will often serve our +purpose equally well. Suppose, for instance, that a point lies +on a line of length~$m.l.$, we may write the alternative `the interval +of length~$l$ on which the point lies is the $x$th~interval of that +length as we move along the line from left to right'~$\equiv\phi(x)$; and +the Principle of Indifference can then be applied safely to the $m$~alternatives +$\phi(1)$, $\phi(2) \ldots \phi(m)$, the number~$m$ increasing as the +length~$l$ of the intervals is diminished. There is no reason why +$l$~should not be of any definite length however small. + +If we deal with the problems of geometrical probability in +%% -----File: 074.png---Folio 63------- +this way, we shall avoid the contradictory conclusions, which +arise from confusing together \emph{distinct} elementary areas. In the +problem, for instance, of the chord drawn at random in a circle, +which is discussed in §\;7, the chord is regarded, not as a one-dimensional +line, but as the limit of an area, the shape of which +is different in each of the variant solutions. In the first solution +it is the limit of a triangle, the length of the base of which tends +to zero; in the second solution it is the limit of a quadrilateral, +two of the sides of which are parallel and at a distance apart +which tends to zero; and in the third solution the area is defined +by the limiting position of a central section of undefined shape. +These distinct hypotheses lead inevitably to different results. If +we were dealing with a strictly linear chord, the Principle of +Indifference would yield us no result, as we could not enunciate +the alternatives in the required form; and if the chord is an +elementary area, we must know the shape of the area of which +it is the limit. So long as we are careful to enunciate the alternatives +in a form to which the Principle of Indifference can be +applied unambiguously, we shall be prevented from confusing +together distinct problems, and shall be able to reach conclusions +in geometrical probability which are unambiguously valid. + +The substance of this explanation can be put in a slightly +different way by saying that it is not a matter of indifference in +these cases in what manner we proceed to the limit. We must +assign the probabilities \emph{before} proceeding to the limit, which +we can do unambiguously. But if the problem in hand does +not stop at small finite lengths, areas, or volumes, and we +have to proceed to the limit, then the final result depends upon +the shape in which the body approaches the limit. Mathematicians +will recognise an analogy between this case and the determination +of potential at points \emph{within} a conductor. Its value +depends upon the shape of the area which in the limit represents +the point. + +\Paragraph{26.} The positive contributions of this chapter to the determination +of valid judgments of equiprobability are two. In the +\index{Equiprobability}% +first place we have stated the Principle of Indifference in a more +accurate form, by displaying its necessary dependence upon +judgments of relevance and so bringing out the hidden element +of direct judgment or intuition, which it has always involved. +It has been shown that the Principle lays down a rule by which +%% -----File: 075.png---Folio 64------- +direct judgments of relevance and irrelevance can lead on to +judgments of preference and indifference. In the second place, +some types of consideration, which are in fact relevant, but which +are in danger of being overlooked, have been brought into prominence. +By this means it has been possible to avoid the various +types of doubtful and contradictory conclusions to which the +Principle seemed to lead, so long as we applied it without due +qualification. +%% -----File: 076.png---Folio 65------- + + +\Chapter{V}{Other Methods of Determining Probabilities} + +\Paragraph{1.} \First{The} recognition of the fact, that not all probabilities are +numerical, limits the scope of the Principle of Indifference. It +has always been agreed that a numerical measure can actually +be obtained in those cases only in which a reduction to a set of +exclusive and exhaustive \emph{equiprobable} alternatives is practicable. +Our previous conclusion that numerical measurement is often +impossible agrees very well, therefore, with the argument of the +preceding chapter that the rules, in virtue of which we can assert +equiprobability, are somewhat limited in their field of application. +\index{Equiprobability}% + +But the recognition of this same fact makes it more necessary +to discuss the principles which will justify comparisons of more +and less between probabilities, where numerical measurement is +theoretically, as well as practically, impossible. We must, for +the reasons given in the preceding chapter, rely in the last resort +on direct judgment. The object of the following rules and +principles is to reduce the judgments of preference and relevance, +\index{Judgments!of preference and relevance}% +which we are compelled to make, to a few relatively simple types.\footnote + {Parts of \Chapref[Chap.]{XV}. are closely connected with the topics of the following + paragraphs, and the discussion which is commenced here is concluded there.} + +\Paragraph{2.} We will enquire first in what circumstances we can expect +a comparison of more and less to be theoretically possible. I +am inclined to think that this is a matter about which, rather +unexpectedly perhaps, we are able to lay down definite rules. +We are able, I think, always to compare a pair of probabilities +which are +\begin{DPalign*} +\text{(i.)}\quad & \text{of the type $ab/h$ and $a/h$}, \\ +\lintertext{or} +\text{(ii.)}\quad & \text{of the type $a/hh_1$ and $a/h$}, +\end{DPalign*} +provided the additional evidence~$h_1$ contains only one independent +piece of relevant information. +%% -----File: 077.png---Folio 66------- +\index{Evidence, and measurement of Probability!addition of}% +\index{Probability, and relevant knowledge!comparison of}% + +(i.) The propositions of \Partref{II}. will enable us to prove that +\[ +ab/h < a/h \text{ unless }b/ah = 1; +\] +that is to say, the probability of our conclusion is diminished by +the addition to it of something, which on the hypothesis of our +argument cannot be inferred from it. This proposition will be +self-evident to the reader. The rule, that the probability of two +propositions jointly is, in general, less than that of either of them +separately, includes the rule that the attribution of a more +specialised concept is less probable than the attribution of a less +specialised concept. + +(ii.) This condition requires a little more explanation. It +states that the probability~$a/hh_1$ is always greater than, equal to, +or less than the probability~$a/h$, if $h_1$~contains no pair of complementary +and independent parts\footnote + {See \Chapref[Chap.]{IV}. §\;14 for the meaning of these terms.} +both relevant to~$a/h$. If $h_1$~is +favourable, $a/hh_1 > a/h$. Similarly, if $h_2$~is favourable to~$a/hh_1$, +$a/hh_1h_2 > a/hh_1$. The reverse holds if $h_1$~and~$h_2$ are unfavourable. +Thus we can compare $a/hh'$~and~$a/h$, in every case in which the +relevant independent parts of the additional evidence~$h'$ are +either all favourable, or all unfavourable. In cases in which our +additional evidence is equivocal, part taken by itself being favourable +and part unfavourable, comparison is not necessarily possible. +In ordinary language we may assert that, according to our rule, +the addition to our evidence of a single fact always has a definite +bearing on our conclusion. It either leaves its probability unaffected +and is irrelevant, or it has a definitely favourable or +unfavourable bearing, being favourably or unfavourably relevant. +It cannot affect the conclusion in an indefinite way, which allows +no comparison between the two probabilities. But if the addition +of one fact is favourable, and the addition of a second is unfavourable, +it is not necessarily possible to compare the probability of +our original argument with its probability when it has been +modified by the addition of \emph{both} the new facts. + +Other comparisons are possible by a combination of these +two principles with the Principle of Indifference. We may +find, for instance, that $a/hh_1 > a/h$, that $a/h = b/h$, that $b/h > b/hh_2$, +and that, therefore, $a/hh_1 > b/hh_2$. We have thus obtained a +comparison between a pair of probabilities, which are not +of the types discussed above, but without the introduction +%% -----File: 078.png---Folio 67------- +\index{Kries, von|inote}% +of any fresh principle. We may denote comparisons of this +type by~(iii.). + +\Paragraph{3.} Whether any comparisons are possible which do not fall +within any of the categories (i.),~(ii.), or~(iii.), I do not feel certain. +We undoubtedly make a number of direct comparisons which +do not seem to be covered by them. We judge it more probable, +for instance, that Caesar invaded Britain than that Romulus +founded Rome. But even in such cases as this, where a reduction +into the regular form is not obvious, it might prove possible if +we could clearly analyse the real grounds of our judgment. We +might argue in this instance that, whereas Romulus's founding of +Rome rests solely on tradition, we have \emph{in addition} evidence of +another kind for Caesar's invasion of Britain, and that, in so +far as our belief in Caesar's invasion rests on tradition, we have +reasons of a precisely similar kind as for our belief in Romulus +\emph{without} the additional doubt involved in the maintenance of a +tradition between the times of Romulus and Caesar. By some +such analysis as this our judgment of comparison might be +brought within the above categories. + +The process of reaching a judgment of comparison in this way +\index{Schematisation}% +may be called `schematisation.'\footnote + {This phrase is used by Von Kries, \textit{op.\ cit.}\ p.~179, in a somewhat similar + connection.} +We take initially an ideal +scheme which falls within the categories of comparison. Let +us represent `the historical tradition~$x$ has been handed down +from a date many years previous to the time of Caesar' by~$\psi_1(x)$; +`the historical tradition~$x$ has been handed down from +the time of Caesar' by~$\psi_2(x)$; `the historical tradition~$x$ has +extra-traditional support' by~$\psi_3(x)$; and the two traditions, +the Romulus tradition and the Caesar tradition respectively, +by $a$~and~$b$. Then if our relevant evidence~$h$ were of the form +$\psi_1(a)\psi_2(b)\psi_3(b)$, it is easily seen that the comparison $a/h<b/h$ +could be justified on the lines laid down above.\footnote + {For \quad $a/\psi_2(a)=b/\psi_2(b)$; + \quad $a/\psi_1(a)<a/\psi_2(a)$; + \quad $b/\psi_2(b)<b/\psi_2(b)\psi_3(b)$; + $a/\psi_1(a)=a/h$; and $b/\psi_2(b)\psi_3(b)=b/h$.} +A further judgment, +that our actual evidence presented no relevant divergence +from this schematic form, would then establish the practical +conclusion. As I am not aware of any plausible judgment of +comparison which we make in common practice, but which is +clearly incapable of reduction to some schematic form, and as +I see no logical basis for such a comparison, I feel justified in +%% -----File: 079.png---Folio 68------- +\index{Analogy, principle of}% +\index{Evidence, and measurement of Probability!addition of}% +\index{Johnson, W. E.!added evidence@{and added evidence}}% +\index{Middle Term, Fallacy of}% +doubting the \emph{possibility} of comparing the probabilities of arguments +dissimilar in form and incapable of schematic reduction. +But the point must remain very doubtful until this part of the +subject has received a more prolonged consideration. + +\Paragraph{4.} Category~(ii.)\ is very wide, and evidently covers a great +variety of cases. If we are to establish general principles of argument +and so avoid excessive dependence on direct individual +judgments of relevance, we must discover some new and more +particular principles included within it. Two of these---those +of Analogy and of Induction---are excessively important, and +\index{Induction!Principle of}% +will be the subject of \Partref{III}. of this book. In addition to these +a few criteria will be examined and established in \Chapref{XIV}., +§§\;4~and 8~(49.1). We must be content here (pending the +symbolic developments of \Partref{II}.) with the two observations +following: + +(1)\Pagelabel{68} The addition of new\footnote + {$h_1$ is \emph{new} evidence so long as $h_1/h \neq 1$.} +evidence~$h_1$ to a doubtful\footnote + {The argument is \emph{doubtful} so long as $a/h$~is neither certain nor impossible.} +argument~$a/h$ +is \emph{favourably} relevant, if either of the following conditions +is fulfilled:---(\textit{a}) if $a/hh_1=0 $; (\textit{b}) if $a/hh_1=1$. Divested of symbolism, +this merely amounts to a statement that a piece of +evidence is favourable if, in conjunction with the previous +evidence, it is either a necessary or a sufficient condition for the +truth of our conclusion. + +(2) It might plausibly be supposed that evidence would be +favourable to our conclusion which is favourable to favourable +evidence---\ie\ that, if $h_1$~is favourable to~$x/h$ and $x$~is favourable to~$a/h$, +$h_1$~is favourable to~$a/h$. Whilst, however, this argument +is frequently employed under conditions, which, if explicitly +stated, would justify it, there are also conditions in which this is +not so, so that it is not necessarily valid. For the very deceptive +fallacy involved in the above supposition, Mr.~Johnson has +suggested to me the name of the \emph{Fallacy of the Middle Term}. The +general question---If $h_1$~is favourable to~$x/h$ and $x$~is favourable to~$a/h$, +in what conditions is $h_1$~favourable to~$a/h$?---will be examined +in \Chapref{XIV}. §§\;4~and 8~(49.1). In the meantime, the intuition +of the reader towards the fallacy may be assisted by the +following observations, which are due to Mr.~Johnson: + +Let $x$,~$x'$,~$x''\ldots$ be exclusive and exhaustive alternatives +under datum~$h$. Let $h_1$~and~$a$ be \emph{concordant} in regard to \emph{each} of +%% -----File: 080.png---Folio 69------- +these alternatives: \ie~any hypothesis which is strengthened by~$h_1$ +will strengthen~$a$, and any hypothesis which is weakened by~$h_1$ +will weaken~$a$. It is obvious that, if $h_1$~strengthens \emph{some} of +the hypotheses $x$,~$x'$,~$x''\ldots$, it will weaken \emph{others}. This fact +helps us to see why we cannot consider the concordance of $h_1$~and~$a$ +in regard to one \emph{single} alternative, but must be able to +assert their concordance with regard to \emph{every one} of the exclusive +and exhaustive alternatives, including the particular one taken. +But a further condition is needed, which (as we shall show) is +obviously satisfied in two typical problems at least. This further +condition is that, for \emph{each} hypothesis $x$,~$x'$,~$x''\ldots$, it shall hold +that, were this hypothesis known to be true, the knowledge of~$h_1$ +would \emph{not weaken} the probability of~$a$. + +These two conditions are \emph{sufficient} to ensure that $h_1$~shall +strengthen~$a$ (independently of knowledge of $x$,~$x'$,~$x''\ldots$); +and, in a sense, they appear to be \emph{necessary}; for, unless they are +satisfied, the dependence of~$h_1$ upon~$a$ would be (so to speak) +\emph{accidental} as regards the `middle terms,' ($x$,~$x'$,~$x''\ldots$). + +The necessity for reference to \emph{all} the alternatives $x$,~$x'$,~$x''\ldots$ +is analogous to the requirement of distribution of the middle +term in ordinary syllogism. Thus, from premises ``All~$P$ is~$x$, +all~$S$ is~$x$,'' the conclusion that ``$S$'s~are~$P$'' does not formally +follow; but given ``all~$P$ is~$x$ and all~$S$ is~$x'$'' it \emph{does} follow that +``no~$S$ are~$P$'', where $x'$~is any contrary to~$x$. The two conditions +taken together would be analogous to the argument: all~$x$ $S$~is~$P$; +all~$x'$ $S$~is~$P$; all~$x''$ $S$~is~$P$;~\ldots\ therefore all~$S$ is~$P$. + +\textit{First Typical Problem.}---An urn contains an unknown proportion +of differently coloured balls. A ball is drawn and replaced. +Then $x$,~$x'$,~$x''\ldots$ stand for the various possible proportions. +Let $h_1$~mean ``a white ball has been drawn''; and let $a$~mean +``a white ball will be again drawn.'' Then any hypothesis which +is strengthened by~$h_1$ will strengthen~$a$; and any hypothesis +which is weakened by~$h_1$ will weaken~$a$. Moreover, were any +one of these hypotheses known to be true, the knowledge of~$h_1$ +would not weaken the probability of~$a$. Hence, in the absence +of definite knowledge as regards $x$,~$x'$,~$x''\ldots$, the knowledge +of~$h_1$ would strengthen the probability of~$a$. + +\textit{Second Typical Problem.}---Let a certain event have taken +place; which may have been $x$,~$x'$,~$x''$ or~\ldots\DPtypo{}{.} Let $h_1$~mean that +$A$~reports so and so; and let $a$~mean that $B$~reports \emph{similarly} or +%% -----File: 081.png---Folio 70------- +\index{Judgments!direct}% +\index{Port Royal logic}% +identically. The phrase \emph{similarly} merely indicates that any +hypothesis as to the actual fact, which would be strengthened by +$A$'s~report, would be strengthened by $B$'s~report. Of course, +even if the reports were verbally \emph{identical}, $A$'s~evidence would not +necessarily strengthen the hypothesis in an \emph{equal} degree with~$B$'s; +because $A$~and~$B$ may be unequally expert or intelligent. +Now, in such cases, we may further affirm (in general), that, were +the actual nature of the event known, the knowledge of $A$'s~report +on it \emph{would not weaken} (though it also need not strengthen) the +probability that $B$~would give a \emph{similar} report. Hence, in the +absence of such knowledge, the knowledge of~$h_1$ would strengthen +the probability of~$a$. + +\Paragraph{5.} Before leaving this part of the argument we must emphasise +the part played by direct judgment in the theory here presented. +The rules for the determination of equality and inequality between +probabilities all depend upon it at some point. This seems to +me quite unavoidable. But I do not feel that we should regard +it as a weakness. For we have seen that most, and perhaps all, +cases can be determined by the application of general principles +to one simple type of direct judgment. No more is asked of the +intuitive power applied to particular cases than to determine +whether a new piece of evidence tells, on the whole, for or against +a given conclusion. The application of the rules involves no +wider assumptions than those of other branches of logic. + +While it is important, in establishing a control of direct +judgment by general principles, not to conceal its presence, yet +the fact that we ultimately depend upon an intuition need not +lead us to suppose that our conclusions have, therefore, no basis +in reason, or that they are as subjective in validity as they are +in origin. It is reasonable to maintain with the logicians of the +Port Royal that we may draw a conclusion which is truly probable +by paying attention to all the circumstances which accompany +the case, and we must admit with as little concern as possible +Hume's taunt that ``when we give the preference to one set of +\index{Hume}% +arguments above another, we do nothing but decide from our +feeling concerning the superiority of their influence.'' +%% -----File: 082.png---Folio 71------- +\index{Evidence, and measurement of Probability!weight of}% + + +\Chapter{VI}{The Weight of Arguments} + +\Paragraph{1.} \First{The} question to be raised in this chapter is somewhat novel; +after much consideration I remain uncertain as to how much +importance to attach to it. The magnitude of the probability +of an argument, in the sense discussed in \Chapref{III}., depends +upon a balance between what may be termed the favourable and +the unfavourable evidence; a new piece of evidence which leaves +this balance unchanged, also leaves the probability of the argument +unchanged. But it seems that there may be another +respect in which some kind of quantitative comparison between +arguments is possible. This comparison turns upon a balance, +not between the favourable and the unfavourable evidence, but +between the \emph{absolute} amounts of relevant knowledge and of +relevant ignorance respectively. + +As the relevant evidence at our disposal increases, the magnitude +of the probability of the argument may either decrease or +increase, according as the new knowledge strengthens the unfavourable +or the favourable evidence; but \emph{something} seems to +have increased in either case,---we have a more substantial basis +upon which to rest our conclusion. I express this by saying that +an accession of new evidence increases the \emph{weight} of an argument. +New evidence will sometimes decrease the probability of +an argument, but it will always increase its `weight.' + +\Paragraph{2.} The measurement of evidential weight presents similar +difficulties to those with which we met in the measurement of +probability. Only in a restricted class of cases can we compare +the weights of two arguments in respect of more and less. But +this must always be possible where the conclusion of the two +arguments is the same, and the relevant evidence in the one includes +and exceeds the evidence in the other. If the new evidence +%% -----File: 083.png---Folio 72------- +is `irrelevant,' in the more precise of the two senses defined in §\;14 +of \Chapref{IV}., the weight is left unchanged. If any part of the +new evidence is relevant, then the value is increased. + +The reason for our stricter definition of `relevance' is now +apparent. If we are to be able to treat `weight' and `relevance' +as correlative terms, we must regard evidence as relevant, part +of which is favourable and part unfavourable, even if, taken as +a whole, it leaves the probability unchanged. With this definition, +to say that a new piece of evidence is `relevant' is the same +thing as to say that it increases the `weight' of the argument. + +A proposition cannot be the subject of an argument, unless +we at least attach some \emph{meaning} to it, and this meaning, even if +it only relates to the form of the proposition, may be relevant +in some arguments relating to it. But there may be no other +relevant evidence; and it is sometimes convenient to term the +probability of such an argument an \textit{à~priori} probability. In +this case the weight of the argument is at its lowest. Starting, +therefore, with minimum weight, corresponding to \textit{à~priori} +probability, the evidential weight of an argument rises, though +its probability may either rise or fall, with every accession of +relevant evidence. + +\Paragraph{3.} Where the conclusions of two arguments are different, or +where the evidence for the one does not overlap the evidence +for the other, it will often be impossible to compare their weights, +just as it may be impossible to compare their probabilities. Some +rules of comparison, however, exist, and there seems to be a close, +though not a complete, correspondence between the conditions +under which pairs of arguments are comparable in respect of +probability and of weight respectively. We found that there were +three principal types in which comparison of probability was +possible, other comparisons being based on a combination of +these:--- + +(i.) Those based on the Principle of Indifference, subject +to certain conditions, and of the form ${\phi a}/{\psi a· h_1}={\phi b}/{\psi b· h_2}$, +where $h_1$~and~$h_2$ are irrelevant to the arguments. + +(ii.) ${a}/{hh_1}\gtrless {a}/{h}$, where $h_1$~is a single unit of information, +containing no independent parts which are relevant. + +(iii.) ${ab}/{h}\leq {a}/{h}$ + +Let us represent the evidential weight of the argument, +whose probability is~${a}/{h}$, by $V({a}/{h})$. Then, corresponding to +%% -----File: 084.png---Folio 73------- +the above, we find that the following comparisons of weight are +possible:--- + +(i.) $V(\phi a/\psi a· h_1) = V(\phi b/\psi b· h_2)$, where $h_1$~and~$h_2$ are irrelevant +in the strict sense. Arguments, that is to say, to which the +Principle of Indifference is applicable, have equal evidential +weights. + +(ii.) $V(a/hh_1) > V(a/h)$, unless $h_1$~is irrelevant, in which case +$V(a/hh_1) = V(a/h)$. The restriction on the composition of~$h_1$, +which is necessary in the case of comparisons of magnitude, is +not necessary in the case of weight. + +There is, however, no rule for comparisons of weight corresponding +to (iii.)~above. It might be thought that $V(ab/h) < V(a/h)$, +on the ground that the more complicated an argument is, relative +to given premisses, the less is its evidential weight. But this +is invalid. The argument~$ab/h$ is further off proof than was the +argument~$a/h$; but it is nearer disproof. For example, if $ab/h=0$ +and $a/h>0$, then $V(ab/h)>V(A/h)$. In fact it would seem to +be the case that the weight of the argument $a/h$ is always +equal to that of $\bar{a}/h$, where $\bar{a}$ is the contradictory of~$a$; \ie, +$V(a/h)=V(\bar{a}/h)$. For an argument is always as near proving or +disproving a proposition, as it is to disproving or proving its +contradictory. + +\Paragraph{4.} It may be pointed out that if $a/h = b/h$, it does not necessarily +follow that $V(a/h)=V(b/h)$. It has been asserted already +that if the first equality follows \emph{directly} from a single application of +the Principle of Indifference, the second equality also holds. But +the first equality can exist in other cases also. If, for instance, +$a$~and~$b$ are members respectively of \emph{different} sets of three equally +probable exclusive and exhaustive alternatives, then $a/h=b/h$; but +these arguments may have very different weights. If, however, +$a$~and~$b$ can each, relatively to~$h$, be inferred from the other, \ie\ if +$a/bh=1$ and $b/ah=1$, then $V(a/h)= V(b/h)$. For in proving or disproving +one, we are necessarily proving or disproving the other. + +Further principles could, no doubt, be arrived at. The above +can be combined to reach results in cases upon which unaided +\DPtypo{common-sense}{common sense} might feel itself unable to pronounce with confidence. +Suppose, for instance, that we have three exclusive +and exhaustive alternatives, $a$,~$b$, and~$c$, and that $a/h = b/h$ +in virtue of the Principle of Indifference, then we have +$V(a/h)=V(b/h)$ and $V(a/h)=V(\bar{a}/h)$, so that $V(b/h)=V(\bar{a}/h)$. It is +%% -----File: 085.png---Folio 74------- +\index{De Morgan}% +also true, since $\bar{a}/(b+c)h = 1$ and $(b+c)/\bar{a}h = 1$, that $V(\bar{a}/h) = +V((b+c)/h)$. Hence $V(b/h)=V((b+c)/h)$. + +\Paragraph{5.} The preceding paragraphs will have made it clear that the +weighing of the \emph{amount} of evidence is quite a separate process +from the \emph{balancing} of the evidence for and against. In so far, +however, as the question of weight has been discussed at all, +attempts have been made, as a rule, to explain the former in +terms of the latter. If $x/h_1h_2 = \frac{2}{3}$ and $x/h_1 = \frac{3}{4}$, it has sometimes +been supposed that it is \emph{more probable} that $x/h_1h_2$ really is~$\frac{2}{3}$ than +that $x/h_1$~really is~$\frac{3}{4}$. According to this view, an increase in the +amount of evidence strengthens the probability of the probability, +or, as De~Morgan would say, the presumption of the +probability. A little reflection will show that such a theory is +untenable. For the probability of~$x$ \emph{on hypothesis}~$h_1$ is independent +of whether as a matter of fact $x$~is or is not true, and if +we find out subsequently that $x$~is true, this does not make it +false to say that on hypothesis~$h_1$ the probability of~$x$ is~$\frac{3}{4}$. Similarly +the fact that $x/h_1h_2$ is~$\frac{2}{3}$ does not impugn the conclusion that +$x/h_1$ is~$\frac{3}{4}$, and unless we have made a mistake in our judgment or +our calculation on the evidence, the two probabilities are $\frac{2}{3}$~and~$\frac{3}{4}$ +respectively. + +\Paragraph{6.} A second method, by which it might be thought, perhaps, +that the question of weight has been treated, is the method of +\emph{probable error}. But while probable error is sometimes connected +\index{Probable error}% +with weight, it is primarily concerned with quite a different question. +`Probable error,' it should be explained, is the name +given, rather inconveniently perhaps, to an expression which +arises when we consider the probability that a given quantity is +measured by one of a number of different magnitudes. Our +\textit{data} may tell us that one of these magnitudes is the most probable +measure of the quantity; but in some cases it will also tell +us how probable each of the other possible magnitudes of the +quantity is. In such cases we can determine the probability +that the quantity will have a magnitude which does not differ +from the most probable by more than a specified amount. The +amount, which the difference between the actual value of the +quantity and its most probable value is as likely as not to exceed, +is the `probable error.' In many practical questions the existence +of a small probable error is of the greatest importance, +if our conclusions are to prove valuable. The probability that +%% -----File: 086.png---Folio 75------- +the quantity has any particular magnitude may be very small; +but this may matter very little, if there is a high probability +that it lies within a certain range. + +Now it is obvious that the determination of probable error +is intrinsically a different problem from the determination of +weight. The method of probable error is simply a summation of +a number of alternative and exclusive probabilities. If we say +that the most probable magnitude is~$x$ and the probable error~$y$, +this is a way, convenient for many purposes, of summing up a +number of probable conclusions regarding a variety of magnitudes +other than~$x$ which, on the evidence, the quantity may +possess. The connection between probable error and weight, such +as it is, is due to the fact that in scientific problems a large +probable error is not uncommonly due to a great lack of evidence, +and that as the available evidence increases there is a tendency +for the probable error to diminish. In these cases the probable +error may conceivably be a good practical measure of the weight. + +It is necessary, however, in a theoretical discussion, to point +out that the connection is casual, and only exists in a limited +class of cases. This is easily shown by an example. We may +have data on which the probability of $x=5$ is~$\frac{1}{3}$, of $x=6$ is~$\frac{1}{4}$, +of $x=7$ is~$\frac{1}{5}$, of $x=8$ is~$\frac{1}{6}$, and of $x=9$ is~$\frac{1}{20}$. Additional evidence +might show that $x$~must either be $5$~or $8$ or~$9$, the probabilities of +each of these conclusions being $\frac{7}{16}$,~$\frac{5}{16}$,~$\frac{4}{16}$. +The evidential weight +of the latter argument is greater than that of the former, but the +probable error, so far from being diminished, has been increased. +There is, in fact, no reason whatever for supposing that the +probable error must necessarily diminish, as the weight of the +argument is increased. + +The typical case, in which there may be a \emph{practical} connection +between weight and probable error, may be illustrated by the +two cases following of balls drawn from an urn. In each case we +require the probability of drawing a white ball; in the first case +we know that the urn contains black and white in equal proportions; +in the second case the proportion of each colour is unknown, +and each ball is as likely to be black as white. It is evident that +in either case the probability of drawing a white ball is~$\frac{1}{2}$ but +that the weight of the argument in favour of this conclusion is +greater in the first case. When we consider the most probable +proportion in which balls will be drawn in the long run, if after +%% -----File: 087.png---Folio 76------- +\index{Bernoulli, Jac.}% +each withdrawal they are replaced, the question of probable +error enters in, and we find that the greater evidential weight of +the argument on the first hypothesis is accompanied by the +smaller probable error. + +This conventionalised example is typical of many scientific +problems. The more we know about any phenomenon, the less +likely, as a rule, is our opinion to be modified by each additional +item of experience. In such problems, therefore, an argument +of high weight concerning some phenomenon is likely to be accompanied +by a low probable error, when the character of a series +of similar phenomena is under consideration. + +\Paragraph{7.} Weight cannot, then, be explained in terms of probability. +An argument of high weight is not `more likely to be right' than +one of low weight; for the probabilities of these arguments only +state relations between premiss and conclusion, and these relations +are stated with equal accuracy in either case. Nor is an +argument of high weight one in which the probable error is small; +for a small probable error only means that magnitudes in the +neighbourhood of the most probable magnitude have a relatively +high probability, and an increase of evidence does not necessarily +involve an increase in these probabilities. + +The conclusion, that the `weight' and the `probability' of an +argument are independent properties, may possibly introduce a +difficulty into the discussion of the application of probability +to practice.\footnote + {See also \Chapref{XXVI}. §\;7.} +For in deciding on a course of action, it seems +plausible to suppose that we ought to take account of the weight +as well as the probability of different expectations. But it is +difficult to think of any clear example of this, and I do not +feel sure that the theory of `evidential weight' has much +practical significance. + +Bernoulli's second maxim, that we must take into account all\Pagelabel{76} +the information we have, amounts to an injunction that we should +be guided by the probability of that argument, amongst those of +which we know the premisses, of which the evidential weight is +the greatest. But should not this be re-enforced by a further +maxim, that we ought to make the weight of our arguments as +great as possible by getting all the information we can?\footnote + {Cf.\ Locke, \textit{Essay concerning Human Understanding}, book~ii.\ chap.~xxi.\ §\;67: +\index{Locke}% + ``He that judges without informing himself to the utmost that he is capable, + cannot acquit himself of judging amiss.''} +It is +%% -----File: 088.png---Folio 77------- +\index{Judgments!disjunctive}% +difficult to see, however, to what point the strengthening of an +argument's weight by increasing the evidence ought to be pushed. +We may argue that, when our knowledge is slight but capable of +increase, the course of action, which will, relative to such knowledge, +probably produce the greatest amount of good, will often +consist in the acquisition of more knowledge. But there clearly +comes a point when it is no longer worth while to spend trouble, +before acting, in the acquisition of further information, and there +is no evident principle by which to determine \emph{how far} we ought +to carry our maxim of strengthening the weight of our argument. +A little reflection will probably convince the reader that this is +a very confusing problem. + +\Paragraph{8.} The fundamental distinction of this chapter may be briefly +repeated. One argument has more \emph{weight} than another if it is +based upon a greater amount of relevant evidence; but it is not +always, or even generally, possible to say of two sets of propositions +that one set embodies \emph{more} evidence than the other. It has +a greater \emph{probability} than another if the balance in its favour, +of what evidence there is, is greater than the balance in favour +of the argument with which we compare it; but it is not always, +or even generally, possible to say that the balance in the one case +is greater than the balance in the other. The weight, to speak +metaphorically, measures the \emph{sum} of the favourable and unfavourable +evidence, the probability measures the \emph{difference}. + +\Paragraph{9.} The phenomenon of `weight' can be described from the +point of view of other theories of probability than that which is +adopted here. If we follow certain German logicians in regarding +probability as being based on the disjunctive judgment, we may +say that the weight is increased when the number of alternatives +is reduced, although the ratio of the number of favourable to +the number of unfavourable alternatives may not have been +disturbed; or, to adopt the phraseology of another German +school, we may say that the weight of the probability is increased, +as the field of possibility is contracted. + +The same distinction may be explained in the language of the +frequency theory.\footnote + {See \Chapref[Chap.]{VIII}.} +We should then say that the weight is increased +if we are able to employ as the class of reference a class +which is contained in the original class of reference. + +\Paragraph{10.} The subject of this chapter has not usually been discussed +%% -----File: 089.png---Folio 78------- +\index{Nitsche, A.}% +by writers on probability, and I know of only two by whom the +question has been explicitly raised:\footnote + {There are also some remarks by Czuber (\textit{Wahrscheinlichkeitsrechnung}, +\index{Czuber}% + vol.~i.\ p.~202) on the \textit{Erkenninisswert} of probabilities obtained by different + methods, which may have been intended to have some bearing on it.} +Meinong, who threw out a +\index{Meinong}% +suggestion at the conclusion of his review of Von Kries' ``Principien,'' +published in the \textit{Göttingische gelehrte Anzeigen} for 1890 +(see especially pp.~70--74), and A.~Nitsche, who took up Meinong's +suggestion in an article in the \textit{Vierteljahrsschrift für wissenschaftliche +Philosophie}, 1892, vol.~xvi.\ pp.~20--35, entitled ``Die Dimensionen +der Wahrscheinlichkeit und die Evidenz der Ungewissheit.'' + +Meinong, who does not develop the point in any detail, distinguishes +probability and weight as `Intensität' and `Qualität,' +and is inclined to regard them as two independent dimensions in +which the judgment is free to move---they are the two dimensions +of the `Urteils-Continuum.' Nitsche regards the weight as being +the measure of the reliability (Sicherheit) of the probability, and +holds that the probability continually approximates to its true +magnitude (reale Geltung) as the weight increases. His treatment +is too brief for it to be possible to understand very clearly what +he means, but his view seems to resemble the theory already +discussed that an argument of high weight is `more likely to be +right' than one of low weight. +%% -----File: 090.png---Folio 79------- +\index{Butler, Bishop}% + + +\Chapter{VII}{Historical Retrospect} + +\Paragraph{1.} The characteristic features of our Philosophy of Probability +must be determined by the solutions which we offer to the +problems attacked in Chapters \Chapref[]{III}~and~\Chapref[]{IV}. Whilst a great part +of the logical calculus, which will be developed in \Partref{II}., would +be applicable with slight modification to several distinct theories +of the subject, the ultimate problems of establishing the premisses +of the calculus bring into the light every fundamental difference +of opinion. + +These problems are often, for this reason perhaps, left on one +side by writers whose interest chiefly lies in the more formal parts +of the subject. But Probability is not yet on so sound a basis +that the formal or mathematical side of it can be safely developed +in isolation, and some attempts have naturally been made to +solve the problem which Bishop Butler sets to the logician in the +concluding words of the brief discussion on probability with +which he prefaces the \emph{Analogy}.\footnote + {``It is not my design to inquire further into the nature, the foundation and + measure of probability; or whence it proceeds that \emph{likeness} should beget that + presumption, opinion and full conviction, which the human mind is formed + to receive from it, and which it does necessarily produce in every one; or to + guard against the errors to which reasoning from analogy is liable. This + belongs to the subject of logic, and is a part of that subject which has not yet + been thoroughly considered.''} + +In this chapter, therefore, we will review in their historical +order the answers of Philosophy to the questions, how we know +relations of probability, what ground we have for our judgments, +and by what method we can advance our knowledge. + +\Paragraph{2.} The natural man is disposed to the opinion that probability +is essentially connected with the inductions of experience and, +if he is a little more sophisticated, with the Laws of Causation +%% -----File: 091.png---Folio 80------- +\index{Butler, Bishop}% +\index{Port Royal logic}% +and of the Uniformity of Nature. As Aristotle says, ``the +\index{Aristotle}% +probable is that which usually happens.'' Events do not always +occur in accordance with the expectations of experience; but +the laws of experience afford us a good ground for supposing +that they usually will. The occasional disappointment of these +expectations prevents our predictions from being more than +probable; but the ground of their probability must be sought in +this experience, and in this experience only. + +This is, in substance, the argument of the authors of the Port +Royal Logic (1662), who were the first to deal with the logic +of probability in the modern manner: ``In order for me to +judge of the truth of an event, and to be determined to believe +it or not believe it, it is not necessary to consider it abstractly, +and in itself, as we should consider a proposition in geometry; +but it is necessary to pay attention to all the circumstances +which accompany it, internal as well as external. I call internal +circumstances those which belong to the fact itself, and external +those which belong to the persons by whose testimony we are led +to believe it. This being done, if all the circumstances are +such that it never or rarely happens that the like circumstances +are the concomitants of falsehood, our mind is led, naturally, +to believe that it is true.''\footnote + {Eng.\ Trans., p.~353.} +Locke follows the Port Royal +\index{Locke}% +Logicians very closely: ``Probability is likeliness to be true\ldots. The +\emph{grounds of it} are, in short, these two following. \emph{First}, the +conformity of anything with our own knowledge, observation, +and experience. \emph{Secondly}, the testimony of others, vouching +their observation and experience'';\footnote + {\textit{An Essay concerning Human Understanding}, book~iv. ``Of Knowledge and + + Opinion.''} +and essentially the same +opinion is maintained by Bishop Butler: ``When we determine +a thing to be probably true, suppose that an event has or will +come to pass, it is from the mind's remarking in it a likeness to +some other event, which we have observed has come to pass. +And this observation forms, in numberless instances, a presumption, +opinion, or full conviction that such event has or will +come to pass.''\footnote + {Introduction to the \textit{Analogy}.} + +Against this view of the subject the criticisms of Hume were +\index{Hume}% +directed: ``The idea of cause and effect is derived from \emph{experience}, +which informs us, that such particular objects, in all past +%% -----File: 092.png---Folio 81------- +\index{Bernoulli, Jac.}% +instances, have been constantly conjoined with each other\ldots. +According to this account of things~\ldots\ probability is founded +on the presumption of a resemblance betwixt those objects, of +which we have had experience, and those, of which we have had +none; and therefore 'tis impossible this presumption can arise +from probability.''\footnote + {\textit{Treatise of Human Nature}, p.~391 (Green's edition).} +``When we are accustomed to see two impressions +conjoined together, the appearance or idea of the one immediately +carries us to the idea of the other\ldots. Thus all probable +reasoning is nothing but a species of sensation. 'Tis not +solely in poetry and music, we must follow our taste and sentiment, +but likewise in philosophy. When I am convinced of any +principle, 'tis only an idea, which strikes more strongly upon me. +When I give the preference to one set of arguments above another, +I do nothing but decide from my feeling concerning the superiority +of their influence.''\footnote + {\textit{Op.\ cit.}\ p.~403.} +Hume, in fact, points out that, while +\index{Hume}% +it is true that past experience gives rise to a psychological anticipation +of some events rather than of others, no ground has been +given for the validity of this superior anticipation. + +\Paragraph{3.} But in the meantime the subject had fallen into the hands +of the mathematicians, and an entirely new method of approach +was in course of development. It had become obvious that +many of the judgments of probability which we in fact make +do not depend upon past experience in a way which satisfies the +canons laid down by the Port Royal Logicians or by Locke. In +particular, alternatives are judged equally probable, without +there being necessarily any actual experience of their approximately +equal frequency of occurrence in the past. And, apart +from this, it is evident that judgments based on a somewhat +indefinite experience of the past do not easily lend themselves +to precise numerical appraisement. Accordingly James +Bernoulli,\footnote + {See especially \textit{Ars Conjectandi}, p.~224. Cf.\ Laplace, \textit{Théorie analytique}, + p.~178.} +the real founder of the classical school of mathematical +probability, while not repudiating the old test of experience, had +based many of his conclusions on a quite different criterion---the +rule which I have named the Principle of Indifference. The +\index{Principle of Indifference}% +traditional method of the mathematical school essentially +depends upon reducing all the possible conclusions to a number +of `\DPchg{equi-probable}{equiprobable} cases.' And, according to the Principle of +%% -----File: 093.png---Folio 82------- +\index{Great Numbers, Law of}% +\index{Locke}% +\index{Succession, Law of}% +Indifference, `cases' are held to be \DPchg{equi-probable}{equiprobable} when there +is no reason for preferring any one to any other, when there is +nothing, as with Buridan's ass, to determine the mind in any one +of the several possible directions. To take Czuber's example +\index{Czuber}% +of dice,\footnote + {\textit{Wahrscheinlichkeitsrechnung,} p.~9.} +this principle permits us to assume that each face is +equally likely to fall, if there is no reason to suppose any particular +irregularity, and it does not require that we should \emph{know} that the +construction is regular, or that each face has, as a matter of fact, +fallen equally often in the past. + +On this Principle, extended by Bernoulli beyond those +problems of gaming in which by its tacit assumption Pascal +\index{Pascal}% +and Huyghens had worked out a few simple exercises, the whole +\index{Huyghens}% +fabric of mathematical probability was soon allowed to rest. +The older criterion of experience, never repudiated, was soon +subsumed under the new doctrine. First, in virtue of Bernoulli's +famous Law of Great Numbers, the fractions representing the +probabilities of events were thought to represent also the actual +proportion of their occurrences, so that experience, if it were +considerable, could be translated into the cyphers of arithmetic. +And next, by the aid of the Principle of Indifference, Laplace +\index{Laplace}% +established his Law of Succession by which the influence of any +experience, \emph{however limited}, could be numerically measured, and +which purported to prove that, if $B$~has been seen to accompany~$A$ +twice, it is two to one that~$B$ will again accompany~$A$ on $A$'s~next +appearance. No other formula in the alchemy of logic +has exerted more astonishing powers, For it has established +the existence of God from the premiss of total ignorance; and it +has measured with numerical precision the probability that the +sun will rise to-morrow. + +Yet the new principles did not win acceptance without +\index{D'Alembert}% +opposition. D'Alembert,\footnote + {D'Alembert's scepticism was directed towards the current mathematical + theory only, and was not, like Hume's, fundamental and far-reaching. His +\index{Hume}% + opposition to the received opinions was, perhaps, more splendid than discriminating.} +\index{Ancillon}% +Hume, and Ancillon\footnote + {Ancillon's communication to the Berlin Academy in 1794, entitled \textit{Doutes + sur les bases du calcul des probabilités}, is not as well known as it deserves to + be. He writes as a follower of Hume, but adds much that is original and + interesting. An historian, who also wrote on a variety of philosophical subjects, + Ancillon was, at one time, the Prussian Minister of Foreign Affairs.} +stand out as +the sceptical critics of probability, against the credulity of +%% -----File: 094.png---Folio 83------- +\index{Bernoulli, Jac.}% +\index{Calculus of Probability|inote}% +\index{Condorcet|inote}% +\index{De Morgan}% +\index{Principle of Indifference|ifoll}% +eighteenth-century philosophers who were ready to swallow +without too many questions the conclusions of a science which +claimed and seemed to bring an entire new field within the +dominion of Reason.\footnote + {French philosophy of the latter half of the eighteenth century was profoundly + affected by the supposed conquests of the Calculus of Probability in + all fields of thought. Nothing seemed beyond its powers of prediction, and + it almost succeeded in men's minds to the place previously occupied by + Revelation. It was under these influences that Condorcet evolved his doctrine + of the perfectibility of the human race. The continuity and oneness of + modern European thought may be illustrated, if such things amuse the + reader, by the reflection that Condorcet derived from Bernoulli, that Godwin + was inspired by Condorcet, that Malthus was stimulated by Godwin's folly + into stating his famous doctrine, and that from the reading of Malthus + on \textit{Population} Darwin received his earliest impulse.} + +The first effective criticism came from Hume, who was also +\index{Hume}% +the first to distinguish the method of Locke and the philosophers +\index{Locke}% +from the method of Bernoulli and the mathematicians. ``Probability,'' +he says, ``or reasoning from conjecture, may be divided +into two kinds, viz.\ that which is founded on \emph{chance} and that which +arises from causes.''\footnote + {\textit{Treatise of Human Nature}, p.~424 (Green's edition).} +By these two kinds he evidently means the +mathematical method of counting the equal chances based on +Indifference, and the inductive method based on the experience +of uniformity. He argues that `chance' alone can be the +foundation of nothing, and ``that there must always be a mixture +of causes among the chances, in order to be the foundation of +any reasoning.''\footnote + {\textit{Op.\ cit.}\ p.~425.} +His previous argument against probabilities, +which were based on an assumption of cause, is thus extended +to the mathematical method also. + +But the great prestige of Laplace and the `verifications' +\index{Laplace}% +of his principles which his more famous results were supposed +to supply had, by the beginning of the nineteenth century, +established the science on the Principle of Indifference in an +almost unquestioned position. It may be noted, however, that +De~Morgan, the principal student of the subject in England, +seems to have regarded the method of actual experiment and +the method of counting cases, which were equally probable +on grounds of Indifference, as alternative methods of equal +validity. + +\Paragraph{4.} The reaction against the traditional teaching during the +past hundred years has not possessed sufficient force to displace +%% -----File: 095.png---Folio 84------- +\index{Ellis, Leslie}% +\index{Kries, von}% +\index{Mathematicians, and probability}% +\index{Pearson, Karl}% +\index{Poincaré, Henri}% +the established doctrine, and the Principle of Indifference is +still very widely accepted in an unqualified form. Criticism +has proceeded along two distinct lines; the one, originated by +Leslie Ellis, and developed by Dr.~Venn, Professor Edgeworth, +\index{Edgeworth}% +\index{Venn}% +and Professor Karl Pearson, has been almost entirely confined +in its influence to England; the other, of which the beginnings +are to be seen in Boole's \textit{Laws of Thought}, has been developed +\index{Boole}% +in Germany, where its ablest exponent has been Von Kries. +France has remained uninfluenced by either, and faithful, on +the whole, to the tradition of Laplace. Even Henri Poincaré, +\index{Laplace}% +who had his doubts, and described the Principle of Indifference +as ``very vague and very elastic,'' regarded it as our only +guide in the choice of that convention, ``which has always +something arbitrary about it,'' but upon which calculation in +probability invariably rests.\footnote + {Poincaré's opinions on Probability are to be found in his \textit{Calcul des Probabilités} + and in his \textit{Science et Hypothèse}. Neither of these books appears + to me to be in all respects a considered work, but his view is sufficiently novel + to be worth a reference. Briefly, he shows that the current mathematical + definition is circular, and argues from this that the choice of the particular + probabilities, which we are to regard as initially equal before the application of + our mathematics, is entirely a matter of `convention.' Much epigram is, + therefore, expended in pointing out that the study of probability is no more + than a polite exercise, and he concludes: ``Le calcul des probabilités offre une + contradiction dans les termes mêmes qui servent à le désigner, et, si je ne craignais + de rappeler ici un mot trop souvent répété, je dirais qu'il nous enseigne + surtout une chose; c'est de savoir que nous ne savons rien.'' On the other + hand, the greater part of his book is devoted to working out instances of practical + application, and he speaks of `metaphysics' legitimising particular conventions. + How this comes about is not explained. He seems to endeavour to + save his reputation as a philosopher by the surrender of probability as a valid + conception, without at the same time forfeiting his claim as a mathematician + to work out probable formulae of practical importance.} + +\Paragraph{5.} Before following up in detail these two lines of development, +I will summarise again the earlier doctrine with which the +leaders of the new schools found themselves confronted. + +The earlier philosophers had in mind in dealing with probability +the application to the future of the inductions of experience, +to the almost complete exclusion of other problems. For the +\textit{data} of probability, therefore, they looked only to their own +experience and to the recorded experiences of others; their +principal refinement was to distinguish these two grounds, and +they did not attempt to make a numerical estimate of the chances. +The mathematicians, on the other hand, setting out from the +simple problems presented by dice and playing cards, and +%% -----File: 096.png---Folio 85------- +\index{Ellis, Leslie}% +\index{Empirical School}% +requiring for the application of their methods a basis of numerical +measurement, dwelt on the negative rather than the positive +side of their evidence, and found it easier to measure equal +degrees of ignorance than equivalent quantities of experience. +This led to the explicit introduction of the Principle of Indifference, +or, as it was then termed, the Principle of Non-Sufficient Reason. +\index{Principle of Non-Sufficient Reason}% +The great achievement of the eighteenth century was, in the eyes +of the early nineteenth, the reconciliation of the two points of +view and the measurement of probabilities, which were grounded +on experience, by a method whose logical basis was the Principle +of Non-Sufficient Reason. This would indeed have been a very +astonishing discovery, and would, as its authors declared, have +gradually brought almost every phase of human activity within +the power of the most refined mathematical analysis. + +But it was not long before more sceptical persons began to +suspect that this theory proved too much. Its calculations, it +is true, were constructed from the \textit{data} of experience, but the +more simple and the less complex the experience the better satisfied +was the theory. What was required was not a wide experience +or detailed information, but a completeness of symmetry in +the little information there might be. It seemed to follow from +the Laplacian doctrine that the primary qualification for one +who would be well informed was an equally balanced ignorance. + +\Paragraph{6.} The obvious reaction from a teaching, which seemed to +derive from abstractions results relevant to experience, was into +the arms of empiricism; and in the state of philosophy at that +time England was the natural home of this reaction. The first +protest, of which I am aware, came from Leslie Ellis in~1842.\footnote + {\textit{On the Foundations of the Theory of Probabilities.}} +At the conclusion of his \textit{Remarks on an alleged proof of the Method +of least squares},\footnote + {Republished in \textit{Miscellaneous Writings.}} +``Mere ignorance,'' he says, ``is no ground +for any inference whatever. \textit{Ex nihilo nihil}.'' In Venn's +\index{Venn!experience@{and experience}}% +\textit{Logic of Chance} Ellis's suggestions are developed into a complete +theory:\footnote + {\textit{Logic of Chance}, p.~74.} ``Experience is our sole guide. If we want to discover +what is in reality a series of \emph{things}, not a series of our own conceptions, +we must appeal to the things themselves to obtain it, for +we cannot find much help elsewhere.'' Professor Edgeworth\index{Edgeworth}\footnote + {\textit{Metretike}, p.~4.} +was an early disciple of the same school: ``The probability,'' he +%% -----File: 097.png---Folio 86------- +\index{Bernoulli, Jac.}% +\index{Intuition \textit{versus} experience}% +\index{Laplace!school of}% +says, ``of head occurring $n$~times if the coin is of the ordinary +make is approximately at least~$(\frac{1}{2})^n$. This value is rigidly deducible +from positive experience, the observations made by gamesters, +the experiments recorded by Jevons and De~Morgan.'' + +The doctrines of the empirical school will be examined in +\index{Empirical School}% +\Chapref{VIII}., and I postpone my detailed criticism to that +chapter. Venn rejects the applications of Bernoulli's theorem, +\index{Venn!Bernoulli@{and Bernoulli}}% +which he describes as ``one of the last remaining relics of Realism,'' +as well as the later Laplacian Law of Succession, thus destroying +the link between the empirical and the \textit{à~priori} methods. But, +apart from this, his view that statements of probability are +simply a particular class of statements about the actual world +of phenomena, would have led him to a closer dependence on +actual experience. He holds that the probability of an event's +having a certain attribute is simply the fraction expressing the +proportion of cases in which, as a matter of actual fact, this +attribute is present. Our knowledge, however, of this proportion +is often reached inductively, and shares the uncertainty to which +all inductions are liable. And, besides, in referring an event to +a series we do not postulate that all the members of the series +should be identical, but only that they should not be \emph{known} to +differ in a relevant manner. Even on this theory, therefore, we +are not solely determined by positive knowledge and the direct +\textit{data} of experience. + +\Paragraph{7.} The Empirical School in their reaction against the pretentious +results, which the Laplacian theory affected to develop +out of nothing, have gone too far in the opposite direction. If +our experience and our knowledge were complete, we should +be beyond the need of the Calculus of Probability. And where +our experience is incomplete, we cannot hope to derive from it +judgments of probability without the aid either of intuition or of +some further \textit{à~priori} principle. Experience, as opposed to intuition, +cannot possibly afford us a criterion by which to judge +whether on given evidence the probabilities of two propositions +are or are not equal. + +However essential the data of experience may be, they cannot +by themselves, it seems, supply us with what we want. Czuber,\index{Czuber}\footnote + {\textit{Wahrscheinlichkeitsrechnung}, p.~11.} +who prefers what he calls the Principle of Compelling Reason +\index{Principle of compelling reason}% +(das Prinzip des zwingenden Grundes), and holds that Probability +%% -----File: 098.png---Folio 87------- +\index{Kries, von!equiprobability@{and equiprobability}}% +has an objective and not merely formal interpretation only when +it is grounded on definite knowledge, is rightly compelled to +admit that we cannot get on altogether without the Principle of +Non-Sufficient Reason. On the grounds both of its own intuitive +plausibility and of that of some of the conclusions for which it +is necessary, we are inevitably led towards this principle as a +necessary basis for judgments of probability. In \emph{some} sense, +judgments of probability do seem to be based on equally balanced +degrees of ignorance. + +\Paragraph{8.} It is from this starting-point that the German logicians +\index{German logicians}% +have set out. They have perceived that there are few judgments +of probability which are altogether independent of some principle +resembling that of Non-Sufficient Reason. But they also apprehend, +with Boole, that this may be a very arbitrary method of +\index{Boole!German logicians@{and German logicians}}% +procedure. + +It was pointed out in §\;18 of \Chapref{IV}. that the cases, in +which the Principle of Indifference (or Non-Sufficient Reason) +\index{Principle of Indifference}% +breaks down, have a great deal in common, and that we break +up the field of possibility into a number of areas, actually unequal, +but indistinguishable on the evidence. Several German logicians, +therefore, have endeavoured to determine some rule by which +it might be possible to postulate actual equality of area for the +fields of the various possibilities. + +By far the most complete and closely reasoned solution on +these lines is that of Von Kries.\Pagelabel{87}\footnote + {\textit{Die Principien der Wahrscheinlichkeitsrechnung. Eine logische Untersuchung.} +Freiburg, 1886.} He is primarily anxious to discover +a proper basis for the numerical measurement of probabilities, +and he is thus led to examine with care the grounds of valid +judgments of equiprobability. His criticisms of the Principle +of Non-Sufficient Reason are searching, and, to meet them, he +elaborates a number of qualifying conditions which are, he +argues, necessary and sufficient. The value of his book, however, +lies, in the opinion of the present writer, in the critical rather +than in the constructive parts. The manner in which his qualifying +conditions are expressed is often, to an English reader at any +rate, somewhat obscure, and he seems sometimes to cover up +difficulties, rather than solve them, by the invention of new +technical terms. These characteristics render it difficult to +expound him adequately in a summary, and the reader must be +%% -----File: 099.png---Folio 88------- +\index{Spielräume, doctrine of}% +referred to the original for a proper exposition of the Doctrine of +\textit{Spielräume}. Briefly, but not very intelligibly perhaps, he may +be said to hold that the hypotheses for the probabilities of which +we wish to obtain a numerical comparison, must refer to `fields' +(\textit{Spielräume}) which are `indifferent,' `comparable' in magnitude, +and `original' (\textit{ursprünglich}). Two fields are `indifferent' if +they are equal before the Principle of Non-Sufficient Reason; +they are `comparable' if it is true that the fields are actually +of equal extent; and they are `original' or ultimate if they are +not derived from some other field. The last condition is exceedingly +obscure, but it seems to mean that the objects with which +we are ultimately dealing must be directly represented by the +`fields' of our hypotheses, and there must not be merely correlation +between these objects and the objects of the fields. The +qualification of comparability is intended to deal with difficulties +such as that connected with the population of different areas of +unknown extent; and the qualification of originality with those +arising from indirect measurement, as in the case of specific +density. + +Von~Kries's solution is highly suggestive, but it does not seem, +so far as I understand it, to supply an unambiguous criterion +for all cases. His discussion of the philosophical character of +probability is brief and inadequate, and the fundamental error +in his treatment of the subject is the \emph{physical}, rather than logical, +bias which seems to direct the formulation of his conditions. +The condition of \textit{Ursprünglichkeit}, for instance, seems to depend +upon physical rather than logical criteria, and is, as a result, +much more restricted in its applicability than a condition, which +will really meet the difficulties of the case, ought to be. But, +although I differ from him in his philosophical conception of +probability, the treatment of the Principle of Indifference, which +fills the greater part of his book, is, I think, along fruitful lines, +and I have been deeply indebted to it in formulating my own +conditions in \Chapref{IV}. + +Of less closely reasoned and less detailed treatments, which +aim at the same kind of result, those of Sigwart and Lotze are +\index{Sigwart}% +worth noticing. Sigwart's\footnote + {Sigwart \textit{Logic} (Eng.\ edition), vol.~ii.\ p.~220.} +position is sufficiently explained by +the following extract: ``The possibility of a mathematical treatment +lies primarily in the fact that in the disjunctive judgment +%% -----File: 100.png---Folio 89------- +the number of terms in the disjunction plays a decisive part. +Inasmuch as a limited number of mutually exclusive possibilities +is presented, of which one alone is actual, the element +of number forms an essential part of our knowledge\ldots. Our +knowledge must enable us to assume that the particular terms of +the disjunction are so far equivalent that they express an equal +degree of specialisation of a general concept, or that they cover +equal parts of the whole extension of the concept\ldots. This +equivalence is most intuitable where we are dealing with equal +parts of a spatial area, or equal parts of a period of time\ldots. +But even where this obvious quality is not forthcoming, we may +ground our expectations upon a hypothetical equivalence, where +we see no reason for considering the extent of one possibility to +be greater than that of the others\ldots.'' + +In the beginning of this passage Sigwart seems to be aware +of the fundamental difficulty, although exception may be taken +to the vagueness of the phrase ``equal degree of specialisation of +a general concept.'' But in the last sentence quoted he surrenders +the advantages he has gained in the earlier part of his explanation, +and, instead of insisting on a knowledge of an equal degree +of specialisation, he is satisfied with an absence of any knowledge +to the contrary. Hence, in spite of his initial qualifications, he +ends unrestrainedly in the arms of Non-Sufficient Reason.\footnote + {Sigwart's treatment of the subject of probability is curiously inaccurate. + Of his four fundamental rules of probability, for instance, three are, as he states + them, certainly false.} + +Lotze,\index{Lotze}\footnote + {Lotze, \textit{Logic} (Eng.\ edition), pp.~364,~365.} +in a brief discussion of the subject, throws out some +remarks well worth quoting: ``We disclaim all knowledge of +the circumstances which condition the real issue, so that when +we talk of equally possible cases we can only mean \emph{coördinated as +equivalent species in the compass of an universal case}; that is to +say, if we enumerate the special forms, which the genus can +assume, we get a disjunctive judgment of the form: if the condition~$B$ +is fulfilled, one of the kinds $f_{1}f_{2}f_{3}~\ldots$ of the universal +consequent~$F$ will occur to the exclusion of the rest. Which of +all those different consequents will, in fact, occur, depends in all +cases on the special form $b_{1}b_{2}b_{3}~\ldots$ in which that universal +condition is fulfilled\ldots. A~\emph{coördinated} case is a case which +answers to one and only one of the mutually exclusive values +$b_{1}b_{2}~\ldots$ of the condition~$B$, and these rival values may occur in +%% -----File: 101.png---Folio 90------- +\index{Certainty!Kahle and|inote}% +\index{Kahle and the Probability relation}% +reality; it does not answer to a more general form B, of this +condition, which can never exist in reality, because it embraces +several of the particular values~$b_{1}b_{2}$\ldots.'' + +This certainly meets some of the difficulties, and its resemblance +to the conditions formulated in \Chapref{IV}. will be evident +to the careful reader. But it is not very precise, and not easily +applicable to all cases, to those, for instance, of the measurement +of continuous quantity. By combining the suggestions of +Von~Kries, Sigwart, and Lotze, we might, perhaps, patch up a +fairly comprehensive rule. We might say, for instance, that if $b_{1}$~and~$b_{2}$ +are classes, their members must be finite in number and +enumerable or they must compose stretches; that, if they are +finite in number, they must be equal in number; and that, if +their members compose stretches, the stretches must be equal +stretches; and that if $b_{1}$~and~$b_{2}$ are concepts, they must represent +concepts of an equal degree of specialisation. But qualifications +so worded would raise almost as many difficulties as they solved. +How, for instance, are we to know when concepts are of an equal +degree of specialisation? + +\Paragraph{9.} That probability is a \emph{relation} has often received incidental +recognition from logicians, in spite of the general failure to place +proper emphasis on it. The earliest writer, with whom I am +acquainted, explicitly to notice this, is Kahle in his \textit{Elementa +logicae Probabilium methodo mathematica in usum Scientiarum +et Vitae adornata} published at Halle in~1735.\footnote + {This work, which seems to have soon fallen into complete neglect and is + now extremely rare, is full of interest and original thought. The following + quotations will show the fundamental position taken up: ``Est cognitio probabilis, + si desunt quaedam requisita ad veritatem demonstrativam (p.~15). + Propositio probabilis esse potest falsa, et improbabilis esse potest vera; ergo + cognitio hodie possibilis, crastina luce mutari potest improbabilem, si accedunt + reliqua requisita omnia, in certitudinem (p.~26)\ldots. Certitudo est terminus + relativus: considerare potest ratione representationum in intellectu nostro\ldots. Incerta nobis dependent a defectu cognitionis (p.~35)\ldots. Actionem + imprudenter et contra regulas probabilitatis susceptam eventus felix sequi + potest. Ergo prudentia actionum ex successu solo non est aestimanda (p.~62)\ldots. + Logica probabilium est scientia dijudicandi gradum certitudinis eorum, + quibus desunt requisita ad veritatem demonstrativam (p.~94).''} +Amongst more +recent writers casual statements are common to the effect that +the probability of a conclusion is relative to the grounds upon +\index{Boole!relation@{and relation of Probability}}% +which it is based. Take Boole\footnote + {``On a General Method in the Theory of Probabilities,'' \textit{Phil.\ Mag}., 4th~Series, + viii., 1854. See also, ``On the Application of the Theory of Probabilities + to the Question of the Combination of Testimonies or Judgments'' (\textit{Edin.\ Phil.\ + Trans.}\ xxi.\ p.~600): ``Our estimate of the probability of an event varies not + absolutely with the circumstances which actually affect its occurrence, but with + our knowledge of those circumstances.''} +for instance: ``It is implied in +the definition that probability is always relative to our actual +%% -----File: 102.png---Folio 91------- +state of information and varies with that state of information.'' +\index{Bradley!relativity@{and relativity of Probability}}% +Or Bradley:\footnote + {\textit{The Principles of Logic}, p.~208.} +``Probability tells us what we ought to believe, +what we ought to believe \emph{on certain data}~\ldots\DPtypo{}{.} Probability is no +more `relative' and `subjective' than is any other act of +logical inference from hypothetical premises. It is relative to +the \textit{data} with which it has to deal, and is not relative in any other +sense.'' Or even Laplace, when he is explaining the diversity +\index{Laplace!relation@{and relation of Probability}}% +of human opinions: +``Dans les choses qui ne sont que vraisemblables, +la différence des données que chaque homme a sur elles, +est une des causes principales de la diversité des opinions que +l'on voit régner sur les mêmes objets~\ldots~c'est ainsi que le +même fait, récité devant une nombreuse assemblée, obtient divers +degrés de croyance, suivant l'étendue des connaissances des +auditeurs.''\footnote + {\textit{Essai philosophique}, p.~7.} + +\Paragraph{10.} Here we may leave this account of the various directions +in which progress has seemed possible, with the hope that it may +assist the reader, who is dissatisfied with the solution proposed in +\Chapref{IV}., to determine the line of argument along which he +is likeliest to discover the solution of a difficult problem. +%% -----File: 103.png---Folio 92------- +\index{Cournot, and frequency theory}% +\index{Ellis, Leslie!frequency@{and frequency theory}}% +\index{Frequency theory|ifoll}% + + +\Chapter{VIII}{The Frequency Theory of Probability} + +\Paragraph{1.} The theory of probability, outlined in the preceding chapters, +has serious difficulties to overcome. There is a theoretical, as +well as a practical, difficulty in measuring or comparing degrees +of probability, and a further difficulty in determining them +\textit{à~priori}. We must now examine an alternative theory which is +much freer from these troubles, and is widely held at the present +time. + +\Paragraph{2.} The theory is in its essence a very old one, Aristotle +\index{Aristotle}% +foreshadowed it when he held that ``the probable is that which +for the most part happens'';\footnote + {\textit{Rhet.}\ i.~2, 1357 a~34.} +and, as we have seen in \Chapref{VII}., +an opinion not unlike this was entertained by those philosophers +of the seventeenth and eighteenth centuries who approached +the problems of probability uninfluenced by the work of mathematicians. +But the underlying conception of earlier writers +received at the hands of some English logicians during the latter +half of the nineteenth century a new and much more complicated +form. + +The theory in question, which I shall call the Frequency +Theory of Probability, first appears\footnote + {I give Ellis the priority because his paper, published in 1843, was read on + Feb.~14, 1842. The same conception, however, is to be found in Cournot's + \textit{Exposition}, also published in 1843: ``La théorie des probabilités a pour objet + certains rapports numériques qui prendraient des valeurs fixes et complétement + déterminées, si l'on pouvait répéter à l'infini les épreuves des mêmes hasards, + et qui, pour un nombre fini d'épreuves, oscillent entre des limites d'autant plus + resserées, d'autant plus voisines des valeurs \emph{finales}, que le nombre des épreuves + est plus grand.''} +as the basis of a proposed logical scheme in a brief essay by Leslie Ellis \textit{On the Foundations +of the Theory of Probabilities}, and is somewhat further developed +in his \textit{Remarks on the Fundamental Principles of the Theory of +%% -----File: 104.png---Folio 93------- +\index{Series of probabilities!frequency@{and frequency theory}}% +\index{Statistical frequency, theory of|ifoll}% +Probabilities}.\footnote + {These essays were published in the \textit{Transactions} of the Camb.\ Phil.\ Soc., the + first in 1843 (vol.~viii.), and the second in 1854 (vol.~ix.). Both were reprinted + in \textit{Mathematical and other Writings} (1863), together with three other brief + papers on Probability and the Method of Least Squares. All five are full of + spirit and originality, and are not now so well known as they deserve to be.} +``If the probability of a given event be correctly +determined,'' he says, ``the event will on a long run of trials tend +to recur with frequency proportional to their probability. This +is generally proved mathematically. It seems to me to be true +\textit{à~priori}\ldots. I have been unable to sever the judgment that +one event is more likely to happen than another from the belief +that in the long run it will occur more frequently.'' Ellis explicitly +introduces the conception that probability is essentially +concerned with a group or series. + +Although the priority of invention must be allowed to Leslie +Ellis, the theory is commonly associated with the name of Venn. +\index{Venn!frequency@{and frequency theory}|ifoll}% +In his \textit{Logic of Chance}\footnote + {The first edition appeared in 1866. Revised editions were issued in 1876 + and 1888. References are given to the third edition of 1888.} +it first received elaborate and systematic +treatment, and, in spite of his having attracted a number of +followers, there has been no other comprehensive attempt to +meet the theory's special difficulties or the criticisms directed +against it. I shall begin, therefore, by examining it in the form +in which Venn has expounded it. Venn's exposition is much +coloured by an empirical view of logic, which is not perhaps as +necessary to the essential part of his doctrine as he himself +implies, and is not shared by all of those who must be classed as +in general agreement with him about probability. It will be +necessary, therefore, to supplement a criticism of Venn by an +account of a more general frequency theory of probability, +divested of the empiricism with which he has clothed it. + +\Paragraph{3.} The following quotations from Venn's \textit{Logic of Chance} will +show the general drift of his argument: The fundamental conception +is that of a series (p.~4). The series is of events which +have a certain number of features or attributes in common (p.~10). +The characteristic distinctive of probability is this,---the occasional +attributes, as distinguished from the permanent, are found +on an examination to tend to exist \emph{in a certain definite proportion +of the whole number of cases} (p.~11). We require that there should +be in nature large classes of objects, throughout all the individual +members of which a general resemblance extends. For this +%% -----File: 105.png---Folio 94------- +\index{Measurement of Probability!frequency@{and frequency theory}}% +purpose the existence of natural kinds or groups is necessary +(p.~55). The distinctive characteristics of probability prevail +principally in the properties of natural kinds, both in the ultimate +and in the derivative or accidental properties (p.~63). The same +peculiarity prevails again in the force and frequency of most +natural agencies (p.~64). There seems reason to believe that it +is in such things only, as distinguished from things artificial, that +the property in question is to be found (p.~65). How, in any +particular case, are we to establish the existence of a probability +series? Experience is our sole guide. If we want to discover +what is in reality a series of \emph{things}, not a series of our own conceptions, +we must appeal to the things themselves to obtain it, +for we cannot find much help elsewhere (p.~174). When probability +is divorced from direct reference to objects, as it substantially +is by not being founded upon experience, it simply resolves +itself into the common algebraical doctrine of Permutations +and Combinations (p.~87). By assigning an expectation in +reference to the individual, we \emph{mean} nothing more than to make +a statement about the average of his class (p.~151). When we say +of a conclusion within the strict province of probability, that it +is not certain, all that we mean is that in some proportion of +cases only will such conclusion be right, in the other cases it will +be wrong (p.~210). + +The essence of this theory can be expressed in a few words. +To say, that the probability of an event's having a certain characteristic +is~$\frac{x}{y}$, is to mean that the event is one of a number of events, +a proportion~$\frac{x}{y}$ of which have the characteristic in question; and +the \emph{fact}, that there \emph{is} such a series of events possessing this +frequency in respect of the characteristic, is purely a matter of +experience to be determined in the same manner as any other +question of fact. That such series do exist happens to be a +characteristic of the real world as we know it, and from this +the practical importance of the calculation of probabilities is +derived. + +Such a theory possesses manifest advantages. There is no +mystery about it---no new indefinables, no appeals to intuition. +Measurement leads to no difficulties; our probabilities or frequencies +are ordinary numbers, upon which the arithmetical +apparatus can be safely brought to bear. And at the same time it +%% -----File: 106.png---Folio 95------- +\index{Probability@{`\textit{Probability}'}!Venn's use of}% +seems to crystallise in a clear, explicit shape the floating opinion +of common sense that an event is or is not probable in certain +supposed circumstances according as it is or is not usual as a +matter of fact and experience. + +The two principal tenets, then, of Venn's system are these,---that +probability is concerned with series or groups of events, +and that all the requisite facts must be determined empirically, +a statement in probability merely summing up in a convenient +way a group of experiences. Aggregate regularity combined +with individual difference happens, he says, to be characteristic +of many events in the real world. It will often be the case, +therefore, that we can make statements regarding the average +of a certain class, or regarding its characteristics in the long run, +which we cannot make about any of its individual members +without great risk of error. As our knowledge regarding the +class as a whole may give us valuable guidance in dealing with an +individual instance, we require a convenient way of saying that +an individual belongs to a class in which certain characteristics +appear on the average with a known frequency; and this the +conventional language of probability gives us. The importance +of probability depends solely upon the actual existence of such +groups or real kinds in the world of experience, and a judgment +of probability must necessarily depend for its validity upon our +empirical knowledge of them. + +\Paragraph{4.} It is the obvious, as well as the correct, criticism of such a +theory, that the identification of probability with statistical +frequency is a very grave departure from the established use of +words; for it clearly excludes a great number of judgments +which are generally believed to deal with probability. Venn +himself was well aware of this, and cannot be accused of supposing +that all beliefs, which are commonly called probable, are really +concerned with statistical frequency. But some of his followers, +to judge from their published work, have not always seen, so +clearly as he did, that his theory is \emph{not} concerned with the same +subject as that with which other writers have dealt under the +same title. Venn justifies his procedure by arguing that no other +meaning, of which it is possible to take strict logical cognisance, +can reasonably be given to the term, and that the other meanings, +with which it has been used, have not enough in common to +permit their reduction to a single logical scheme. It is useless, +%% -----File: 107.png---Folio 96------- +\index{Probability@{`\textit{Probability}'}!Edgeworth's use of|inote}% +therefore, for a critic of Venn to point out that many supposed +judgments of probability are \emph{not} concerned with statistical +frequency; for, as I understand the \textit{Logic of Chance}, he admits +it; and the critic must show that the sense different from Venn's +in which the term probability is often employed \emph{has} an important +logical interpretation about which we can generalise. This +position I seek to establish. It is, in my opinion, this other sense +\emph{alone} which has importance; Venn's theory by itself has few +practical applications, and if we allow it to hold the field, we must +admit that probability is \emph{not} the guide of life, and that in following +it we are not acting according to reason. + +\Paragraph{5.} Part of the plausibility of Venn's theory is derived, I +think, from a failure to recognise the narrow limits of its applicability, +or to notice his own admissions regarding this. ``In +every case,'' he says (p.~124), ``in which we extend our inferences +by Induction or Analogy, or depend upon the witness of others, +or trust to our own memory of the past, or come to a conclusion +through conflicting arguments, or even make a long and complicated +deduction by mathematics or logic, we have a result of +which we can scarcely feel as certain as of the premisses from +which it was obtained. In all these cases, then, we are conscious +of varying quantities of belief, but are the laws according to which +the belief is produced and varied the same? If they cannot be +reduced to one harmonious scheme, if, in fact, they can at best be +brought to nothing but a number of different schemes, each with +its own body of laws and rules, then it is vain to endeavour to +force them into one science.'' All these cases, therefore, in which +we are `not certain,' Venn explicitly excludes from what he +chooses to call the science of probability, and he pays no further +attention to them. The science of probability is, according to +him, \emph{no more} than a method which enables us to express in a +convenient form statistical statements of frequency. ``The +province of probability,'' he says again on page~160, ``is not so +extensive as that over which variation of belief might be observed. +Probability only considers the case in which this variation is +brought about in a certain definite statistical way.''\footnote + {Edgeworth uses the term `probability' widely, as I do; but he makes +\index{Edgeworth!use of `\textit{Probability}'|inote}% + a distinction corresponding to Venn's by limiting the subject-matter of the + \emph{Calculus} of Probabilities. He writes (`Philosophy of Chance,' \textit{Mind}, 1884, + p.~223): ``The Calculus of Probabilities is concerned with the estimation of + degrees of probability; not every species of estimate, but that which is founded + on a particular standard. That standard is the phenomenon of statistical + uniformity: the fact that a genus can very frequently be subdivided into species + such that the number of individuals in each species bears an approximately + constant ratio to the number of individuals in the genus.'' This use of terms is + legitimate, though it is not easy to follow it consistently. But, like Venn's, + it leaves aside the most important questions. The Calculus of Probabilities, + thus interpreted, is no guide by itself as to which opinion we ought + to follow, and is not a measure of the weight we should attach to conflicting + arguments.} +He points +%% -----File: 108.png---Folio 97------- +\index{Probability, and relevant knowledge!rational@{and rational belief}}% +out on p.~194 that for the purposes of probability we must take +the statistical frequency from which we start \emph{ready made} and +ask no questions about the process or completeness of its manufacture: +``It may be obtained by any of the numerous rules +furnished by Induction, or it may be inferred deductively, or +\index{Induction}% +given by our own observation; its value may be diminished by +its depending upon the testimony of witnesses, or its being +recalled by our own memory. Its real value may be influenced +by these causes or any combinations of them; but all these are +preliminary questions with which we have nothing directly to do. +We assume our statistical proposition to be true, neglecting the +diminution of its value by the processes of attainment.'' + +It must be recognised, therefore, that Venn has deliberately +excluded from his survey almost all the cases in which we regard +our judgments as `only probable'; and, whatever the value or +consistency of his own scheme, he has left untouched a wide +field of study for others. + +\Paragraph{6.} The main grounds, which have induced Venn to regard +judgments based on statistical frequency as the only cases of +probability which possess logical importance, seem to be two: +(i.)~that other cases are mainly subjective, and (ii.)~that they +are incapable of accurate measurement. + +With regard to the first it must be admitted that there are +many instances in which variation of belief is occasioned by purely +psychological causes, and that his argument is valid against those +who have defined probability as measuring the degree of subjective +belief. But this has not been the usual way of +looking at the subject. Probability is the study of the +grounds which lead us to entertain a \emph{rational} preference for +one belief over another. That there are rational grounds other +than statistical frequency, for such preferences, Venn does +not deny; he admits in the quotation given above that the +`\emph{real value}' of our conclusion is influenced by many other considerations +%% -----File: 109.png---Folio 98------- +\index{Modality and probability!Venn and}% +\index{Probability, and relevant knowledge!statistical@{and statistical frequency}}% +than that of statistical frequency. Venn's theory, +therefore, cannot be fairly propounded by his disciples as \emph{alternative} +to such a theory as is propounded here. For my Treatise is +concerned with the general theory of arguments from premisses +leading to conclusions which are reasonable but not certain; +and this is a subject which Venn has, deliberately, not treated +in the \textit{Logic of Chance}. + +\Paragraph{7.} Apart from two circumstances, it would scarcely be necessary +to say anything further; but in the first place some writers +have believed that Venn has propounded a \emph{complete} theory +of probability, failing to realise that he is not at all concerned +with the sense in which we may say that one induction or analogy, +\index{Induction!frequency@{and frequency theory}}% +or testimony, or memory, or train of argument is more probable +than another; and in the second place he himself has not always +kept within the narrow limits, which he has himself laid down +as proper to his theory. + +For he has not remained content with defining a probability +as identical with a statistical frequency, but has often spoken +as if his theory told us which alternatives it is \emph{reasonable to prefer.} +When he states, for instance, that modality ought to be banished +from Logic and relegated to Probability (p.~296), he forgets his +own dictum that of premisses, the distinctive characteristic of +which is their lack of certainty, Probability takes account of +\emph{one class only}, Induction concerning itself with another class, and +so forth (p.~321). He forgets also that, when he comes to consider +the practical use of statistical frequencies, he has to admit that +an event may possess more than one frequency, and that we must +decide which of these to prefer on extraneous grounds (p.~213). +The device, he says, must be to a great extent arbitrary, and there +are no logical grounds of decision; but would he deny that it is +often reasonable to found our probability on one statistical +frequency rather than on another? And if our grounds are +reasonable, are they not in an important sense logical? + +Even in those cases, therefore, in which we derive our preference +for one alternative over another from a knowledge of statistical +frequencies, a statistical frequency by itself is insufficient +to determine us. We may call a statistical frequency a probability, +if we choose; but the fundamental problem of determining +which of several alternatives is logically preferable still awaits +solution. We cannot be content with the only counsel Venn +%% -----File: 110.png---Folio 99------- +\index{Principle of Indifference!induction@{and induction}}% +can offer, that we should choose a frequency which is derived +from a series neither too large nor too small. + +The same difficulty, that a probability in Venn's sense is +insufficient to determine which alternative is logically preferable, +arises in another connection. In most cases the statistical +frequency is not given in experience for certain, but is arrived +at by a process of \emph{induction}, and inductions, he admits, are not +\index{Induction!frequency@{and frequency theory}}% +certain. If, in the past, three infants out of every ten have +died in their first four years, induction may base on this the +doubtful assertion, All infants die in that proportion. But we +cannot assert on this ground, as Venn wishes to do, that the probability +of the death of an infant in its first four years \emph{is}~$\frac{3}{10}$ths. +We can say no more than that it is probable (in my sense) that +there is such a probability (in his sense). For the purpose of +coming to a decision we cannot compare the value of this +conclusion with that of others until we know the probability +(in my sense) that the statistical frequency really is~$\frac{3}{10}$ths. +The cases in which we can determine the logical value of a +conclusion entirely on grounds of statistical frequency would +seem to be extremely few in number. + +\Paragraph{8.} The second main reason which led Venn to develop his +theory is to be found in his belief that probabilities which are +based on statistical frequencies are alone capable of accurate +measurement. The term `probabilities,' he argues, is properly +confined to the case of chances which can be calculated, and all +calculable chances can be made to depend upon statistical +frequency. In attempting to establish this latter contention +he is involved in some paradoxical opinions. ``In many cases,'' +he admits, ``it is undoubtedly true that we do not resort to direct +experience at all. If I want to know what is my chance of +holding ten trumps in a game of whist, I do not enquire how +often such a thing has occurred before\ldots. In practice, \textit{à~priori} +determination is often easy, whilst \textit{à~posteriori} appeal to experience +would be not merely tedious but utterly impracticable.'' +But these cases which are usually based on the Principle of +Indifference can, he maintains, be justified on statistical grounds. +In the case of coin tossing there is a considerable experience of +the equally frequent occurrence of heads and tails; the experience +gained in this simple case is to be extended to the complex +cases by ``Induction and Analogy.'' In one simple case the +%% -----File: 111.png---Folio 100------- +\index{Experience and the Principle of!Indifference}% +\index{Pearson, Karl!frequency@{and frequency theory}}% +result to which the Principle of Indifference would lead is that +which experience recommends. Therefore in complex cases, +where there is no basis of experiment at all, we may assume that +Experience, if experience there was, would speak with the same +voice as Indifference. This is to assert that, because in one case, +where there is no known reason to the contrary, there actually +is none, therefore in other cases incapable of verification the +absence of known reason to the contrary proves that \emph{actually} +there is none. + +The attempt to justify the rules of inverse probability on +\index{Inverse Probability!Venn@{and Venn}}% +statistical grounds I have failed to understand; and after a careful +reading, I am unable to produce an intelligible account of +the argument involved in the latter part of chapter~vii.\ of the +\textit{Logic of Chance}.\footnote + {Let the reader, who is acquainted with this chapter, consider what precise + assumption Venn's reasoning requires on p.~187 in the example which seeks to +\index{Venn!inverse probability@{and inverse probability}}% + show the efficacy of Lord Lister's antiseptic treatment \textit{à~posteriori}. + What is the `inevitable assumption about the bags' when it is translated into the + language of this example?} +I am doubtful whether Venn should not have +excluded \textit{à~posteriori} arguments in probability from his scheme +as well as inductive arguments. The attempt to include them +may have been induced by a desire to deal with all cases +in which numerical calculation has been commonly thought +possible. + +\Paragraph{9.} The argument so far has been solely concerned with the +case for the frequency theory developed in the \textit{Logic of Chance.} +The criticisms which follow will be directed against a more +general form of the same theory which may conceivably have +recommended itself to some readers. It is unfortunate that no +adherent of the doctrine, with the exception of Venn, has attempted +to present the theory of it in detail. Professor Karl +Pearson, for instance, probably agrees with Venn in a general +way only, and it is very likely that many of the foregoing remarks +do not apply to his view of probability; but while I generally +disagree with the fundamental premisses upon which his work +in probability and statistics seems to rest, I am not clearly +aware of the nature of the philosophical theory from which he +thinks that he derives them and which makes them appear to +him to be satisfactory. A careful exposition of his logical presuppositions +would greatly add to the completeness of his work. +In the meantime it is only possible to raise general objections to +%% -----File: 112.png---Folio 101------- +\index{Probability, and relevant knowledge!truth frequency@{and truth frequency}|ifoll}% +\index{Proposition, characterisation of!classes of|ifoll}% +\index{Statistical frequency, theory of!generalisation of}% +\index{Truth frequency}% +any theory of probability which seeks to found itself upon the +conception of statistical frequency. + +The generalised frequency theory which I propose to put +forward, as perhaps representative of what adherents of this +doctrine have in mind, differs from Venn's in several important +respects.\footnote + {In what follows I am much indebted for some suggestions in favour of the + frequency theory communicated to me by Dr.~Whitehead; but it is not to be + supposed that the exposition which follows represents his own opinion.} +In the first place, it does not regard probability as +being \emph{identical} with statistical frequency, although it holds that +all probabilities must be based on statements of frequency, and +can be defined in terms of them. It accepts the theory that +propositions rather than events should be taken as the subject-matter +of probability; and it adopts the comprehensive view +of the subject according to which it includes induction and all +other cases in which we believe that there are \emph{logical} grounds for +preferring one alternative out of a set none of which are certain. +Nor does it follow Venn in supposing any special connection to +exist between a frequency theory of probability and logical +empiricism. + +\Paragraph{10.} A proposition can be a member of many distinct classes +of propositions, the classes being merely constituted by the +existence of particular resemblances between their members +or in some such way. We may know of a given proposition that +it is one of a particular class of propositions, and we may also +know, precisely or within defined limits, what \emph{proportion} of this +class are true, without our being aware whether or not the given +proposition is true. Let us, therefore, call the actual proportion +of true propositions in a class the truth-frequency\footnote + {This is Dr.~Whitehead's phrase.} +of the class, +and define the measure of the probability of a proposition relative +to a class, of which it is a member, as being equal to the truth-frequency +of the class. + +The fundamental tenet of a frequency theory of probability +is, then, that the probability of a proposition always depends +upon referring it to some class whose truth-frequency is known +within wide or narrow limits. + +Such a theory possesses most of the advantages of Venn's, +but escapes his narrowness. There is nothing in it so far which +could not be easily expressed with complete precision in the +terms of ordinary logic. Nor is it necessarily confined to probabilities +%% -----File: 113.png---Folio 102------- +\index{Relativity, of knowledge!of probabilities}% +which are numerical. In some cases we may know the +exact number which expresses the truth-frequency of our class; +but a less precise knowledge is not without value, and we may +say that one probability is greater than another, without knowing +how much greater, and that it is large or small or negligible, if +we have knowledge of corresponding accuracy about the truth-frequencies +of the classes to which the probabilities refer. The +magnitudes of some pairs of probabilities we shall be able to +compare numerically, others in respect of more and less only, +and others not at all. A great deal, therefore, of what has been +said in \Chapref{III}. would apply equally to the present theory, +with this difference that the probabilities would, as a matter of +fact, have numerical values \emph{in all cases}, and the less complete +comparisons would only hold the field in cases where the real +probabilities were partially unknown. On the frequency theory, +therefore, there is an important sense in which probabilities can +be unknown, and the relative vagueness of the probabilities +employed in ordinary reasoning is explained as belonging not +to the probabilities themselves but only to our knowledge of +them. For the probabilities are relative, not to our knowledge, +but to some objective class, possessing a perfectly definite truth-frequency, +to which we have chosen to refer them. + +The frequency theory expounded in this manner cannot easily +avoid mention of the relativity of probabilities which is implicit +here, as it is in Venn's. Whether or not the probability of a +proposition is relative to given \textit{data}, it is clearly relative to the +particular class or series to which we choose to refer it. A given +proposition has a great variety of different probabilities corresponding +to each of the various distinct classes of which it is a +member; and before an intelligible meaning can be given to a +statement that the probability of a proposition is so-and-so, the +class must be specified to which the proposition is being referred. +Most adherents of the frequency theory would probably go +further, and agree that the class of reference must be determined +in any particular case by the \textit{data} at our disposal. Here, then, +is another point on which it is not necessary for the frequency +theory to diverge from the theory of this Treatise. It should, +I think, be generally agreed by every school of thought that the +probability of a conclusion is in an important sense \emph{relative to +given premisses}. On this issue and also on the point that our +%% -----File: 114.png---Folio 103------- +\index{Statistical frequency, theory of!criticism of}% +knowledge of many probabilities is not numerically definite, +there might well be for the future an end of disagreement, and +disputation might be reserved for the philosophical interpretation +of these settled facts, which it is unreasonable to deny, however +we may explain them. + +\Paragraph{11.} I now proceed to those contentions upon which my +fundamental criticism of the frequency theory is founded. The +first of these relates to the method by which the class of reference +is to be determined. The magnitude of a probability is always +to be measured by the truth-frequency of some class; and this +class, it is allowed, must be determined by reference to the +premisses, on which the probability of the conclusion is to be +determined. But, as a given proposition belongs to innumerable +different classes, how are we to know which class the premisses +indicate as appropriate? What substitute has the frequency +theory to offer for judgments of relevance and indifference? +And without something of this kind, what principle is there for +uniquely determining the class, the truth-frequency of which is +to measure the probability of the argument? Indeed the +difficulties of showing how given premisses determine the class +of reference, by means of rules expressed in terms of previous +ideas, and without the introduction of any notion, which is new +and peculiar to probability, appear to me insurmountable. + +Whilst no general criterion of choice seems to exist, where of +two alternative classes neither includes the other, it might be +thought that where one does include the other, the obvious +course would be to take the narrowest and most specialised class. +This procedure was examined and rejected by Venn\DPtypo{:}{;} though the +objection to it is due, not, as he supposed, to the lack of sufficient +statistics in such cases upon which to found a generalisation, +but to the inclusion in the class-concept of marks characteristic +of the proposition in question, but nevertheless \emph{not relevant} +to the matter in hand. If the process of narrowing the class +were to be carried to its furthest point, we should generally be +left with a class whose \emph{only} member is the proposition in question, +for we generally know something about it which is true of no +other proposition. We cannot, therefore, define the class of +reference as being the class of propositions of which everything +is true which is \emph{known} to be true of the proposition whose probability +we seek to determine. And, indeed, in those examples +%% -----File: 115.png---Folio 104------- +\index{Addition, of probabilities!Theorem of}% +\index{Relevance, judgments of!frequency@{and frequency theory}}% +for which the frequency theory possesses the greatest \textit{prima facie} +plausibility, the class of reference is selected by taking account +of \emph{some only} of the known characteristics of the \textit{quaesitum}, those +characteristics, namely, which are \emph{relevant} in the circumstances. +In those cases in which one can admit that the probability can be +measured by reference to a known truth-frequency, the class of +reference is formed of propositions about which our \emph{relevant} +knowledge is the same as about the proposition under consideration. +In these special cases we get the same result from the +frequency theory as from the Principle of Indifference. But +\index{Principle of Indifference}% +this does not serve to rehabilitate the frequency theory as a +\emph{general} explanation of probability, and goes rather to show that +the theory of this Treatise is the generalised theory, comprehending +within it such applications of the idea of statistical truth-frequency +as have validity. + +`Relevance' is an important term in probability, of which +the meaning is readily intelligible. I have given my own definition +of it already. But I do not know how it is to be explained +in terms of the frequency theory. Whether supporters of this +theory have fully appreciated the difficulty I much doubt. It is +a fundamental issue involving the essence of the \emph{peculiarity} of +probability, which prevents its being explained away in terms +of statistical frequency or anything else. + +\Paragraph{12.} Yet perhaps a modified view of the frequency theory +could be evolved which would avoid this difficulty, and I proceed, +therefore, to some further criticisms. It might be agreed that a +novel element must be admitted at this point, and that relevancy +must be determined in some such manner as has been explained +in earlier chapters. With this admission, it might be argued, the +theory would still stand, divested, it is true, of some of its original +simplicity, but nevertheless a substantial theory differing in +important respects, although not quite so fundamentally as +before, from alternative schemes. + +The next important objection, then, is concerned with the +manner in which the principal theorems of probability are to be +established on a theory of frequency. This will involve an +anticipation in some part of later arguments; and the reader +may be well advised to return to the following paragraph after +he has finished Part II. + +\Paragraph{13.} Let us begin by a consideration of the `Addition Theorem.' +%% -----File: 116.png---Folio 105------- +If $a/h$~denotes the probability of~$a$ on hypothesis~$h$, this theorem +may be written $(a+b)/h=a/h+b/h-ab/h$, and may be read +`On hypothesis~$h$ the probability of ``$a$~or~$b$'' is equal to the +probability of~$a+{}$ the probability of~$b-{}$ the probability of +``both $a$~and~$b$.''\,' This theorem, interpreted in some way or +other, is universally assumed; and we must, therefore, inquire +what proof of it the frequency theory can afford. A little +symbolism will assist the argument: Let $\DPtypo{a}{\alpha}_f$~represent the truth-frequency +of any class~$\DPtypo{a}{\alpha}$, and let $a_\alpha/h$~stand for `the probability +of~$a$ on hypothesis~$h$, $\DPtypo{a}{\alpha}$~being the class of reference determined +by this hypothesis.'\footnote + {The question, previously at issue, as to \emph{how} the class of reference is determined + by the hypothesis, is now ignored.} +We then have~$a_\alpha/h = \DPtypo{a}{\alpha}_f$, and we require to +prove a proposition, for values of $\gamma$~and~$\delta$ not yet determined, +which will be of the form: +\[ +(a+b)_\delta/h = a_\alpha/h + b_\beta/h - ab_\gamma/h. +\] +Now if $\delta'$~is the class of propositions $(a+b)$ such that $a$~is an~$\alpha$ +and $b$~a~$\beta$, it is easily shown by the ordinary arithmetic of classes +that $\delta'_f = \alpha_f + \beta_f - \alpha\beta_f$ where $\alpha\beta$~is the class of propositions which +are members of both $\alpha$~and~$\beta$. In the case, therefore, where +$\delta=\delta'$ and $\gamma=\alpha\beta$, an addition theorem of the required kind has +been established. + +But it does not follow by any reasonable rule that, if $h$~determines +$\alpha$~and~$\beta$ as the appropriate classes of reference for $a$~and~$b$, +$h$~must necessarily determine $\delta'$~and~$\alpha\beta$ as the appropriate classes +of reference for $(a + b)$ and~$ab$; it may, for instance, be the case +that~$h$, while it renders $\alpha$~and~$\beta$ determinate, yields no information +whatever regarding~$\alpha\beta$, and points to some quite different +class~$\mu$ as the suitable class of reference for~$ab$. On the frequency +theory, therefore, we cannot maintain that the addition theorem +is true in general, but only in those special cases where it happens +that $\delta=\delta'$ and $\gamma=\alpha\beta$. + +The following is a good example: We are given +that the proportion of black-haired men in the population +is~$\dfrac{p_1}{q}$ and the proportion of colour-blind men~$\dfrac{p_2}{q}$, and there is no +\emph{known} connection between black-hair and colour-blindness: +what is the probability that a man, about whom nothing special +%% -----File: 117.png---Folio 106------- +\index{Probability, and relevant knowledge!Inverse}% +\index{Venn|inote}% +is known, is\footnote + {In the course of the present discussion the disjunctive $a+b$ is never interpreted + so as to \emph{exclude} the conjunctive~$ab$.} +\emph{either} black-haired \emph{or} colour-blind? If we represent +the hypotheses by~$h$ and the alternatives by $a$~and~$b$, it would +usually be held that, colour-blindness and black hair being +\emph{independent for knowledge}\footnote + {For a discussion of this term see \Chapref{XVI}. §\;2.} +relative to the given data, $ab/h = \dfrac{p_1p_2}{q^2}$, +and that, therefore, by the addition theorem, $(a+b)/h = \dfrac{p_1}{q} + +\dfrac{p_2}{q} - \dfrac{p_1p_2}{q^2}$. But, on the frequency theory, this result might be +invalid; for $\alpha\beta_f = \dfrac{p_1p_2}{q^2}$, \emph{only} if this is the \emph{actual} proportion in fact +of persons who are both colour-blind and black-haired, and that +this is the actual proportion cannot possibly be inferred from +the \emph{independence for knowledge} of the characters in question.\footnote + {Venn argues (\textit{Logic of Chance}, pp.~173,~174) that there is an inductive + ground for making this inference. The question of extending the fundamental + theorems of a frequency theory of probability by means of induction is discussed + in §\;14~below.} + +Precisely the same difficulty arises in connection with the +multiplication theorem $ab/h = a/bh· b/h$.\footnote + {\textit{Vide} \Chapref{XII}. §\;6, and \Chapref{XIV}. §\;4.} +In the frequency notation, +which is proposed above, the corresponding theorem will +be of the form $ab_\delta/h = a_\gamma/bh· b_\beta/h$. For this equation to be satisfied +it is easily seen that $\delta$~must be the class of propositions~$xy$ such +that $x$~is a member of~$\alpha$ and $y$ of~$\beta$, and $\gamma$~the class of propositions~$xb$ +such that $x$~is a member of~$\alpha$; and, as in the case of the addition +theorem, we have no guarantee that these classes $\gamma$~and~$\delta$ will be +those which the hypotheses $bh$~and~$h$ will respectively determine +as the appropriate classes of reference for $a$~and~$ab$. + +In the case of the theorem of inverse probability\index{Inverse Probability!frequency@{and frequency theory}}\footnote + {\textit{Vide} \Chapref{XIV}. §\;5.} +\[ +\frac{b/ah}{c/ah} = \frac{a/bh}{a/ch}· \frac{b/h}{c/h} +\] +the same difficulty again arises, with an additional one when +practical applications are considered. For the relative probabilities +of our \textit{à~priori} hypotheses, $b$~and~$c$, will scarcely ever be +capable of determination by means of known frequencies, and in +the most legitimate instances of the inverse principle's operation +%% -----File: 118.png---Folio 107------- +\index{Independence, for knowledge}% +we depend either upon an inductive argument or upon the +Principle of Indifference. It is hard to think of an example in +\index{Principle of Indifference}% +which the frequency conditions are even approximately satisfied. + +Thus an important class of case, in which arguments in probability, +generally accepted as satisfactory, do not satisfy the +frequency conditions given above, are those in which the notion +is introduced of two propositions being, on certain \textit{data}, independent +for knowledge. The meaning and definition of this +expression is discussed more fully in \Partref{II}.; but I do not see +what interpretation the frequency theory can put upon it. Yet +if the conception of `independence for knowledge' is discarded, +we shall be brought to a standstill in the vast majority of problems, +which are ordinarily considered to be problems in probability, +simply from the lack of sufficiently detailed data. Thus the +frequency theory is not adequate to explain the processes of +reasoning which it sets out to explain. If the theory restricts its +operation, as would seem necessary, to those cases in which we +\emph{know} precisely how far the true members of $\alpha$~and~$\beta$ overlap, +the vast majority of arguments in which probability has been +employed must be rejected. + +\Paragraph{14.} An appeal to some further principle is, therefore, required +before the ordinary apparatus of probable inference can be established +on considerations of statistical frequency; and it may +have occurred to some readers that assistance may be obtained +from the principles of induction. Here also it will be necessary +\index{Induction!frequency@{and frequency theory}}% +to anticipate a subsequent discussion. If the argument of \Partref{III}. +is correct, nothing is more fatal than Induction to the theory +now under criticism. For, so far from Induction's lending +support to the fundamental rules of probability, it is itself +dependent on them. In any case, it is generally agreed that +an inductive conclusion is only probable, and that its probability +increases with the number of instances upon which it is founded. +According to the frequency theory, this belief is only justified if +the majority of inductive conclusions actually are true, and it +will be false, even on our existing data, that \emph{any} of them are even +probable, if the acknowledged possibility that a majority are +false is an actuality. Yet what possible reason can the frequency +theory offer, which does not beg the question, for supposing that +a majority \emph{are true}? And failing this, what ground have we +for believing the inductive process to be reasonable? Yet we +%% -----File: 119.png---Folio 108------- +invariably assume that with our existing knowledge it is logically +reasonable to attach some weight to the inductive method, even +if future experience shows that \emph{not one} of its conclusions is verified +in fact. The frequency theory, therefore, in its present form at +any rate, entirely fails to explain or justify the most important +source of the most usual arguments in the field of probable +inference. + +\Paragraph{15.} The failure of the frequency theory to explain or justify +arguments from induction or analogy suggests some remarks of a +more general kind. While it is undoubtedly the case that many +valuable judgments in probability are partly based on a knowledge +of statistical frequencies, and that many more can be held, +with some plausibility, to be indirectly derived from them, there +remains a great mass of probable argument which it would be +paradoxical to justify in the same manner. It is not sufficient, +therefore, even if it is possible, to show that the theory can be +developed in a self-consistent manner; it must also be shown +how the body of probable argument, upon which the greater +part of our generally accepted knowledge seems to rest, can +be explained in terms of it; for it is certain that much of +it does not appear to be derived from premisses of statistical +frequency. + +Take, for instance, the intricate network of arguments upon +which the conclusions of \textit{The Origin of Species} are founded: +how impossible it would be to transform them into a shape in +which they would be seen to rest upon statistical frequency! +Many individual arguments, of course, are explicitly founded +upon such considerations; but this only serves to differentiate +them more clearly from those which are not. Darwin's own +\index{Darwin}% +account of the nature of the argument may be quoted: ``The +belief in Natural Selection must at present be grounded entirely +on general considerations: (1)~on its being a \textit{vera causa}, from +the struggle for existence and the certain geological fact that +species do somehow change; (2)~from the analogy of change +under domestication by man's selection; (3)~and chiefly from +this view connecting under an intelligible point of view a host +of facts. When we descend to details~\ldots\ we cannot prove that +a single species has changed; nor can we prove that the supposed +changes are beneficial, which is the groundwork of the theory; +nor can we explain why some species have changed and others +%% -----File: 120.png---Folio 109------- +have not.''\footnote + {Letter to G.~Bentham, \textit{Life and Letters}, vol.~iii.\ p.~25.} +Not only in the main argument, but in many of the +subsidiary discussions,\footnote + {\Eg\ in the discussion on the relative effect of disuse and selection in + reducing unnecessary organs to a rudimentary condition.} +an elaborate combination of induction +and analogy is superimposed upon a narrow and limited knowledge +of statistical frequency. And this is equally the case in +almost all everyday arguments of any degree of complexity. +The class of judgments, which a theory of statistical frequency +can comprehend, is too narrow to justify its claim to present a +complete theory of probability. + +\Paragraph{16.} Before concluding this chapter, we should not overlook +the element of truth which the frequency theory embodies and +which provides its plausibility. In the first place, it gives a +true account, so long as it does not argue that probability and +frequency are \emph{identical}, of a large number of the most \emph{precise} +arguments in probability, and of those to which mathematical +treatment is easily applicable. It is this characteristic which +has recommended it to statisticians, and explains the large +measure of its acceptance in England at the present time; for +the popularity in this country of an opinion, which has, so far +as I know, no thorough supporters abroad, may reasonably be +attributed to the chance which has led most of the English +writers, who have paid much attention to probability in recent +years, to approach the subject from the statistical side. + +In the second place, the statement that the probability of an +event is measured by its actual frequency of occurrence `in the +long run' has a very close connection with a valid conclusion +which can be derived, \emph{in certain cases}, from Bernoulli's theorem. +\index{Bernoulli's Theorem}% +This theorem and its connection with the theory of frequency will +be the subject of \Chapref{XXIX}. + +\Paragraph{17.} The absence of a recent exposition of the logical basis of +the frequency theory by any of its adherents has been a great +disadvantage to me in criticising it. It is possible that some +of the opinions, which I have examined at length, are now held +by no one; nor am I absolutely certain, at the present stage of +the inquiry, that a partial rehabilitation of the theory may not +be possible. But I am sure that the objections which I have +raised cannot be met without a great complication of the theory, +and without robbing it of the simplicity which is its greatest +%% -----File: 121.png---Folio 110------- +preliminary recommendation. Until the theory has been given +new foundations, its logical basis is not so secure as to permit +controversial applications of it in practice. A good deal of +modern statistical work may be based, I think, upon an inconsistent +logical scheme, which, avowedly founded upon a theory +of frequency, introduces principles which this theory has no +power to justify. +%% -----File: 122.png---Folio 111------- + + +\Chapter{IX}{The Constructive Theory of Part I. Summarized} + +\Paragraph{1.} \First{That} part of our knowledge which we obtain directly, +supplies the premisses of that part which we obtain by argument. +From these premisses we seek to justify some degree of rational +belief about all sorts of conclusions. We do this by perceiving +certain logical relations between the premisses and the +conclusions. The kind of rational belief which we \emph{infer} in +this manner is termed \emph{probable} (or in the limit \emph{certain}), and the +logical relations, by the perception of which it is obtained, we +term \emph{relations of probability}. + +The probability of a conclusion~$a$ derived from premisses~$h$ +we write~$a/h$; and this symbol is of fundamental importance. + +\Paragraph{2.} The object of the Theory or Logic of Probability is to +systematise such processes of inference. In particular it aims +at elucidating rules by means of which the probabilities of different +arguments can be compared. It is of great practical importance +to determine which of two conclusions is on the evidence the +more probable. + +The most important of these rules is the Principle of +Indifference. According to this Principle we must rely upon +direct judgment for discriminating between the relevant and +the irrelevant parts of the evidence. We can only discard +those parts of the evidence which are irrelevant by \emph{seeing} that +they have no logical bearing on the conclusion. The irrelevant +evidence being thus discarded, the Principle lays it down that +if the evidence for either conclusion is the same (\ie~symmetrical), +then their probabilities also are the same (\ie~equal). + +If, on the other hand, there is additional evidence (\ie~in +addition to the symmetrical evidence) for one of the conclusions, +and this evidence is \emph{favourably relevant}, then that conclusion is +%% -----File: 123.png---Folio 112------- +the more probable. Certain rules have been given by which to +judge whether or not evidence is favourably relevant. And by +combinations of these judgments of preference with the judgments +of indifference warranted by the Principle of Indifference +more complicated comparisons are possible. + +\Paragraph{3.} There are, however, many cases in which these rules +furnish no means of comparison; and in which it is certain that +it is not actually within our power to make the comparison. It +has been argued that in these cases the probabilities are, in fact, +\emph{not comparable}. As in the example of similarity, where there +are different orders of increasing and diminishing similarity, but +where it is not possible to say of every pair of objects which of +them is on the whole the more like a third object, so there are +different orders of probability, and probabilities, which are not +of the same order, cannot be compared. + +\Paragraph{4.} It is sometimes of practical importance, when, for example, +we wish to evaluate a chance or to determine the amount of +our expectation, to say not only that one probability is greater +than another, but by how much it is greater. We wish, that is +to say, to have a numerical measure of degrees of probability. + +This is only occasionally possible. A rule can be given for +numerical measurement when the conclusion is one of a number +of equiprobable, exclusive, and exhaustive alternatives, but not +otherwise. + +\Paragraph{5.} In \Partref{II}. I proceed to a symbolic treatment of the +subject, and to the greater systematisation, by symbolic methods +on the basis of certain axioms, of the rules of probable argument. + +In Parts \Partref[]{III}., \Partref[]{IV}., and~\Partref[]{V}. the nature of certain very important +types of probable argument of a complex kind will be treated +in detail; in \Partref{III}. the methods of Induction and Analogy, +in \Partref{IV}. certain semi-philosophical problems, and in \Partref{V}. +the logical foundations of the methods of inference now commonly +known as \emph{statistical}. +%% -----File: 124.png---Folio 113------- + + +\Part{II}{Fundamental Theorems} +%% -----File: 125.png---Folio 114------- +%[Blank Page] +%% -----File: 126.png---Folio 115------- +\index{Russell, Bertrand}% + + +\Chapter{X}{Introductory} + +\Paragraph{1.} In \Partref{I}. we have been occupied with the epistemology of our +subject, that is to say, with what we know about the characteristics +and the justification of probable Knowledge. In \Partref{II}. I pass +to its Formal Logic. I am not certain of how much positive value +this Part will prove to the reader. My object in it is to show +that, starting from the philosophical ideas of \Partref{I}., we can +deduce by rigorous methods out of simple and precise definitions +the usually accepted results, such as the theorems of the addition +and multiplication of probabilities and of inverse probability. +The reader will readily perceive that this Part would never have +been written except under the influence of Mr.~Russell's \textit{Principia +Mathematica}. But I am sensible that it may suffer from the +over-elaboration and artificiality of this method without the +justification which its grandeur of scale affords to that great work. +In common, however, with other examples of formal method, +this attempt has had the negative advantage of compelling the +author to make his ideas precise and of discovering fallacies and +mistakes. It is a part of the spade-work which a conscientious +author has to undertake; though the process of doing it may +be of greater value to him than the results can be to the reader, +who is concerned to know, as a safeguard of the reliability of the +rest of the construction, that the thing can be done, rather than +to examine the architectural plans in detail. In the development +of my own thought, the following chapters have been of great +importance. For it was through trying to prove the fundamental +theorems of the subject on the hypothesis that Probability was +a \emph{relation} that I first worked my way into the subject; and the +rest of this Treatise has arisen out of attempts to solve the +successive questions to which the ambition to treat Probability +as a branch of Formal Logic first gave rise. +%% -----File: 127.png---Folio 116------- +\index{Johnson, W. E.}% +\index{Probability, and relevant knowledge!truth@{and truth}}% +\index{Spinoza|inote}% +\index{Truth and probability|inote}% + +A further occasion of diffidence and apology in introducing +this Part of my Treatise arises out of the extent of my debt to +Mr.~W.~E. Johnson. I worked out the first scheme in complete +independence of his work and ignorant of the fact that he had +thought, more profoundly than I had, along the same lines; I +have also given the exposition its final shape with my own hands. +But there was an intermediate stage, at which I submitted what +I had done for his criticism, and received the benefit not only of +criticism but of his own constructive exercises. The result is +that in its final form it is difficult to indicate the exact extent of +my indebtedness to him. When the following pages were first +in proof, there seemed little likelihood of the appearance of any +work on Probability from his own pen, and I do not now proceed +to publication with so good a conscience, when he is announcing +the approaching completion of a work on Logic which will include +``Problematic Inference.'' + +I propose to give here a brief summary of the five chapters +following, without attempting to be rigorous or precise. I shall +then be free to write technically in Chapters \Chapref[]{XI}.--\Chapref[]{XV}., inviting +the reader, who is not specially interested in the details of this +sort of technique, to pass them by. + +\Paragraph{2.} Probability is concerned with \emph{arguments}, that is to say, +with the ``bearing'' of one set of propositions upon another set. +If we are to deal formally with a generalised treatment of this +subject, we must be prepared to consider relations of probability +between \emph{any} pair of sets of propositions, and not only between +sets which are actually the subject of knowledge. But we soon +find that some limitation must be put on the character of sets of +propositions which we can consider as the hypothetical subject +of an argument, namely, that they must be \emph{possible} subjects of +knowledge. We cannot, that is to say, conveniently apply our +theorems to premisses which are self-contradictory and formally +inconsistent with themselves. + +For the purpose of this limitation we have to make a distinction +between a set of propositions which is merely false in fact +and a set which is formally inconsistent with itself.\footnote + {Spinoza had in mind, I think, the distinction between Truth and Probability + in his treatment of Necessity, Contingence, and Possibility. \textit{Res + enim omnes ex data Dei natura necessario sequutae sunt, et ex necessitate naturae + Dei determinatae sunt ad certo modo existendum et operandum} (\textit{Ethices} i.~33). + That is to say, everything is, without qualification, true or false. \textit{At res + aliqua nulla alia de causa contingens dicitur, nisi respectu defectus nostrae + cognitionis} (\textit{Ethices} i.~33, scholium). That is to say, Contingence, or, as I + term it, Probability, solely arises out of the limitations of our knowledge. + Contingence in this wide sense, which includes every proposition which, in + relation to our knowledge, is only probable (this term covering all intermediate + degrees of probability), may be further divided into Contingence in the strict + sense, which corresponds to an \textit{à~priori} or formal probability exceeding zero, + and Possibility; that is to say, into formal possibility and empirical possibility. + \textit{Res singulares voco contingentes, quatenus, dum ad earum solam essentiam + attendimus, nihil invenimus, quod earum existentiam necessario ponat, vel + quod ipsam necessario secludat. Easdem res singulares voco possibiles, quatenus + dum ad causas, ex quibus produci debent, attendimus, nescimus, an ipsae + determinatae sint ad easdem producendum} (\textit{Ethices} iv.\ Def~3,~4).} +This leads +%% -----File: 128.png---Folio 117------- +\index{Groups, of propositions}% +\index{Proposition, characterisation of!groups of}% +\index{Russell, Bertrand!inference@{and inference}}% +\index{Universe of reference}% +us to the conception of a \emph{group} of propositions, which is defined +as a set of propositions such that---(i.)~if a logical principle +belongs to it, all propositions which are instances of that logical +principle also belong to it; (ii.)~if the proposition~$p$ and the +proposition `not-$p$~or~$q$' both belong to it, then the proposition~$q$ +also belongs to it; (iii.)~if any proposition~$p$ belongs to it, then +the contradictory of~$p$ is \emph{excluded} from it. If the group defined +by one part of a set of propositions excludes a proposition which +belongs to a group defined by another part of the set, then the +set taken as a whole is \emph{inconsistent with itself} and is incapable of +forming the premiss of an argument. + +The conception of a group leads on to a precise definition of +one proposition \emph{requiring} another (which in the realm of assertion +corresponds to \emph{relevance} in the realm of probability), and of logical +priority as being an order of propositions arising out of their +relation to those special groups, or \emph{real groups}, which are in fact +the subject of knowledge. Logical priority has no absolute +signification, but is relative to a specific body of knowledge, or, +as it has been termed in the traditional logic, to the \emph{Universe of +Reference}. + +It also enables us to reach a definition of \emph{inference} distinct from +\emph{implication}, as defined by Mr.~Russell. This is a matter of very +great importance. Readers who are acquainted with the work +of Mr.~Russell and his followers will probably have noticed that +the contrast between his work and that of the traditional logic +is by no means wholly due to the greater precision and more +mathematical character of his technique. There is a difference +also in the design. His object is to discover what assumptions +are required in order that the formal propositions generally +accepted by mathematicians and logicians may be obtainable +%% -----File: 129.png---Folio 118------- +as the result of successive steps or substitutions of a few very +simple types, and to lay bare by this means any inconsistencies +which may exist in received results. But beyond the fact that +the conclusions to which he seeks to lead up are those of common +sense, and that the uniform type of argument, upon the validity +of which each step of his system depends, is of a specially obvious +kind, he is not concerned with analysing the methods of valid +reasoning which we actually employ. He concludes with +familiar results, but he reaches them from premisses, which have +never occurred to us before, and by an argument so elaborate that +our minds have difficulty in following it. As a method of setting +forth the system of formal truth, which shall possess beauty, +\DPchg{inter-dependence}{interdependence}, and completeness, his is vastly superior to +any which has preceded it. But it gives rise to questions about +the relation in which ordinary reasoning stands to this ordered +system, and, in particular, as to the precise connection between +the process of inference, in which the older logicians were principally +interested but which he ignores, and the relation of implication +on which his scheme depends. + +`$p$~implies~$q$' is, according to his definition, exactly equivalent +to the disjunction `$q$~is true or $p$~is false.' If $q$~is true, `$p$~implies~$q$' +holds for all values of~$p$; and similarly if $p$~is false, the implication +holds for all values of~$q$. This is not what we mean +when we say that $q$~can be inferred or follows from~$p$. For whatever +the exact meaning of inference may be, it certainly does not +hold between \emph{all} pairs of true propositions, and is not of such a +character that \emph{every} proposition follows from a false one. It is +not true that `A male now rules over England' follows or can be +inferred from `A male now rules over France'; or `A female now +rules over England' from `A female now rules over France'; +whereas, on Mr.~Russell's definition, the corresponding implications +hold simply in virtue of the facts that `A male now rules +over England' is true and `A female now rules over France' +is false. + +The distinction between the Relatival Logic of Inference and +Probability, and Mr.~Russell's Universal Logic of Implication, +seems to be that the former is concerned with the relations of +propositions in general to a particular limited \emph{group}. Inference +and Probability depend for their importance upon the fact that +in actual reasoning the limitation of our knowledge presents us +%% -----File: 130.png---Folio 119------- +with a particular set of propositions, to which we must relate any +other proposition about which we seek knowledge. The course +of an argument and the results of reasoning depend, not simply +on what is true, but on the particular body of knowledge from +which we have set out. Ultimately, indeed, Mr.~Russell cannot +avoid concerning himself with groups. For his aim is to discover +the smallest set of propositions which specify our formal knowledge, +and then to show that they do in fact specify it. In this +enterprise, being human, he must confine himself to that part of +formal truth which we know, and the question, how far his +axioms comprehend \emph{all} formal truth, must remain insoluble. +But his object, nevertheless, is to establish a train of implications +between formal truths; and the character and the justification of +rational argument as such is not his subject. + +\Paragraph{3.} Passing on from these preliminary reflections, our first +task is to establish the axioms and definitions which are to make +operative our symbolical processes. These processes are almost +entirely a development of the idea of representing a probability +by the symbol~$a/h$, where $h$~is the premiss of an argument and $a$~its +conclusion. It might have been a notation more in accordance +with our fundamental ideas, to have employed the symbol~$a/h$ +to designate the \emph{argument} from $h$~to~$a$, and to have represented +the probability of the argument, or rather the degree of rational +belief about~$a$ which the argument authorises, by the symbol +$P(a/h)$. This would correspond to the symbol $V(a/h)$ which has +been employed in \Chapref{VI}. for the evidential value of the +argument as distinct from its probability. But in a section +where we are only concerned with probabilities, the use of $P(a/h)$ +would have been unnecessarily cumbrous, and it is, therefore, +convenient to drop the prefix~$P$ and to denote the probability +itself by~$a/h$. + +The discovery of a convenient symbol, like that of an essential +word, has often proved of more than verbal importance. Clear +thinking on the subject of Probability is not possible without a +symbol which takes an explicit account of the premiss of the +argument as well as of its conclusion; and endless confusion has +arisen through discussions about the probability of a conclusion +without reference to the argument as a whole. I claim, therefore, +the introduction of the symbol~$a/h$ as an essential step towards +any progress in the subject. +%% -----File: 131.png---Folio 120------- +\index{Addition, of probabilities!definition of}% +\index{Equivalence, definition of}% +\index{Groups, of propositions!definition of}% +\index{Inconsistency, definition of}% +\index{Independence, for knowledge!definition of}% + +\Paragraph{4.} Inasmuch as relations of Probability cannot be assumed +to possess the properties of numbers, the terms \emph{addition} and +\emph{multiplication} of probabilities have to be given appropriate +\index{Multiplication!definition of}% +meanings by definition. It is convenient to employ these +familiar expressions, rather than to invent new ones, because the +properties which arise out of our definitions of addition and +\index{Definitions!summary of}% +multiplication in Probability are analogous to those of addition +and multiplication in Arithmetic. But the process of establishing +these properties is a little complicated and occupies the greater +part of \Chapref{XII}. + +The most important of the definitions of \Chapref{XII}. are the +following (the numbers referring to the numbers of \Chapref{XII}.): + +II\@. The Definition of \emph{Certainty}: $a/h=1$. +\index{Certainty!definition of}% + +III\@. The Definition of \emph{Impossibility}: $a/h=0$. +\index{Impossibility!definition of}% + +VI\@. The Definition of \emph{Inconsistency}: $ah$~is inconsistent if +$a/h=0$. + +VII\@. The Definition of a \emph{Group}: the class of propositions~$a$ +such that $a/h=1$ is the group~$h$. + +VIII\@. The Definition of \emph{Equivalence}: if $b/ah=1$ and $a/bh=1$ +$(a\equiv b)/h=1$. + +IX\@. The Definition of \emph{Addition}: $ab/h + ab/h\footnotemark = a/h$. +\footnotetext{$\bar{b}$~stands for the contradictory of~$b$.} + +X\@. The Definition of \emph{Multiplication}: $ab/h=a/bh· b/h=b/ah· a/h$. +The symbolical development of the subject largely +proceeds out of these definitions of Addition and Multiplication. +It is to be observed that they give a meaning, not to the addition +and multiplication of \emph{any} pairs of probabilities, but only to pairs +which satisfy a certain form. The definition of Multiplication +may be read: `the probability of both $a$~and~$b$ given~$h$ is equal +to the probability of~$a$ given~$bh$, multiplied by the probability of~$b$ +given~$h$.' + +XI\@. The Definition of \emph{Independence}: if $a_1/a_2h = a_1/h$ and +$a_2/a_1h = a_2/h$, $a_1/h$~and~$a_2/h$ are independent. + +XII\@. The Definition of \emph{Irrelevance}: if $a_1/a_2h = a_1/h$, $a_2$ is +\index{Irrelevance!definition of}% +irrelevant to~$a_1/h$. + +\Paragraph{5.} In \Chapref{XIII}. these definitions, supplemented by a few +axioms, are employed to demonstrate the fundamental theorems +of \emph{Certain} or \emph{Necessary Inference}. The interest of this chiefly +\index{Inference!necessary}% +lies in the fact that these theorems include those which the +%% -----File: 132.png---Folio 121------- +\index{Addition, of probabilities!Theorem of}% +\index{Independence, for knowledge!Theorem of}% +\index{Johnson, W. E.!cumulative@{and cumulative formula}}% +traditional Logic has termed the \emph{Laws of Thought}, as for example +the Law of Contradiction and the Law of Excluded Middle. +These are here exhibited as a part of the generalised theory +of Inference or Rational Argument, which includes probable +Inference as well as certain Inference. The object of this chapter +is to show that the ordinarily accepted rules of Inference can in +fact be deduced from the definitions and axioms of \Chapref{XII}. + +\Paragraph{6.} In \Chapref{XIV}. I proceed to the fundamental Theorems +of Probable Inference, of which the following are the most +interesting: + +\textit{Addition Theorem}: $(a + b)/h = a/h + b/h - ab/h$, which reduces +to $(a + b)/h = a/h + b/h$, where $a$~and~$b$ are mutually exclusive; +and, if $p_1p_2 \ldots p_n$ form, relative to $h$ a set of exclusive and +exhaustive alternatives, $a/h = \Sum_1^n p_r a/h$. + +\textit{Theorem of Irrelevance}: If $a/h_1h_2 = a/h_1$ then $a/h_1\bar{h}_2 = a/h_1$; +\index{Irrelevance!Theorem of}% +\ie\ if a proposition is irrelevant, its contradictory also is irrelevant. + +\textit{Theorem of Independence}: If $a_2/a_1h = a_2/h$, $a_1/a_2h = a_1/h$; \ie\ +if $a_1$~is irrelevant to~$a_2/h$, it follows that $a_2$~is irrelevant to~$a_1/h$ +and that $a_1/h$~and~$a_2/h$ are independent. + +\textit{Multiplication Theorem}: If $a_1/h$ and $a_2/h$ are independent, +\index{Multiplication!theorems of}% +$a_1a_2/h = a_1/h· a_2/h$. + +\textit{Theorem of Inverse Probability}: $\dfrac{a_1/bh}{a_2/bh} = \dfrac{b/a_1h}{b/a_2h}· \dfrac{a_1/h}{a_2/h}$. Further, +\index{Inverse Probability!Theorem of}% +if $a_1/h = p_1$, $a_2/h = p_2$, $b/a_1h = q_1$, $b/a_2h = q_2$, and $a_1/bh + a_2/bh = 1$, +then $a_1/bh = \dfrac{p_1q_1}{p_1q_1 + p_2q_2}$; and if $a_1/h = a_2/h$, $a_1/bh = \dfrac{q_1}{q_1 + q_2}$, which +is equivalent to the statement that the probability of~$a_1$ when +we know $b$~is equal to $\dfrac{q_1}{q_1 + q_2}$, where $q_1$~is the probability of~$b$ when +we know~$a_1$ and $q_2$~its probability when we know~$a_2$. This +theorem enunciated with varying degrees of inaccuracy appears +in all Treatises on Probability, but is not generally proved. + +\Chapref{XIV}. concludes with some elaborate theorems on the +combination of premisses based on a technical symbolic device, +known as the \emph{Cumulative Formula}, which is the work of Mr.~W.~E. +\index{Cumulative Formula!Johnson and}% +Johnson. + +\Paragraph{7.} In \Chapref{XV}. I bring the non-numerical theory of +probability developed in the preceding chapters into connection +with the usual numerical conception of it, and demonstrate how +%% -----File: 133.png---Folio 122------- +and in what class of cases a meaning can be given to a numerical +measure of a relation of probability. This leads on to what +may be termed numerical approximation, that is to say, the +relating of probabilities, which are not themselves numerical, +to probabilities, which are numerical, by means of \emph{greater} and \emph{less}, +by which in some cases numerical limits may be ascribed to +probabilities which are not capable of numerical measures. +%% -----File: 134.png---Folio 123------- +\index{Variables in Probability}% + + +\Chapter{XI}{The Theory of Groups, with special reference to +Logical Consistence, Inference, and Logical Priority} + +\Paragraph{1.} The Theory of Probability deals with the relation between +two sets of propositions, such that, if the first set is known to be +true, the second can be known with the appropriate degree of +probability by argument from the first.\footnote + {Or more strictly, ``perception of which, together with knowledge of the + first set, justifies an appropriate degree of rational belief about the second.''} +The relation, however, +also exists when the first set is not known to be true and is hypothetical. + +In a symbolical treatment of the subject it is important +that we should be free to consider \emph{hypothetical} premisses, and +to take account of relations of probability as existing between +\emph{any} pair of sets of propositions, whether or not the premiss is +actually part of knowledge. But in acting thus we must be +careful to avoid two possible sources of error. + +\Paragraph{2.} The first is that which is liable to arise wherever \emph{variables} +are concerned. This was mentioned in passing in §\;18~of \Chapref{IV}\@. +We must remember that whenever we substitute for a +variable some particular value of it, this may so affect the relevant +evidence as to modify the probability. This danger is always +present except where, as in the first half of \Chapref{XIII}., the +conclusions respecting the variable are \emph{certain}. + +\Paragraph{3.} The second difficulty is of a different character. Our +premisses may be hypothetical and not actually the subject of +knowledge. But must they not be \emph{possible} subjects of knowledge? +How are we to deal with hypothetical premisses which +are self-contradictory or formally inconsistent with themselves, +and which cannot be the subject of rational belief of any degree? +%% -----File: 135.png---Folio 124------- +\index{Consistence and group theory}% +\index{Groups, of propositions}% +\index{Johnson, W. E.!groups@{and groups}}% +\index{Proposition, characterisation of!groups of}% +\index{Russell, Bertrand|inote}% +\index{Russell, Bertrand!implication@{and implication}}% + +Whether or not a relation of probability can be held to exist +between a conclusion and a self-inconsistent premiss, it will be +convenient to exclude such relations from our scheme, so as to +avoid having to provide for anomalies which can have no interest +in an account of the actual processes of valid reasoning. Where +a premiss is inconsistent with itself it cannot be required. + +\Paragraph{4.} Let us term the collection of propositions, which are +logically involved in the premisses in the sense that they follow +from them, or, in other words, stand to them in the relation of +certainty,\footnote + {`$a$~can be inferred from~$b$,' `$a$~follows from~$b$,' `$a$~is certain in relation to~$b$,' + `$a$~is logically involved in~$b$,' I regard as equivalent expressions, the precise + meaning of which will be defined in succeeding paragraphs. `$a$~is implied by~$b$,' + I use in a different sense, namely, in Mr.~Russell's sense, as the equivalent of + `$b$~or not-$a$.'} +the \emph{group} specified by the premisses. That is to say, +we define a group as \emph{containing} all the propositions logically +involved in any of the premisses or in any conjunction of them; +and as \emph{excluding} all the propositions the contradictories of which +are logically involved in any of the premisses or in any conjunction +of them.\footnote + {For the conception of a \emph{group}, and for many other notions and definitions + in the course of this chapter---those, for example, of a real group and of + logical priority---I am largely indebted to Mr.~W.~E. Johnson. The origination + of the theory of groups is due to him.} +To say, therefore, that a proposition follows +from a premiss, is the same thing as to say that it belongs to the +group which the premiss specifies. + +The idea of a `group' will then enable us to define `logical +consistency.' If any part of the premisses specifies a group +containing a proposition, the contradictory of which is contained +in a group specified by some other part, the premisses are \emph{logically +inconsistent}; otherwise they are logically consistent. In short, +premisses are inconsistent if a proposition `follows from' one +part of them, and its contradictory from another part. + +\Paragraph{5.} We have still, however, to make precise what we mean in +this definition by one proposition \emph{following from} or being \emph{logically +involved in} the truth of another. We seem to intend by these +expressions some kind of transition by means of a \emph{logical principle.} +A logical principle cannot be better defined, I think, than in terms +of what in Mr.~Russell's \emph{Logic of Implication} is termed a formal +\index{Implication}% +implication. `$p$~implies~$q$' is a \emph{formal implication} if `not-$p$~or~$q$' +is formally true; and a proposition is formally true, if it is a value +of a propositional function, in which all the constituents other +%% -----File: 136.png---Folio 125------- +\index{Groups, of propositions!definition of}% +than the arguments are logical constants, and of which all the +values are true. + +We might define a \emph{group} in such a way that all logical principles +belonged to every group. In this case \emph{all} formally true propositions +would belong to every group. This definition is logically +precise and would lead to a coherent theory. But it possesses +the defect of not closely corresponding to the methods of reasoning +we actually employ, because all logical principles are not in fact +known to us. And even in the case of those which we do know, +there seems to be a logical order (to which on the above definition +we cannot give a sense) amongst propositions, which are about +logical constants and are formally true, just as there is amongst +propositions which are not formally true. Thus, if we were to +assume the premisses in every argument to include all formally +true propositions, the sphere of probable argument would be +limited to what (in contradistinction to formally true propositions) +we may term empirical propositions. + +\Paragraph{6.} For this reason, therefore, I prefer a narrower definition---which +shall correspond more exactly to what we seem to mean +when we say that one proposition follows from another. Let us +define a \emph{group} of propositions as a set of propositions such that: + +(i.)~if the proposition `$p$~is formally true' belongs to the group, +all propositions which are instances of the same formal propositional +function also belong to it; + +(ii.)~if the proposition~$p$ and the proposition `$p$~implies~$q$' +both belong to it, then the proposition~$q$ also belongs to it; + +(iii.)~if any proposition~$p$ belongs to it, then the contradictory +of~$p$ is excluded from it. + +According to this definition all processes of certain inference +are wholly composed of steps each of which is of one of two simple +types (and if we like we might perhaps regard the first as comprehending +the other). I do not feel certain that these conditions +may not be narrower than what we mean when we say that one +proposition follows from another. But it is not necessary for the +purpose of defining a group, to dogmatise as to whether any other +additional methods of inference are, or are not, open to us. If +we define a group as the propositions logically involved in the +premisses in the above sense, and prescribe that the premisses of +an argument in probability must specify a group not \emph{less} extensive +than this, we are placing the \emph{minimum} amount of restriction upon +%% -----File: 137.png---Folio 126------- +%[** TN: "sub-group" hyphenated in original index; regularized.] +\index{Proposition, characterisation of!subgroups of}% +\index{Russell, Bertrand}% +\index{Subgroups of propositions}% +the form of our premisses. If, sometimes or as a rule, our +premisses in fact include some more powerful principle of argument, +so much the better. + +In the formal rules of probability which follow, it will be +postulated that the set of propositions, which form the premiss +of any argument, must not be inconsistent. The premiss must, +that is to say, specify a `group' in the sense that no part of the +premiss must exclude a proposition which follows from another +part. But for this purpose we do not need to dogmatise as to +what the \emph{criterion} is of inference or certainty. + +\Paragraph{7.} It will be convenient at this point to define a term which +expresses the relation converse to that which exists between a +set of propositions and the group which they specify. The propositions +$p_1, p_2 \ldots p_n$ are said to be \emph{fundamental} to the group~$h$ +if (i.)~they themselves belong to the group (which involves their +being consistent with one another); (ii.)~if between them they +completely specify the group; and (iii.)~if none of them belong +to the group specified by the rest (for if $p_r$~belongs to the group +specified by the rest, this term is redundant). + +When the fundamental set is \emph{uniquely} determined, a group~$h'$ +is a \DPchg{sub-group}{subgroup} to the group~$h$, if the set fundamental to~$h'$ is +included in the set fundamental to~$h$. + +Logically there can be more than one distinct set of propositions +fundamental to a given group; and some extra-logical test +must be applied before the fundamental set is determined uniquely. +On the other hand, a group is completely determined when the +constituent propositions of the fundamental set are given. +Further, any consistent set of propositions evidently specifies +some group, although such a set may contain propositions +\emph{additional} to those which are fundamental to the group it specifies. +It is clear also that only one group can be specified by a given +set of consistent propositions. The members of a group are, +we may say, \emph{rationally bound up} with the set of propositions +fundamental to it. + +\Paragraph{8.} If Mr.~Bertrand Russell is right, the whole of pure +mathematics and of formal logic follows, in the sense defined +above, from a small number of primitive propositions. The +group, therefore, which is specified by these primitive propositions, +includes the most remote deductions not only amongst +those known to mathematicians, but amongst those which time +%% -----File: 138.png---Folio 127------- +and skill have not yet served to solve. If we define certainty +\index{Certainty}% +in a logical and not a psychological sense, it seems necessary, +if our premisses include the essential axioms, to regard as +certain all propositions which follow from these, whether or +not they are known to us. Yet it seems as if there must +be some logical sense in which unproved mathematical +theorems---some of those, for instance, which deal with the +theory of numbers---can be likely or unlikely, and in which a +proposition of this kind, which has been suggested to us by +analogy or supported by induction, can possess an intermediate +degree of probability. + +There can be no doubt, I think, that the logical relation of +certainty does exist in these cases in which lack of skill or insight +prevents our apprehending it, in spite of the fact that sufficient +premisses, including sufficient logical principles, are known to us. +In these cases we must say, what we are not permitted to say +when the indeterminacy arises from lack of premisses, that the +probability is \emph{unknown}. There is still a sense, however, in which +in such a case the knowledge we actually possess can be, in a +logical sense, only probable. While the relation of certainty +exists between the fundamental axioms and every mathematical +hypothesis (or its contradictory), there are other data in relation +to which these hypotheses possess intermediate degrees of +probability. If we are unable through lack of skill to discover +the relation of probability which an hypothesis does in fact bear +towards one set of data, this set is practically useless, and we must +fix our attention on some other set in relation to which the probability +is not unknown. When Newton held that the binomial +theorem possessed for empirical reasons sufficient probability +to warrant a further investigation of it, it was not in relation to +the axioms of mathematics, whether he knew them or not, that +the probability existed, but in relation to his empirical evidence +combined, perhaps, with \emph{some} of the axioms. There is, in short, +an exception to the rule that we must always consider the probability +of any conclusion in relation to the whole of the data in +our possession. When the relation of the conclusion to the whole +of our evidence cannot be known, then we must be guided by +its relation to some part of the evidence. When, therefore, in +later chapters I speak of a formal proposition as possessing an +intermediate degree of probability, this will always be in relation +%% -----File: 139.png---Folio 128------- +to evidence from which the proposition does not logically follow +in the sense defined in~§\;6. + +\Paragraph{9.} It follows from the preceding definitions that a proposition +is \emph{certain} in relation to a given premiss, or, in other words, \emph{follows +from} this premiss if it is included in the group which that premiss +specifies. It is \emph{impossible} if it is excluded from the group---if, +that is to say, its contradictory follows from the premiss. We +often say, somewhat loosely, that two propositions are contradictory +to one another, when they are inconsistent in the sense +that, relative to our evidence, they cannot belong to the same +group. On the other hand, a proposition, which is not itself +included in the group specified by the premiss and whose contradictory +is not included either, has in relation to the premiss an +intermediate degree of probability. + +If $a$~follows from~$h$ and is, therefore, included in the group +specified by~$h$, this is denoted by $a/h = 1$. The relation of certainty, +\index{Certainty}% +that is to say, is denoted by the symbol of unity. The reason +why this notation is useful and has been adopted by common +consent will appear when the meaning of the \emph{product} of a pair +of relations of probability has been explained. If we represent +the relation of certainty by~$\gamma$ and any other probability by~$\DPtypo{a}{\alpha}$, +the product $\DPtypo{a}{\alpha}·\gamma = \DPtypo{a}{\alpha}$. Similarly, if $a$~is excluded from the +group specified by~$h$ and is impossible in relation to it, this is +denoted by $a/h = 0$. The use of the symbol zero to denote +impossibility arises out of the fact that, if $\omega$~denotes impossibility +and $\DPtypo{a}{\alpha}$~any other relation of probability, then, in the senses of +multiplication and addition to be defined later, the product +$\DPtypo{a}{\alpha}·\omega = \omega$, and the sum $\DPtypo{a}{\alpha} + \omega = \DPtypo{a}{\alpha}$. Lastly, if $a$~is not included +in the group specified by~$h$, this is written $a/h\neq 1$ or $a/h < 1$; +and if it is not excluded, this is written $a/h\neq 0$ or $a/h > 0$. + +\Paragraph{10.} The theory of groups now enables us to give an account, +with the aid of some further conceptions, of logical priority and +of the true nature of inference. The groups, to which we refer +the arguments by which we actually reason, are not arbitrarily +chosen. They are determined by those propositions of which +we have direct knowledge. Our group of reference is specified +by those direct judgments in which we personally \emph{rationally +certify} the truth of some propositions and the falsity of others. +So long as it is undetermined, or not determined uniquely, +which propositions are fundamental, it is not possible to discover +%% -----File: 140.png---Folio 129------- +%[** TN: "sub-group" hyphenated in original index; regularized.] +\index{Groups, of propositions!real and hypothetical}% +\index{Proposition, characterisation of!subgroups of}% +\index{Subgroups of propositions}% +a necessary order amongst propositions or to show in what way +a true proposition `follows from' one true premiss rather than +another. But when we have determined what propositions are +fundamental, by selecting those which we know directly to be true, +or in some other way, then a meaning can be attached to priority +and to the distinction between inference and implication. When +\index{Inference}% +the propositions which we know directly are given, there is a +logical order amongst those other propositions which we know +indirectly and by argument. + +\Paragraph{11.} It will be useful to distinguish between those groups which +are hypothetical and those of which the fundamental set is known +to be true. We will term the former \emph{hypothetical groups} and the +latter \emph{real groups}. To the real group, which contains all the +propositions which are known to be true, we may assign the old +logical term \emph{Universe of Reference}. While knowledge is here +\index{Universe of reference}% +taken as the criterion of a real group, what follows will be equally +valid whatever criterion is taken, so long as the fundamental set +is in some manner or other determined uniquely. + +If it is impossible for us to know a proposition~$p$ except by +inference from a knowledge of~$q$, so that we cannot know $p$~to be +true unless we already know~$q$, this may be expressed by saying +that `$p$~requires~$q$'. More precisely \emph{requirement} is defined as +\index{Requirement}% +follows: + +\textit{$p$~does not require~$q$} if there is some real group to which $p$~belongs +and $q$~does not belong, \ie\ if there is a real group~$h$ +such that $p/h = 1$, $q/h \neq 1$; hence + +\textit{$p$~requires~$q$} if there is \emph{no} real group to which $p$~belongs +and $q$~does not belong. + +\textit{$p$~does not require~$q$ within the group~$h$}, if the group~$h$, to which +$p$~belongs, contains a subgroup\footnotemark~$h'$ +\footnotetext{Subgroups have only been defined, it must be noticed (see §\;7 above) when + the fundamental set of the group has been, in some way, uniquely determined.} +to which $p$~belongs and $q$~does +not belong; \ie\ if there is a group~$h'$ such that $h'/h = 1$, $p/h' = 1$, +$q/h' \neq 1$. This reduces to the proposition next but one above +if $h$~is the Universe of Reference. In §\;13 these definitions +will be generalised to cover intermediate degrees of probability. + +\Paragraph{12.} Inference and logical priority can be defined in terms of +\index{Logical priority}% +requirement and real groups. It is convenient to distinguish +two types of inference corresponding to hypothetical and real +%% -----File: 141.png---Folio 130------- +groups---\ie\ to cases where the argument is only hypothetical, +and cases where the conclusion can be asserted: + +\textit{Hypothetical Inference.}---`If $p$,~$q$,' which may also be read +\index{Inference!hypothetical and assertoric}% +`$q$~is hypothetically inferrible from~$p$,' means that there is a +real group~$h$ such that $q/ph = 1$, and $q/h \neq 1$. In order that this +may be the case, $ph$~must specify a group; \ie\ $p/h \neq 0$, or in +other words $p$~must not be excluded from~$h$. Hypothetical +inference is also equivalent to: `$p$~implies~$q$' and `$p$~implies~$q$' +does not require~`$q$'. In other words, $q$~is hypothetically +inferrible from~$p$, if we know that $q$~is true or $p$~is false and if +we can know this without first knowing either that $q$~is true or +that $p$~is false. + +\textit{Assertoric Inference.}---`$p \therefore q$,' which may be read `$p$~therefore~$q$' +or `$q$~may be asserted by inference from~$p$,' means that `if~$p$,~$q$' +is true, and in addition `$p$'~belongs to a real group; \ie\ there +are proper groups $h$~and~$h'$ such that $p/h = 1$, $q/ph' = 1$, $q/h' \neq 1$, +and $p/h' \neq 0$. + +\textit{$p$~is prior to~$q$} when $p$~does not require~$q$, and $q$~requires~$p$, +when, that is to say, we can know~$p$ without knowing~$q$, but +not~$q$ unless we first know~$p$. + +\textit{$p$~is prior to~$q$ within the group~$h$} when $p$~does not require~$q$ +within the group, and $q$~does require~$p$ within the group. + +It follows from this and from the preceding definitions that, +if a proposition is fundamental in the sense that we can only +know it directly, there is no proposition prior to it; and, more +generally, that, if a proposition is fundamental to a given +group, there is no proposition prior to it within the group. + +\Paragraph{13.} We can now apply the conception of requirement to +intermediate degrees of probability. The notation adopted is, +it will be remembered, as follows: + +$p/h = \alpha$ means that the proposition~$p$ has the probable relation +of degree~$\alpha$ to the proposition~$h$; while it is postulated that $h$~is +self-consistent and therefore specifies a group. + +$p/h = 1$ means that $p$~follows from~$h$ and is, therefore, included +in the group specified by~$h$. + +$p/h = 0$ means that $p$~is excluded from the group specified by~$h$. + +If $h$~specifies the Universe of Reference, \ie\ if its group comprehends +\index{Universe of reference}% +the whole of our knowledge, $p/h$~is called \emph{the absolute +probability of~$p$}, or (for short) \emph{the probability of~$p$}; and if $p/h = 1$ +and $h$~specifies any real group, $p$~is said to be \emph{absolutely certain} +%% -----File: 142.png---Folio 131------- +or (for short) \emph{certain}. Thus $p$~is `certain' if it is a member of a +real group, and a `certain' proposition is one which we know +to be true. Similarly if $p/h = 0$ under the same conditions, $p$~is +\emph{absolutely impossible}, or (for short) \emph{impossible}. Thus an `impossible' +proposition is one which we know to be false. + +The definition of requirement, when it is generalised so as to +take account of intermediate degrees of probability, becomes, it +will be seen, equivalent to that of relevance: + +\emph{The probability of~$p$ does not require~$q$ within the group~$h$}, if +there is a subgroup~$h'$ such that, for every subgroup~$h''$ which +includes~$h'$ and is included in~$h$ (\ie\ $h'/h'' = 1$, $h''/h = 1$), $p/h'' = p/h'$, +and $q/h' \neq q/h$. + +When $p$~is included in the group~$h$, this definition reduces to +the definition of requirement given in~§\;11. + +\Paragraph{14.} The importance of the theory of groups arises as soon as +we admit that there are \emph{some} propositions which we take for +granted without argument, and that all arguments, whether +demonstrative or probable, consist in the relating of other conclusions +to these as premisses. + +The particular propositions, which are in fact fundamental +to the Universe of Reference, vary from time to time and from +person to person. Our theory must also be applicable to hypothetical +Universes. Although a particular Universe of Reference +may be defined by considerations which are partly psychological, +when once the Universe is given, our theory of the relation in +which other propositions stand towards it is entirely logical. + +The formal development of the theory of argument from +imposed and limited premisses, which is attempted in the following +chapters, resembles in its general method other parts of formal +logic. We seek to establish implications between our primitive +axioms and the derivative propositions, without specific reference +to what particular propositions are fundamental in our actual +Universe of Reference. + +It will be seen more clearly in the following chapters that the +laws of inference are the laws of probability, and that the former +is a particular case of the latter. The relation of a proposition to +a group depends upon the relevance to it of the group, and a +group is relevant in so far as it contains a necessary or sufficient +condition of the proposition, or a necessary or sufficient condition +of a necessary or sufficient condition, and so on; a condition +%% -----File: 143.png---Folio 132------- +being necessary if every hypothetical group, which includes the +proposition together with the Universe of Reference, includes +the condition, and sufficient if every hypothetical group, which +includes the condition together with the Universe of Reference, +includes the proposition. +%% -----File: 144.png---Folio 133------- + + +\Chapter{XII}{The Definitions and Axioms of Inference and Probability} + +\Paragraph{1.} \First{It} is not necessary for the validity of what follows to decide +in what manner the set of propositions is determined, which is +fundamental to our Universe of Reference, or to make definite +assumptions as to what propositions are included in the group +which is specified by the \textit{data}. When we are investigating an +empirical problem, it will be natural to include the whole of +our logical apparatus, the whole body, that is to say, of +formal truths which are known to us, together with that part +of our empirical knowledge which is relevant. But in the +following formal developments, which are designed to display +the logical rules of probability, we need only assume that our data +always include those logical rules, of which the steps of our +proofs are instances, together with the axioms relating to probability +which we shall enunciate. + +The object of this and the chapters immediately following is +to show that all the usually assumed conclusions in the fundamental +logic of inference and probability follow rigorously from +a few axioms, in accordance with the fundamental conceptions +expounded in \Partref{I}\@. This body of axioms and theorems +corresponds, I think, to what logicians have termed the \emph{Laws of +Thought}, when they have meant by this something narrower than +the whole system of formal truth. But it goes beyond what has +been usual, in dealing at the same time with the laws of probable, +as well as of necessary, inference. + +\Paragraph{2.} This and the following chapters of \Partref{II}. are largely +independent of many of the more controversial issues raised in +the preceding chapters. They do not prejudge the question as +%% -----File: 145.png---Folio 134------- +\index{Definitions|ifoll}% +\index{Equivalence, definition of}% +\index{Probability relation}% +\index{Proposition, characterisation of!disjunction and conjunction of}% +to whether or not all probabilities are theoretically measurable; +and they are not dependent on our theories as to the part played +by direct judgment in establishing relations of probability or +inference between particular propositions. Their premisses are +all \emph{hypothetical}. \emph{Given} the existence of certain relations of +probability, others are inferred. Of the conclusions of \Chapref{III}., +of the criteria of equiprobability and of inequality discussed +in Chapters \Chapref[]{IV}.~and~\Chapref[]{V}., and of the criteria of inference discussed +in §§\;5,~6 of \Chapref{XI}., they are, I think, wholly independent. +They deal with a different part of the subject, not so closely +connected with epistemology. + +\Paragraph{3.} In this chapter I confine myself to Definitions and Axioms. + +Propositions will be denoted by small letters, and relations +by capital letters. In accordance with common usage, a disjunctive +combination of propositions is represented by the sign +of addition, and a conjunctive combination by simple juxtaposition +(or, where it is necessary for clearness, by the sign of +multiplication): \eg\ `$a$~or $b$ or~$c$' is written `$a + b + c$,' and `$a$~and +$b$ and~$c$' is written~`$abc$.' `$a + b$'~is not so interpreted as to +exclude `$a$~and~$b$.' The contradictory of~$a$ is written~$\bar{a}$. + +\Paragraph{4.} \textit{Preliminary Definitions}: + +I\@. If there exists a relation of probability~$P$ between the +proposition~$a$ and the premiss~$h$ +\begin{DPgather*} +a/h = P\DPtypo{}{.} +\rintertext{Def.} +\end{DPgather*} + +II\@. If $P$~is the relation of certainty\index{Certainty!relation of}\footnote + {These symbols were first employed by Leibnitz. See p.~155 below.} +\begin{DPgather*} +P = 1\DPtypo{}{.} +\rintertext{Def.} +\end{DPgather*} + +III\@. If $P$~is the relation of impossibility\footnotemark[1] +\index{Impossibility!relation of}% +% {These symbols were first employed by Leibnitz. See p.~155 below.} +\begin{DPgather*} +P = 0\DPtypo{}{.} +\rintertext{Def.} +\end{DPgather*} + +IV\@. If $P$~is a relation of probability, but not the relation of +certainty +\begin{DPgather*} +P < 1. +\rintertext{Def.} +\end{DPgather*} + +V\@. If $P$~is a relation of probability, but not the relation of +impossibility +\begin{DPgather*} +P > 0. +\rintertext{Def.} +\end{DPgather*} + +VI\@. If $a/h = 0$, the conjunction~$ah$ is \emph{inconsistent}. \hfill Def. + +VII\@. The class of propositions~$a$ such that $a/h = 1$ \emph{is the +group specified} by~$h$ or (for short) \emph{the group~$h$}. \hfill Def. + +VIII\@. If $b/ah = 1$ and $a/bh = 1$, $(a \equiv b)/h = 1$. \hfill Def. + +This may be regarded as the definition of \emph{Equivalence}. Thus +we see that equivalence is relative to a premiss~$h$. $a$~is equivalent +to~$b$, given~$h$, if $b$~follows from~$ah$, and $a$~from~$bh$. +%% -----File: 146.png---Folio 135------- +\index{Addition, of probabilities}% +\index{Axioms|ifoll}% +\index{Equivalence, definition of!axiom of}% + +\Paragraph{5.} \textit{Preliminary Axioms}: + +We shall assume that there is included in every premiss with +which we are concerned the formal implications which allow us +to assert the following axioms: + +(i.)\ Provided that $a$~and~$h$ are propositions or conjunctions +of propositions or disjunctions of propositions, and that $h$~is not +an inconsistent conjunction, there exists one and only one relation +of probability~$P$ between $a$~as conclusion and $h$~as premiss. +Thus any conclusion~$a$ bears to any consistent premiss~$h$ one and +only one relation of probability. + +(ii.)\ If $(a \equiv b)/h = 1$, and $x$~is a proposition, $x/ah=x/bh$. This +is the Axiom of Equivalence. +\begin{DPalign*} +\lintertext{\indent(iii.)} +(\,\overline{a+b} \equiv \bar{a} \bar{b}\,)/h &= 1 \\ + (aa \equiv a)/h &= 1 \\ + (\,\overline{\bar{a}} \equiv a\,)/h &= 1 \\ + (ab + \bar{a}b \equiv b)/h &= 1. \\ + \text{If }a/h &= 1, ah \equiv h. +\end{DPalign*} +That is to say, %[** TN: Not preserving book's formatting] +if $a$~is included in the group specified by~$h$, $h$~and~$ah$ are +equivalent. + +\Paragraph{6.} \textit{Addition and Multiplication}.---If we were to assume that +\index{Multiplication}% +probabilities are numbers or ratios, these operations could be +given their usual arithmetical signification. In adding or +multiplying probabilities we should be simply adding or multiplying +numbers. But in the absence of such an assumption, it +is necessary to give a meaning by definition to these processes. +I shall define the addition and multiplication of relations of +probabilities only for certain types of such relations. But it +will be shown later that the limitation thus placed on our operations +is not of practical importance. + +We define the \emph{sum} of the probable relations $ab/h$ and~$a\bar{b}/h$ +as being the probable relation~$a/h$; and the \emph{product} of the probable +relations $a/bh$~and~$b/h$ as being the probable relation~$ab/h$. That +is to say: +\begin{DPgather*} +\lintertext{\indent IX\@.} +ab/h + a\bar{b}/h = ah. +\rintertext{Def.} \\ +\lintertext{\indent X\@.} +ab/h = a/bh· b/h = b/ah· a/h. +\rintertext{Def.} +\end{DPgather*} + +Before we proceed to the axioms which will make these symbols +operative, the definitions may be restated in more familiar +language. IX.~may be read: ``The sum of the probabilities +of `both $a$~and~$b$' and of `$a$~but not~$b$,' relative to the same +hypothesis, is equal to the probability of~`$a$' relative to this hypothesis.'' +%% -----File: 147.png---Folio 136------- +X.~may be read: ``The probability of `both $a$~and~$b$,' +assuming~$h$, is equal to the product of the probability of~$b$, assuming~$h$, +and the probability of~$a$, assuming both $b$~and~$h$.'' Or in +the current terminology\footnote + {\Eg\ Bertrand, \textit{Calcul des probabilités}, p.~26.} +\index{Bertrand!on multiplication}% +we should have: ``The probability +that both of two events will occur is equal to the probability of +the first multiplied by the probability of the second, assuming +the occurrence of the first.'' It is, in fact, the ordinary rule for +the multiplication of the probabilities of events which are not +`independent.' It has, however, a much more central position +in the development of the theory than has been usually recognised. + +Subtraction and division are, of course, defined as the inverse +operations of addition and multiplication: + +XI\@. If $PQ = R$, $P = \dfrac{R}{Q}$.\hfill Def. + +XII\@. If $P + Q = R$, $P = R - Q$.\hfill Def. + +Thus we have to introduce as definitions what would be axioms +if the meaning of addition and multiplication were already defined. +In this latter case we should have been able to apply the ordinary +processes of addition and multiplication without any further +axioms. As it is, we need axioms in order to make these symbols, +to which we have given our own meaning, operative. When +certain properties are associated, it is often more or less arbitrary +which we take as defining properties and which we associate +with these by means of axioms. In this case I have found it +more convenient, for the purposes of formal development, to +reverse the arrangement which would come most natural to +\DPtypo{commonsense}{common sense}, full of preconceptions as to the meaning of addition +and multiplication. I define these processes, for the theory of +probability, by reference to a comparatively unfamiliar property, +and associate the more familiar properties with this one by means +of axioms. These axioms are as follows: + +(iv.)\ If $P$,~$Q$,~$R$ are relations of probability such that the +products $PQ$,~$PR$ and the sums $P + Q$, $P + R$ exist, then: + +(iv.\textit{a}) If $PQ$~exists, $QP$~exists, and $PQ = QP$. If $P + Q$ exists, +$Q + P$ exists and $P + Q = Q + P$. + +(iv.\textit{b}) $PQ < P$ unless $Q = 1$ or $P=0$; $P + Q > P$ unless $Q = 0$. + +\PadTxt{(iv.\textit{b})}{} $PQ = P$ \PadTxt{unless}{if} $Q = 1$ or $P=0$; $P + Q = P$ \PadTxt{unless}{if} $Q = 0$. + +(iv.\textit{c}) If $PQ \lesseqqgtr PR$, then $Q\lesseqqgtr R$ unless $P = 0$. If $P + Q\lesseqqgtr P + R$, +then $Q \lesseqqgtr R$ and conversely. +%% -----File: 148.png---Folio 137------- + +A meaning has not been given, it is important to notice, to +the signs of addition and multiplication between probabilities +\emph{in all cases}. According to the definitions we have given, $P + Q$ +and~$PQ$ have not an interpretation whenever $P$~and~$Q$ are +relations of probability, but in certain conditions only. Furthermore, +if $P + Q = R$ and $Q = S + T$, it does not follow that +$P + S + T = R$, since no meaning has been assigned to such an +expression as $P + S + T$. The equation must be written $P + (S + T) += R$, and we cannot infer from the foregoing axioms that +$(P + S) + T = R$. The following axioms allow us to make this +and other inferences in cases in which the sum~$P+S$ exists, \ie\ +when $P + S = A$ and $A$~is a relation of probability. +\begin{DPalign*} %[** TN: Reformatting] +\lintertext{\indent (v.)} +[±P±Q] + [±R±S] &= [±P±R] - [\mp Q\mp S] \\ +&= [±P±R] + [±Q±S] \\ +&= [±P±Q] - [\mp R\mp S] +\end{DPalign*} +in every case in which the probabilities $[±P±Q]$, $[±R±S]$, +$[±P±R]$, etc., exist, \ie, in which these sums satisfy the conditions +necessary in order that a meaning may be given to them +in the terms of our definition. + +(vi.)\ $P(R±S) = PR ± PS$, if the sum~$R±S$ and the products +$PR$~and~$PS$ exist as probabilities. + +\Paragraph{7.} From these axioms it is possible to derive a number of +propositions respecting the addition and multiplication of probabilities. +They enable us to prove, for instance, that if $P + Q = +R + S$ then $P - R = S - Q$, provided that the differences $P-R$~and~$S-Q$ +exist; and that $(P + Q)(R + S) = (P + Q)R + (P + Q)S = +[PR + QR] + [PS + QS] = [PR + QS] + [QR + PS]$, provided that +the sums and products in question exist. In general any rearrangement +which would be legitimate in an equation between +arithmetic quantities is also legitimate in an equation between +probabilities, provided that our initial equation and the equation +which finally results from our symbolic operations can both be +expressed in a form which contains only products and sums which +have an interpretation as probabilities in accordance with the +definitions. If, therefore, this condition is observed, we need not +complicate our operations by the insertion of brackets at every +stage, and no result can be obtained as a result of leaving them +out, if it is of the form prescribed above, which could not be +obtained if they had been rigorously inserted throughout. We +can only be interested in our final results when they deal with +actually existent and intelligible probabilities---for our object is, +%% -----File: 149.png---Folio 138------- +\index{Independence, for knowledge!definition of}% +always, to compare one probability with another---and we are +not incommoded, therefore, in one symbolic operations by the +circumstance that sums and products do not exist between +every pair of probabilities. + +\Paragraph{8.} \textit{Independence}: + +XIII\@. If $a_1/a_2h = a_1/h$ and $a_{2}/a_{1}h=a_{2}/h$, the probabilities +$a_1/h$~and~$a_2/h$ are \emph{independent}. \hfill Def. + +Thus the probabilities of two arguments having the same +premisses are independent, if the addition to the premisses of the +conclusion of either leaves them unaffected. + +\textit{Irrelevance}:\index{Irrelevance!definition of}\footnote + {This is repeated for convenience of reference from \Chapref{IV}. §\;14. It is + only necessary here to take account of \emph{irrelevance on the whole}, not of the more + precise sense.} + +XIV\@. If $a_1/a_2h = a_1/h$, $a_2$~is \emph{irrelevant on the whole}, or, for +short, \emph{irrelevant} to~$a_1/h$. \hfill Def. +%% -----File: 150.png---Folio 139------- +\index{De Morgan!inference@{and inference}}% +\index{Probability, and relevant knowledge!negative}% + + +\Chapter{XIII}{The Fundamental Theorems of Necessary Inference} +\index{Inference!necessary}% + +\Paragraph{1.} \First{In} this chapter we shall be mainly concerned with deducing +the existence of relations of certainty or impossibility, given other +relations of certainty or impossibility,---with the rules, that is to +say, of \emph{Certain} or, as De~Morgan termed it, of \emph{Necessary} Inference. +But it will be convenient to include here a few theorems dealing +with intermediate degrees of probability. Except in one or two +important cases I shall not trouble to translate these theorems +from the symbolism in which they are expressed, since their +interpretation presents no difficulty. + +\Paragraph{2.} (1) $a/h + \bar{a}/h = 1$. +\begin{DPgather*} +\lintertext{For} + ab/h + \bar{a}b/h = b/h + \rintertext{by IX.,}\\ +% + a/bh · b/h + \bar{a}/bh · b/h = b/h + \rintertext{by X.}\\ +% +\lintertext{Put} + b/h = 1, \text{ then }a/bh + \bar{a}/bh = 1 + \rintertext{by (iv.~\textit{b}),}\\ +% +\lintertext{since} + b/h = 1,\quad bh\equiv h + \rintertext{by (iii.).}\\ +% +\lintertext{Thus} + a/h + \bar{a}/h = 1 + \rintertext{by (ii.).} +\end{DPgather*} + +(1.1) If $a/h = 1$, $\bar{a}/h = 0$, +\begin{DPgather*} +a/h + \bar{a}/h = 1 + \rintertext{by (1),}\\ +% +\therefore a/h + \bar{a}/h = a/h = a/h + 0 + \rintertext{by (iv.~\textit{b}),}\\ +% +\therefore \bar{a}/h = 0 + \rintertext{by (iv.~\textit{c}).} +\end{DPgather*} + +(1.2) Similarly, if $\bar{a}/h=1$, $a/h=0$. + +(1.3) If $a/h=0$, $\bar{a}/h=1$, +\begin{DPgather*} + a/h + \bar{a}/h = 1 + \rintertext{by (1),}\\ +% +\therefore 0 + \bar{a}/h = 0 + 1 + \rintertext{by (iv.~\textit{b}),}\\ +% +\therefore \bar{a}/h = 1 + \rintertext{by (iv.~\textit{c}).} +\end{DPgather*} + +(1.4) Similarly, if $\bar{a}/h = 0$, $a/h = 1$. +\begin{DPgather*} +\lintertext{\indent(2)} + a/h < 1 \text{ or } a/h = 1 + \rintertext{by IV.}\\ +% +\lintertext{\indent(3)} + a/h > 0 \text{ or } a/h = 0 + \rintertext{by V.,} +\end{DPgather*} +\ie\ there are no negative probabilities. +%% -----File: 151.png---Folio 140------- +\begin{DPgather*} +\lintertext{\indent(4)} + \text{$ab/h < b/h$ or $ab/h = b/h$} + \rintertext{by X. and (iv.~\textit{b}).} +\end{DPgather*} +If $P$~and~$Q$ are relations of probability and $P + Q = 0$, +then $P=0$ and $Q=0$. +\begin{DPgather*} +\text{$P+Q > P$ unless $Q = 0$} + \rintertext{by (iv.~\textit{b}),}\\ +% +\lintertext{and} + \text{$P>0$ unless $P = 0$} + \rintertext{by V.}\\ +% +\text{$\therefore P+Q>0$ unless $Q = 0$.} +\end{DPgather*} +Hence, if $P + Q = 0$, $Q = 0$ and similarly $P = 0$. + +(6)~If $PQ = 0$, $P = 0$ or $Q = 0$, +\begin{DPgather*} +\text{$Q > 0$ unless $Q = 0$} + \rintertext{by V.}\\ +% +\lintertext{Hence} + \text{$PQ > P· 0$ unless $Q = 0$ or $P=0$} + \rintertext{by (iv.~\textit{c}),}\\ +% +\lintertext{\ie} + \text{$PQ > 0$ unless $Q = 0$ or $P = 0$} + \rintertext{by (iv.~\textit{b}).}\\ +% +\intertext{Whence, if $PQ = 0$, the result follows.} +% +\intertext{\indent(7)~If $PQ=1$, $P=1$ and $Q=1$,}\\ +% +\text{$PQ < P$ unless $P = 0$ or $Q = 1$} + \rintertext{by (iv.~\textit{b}),}\\ +% +\text{$PQ = P$ if $P = 0$ or $Q = 1$} + \rintertext{by (iv.~\textit{b}),}\\ +% +\lintertext{and} + \text{$P < 1$ unless $P = 1$} + \rintertext{by IV.,}\\ +% +\text{$\therefore PQ < 1$ unless $P = 1$.} \\ +% +\intertext{Hence $P = 1$; similarly $Q = 1$.} +\end{DPgather*} + +(8)~If $a/h = 0$, $ab/h = 0$ and $a/bh = 0$ if $bh$ is not inconsistent. +\begin{DPgather*} +\lintertext{For} + \text{$ab/h = b/ah · a/h = a/bh · b/h$} + \rintertext{by X.,}\\ +% +\lintertext{\PadTxt[l]{and since}{and since $a/h = 0$,}} + \text{$b/ah · a/h = 0$} + \rintertext{by (iv.~\textit{b}),}\\ +% +\text{$\therefore ab/h = 0$ and $a/bh · b/h = 0$,}\\ +\text{$\therefore$ unless $b/h = 0$, $a/bh = 0$} + \rintertext{by (5),} +\end{DPgather*} +whence the result by~VI\@. + +Thus, if a conclusion is impossible, we may add to the conclusion +or add consistently to the premisses without affecting the +argument. + +(9)~If $a/h = 1$, $a/bh = 1$ if $bh$ is not inconsistent. +\begin{DPgather*} +\lintertext{Since $a/h = 1$,} + \bar{a}/h = 0 + \rintertext{by (1.1),}\\ +% +\intertext{$\therefore \bar{a}/bh = 0$ by~(8) if $bh$ is not inconsistent,}\\ +\lintertext{whence} + a/bh = 1 + \rintertext{by (1.4).} +\end{DPgather*} + +Thus we may add to premisses, which make a conclusion +certain, any other premisses not inconsistent with them, without +affecting the result. + +(10)~If $a/h = 1$, $ab/h = b/ah = b/h$, +\begin{DPgather*} +ab/h = b/ah · a/h = a/bh · b/h + \rintertext{by X.}\\ +% +\intertext{Since $a/h = 1$, $a/bh = 1$ by~(9) unless $b/h = 0$,}\\ +% +\text{$\therefore b/ah · a/h = b/ah$ and $a/bh · b/h = b/h$} + \rintertext{by (iv.~\textit{b}),} +\end{DPgather*} +whence the result, unless $b/h = 0$. +%% -----File: 152.png---Folio 141------- +\index{Equivalence, definition of!principle of}% +If $b/h = 0$, the result follows from (8). + +(11)~If $ab/h = 1$, $a/h = 1$. +\begin{DPgather*} +\lintertext{For} ab/h = b/ah · a/h \rintertext{by X.,} \\ + \therefore\: a/h = 1 \rintertext{by (7).} +\end{DPgather*} + +(12)~If $(a \equiv b)/h = 1$, $a/h = b/h$, +\begin{DPgather*} + b/ah · a/h = a/bh · b/h \rintertext{by X.}\\ +\lintertext{and} b/ah = 1, a/bh = 1 \rintertext{by VIII.,}\\ + \therefore a/h= b/h \rintertext{by (iv.~b).} +\end{DPgather*} + +(12.1)~If $(a \equiv b)/h = 1$ and $hx$ is not inconsistent, +\begin{DPgather*} + a/hx = b/hx.\\ + a/hx· x/h = x/ah · a/h,\\ +\lintertext{and} b/hx · x/h = x/bh · b/h \rintertext{by X.,}\\ + x/ah = x/bh \rintertext{by (ii.),}\\ +\lintertext{and} a/h = b/h \rintertext{by (12),}\\ + \therefore a/hx = b/hx\text{ unless }x/h = 0. +\end{DPgather*} + +This is the \emph{principle of equivalence}. In virtue of it and of +axiom (ii.), if $(a \equiv b)/h = 1$, we can substitute $a$~for~$b$ and \textit{vice versa}, +wherever they occur in a probability whose premisses include $h$. + +(13)~$a/a= 1$, unless $a$ is inconsistent. +\begin{DPgather*} +\lintertext{For} a/a = aa/a = a/aa· a/a \rintertext{by (iii.), (12), and X.,} +\end{DPgather*} +whence $a/aa = 1$ by (ii.), unless $a/a = 0$, +\begin{DPgather*} +\lintertext{\ie} a/a = 1, \text{ unless $a$ is inconsistent} \rintertext{by (iii.), (12), and VI.} +\end{DPgather*} + +(13.1)~$\bar{a}/a = 0$, unless $a$ is inconsistent. This follows from +(13)~and~(1.1). + +(13.2)~$a/\bar{a} = 0$, unless $\bar{a}$ is inconsistent. This follows from +(iii.)\ by writing $\bar{a}$ for $a$ in (13.1). + +(14)~If $a/b = 0$ and $a$~is not inconsistent, $b/a = 0$. + +Let $f$ be the group of assumptions, common to $a$~and~$b$, which +we have supposed to be included in every real group; +\begin{DPgather*} +\lintertext{then} a/b = a/bf \text{ and } b/a = b/af \rintertext{by (iii.)\ and (12),}\\ +\lintertext{and} a/bf · b/f = b/af · a/f \rintertext{by X.}\\ +\lintertext{Since} a/bf = 0 \text{ by hypothesis,}\\ +\lintertext{and} a/f \neq 0, \text{ since $a$ is not inconsistent,}\\ + \therefore b/af = 0,\\ +\lintertext{whence} b/a = 0. +\end{DPgather*} +Thus, if $a$ is impossible given~$b$, then $b$~is impossible given~$a$. + +(15)~If $h_1/h_2 = 0$, $h_1h_2/h = 0$, +\begin{DPgather*} + h_1/h_2/h = h_1/h_2h · h_2/h \rintertext{by X.,} +\end{DPgather*} +and since $h_1/h_2 = 0$, $h/h_2h = 0$ by~(8), unless $h/h_2 = 0$, whence +the result by~(iv.~b), unless $h/h_2 = 0$. +%% -----File: 153.png---Folio 142------- +\begin{DPgather*} +\lintertext{\indent If} +h/h_2 = 0,\quad h_2/h = 0 + \rintertext{by~(14),}\\ +\intertext{since we assume that $h$~is not inconsistent, and hence} +h_1h_2/h = 0 + \rintertext{by~(8).} +\end{DPgather*} +Thus, if $h_1$ is impossible given~$h_2$, $h_1h_2$~is always impossible and is +excluded from every group. + +(15.1) If $h_1h_2/h = 0$ and $h_2h$~is not inconsistent, $h_1/h_2h = 0$. +This, which is the converse of~(15), follows from X.~and~(6). + +(16) If $h_1/h_2 = 1$, $(h_1 + \bar{h}_2)/h = 1$, +\begin{DPgather*} +\bar{h}_1/h_2 = 0 + \rintertext{by~(1),}\\ +\therefore \bar{h}_1h_2/h = 0 + \rintertext{by~(15),}\\ +\therefore \overline{h_1 h_2}/h = 1 + \rintertext{by~(1.3),}\\ +\therefore (h_1 + \bar{h}_2)/h = 1 + \rintertext{by (12)~and~(iii.).} +\end{DPgather*} + +(16.1) We may write~(16): + +If $h_1/h_2 = 1$, $(h_2 \supset h_1)/h = 1$, where `$\supset$'~symbolises `implies.' +Thus if $h_1$~follows from~$h_2$, then it is always certain that +$h_2$~implies~$h_1$. + +(16.2) If $(h_1 + \bar{h}_2)/h = 1$ and $h_2h$~is not inconsistent, +$h_1/h_2h = 1$. +\begin{DPgather*} +\bar{h}_1h_2/h = 0,\quad \text{as in~(16),}\\ +\therefore \bar{h}_1/h_2h = 0\quad \text{by~(15.1), since $h_2h$~is not inconsistent,}\\ +\therefore h_1/h_2h = 1 + \rintertext{\llap{by~(1.4).}} +\end{DPgather*} +This is the converse of~(14). + +(16.3) We may write~(16.2): + +If $(h_2 \supset h_1)/h = 1$ and $h_2h$~is not inconsistent, $h_1/h_2h = 1$. +Thus, if we define a `group' as a set of propositions, which follow +from and are certain relatively to the proposition which specifies +them, this proposition proves that, if $h_2 \supset h_1$ and~$h_2$ belong to a +group~$h_2h$, then $h_1$~also belongs to this group. + +(17) If $(h_1 \supset: a \equiv b)/h = 1$ and $h_1h$~is not inconsistent, $a/h_1h += b/h_1h$. This follows from (16.3)~and~(12). + +(18) $a/a = 1$ or $\bar{a}/\bar{a} = 1$. +\begin{DPgather*} +\bar{a}/a = 1,\text{ unless $a$~is inconsistent,} + \rintertext{by~(13).}\\ +\intertext{\indent If $a$ is inconsistent, $a/h = 0$, where $h$~is not inconsistent, and therefore} +\bar{a}/h = 1 + \rintertext{by~(1.3).}\\ +\intertext{Thus unless $a$ is inconsistent, $\bar{a}$~is not inconsistent, and therefore} +\bar{a}/\bar{a} = 1 + \rintertext{by~(13).}\\ +\lintertext{\indent (19) \rlap{$a\bar{a}/h = 0$,}} \\ +\bar{a}/\bar{a} = 1 \text{ or } a/a = 1 + \rintertext{by~(18),}\\ +\therefore a/\bar{a} = 0 \text{ or } \bar{a}/a = 0 + \rintertext{\llap{by (1.1)~and~(1.2).}} +\end{DPgather*} +In either case $a\bar{a}/h = 0$ by~(15). +%% -----File: 154.png---Folio 143------- +\index{Excluded Middle, Law of}% + +Thus it is impossible that both $a$ and its contradictory +should be true. This is the Law of Contradiction. +\index{Contradiction}% + +(20) $(a + \bar{a})/h = 1$. +\begin{DPgather*} +\lintertext{Since} +(a\bar{a} \equiv \overline{a + \bar{a}})/h = 1 + \rintertext{by~(iii.),} \\ +% +\overline{a + \bar{a}}/h = 0 + \rintertext{by (19)~and~(12),} \\ +% +\therefore (a + \bar{a})/h = 1 + \rintertext{by~(1.3).} +\end{DPgather*} +Thus it is certain that either $a$~or its contradictory is true. This +is the Law of Excluded Middle. + +(21) If $a/h_1 = 1$ and $a/h_2 = 0$, $h_1h_2/h = 0$. +\begin{DPgather*} +\lintertext{For} +a/h_1h_2 · h_1/h_2 = h_1/ah_2 · ah_2, \\ +% +\lintertext{and} +\bar{a}/h_1h_2 · h_2/h_1 = h_2/\bar{a}h_1 · \bar{a}/h_1 + \rintertext{by~X.,} \\ +% +\therefore a/h_1h_2 · h_1/h_2 = 0 \text{ and } \bar{a}/h_1h_2 · h_2/h_1 = 0, \\ +\intertext{since, by hypothesis and~(1), $\bar{a}/h_1 = 0$ and $a/h_2 = 0$,} +% +\therefore a/h_1h_2 = 0 \text{ or } h_1/h_2 = 0, \\ +\lintertext{and} +a/h_1h_2 = 1 \text{ or } h_2/h_1 = 0, \\ +% +\therefore h_1/h_2 = 0 \text{ or } h_2/h_1 = 0. +\end{DPgather*} +In either case $h_1h_2/h = 0$ by~(15). + +Thus, if a proposition is certain relatively to one set of +premisses, and impossible relatively to another set, the two sets +are incompatible. + +(22) If $a/h_1 = 0$ and $h_1/h = 1$, $a/h = 0$, +\begin{gather*} +ah_1/h = 0 \text{ by~(15),}\quad \therefore h_1/ah · a/h = 0, \\ +h_1/ah = 1 \text{ by~(9), unless } a/h = 0. \\ +\therefore \text{ in any case $a/h = 0$.} +\end{gather*} + +(23) If $b/a = 0$ and $b/\bar{a} = 0$, $b/h = 0$. +\begin{DPgather*} +ab/h = 0 \text{ and } \bar{a}b/h = 0 + \rintertext{by~(15),} \\ +% +\therefore a/bh = 0 \text{ or } b/h = 0, \\ +\lintertext{and} +\bar{a}/bh = 0 \text{ or } b/h = 0 + \rintertext{by II.~and~(iv.),} \\ +% +\lintertext{whence} +b/h = 0 + \rintertext{by~(1.4).} +\end{DPgather*} +%% -----File: 155.png---Folio 144------- +\index{Addition, of probabilities!Theorem of}% + + +\Chapter{XIV}{The Fundamental Theorems of Probable Inference} + +\Paragraph{1.} \First{I shall} give proofs in this chapter of most of the fundamental +theorems of Probability, with very little comment. The bearing +of some of them will be discussed more fully in \Chapref{XVI}\@. + +\Paragraph{2.} \textit{The Addition Theorems}: + +(24) $(a + b)/h = a/h + b/h - ab/h$. + +In~IX. write $(a + b)$ for~$a$, and $\bar{a}b$ for~$b$. +\begin{DPgather*} +\lintertext{Then} + (a + b) \bar{a}b/h + (a + b) \overline{\bar{a}b}/h = (a + b)/h, \\ +% +\lintertext{whence} + \bar{a}b/h + (a + b)(a + \bar{b})/h = (a + b)/h + \rintertext{by~(iii.),}\\ +% +\bar{a}/bh · b/h + a/h = (a + b)/h + \rintertext{\llap{by (iii.)\ and~IX.}} +\end{DPgather*} +\begin{DPalign*} +\lintertext{That is to say,} +(a + b)/h + &= a/h + (1 - a/bh) · b/h,\\ + &= a/h + b/h - ab/h. +\end{DPalign*} + +In accordance with the principles of \Chapref{XII}. §\;6, this +should be written, strictly, in the form $a/h + (b/h - ab/h)$, or in +the form $b/h + (a/h - ab/h)$. The argument is valid, since the +probability $(b/h - ab/h)$ is equal to $\bar{a}b/h$, as appears from the +preceding proof, and, therefore, exists. This important theorem +gives the probability of `$a$~or~$b$' relative to a given hypothesis +in terms of the probabilities of `$a$,'~`$b$,' and `$a$~and~$b$' relative to +the same hypothesis. + +(24.1) If $ab/h = 0$, \ie\ if $a$~and~$b$ are exclusive alternatives +relative to the hypothesis, then +\[ +(a + b)/h = a/h + b/h. +\] +This is the ordinary rule for the addition of the probabilities of +exclusive alternatives. + +(24.2) $ab/h + \bar{a}b/h = b/h$, +\begin{DPalign*} +\lintertext{since} + &ab + \bar{a}b \equiv b + \rintertext{by (iii.),}\\ +% +\lintertext{and} + &a\bar{a}b/h = 0 + \rintertext{by (19)~and~(8).} +\end{DPalign*} + +(24.3) $(a + b)/h = a/h + b\bar{a}/h$. This follows from (24)~and~(24.2). +%% -----File: 156.png---Folio 145------- +\begin{DPalign*}[m] +\lintertext{\rlap{\indent(24.4)}} +(a + b + c)/h &= (a + b)/h + c/h - (ac + bc)/h\\ + &= a/h + b/h + c/h \\ %[** TN: Added another line break] + &\qquad - ab/h - bc/h - ca/h + abc/h. +\end{DPalign*} + +(24.5) And in general +\begin{multline*} +(p_1 + p_2 + \ldots + p_n)/h + = \Sum p_r/h - \Sum p_s p_t/h + + \Sum p_rp_sp_t/h \ldots \\ + + (-1)^n - 1 p_1p_2 \ldots p_n/h. +\end{multline*} + +(24.6) If $p_s p_t/h = 0$ for all pairs of values of $s$~and~$t$, it follows +by repeated application of~X. that +\[ +(p_1 + p_2 + \ldots + p_n)/h = \Sum_1^np_r/h. +\] + +(24.7) If $p_s p_t/h = 0$, etc., and $(p_1 + p_2 + \ldots + p_n)/h = 1$, \ie\ +if $p_1p_2 \ldots p_n$ form, relatively to~$h$, a set of exclusive and +exhaustive alternatives, then +\[ +\Sum_1^n p_r/h = 1. +\] + +(25) If $p_1p_2 \ldots p_n$ form, relative to~$h$, a set of exclusive +and exhaustive alternatives, +\[ +a/h = \Sum_1^n p_r a/h. +\] +Since $(p_1 + p_2 + \ldots + p_n)/h = 1$ by hypothesis, +\[ +\therefore (p_1 + p_2 + \ldots + p_n)/ah = 1\quad + \text{ by~(9) if $ah$ is not inconsistent;} +\] +and since $p_s p_t/h = 0$ by hypothesis, +\[ +\therefore p_s p_t/ah = 0\quad + \text{by~(9), if $ah$ is not inconsistent.} +\] +\begin{DPalign*} +\lintertext{Hence} + \Sum_1^n p_r/ah &= (p_1 + p_2 + \ldots + p_n)/ah + \rintertext{by~(24.6)}\\ + &=1 \\ +%[** TN: Continuing the alignment, not as in original] +\lintertext{Also} +p_r a/h &= p_r/ah · a/h.\\ +% +\lintertext{Summing} +\Sum_1^n p_r a/h &= a/h · \Sum_1^n p_r/ah, +\end{DPalign*} +\[ +\therefore a/h = \Sum_1^n p_r a/h,\quad \text{if $ah$ is not inconsistent}. +\] + +If $ah$ is inconsistent, \ie\ if $a/h = 0$ (for $h$~is by hypothesis consistent), +the result follows at once by~(8). + +(25.1) If $p_ra/h = X_r$, the above may be written +\[ +p_r/ah = \frac{X_r}{\Sum_1^n X_r}. +\] +%% -----File: 157.png---Folio 146------- +\index{Independence, for knowledge!Theorem of}% + +(26) $a/h = (a + \bar{h})/h$. +\begin{DPalign*} +\lintertext{\indent For} +(a + \bar{h})/h &= a/h + \bar{h}/h - a\bar{h}/h + \rintertext{by~(24),}\\ +% + &= a/h + \rintertext{\llap{by (13.1)~and~(8).}} +\end{DPalign*} + +(26.1) This may be written +\[ +a/h = (h\supset a)/h. +\] + +(27) If $(a + b)/h = 0$, $a/h = 0$. +\begin{DPgather*}[m] +a/h + [b/h - ab/h] = 0, \rintertext{\llap{by~(24) and hypothesis}}\\ +\therefore a/h = 0 \rintertext{by~(5)}. +\end{DPgather*} + +(27.1) If $a/h = 0$ and $b/h = 0$, $(a + b)/h = 0$. This follows +from~(24). + +(28) If $a/h = 1$, $(a + \bar{b})/h = 1$, +\begin{DPgather*} +(a + \bar{b})/h = a/h + \bar{b}\bar{a}/h \rintertext{by~(24.3)}, +\end{DPgather*} +whence $(a +\bar{b})/h = a/h = 1$ by (1.1)~and~(8), together with the +hypothesis. That is to say, a certain proposition is implied by +every proposition. + +(28.1) If $a/h = 0$, $(\bar{a}+ b)/h = 1$ by substituting~$\bar{a}$ for~$a$ and~$b$ +for~$\bar{b}$ in~(28). That is to say, a certainly false proposition +implies every proposition. + +(29) If $a/(h_1 + h_2) = 1$, $a/h_1 = 1$, $a/h_2 = 1$. +\begin{DPgather*} +\bar{a}/(h_1 + h_2) = 0, \\ +\lintertext{and} + \therefore \bar{a}(h_1 + h_2)/h_1 = 0 + \rintertext{by~(15).}\\ +% +\lintertext{Hence} +\bar{a}h_1/h_1 = 0 + \rintertext{by~(27),} +\end{DPgather*} +whence the result. + +(29.1) $If a/h_1 = 1$ and $a/h_2 = 1$, $a/(h_1 + h_2) = 1$. +\begin{DPgather*} +\lintertext{As in~(20)} +\bar{a}h_1/(h_1 + h_2) = 0 \text{ and } \bar{a}h_2/(h_1 + h_2) = 0.\\ +% +\lintertext{Hence} +\bar{a}(h_1 + h_2)/(h_1 + h_2) = 0 + \rintertext{by~(27.1),} +\end{DPgather*} +whence the result. + +(29.2) If $a/(h_1 + h_2) = 0$, $a/h_1 = 0$. This follows from~(29). + +(29.3) If $a/h_1 = 0$ and $a/h_2 = 0$, $a/(h_1 + h_2) = 0$. This follows +from~(29.1). + +\Paragraph{3.} \textit{Irrelevance and Independence}: +\index{Irrelevance!Theorem of}% + +(30) If $a/h_1h_2 = a/h_1$, then $a/h_1\bar{h}_2 = a/h_1$, if $h_1\bar{h}_2$ is not inconsistent. +\begin{DPalign*} +a/h_1 &= ah_2/h_1 + a\bar{h}_2/h_1 \rintertext{by~(24.2),}\\ + &= a/h_1h_2 · h_2/h_1 + a/h_1\bar{h}_2 · \bar{h}_2/h_1,\\ + &= a/h_1 · h_2/h_1 + a/h_1\bar{h}_2 · \bar{h}_2/h_1, +\end{DPalign*} +\[ +\therefore a/h_1 · \bar{h}_2/h_1 = a/h_1\bar{h}_2 · \bar{h}_2/h_1, +\] +whence $a/h_1 = a/h_1\bar{h}_2$, unless $\bar{h}_2/h_1 = 0$, \ie\ if $h_1\bar{h}_2$ is not inconsistent. +%% -----File: 158.png---Folio 147------- +\index{Relevance, judgments of!theorems of}% + +Thus, if a proposition is irrelevant to an argument, then the +contradictory of the proposition is also irrelevant. + +(31)~If $a_2/a_1h = a_2/h$ and~$a_2h$ is not inconsistent, $a_1/a_2h = a_1/h$. + +This follows by (iv.~\textit{c}), since $a_2/a_1h · a_1/h = a_1/a_2h · a_2/h$ by~X. +If, that is to say, $a_1$~is irrelevant to the argument~$a_2/h$ (see~XIV.), +and $a_2$~is not inconsistent with~$h$: then $a_2$~is irrelevant +to the argument~$a_1/h$; and $a_1/h$~and~$a_2/h$ are independent +(see~XIII.). + +\Paragraph{4.}\Pagelabel{147} \textit{Theorems of Relevance:} + +(32)~If $a/hh_1 > a/h$, $h_1/ah > h_1/h$. \\ +$ah$~is consistent since, otherwise, $a/hh_1 = a/h = 0$. +\begin{DPgather*} +\lintertext{Therefore} + a/h · h_1/ah = a/hh_1 · h_1/h + \rintertext{by~X.,}\\ +% +> a/h · h_1/h + \rintertext{by hypothesis;}\\ +% +\lintertext{so that} + h_1/ah > h_1/h. +\end{DPgather*} + +Thus if $h_1$~is favourably relevant to the argument~$a/h$, $a$~is +favourably relevant to the argument~$h_1/h$. + +This constitutes a formal demonstration of the generally +accepted principle that if a hypothesis helps to explain a +phenomenon, the fact of the phenomenon supports the reality +of the hypothesis. + +In the following theorems $p$~will be said to be more +favourable to~$a/h$, than $q$~is to~$b/h$, if $\dfrac{a/ph}{a/h} > \dfrac{b/qh}{b/h}$, \ie~if, in the +language of §\;8 below, the coefficient of influence of~$p$ on~$a/h$ +is greater than the coefficient of influence of~$q$ on~$b/h$. + +(33) If $x$~is favourable to~$a/h$, and $h_1$~is not more favourable +to~$a/hx$ than $x$~is to~$a/hh_1$, then $h_1$~is favourable to~$a/h$. + +For $a/hh_1 = a/h · \dfrac{a/hx}{a/h} · \dfrac{a/hh_1x}{a/hx} · \dfrac{a/hh_1}{a/hh_1x}$; and by hypothesis the +second term on the right is greater than unity and the product +of the third and fourth terms is greater than or equal +to unity. + +(33.1) \textit{A~fortiori}, if $x$~is favourable to~$a/h$ and not favourable +to~$a/hh_1$, and if $h_1$~is not unfavourable to~$a/hx$, then $h_1$~is +favourable to~$a/h$. + +(34) If $x$~is favourable to~$a/h$, and $h_1$~is not less favourable +to~$x/ha$ than $x$~is to~$h_1/ha$, then $h_1$~is favourable to~$a/h$. + +This follows by the same reasoning as~(33), since by an +application of the Multiplication Theorem +%% -----File: 159.png---Folio 148------- +\[ +\frac{a/hh_1x}{a/hx} · \frac{a/hh_1}{a/hh_1x} +=\frac{x/hh_1a}{x/ha} · \frac{h_1/ha}{h_1/hax}. +\] + +(35)~If $x$ is favourable to~$a/h$, but not more favourable to it +than $h_1x$~is, and not less favourable to it than to~$a/hh_1$, then +$h_1$ is favourable to~$a/h$. + +\begin{DPgather*}[m] +\lintertext{\indent For} +a/hh_1 += a/h · \left\{\frac{a/h}{a/hx} · \frac{a/hh_1x}{a/h}\right\} + · \left\{\frac{a/hx}{a/h} · \frac{a/hh_1}{a/hh_1x}\right\}. +\end{DPgather*} + +This result is a little more substantial than the two +preceding. By judging the influence of $x$~and~$h_1x$ on the +arguments $a/h$~and~$a/hh_1$, we can infer the influence of~$h_1$ by +itself on the argument~$a/h$. + +\Paragraph{5.} \textit{The Multiplication Theorems:} +\index{Multiplication!theorems of}% + +(36)~If $a_1/h$~and~$a_2/h$ are independent, $a_1a_2/h=a_1/h · a_2/h$. +\begin{DPgather*} +\lintertext{\indent For} +a_1a_2/h += a_1/a_2h · a_2/h=a_2/a_1h · a_1/h \rintertext{by X.,}\\ +\intertext{and since $a_1/h$ and $a_2/h$ are independent,} + a_1/a_2h=a_1/h \text{ and } a_2/a_1h=a_2/h \rintertext{by XIII.}\\ +\lintertext{Therefore} a_1a_2/h=a_1/h· a_2/h. +\end{DPgather*} +Hence, when $a_1/h$~and~$a_2/h$ are independent, we can arrive at the +probability of $a_1$~and~$a_2$ jointly on the same hypothesis by simple +multiplication of the probabilities $a_1/h$~and~$a_2/h$ taken separately. + +(37)~If $p_1/h=p_2/p_1h=p_3/p_1p_2h=\ldots$, +\[ +p_1p_2p_3\ldots p_n/h=\left\{p_1/h\right\}^n. +\] + +For $p_1p_2p_3\ldots /h=p_1/h · p_2/p_1h · p_3/p_1p_2h\ldots$ by repeated +applications of~X. + +\Paragraph{6.} \textit{The Inverse Principle:} + +(38)~$\dfrac{a_1/bh}{a_2/bh}=\dfrac{b/a_1h}{b/a_2h}· \dfrac{a_1/h}{a_2/h}$, provided $bh$, $a_1h$, and~$a_2h$ are +each consistent. +\begin{DPgather*} +\lintertext{For} a_1/bh · b/h=b/a_1h · a_1/h,\\ +\lintertext{and} a_2/bh · b/h=b/a_2h · a_2/h \rintertext{by X.,} +\end{DPgather*} +whence the result follows, since $b/h\neq 0$, unless $bh$ is inconsistent. + +(38.1)~If \quad$a_1/h=p_1$,\quad $a_2/h=p_2$,\quad $b/a_1h=q_1$,\quad $b/a_2h=q_2$,\quad and +$a_1/bh+a_2/bh=1$ then it easily follows that +\begin{DPalign*} +a_1/bh &= \frac{p_1q_1}{p_1q_1+p_2q_2},\\ +%% -----File: 160.png---Folio 149------- +\index{Inverse Probability}% +\index{Probability, and relevant knowledge!Inverse}% +\lintertext{and} a_2/bh &= \frac{p_2q_2}{p_1q_1+p_2q_2}. +\end{DPalign*} + +(38.2)~If $a_1/h=a_2/h$ the above reduces to +\begin{DPalign*} + a_1/bh &=\frac{q_1}{q_1+q_2},\\ +\lintertext{and} a_2/bh &=\frac{q_2}{q_1+q_2}, +\end{DPalign*} +since $a_1/h\neq 0$, unless $a_1h$ is inconsistent. + +The proposition is easily extended to the cases in which the +number of~$a$'s is greater than two. + +It will be worth while to translate this theorem into familiar +language. Let $b$~represent the occurrence of an event~$B$, $a_1$~and~$a_2$ +the hypotheses of the existence of two possible causes +$A_1$~and~$A_2$ of~$B$, and $h$~the general data of the problem. Then $p_1$~and~$p_2$ +are the \textit{à~priori} probabilities of the existence of $A_1$~and~$A_2$ +respectively, when it is not known whether or not the event~$B$ +has occurred; $q_1$~and~$q_2$ the probabilities that each of the causes +$A_1$~and~$A_2$, if it exists, will be followed by the event~$B$. Then +$\dfrac{p_1q_1}{p_1q_1+p_2q_2}$ and $\dfrac{p_2q_2}{p_1q_1+p_2q_2}$ are the probabilities of the existence +of $A_1$~and~$A_2$ respectively \emph{after} the event, \ie\ when, in addition +to our other data, we know that the event~$B$ has occurred. The +initial condition, that $bh$ must not be inconsistent, simply ensures +that the problem is a possible one, \ie\ that the occurrence of the +event~$B$ is on the initial data at least possible. + +The reason why this theorem has generally been known as +the Inverse Principle of Probability is obvious. The causal +problems to which the Calculus of Probability has been applied +\index{Calculus of Probability}% +are naturally divided into two classes---the direct in which, given +the cause, we deduce the effect; the indirect or inverse in which, +given the effect, we investigate the cause. The Inverse Principle +has been usually employed to deal with the latter class of +problem. + +\Paragraph{7.} \textit{Theorems on the Combination of Premisses:} +\index{Combination of premisses}% + +The Multiplication Theorems given above deal with the combination +of \emph{conclusions}; given $a/h_1$~and~$a/h_2$ we considered the +relation of~$a_1a_2/h$ to these probabilities. In this paragraph the +corresponding problem of the combination of \emph{premisses} will be +%% -----File: 161.png---Folio 150------- +\index{Johnson, W. E.!cumulative@{and cumulative formula}}% +treated; given $a/h_1$~and~$a/h_2$ we shall consider the relation of~$a/h_1h_2$ +to these probabilities. +\begin{DPalign*} +\lintertext{\indent(39)} +a/h_1h_2h + &= \frac{ah_1h_2/h}{h_1h_2/h} + = \frac{ah_1h_2/h}{ah_1h_2/h + \bar ah_1h_2/h} + \rintertext{by X. and (24.2)}\\ + &= \frac{u}{u + v}, +\end{DPalign*} +where $u$ is the \textit{à~priori} probability of the conclusion~$a$ and both +hypotheses $h_1$~and~$h_2$ jointly, and $v$~is the \textit{à~priori} probability +of the contradictory of the conclusion and both hypotheses $h_1$~and~$h_2$ jointly. +\begin{DPalign*}[m] +\lintertext{\indent(40)} a/h_1h_2 = \frac{ah_1/h_2}{ah_1/h_2+\bar ah_1/h_2} &= + \frac{h_1/ah_2·q}{h_1/ah_2·q+h_1/\bar ah_2·(1-q)},\\ + &= \frac{h_2/ah_1·p}{h_2/ah_1·p + h_2/\bar ah_1·(1-p)}, +\end{DPalign*} +where $p=a/h_1$ and $q=a/h_2$. +\begin{DPalign*} +\lintertext{\indent(40.1)~If $p = \frac{1}{2}$,} a/h_1h_2 & + = \frac{h_2/ah_1}{h_2/ah_1+h_2/\bar ah_1},\\ +\lintertext{and increases with} &\phantom{=~} \frac{h_2/ah_1}{h_2/\bar ah_1}. +\end{DPalign*} + +These results are not very valuable and show the need of an +original method of reduction. This is supplied by Mr.~W.~E. +Johnson's \textit{Cumulative Formula}, which is at present unpublished +\index{Cumulative Formula}% +but which I have his permission to print below.\footnote + {The substance of propositions (41)~to~(49) below is derived in its entirety + from his notes,---the exposition only is mine.} + +\Paragraph{8.} It is first of all necessary to introduce a new symbol. Let +us write +\begin{DPgather*} +\lintertext{\indent XV.} a/bh = \{a^hb\}a/h \rintertext{Def.} +\end{DPgather*} +We may call $\{a^hb\}$ \emph{the coefficient of influence} of $b$ upon~$a$ on +hypothesis~$h$. +\begin{DPalign*} +\lintertext{\indent XVI.} \{a^hb\} · \{ab^hc\} &=\{a^hb^hc\} \rintertext{Def.}\\ +\lintertext{and similarly} \{a^hb\} · \{ab^hcd^he\} &=\{a^hb^hcd^he\}. +\end{DPalign*} +These coefficients thus belong by definition to a general class of +operators, which we may call \emph{separative factors}. +\noindent +\begin{DPgather*} +\lintertext{\indent (41)} ab/h=\{a^hb\} · a/h · b/h,\\ +\lintertext{since} ab/h=a/bh · b/h. +\end{DPgather*} +%% -----File: 162.png---Folio 151------- +Thus we may also call $\{a^{h}b\}$ \emph{the coefficient of dependence} between +$a$~and~$b$ on hypothesis~$h$. +\begin{DPalign*} +\lintertext{\indent(41.1)} abc/h & = \{a^{h}b^{h}c\} · a/h · b/h · c/h. \\ +\lintertext{For} abc/h & = \{ab^{h}c\}ab/h · c/h \rintertext{by (41),}\\ + & = \{ab^{h}c\} · \{a^{h}b\} · a/h · b/h · c/h \rintertext{by (41).} +\end{DPalign*} + +(41.2)~And in general +\begin{DPalign*} +abcd \ldots /h = \{a^{h}b^{h}c^{h}d^{h} \ldots \} + &{} · a/h · b/h · c/h · d/h \ldots \\ +\lintertext{\indent(42)} +\{a^{h}b\} & = \{b^{h}a\}, \\ +\lintertext{since} +a/bh · b/h & = b/ah · a/h. \\ +\lintertext{\indent(42.1)} +\{a^{h}b^{h}c\} &= {a^{h}c^{h}b}, \\ +\lintertext{since} +a/h · b/h · c/h & = a/h · c/h · b/h. +\end{DPalign*} + +(42.2)~And in general we have \emph{a commutative rule}, by which +the order of the terms may be always commuted--- +\begin{DPalign*} +\lintertext{\eg} \{a^{h}bc^{h} def^{h} g\} & = \{bc^{h}a^{h}g^{h} def\} \\ + \{a^{h}bc^{h} def^{h} g\} & = \{a^{h}cb^{h} fed^{h} g\}. +\end{DPalign*} + +(43)~As a multiplier the separative factor operates so as to +separate the terms that may be associated (or joined) in the +multiplicand. + +\begin{DPalign*} +\lintertext{Thus} + & \{ab^{h}cd^{h}e\} · \{a^{h}b\} = \{a^{h}b^{h}cd^{h}e\}, \\ +\lintertext{for} +abcde/h + & = \{ab^{h}cd^{h}e\} · ab/h · cd/h · e/h \\ + & = \{ab^{h}cd^{h}e\} · \{a^{h}b\} · a/h · b/h · cd/h · e/h, \\ +\lintertext{and \rlap{also}} +abcde/h + & = \{a^{h}b^{h}cd^{h}e\} · a/h · b/h · cd/h · e/h. +\end{DPalign*} +Similarly (for example) +\begin{DPgather*} +\{abc^{h}d^{h}ef\} · \{ab^{h}c\} · \{a^{h}b\} + = \{a^{h}b^{h}c^{h}d^{h}ef\}.\\ +\lintertext{(44)} +\{a^{h}b\} · \{ab\} = \{a^{h}b\}.\\ +\lintertext{For} +ab/h = \{ab\}ab/h. +\end{DPgather*} +By a symbolic convention, therefore, we may put $\{ab\} = 1$. + +(44.1)~If $\{a^{h}b\} = 1$, it follows that $a/h$~and~$b/h$ are independent +arguments; and conversely. + +(45)~Rule of Repetition $\{aa^{h}b\} = \{a^{h}b\}$. +\begin{DPgather*} +\lintertext{For} aab/h = ab/h \rintertext{by (vi.)\ and (12).} +\end{DPgather*} + +(46) \textit{The Cumulative Formula:} +{\setlength{\multlinegap}{0pt} +\begin{multline*} +x/ah : x'/ah : x''/ah : \ldots \\ + = x/h · a/xh : x'/h · a/x'h : x''/h · a/x''h : \ldots \qquad\text{by (38).} +\end{multline*} +} +Take $n + 1$ propositions $a, b, c \ldots$ \quad Then by repetition +{\setlength{\multlinegap}{0pt} +\begin{multline*} +x/ah · x/bh · x/ch \ldots : x'/a · \DPtypo{x'b/}{x'/b} +· x'/c \ldots : x''/a · x''/b · x''/c \ldots : \ldots \\ +\shoveleft{ = (x/h)^{n + 1} a/xh · b/xh \ldots : (x'/h)^{n + 1}a/x'h · b/x'h \ldots} \\ + : (x''/h)^{n + 1} a/x''h · b/x''h \ldots +\end{multline*} +}% +which may be written +%% -----File: 163.png---Folio 152------- +\index{Mathematicians, and probability!cumulative formula@{and cumulative formula}}% +{\setlength{\multlinegap}{0pt} +\begin{multline*} +\Prod^{n+1} x/ah : \Prod^{n+1} x'/ah : \Prod^{n+1} x''/ah: \ldots {}\\ + = (x/h)^{n+1} \Prod^{n+1} a/xh : (x'/h)^{n+1} \Prod^{n+1} a/x'h: \ldots +\end{multline*} +} +Now +{\setlength{\multlinegap}{0pt} +\begin{multline*} +x/habc \ldots: x'/habc \ldots: x''/habc \ldots \\ + = x/h · (abc \ldots) /xh : x'/h · (abc \ldots) /x'h: \ldots \text{by (38),} +\end{multline*} +} +and +\begin{DPgather*} +abc \ldots /xh = \{a^{xh}b^{xh}c \ldots\} \Prod^{n+1}a/xh \rintertext{\; by (41.2),} +\end{DPgather*} +\begin{multline*} +\therefore +(x/h)^{n} · x/habc \ldots : +(x'/h)^{n} · x'/habc \ldots: +(x''/h)^{n} · x''/habc \ldots: \ldots \\ + = \{a^{xh}b^{xh}c \ldots\} x/ah · x/bh · x/ch \ldots: + \{a^{x'h}b^{x'h}c \ldots\} x'/ah · x'/bh \\ + {} · x'/ch \ldots : \ldots +\end{multline*} +which may be written +\[ +(x/h)^{n}· x/habc \ldots \propto\{a^{xh}b^{xh}c \ldots\}· x/ah· x/bh· x/ch\ldots +\] +where variations of~$x$ are involved. + +The cumulative formula is to be applied when, having accumulated +the evidence $a, b, c \ldots$, we desire to know the comparative +probabilities of the various possible inferences $x, x' \ldots$ which +may be drawn, and already know determinately the force of +each of the items $a, b, c \ldots$ separately as evidence for $x, x' \ldots$. + +Besides the factors $x/ah$, $x/bh$, etc., we require to know two +other sets of values, viz.: (1)~$x/h$, etc., \ie\ the \textit{à~priori} +probabilities of~$x$, etc., and (2)~$\{a^{xh}b^{xh}c \ldots\}$, etc., \ie\ the +coefficients of dependence between $a$,~$b$, and~$c\ldots$ on hypotheses~$xh$, +etc. It may be remarked that the values $\{a^{xh}b^{xh}c \ldots\}$, +$\{a^{x'h}b^{x'h}c \ldots\} \ldots$ are not in any way related, even when $x' \equiv \bar{x}$. + +What corresponds to the cumulative formula has been employed, +sometimes, by mathematicians in a simplified form +which is, except under special conditions, incorrect. First, it +has been tacitly assumed that $\{a^{xh}b^{xh}c \ldots\}$, $\{a^{x'h}b^{x'h}c \ldots\} \ldots$ +are all unity: so that +\[ +(x/h)^{n}x/habc \ldots \propto x/ah · x/bh · x/ch \ldots +\] +Secondly, the factor $(x/h)^{n}$ has been omitted, so that +\[ +x/habc \ldots \propto x/ah · x/bh · x/ch \ldots +\] + +It is this second incorrect statement of the formula which +leads to the fallacious rule for the combination of the testimonies +of independent witnesses ordinarily given in the text-books.\footnote + {See \Pageref{180} below.} + +(46.1). If $abc \ldots /xh = \{a^{xh}b^{xh}c \ldots\} a/xh · b/xh · c/xh \ldots$ +\begin{DPgather*} +\lintertext{then} x/habc \ldots \propto \{a^{xh}b^{xh}c \ldots\} x/ah · x/bh · x/ch \ldots\DPtypo{}{.} +\end{DPgather*} +%% -----File: 164.png---Folio 153------- +\index{Johnson, W. E.!cumulative@{and cumulative formula}}% +This result is exceedingly interesting. Mr.~Johnson is the first to +arrive at the simple relation, expressed above, between the direct +and the inverse formulæ: viz.\ that the \emph{same} coefficient is required +for correcting the simple formulæ of multiplication in +both cases. As he remarks, however, while the direct formula +gives the required probability directly by multiplication, the +inverse formula gives only the \emph{comparative} probability. + +(46.2) If $x, x', x'' \ldots$ are exclusive and exhaustive alternatives, +\begin{DPgather*} +x/habc\ldots = \frac + {(x/h)^{-n} · \{a^{xh}b^{xh}c \ldots\}\Prod\limits^{n-1} x/ah} + {\Sum\left[(x'/h)^{-n}·\{a^{x'h}b^{x'h}c \ldots\}\Prod\limits^{n-1}x'/ah\right]},\\ +% +\lintertext{since} x/habc \ldots\propto (x/h)^{-n}\{a^{xh}b^{xh}c \ldots\}\Prod\limits^{n-1}x/ah,\\ +% +\lintertext{and} \Sum x'/habc \ldots = 1 \rintertext{\llap{by (24.7).}} +\end{DPgather*} +\begin{DPalign*}[m] +\lintertext{\indent(47).} +\frac{x/habc \ldots}{x/h} + &= \frac{a/h · b/h · c/h \ldots}{abc \ldots/h} \\ + &\qquad{} · \frac{abc \ldots /xh}{a/xh · b/xh · c/xh \ldots} + · \left[\frac{x/ah}{x/h} · \frac{x/bh}{x/h} \ldots\right]. +\end{DPalign*} +\begin{DPgather*} +\lintertext{For} abc \ldots x/h = x/h · abc \ldots/xh, +\end{DPgather*} +\begin{multline*} +\therefore \frac{abc \ldots x/h}{abc \ldots/h· x/h} + = \frac{abc \ldots /xh}{abc \ldots /h} + = \frac{a/h · b/h · c/h \ldots}{abc \ldots /h} \\ + {} · \frac{abc \ldots /xh}{a/xh · b/xh · c/xh\ldots} + · \left[\frac{a/xh}{a/h} · \frac{b/xh}{b/h}\ldots\right], +\end{multline*} +whence the result, since $\dfrac{a/xh}{a/h} = \dfrac{x/ah}{x/h}$, etc. + +(47.1) The above formula may be written in the condensed +form +\[ +\{abc \ldots ^{h}x\} + = \frac{1}{\{a^{h}b^{h}c^{h} \ldots\}}· + \{a^{xh}b^{xh}c^{xh} \ldots\} · + [\{a^{h}x\} · \{b^{h}x\} · \{c^{h}x\} \ldots]. +\] +\begin{DPgather*}[m] +\lintertext{\indent(48.)} +\frac{\{x/h\}^{n}x/habc \ldots}{\{\bar{x}/h\}^{n}\bar{x}/habc \ldots} + = \frac{\{a^{xh}b^{xh}c^{xh} \ldots\}} + {\{a^{\bar{x}h}b^{\bar{x}h}c^{\bar{x}h} \ldots\}} + · \frac{x/ah · x/bh · x/ch \ldots} + {\bar{x}/ah · \bar{x}/bh · \bar{x}/ch \ldots}. +\end{DPgather*} +This follows at once from~(46.2), since $x$~and~$\bar{x}$ are exclusive and +exhaustive alternatives. (It is assumed that $xh$,~$\bar{x}h$, and~$ah$, +etc., are not inconsistent.) +%% -----File: 165.png---Folio 154------- + +This formula gives $x/habc \ldots$ in terms of $x/ah$, $x/bh$, etc., +together with the three values $x/h$, $\{a^{xh}b^{xh}c^{xh} \ldots\}$, and +$\{a^{\bar{x}h}b^{\bar{x}h}c^{\bar{x}h} \ldots\}$. +\begin{DPgather*}[m] +\lintertext{\indent(48.1)} \dfrac{x/habcd \ldots}{\bar{x}/habcd \ldots} : \dfrac{x/hbcd \ldots}{\bar{x}/hbcd \ldots} += \dfrac{\{a^{xh}bcd \ldots\} · x/ah}{\{a^{\bar{x}h}bcd \ldots\} · \bar{x}/ah} : \dfrac{x/h}{\bar{x}/h}. +\end{DPgather*} +This gives the effect on the odds ($\text{prob.\ } x : \text{prob.\ } \bar{x}$) of the extra +knowledge~$a$. + +(49) When several data co-operate as evidence in favour of a +proposition, they continually strengthen their own mutual +probabilities, on the assumption that when the proposition +is known to be true or to be false the data jointly are not +counterdependent. + +\Ie\ if $\{a^{xh}b^{xh}c \ldots\}$ and $\{a^{\bar{x}h}b^{\bar{x}h}c \ldots\}$ are not less than +unity, and $x/kh > x/h$ where $k$ is any of the data $a, b, c \ldots$, then +$\{a^hb^hc^hd \ldots\}$ beginning with unity, continually increases, as +the number of its terms is increased. +\begin{DPalign*}[m] +abc \ldots/h + & = xabc \ldots/h + \bar{x}abc \ldots/h +\rintertext{\llap{by (24.2).}} \\ + & = x/h · abc \ldots/xh + \bar{x}/h · abc \ldots/\bar{x}h.\\ + & \geq x/h · \Prod a/xh · b/xh \ldots + + \bar{x}h\Prod a/\bar{x}h · b/\bar{x}h \ldots \\ +\intertext{\centering(since $\{a^{xh}b^{xh}c \ldots\}$ and $\{a^{\bar{x}h}b^{\bar{x}h}c \ldots\}$ are not less than unity),} + & \geq x/h · + \Prod\left[\frac{ax/h}{x/h} + · \frac{bx/h}{x/h} \ldots\right] + + \bar{x}/h · + \Prod \left[\frac{a\bar{x}/h}{\bar{x}/h} + · \frac{b\bar{x}/h}{\bar{x}/h} \ldots\right], +\end{DPalign*} +\begin{multline*} +\therefore \frac{abc \ldots/h}{\Prod[a/h · b/h \ldots]} + \geq x/h · \Prod\left[\frac{x/ah}{x/h} + · \frac{x/bx}{x/h} \ldots\right] \\ %[** TN: Moved line break] + + \bar{x}/h · \Prod\Big[\frac{\bar{x}/ah}{\bar{x}/h} + · \frac{\bar{x}/bh}{\bar{x}/h} \ldots\Big]. +\end{multline*} + +We can show that each additional piece of evidence $a, b, c \ldots$ +increases the value of this expression. For let $x/h · G + \bar{x}/h · G'$ be +its value when all the evidence up to~$k$ exclusive is taken, so that +\[ +x/kh · G + \bar{x}/kh · G' +\] +is its value when $k$~is taken. Now $G > G'$ since $x/ah > x/h$, etc., +and $\bar{x}/ah < \bar{x}/h$, etc., by the hypothesis that the evidence +favours~$x$; and for the same reason $x/kh - x/h$, which is equal +to $\bar{x}/h - \bar{x}/kh$, is positive. +\begin{DPalign*} +\therefore G(x/kh - x/h) & > G'(\bar{x}/h - \bar{x}/kh),\\ +\lintertext{\ie} +x/kh · G + \bar{x}/kh · G' + & > x/h · G + \bar{x}/h · G', +\end{DPalign*} +whence the result. +%% -----File: 166.png---Folio 155------- +\index{Johnson, W. E.!cumulative@{and cumulative formula}}% +\index{McColl, and symbolic probability}% +\index{Middle Term, Fallacy of}% + +(49.1) The above proposition can be generalised for the +case of exclusive alternatives $x, x', x'' \ldots$ (in place of $x$,~$\bar{x}$). +\begin{DPalign*} +\lintertext{\rlap{For $\{a^h b^h c^h \ldots\}$}}\\ +&\indent = x/h . \{a^{xh} b^{xh} c \ldots\} \{a^h x\} \{b^h x\} \{c^h x\} \ldots\\ +&\indent + x'/h . \{a^{x'h} b^{x'h} c \ldots\} \{a^h x'\} \{b^h x'\} \{c^h x'\} \ldots\\ +&\indent + x''/h . \{a^{x''h} b^{x''h} c \ldots\} \{a^h x''\} \{b^h x''\} \{c^h x''\} \ldots + \ldots; +\end{DPalign*} +from which, it follows that, if $\{a^{xh} b^{xh} c \ldots\} \text{etc.} \nless 1$, and if +$\DPtypo{\{a^h x\} - 1}{\{a^hx - 1\}}$, $\{b^h x - 1\}$, $\{c^h x - 1\}$, etc., have the same sign, then +$\{a^h b^h c \ldots\}$ is increasing (with the number of letters) from unity. + +Mr.~Johnson describes this result as a generalisation of +the corrected ``middle term fallacy'' (see \Chapref[Chap.]{V}. §\;4). + + +\Appendix{ON SYMBOLIC TREATMENTS OF PROBABILITY} + +\First{The} use of the symbol~$0$ for impossibility and $1$~for certainty was +first introduced by Leibnitz in a very early pamphlet, entitled +\textit{Specimen certitudinis seu demonstrationum in jure, exhibitum in +doctrina conditionum}, published in 1665 (\textit{vide} Couturat, \textit{Logique de +Leibnitz}, p.~553). Leibnitz represented intermediate degrees of +probability by the sign~$\frac{1}{2}$, meaning, however, by this symbol a +\emph{variable} between $0$~and~$1$. + +Several modern writers have made some attempt at a symbolic +treatment of Probability. But with the exception of Boole, whose +\index{Boole!symbolic probability@{and symbolic probability}}% +methods I have discussed in detail in Chapters \Chapref[]{XV}., \Chapref[]{XVI}., and +\Chapref[]{XVII}., no one has worked out anything very elaborate. + +Mr.~McColl published a number of brief notes on Probability of +considerable interest---see especially his \textit{Symbolic Logic}, \textit{Sixth Paper +on the Calculus of Equivalent Statements}, and \textit{On the Growth and Use +of a Symbolical Language}. The conception of probability as a relation +between propositions underlies his symbolism, as it does mine.\footnote + {I did not come across these notes until my own method was considerably + developed. Mr.~McColl has been the first to use the fundamental symbol of + Probability.} +The probability of $a$, relative to the \textit{à~priori} premiss~$h$, he writes~$\dfrac{a}{\epsilon}$; and +the probability, given $b$~in addition to the \textit{à~priori} premiss, he writes~$\dfrac{a}{b}$. +Thus $\dfrac{a}{\epsilon} = \dfrac{a}{h}$, and $\dfrac{a}{b} = a/bh$. The difference $\dfrac{a}{b} - \dfrac{a}{\epsilon}$, \ie\ the change +in the probability of~$a$ brought about by the addition of~$b$ to the +evidence, he calls `the dependence of the statement $a$ upon the statement~$b$,' +%% -----File: 167.png---Folio 156------- +\index{Gilman, B. I., and symbolic probability}% +\index{Lammel@{Lämmel}!symbolic probability@{and symbolic probability}}% +and denotes it by~$\delta\dfrac{a}{b}$. Thus $\delta\dfrac{a}{b}=0$, where, in my terminology, +$b$ is \emph{irrelevant} to~$a$ on evidence~$h$. The multiplication and +addition formulæ he gives as follows: +$\dfrac{ab}{\epsilon} = \dfrac{a}{\epsilon}·\dfrac{b}{a} + = \dfrac{b}{\epsilon}·\dfrac{a}{b}$. +\begin{DPgather*} +\frac{a+b}{\epsilon} + = \frac{a}{\epsilon} + \frac{b}{\epsilon} - \frac{ab}{\epsilon}.\\ +\lintertext{Also} +\delta\frac{a}{b} + = \frac{A}{B}\delta \frac{b}{a}, \text{ where } A = \frac{a}{\epsilon}. +\end{DPgather*} + +It is surprising how little use he succeeds in making of these good +results. He arrives, however, at the inverse formula in the shape--- +\[ +\frac{c_r}{v} + = \frac{\dfrac{c_r}{\epsilon}\dfrac{v}{c_r}} + {\Sum_{r=1}^{r=n}\dfrac{c_r}{\epsilon}·\dfrac{v}{c_r}}, +\] +where $c_{1} \ldots c_{n}$ are a series of mutually exclusive causes of the event~$v$ +and include all possible causes of it; reaching it as a generalisation +of the proposition +\[ +\frac{a}{b} = \frac{\dfrac{a}{\epsilon}·\dfrac{b}{a}} + {\dfrac{a}{\epsilon}·\dfrac{b}{a}+\dfrac{\bar{a}}{\epsilon}·\dfrac{b}{\bar{a}}} +\] + +In a paper entitled ``Operations in Relative Number with Applications +to the Theory of Probabilities,''\footnote + {Published in the volume of Johns Hopkins \textit{Studies in Logic}.} +Mr.~B.~I. Gilman attempted +a symbolic treatment based on a frequency theory similar to Venn's, +but made more precise and more consistent with itself: ``Probability +has to do, not with individual events, but with classes of events; and +not with one class, but with a pair of classes,---the one containing, +the other contained. The latter being the one with which we are +principally concerned, we speak, by an ellipsis, of its probability +without mentioning the containing class; but in reality probability +is a ratio, and to define it we must have both correlates given.'' But +Mr.~Gilman's symbolic treatment leads to very little. More recently +R.~Laemmel, in his \textit{Untersuchungen über die Ermittlung von Wahrscheinlichkeiten}, +made a beginning on somewhat similar lines; but +in his case also the symbolic treatment leads to no substantial results. + +Apart from the writers mentioned above, there are a few who +have incidentally made use of a probability symbol. It will be +\index{Czuber!symbolic probability@{and symbolic probability}}% +sufficient to cite Czuber.\footnote + {\textit{Wahrscheinlichkeitsrechnung}, vol.~i.\ pp.~43--48.} +He denotes the probability of an event~$E$ +%% -----File: 168.png---Folio 157------- +\index{Poretzki, Platon S., and symbolic!probability}% +\index{Schröder and symbolic probability}% +by~$W(E)$, and the probability of the event~$E$ given the occurrence of +an event~$F$ by~$W_{F}(E)$. He uses this symbol to give $W_{F}(E) = W_{\bar{F}}(E)$ +as the criterion of the independence of the events $E$~and~$F$ ($\bar{F}$ denoting +the non-occurrence of~$F$); $W_{F}(E)=1$, as the expression of the fact +that $E$ is a necessary consequence of~$F$; and one or two other similar +results. + +Finally there is in the \textit{Bulletin of the Physico-mathematical Society +of Kazan} for 1887 a memoir in Russian by Platon S. Porotzki entitled +``A Solution of the General Problem of the Theory of Probability by +Means of Mathematical Logic.'' I have seen it stated that Schröder +intended to publish ultimately a symbolic treatment of Probability. +Whether he had prepared any manuscript on the subject before his +death I do not know. +%% -----File: 169.png---Folio 158------- +\index{Addition, of probabilities!measurement@{and measurement}}% +\index{Measurement of Probability}% + + +\Chapter{XV}{Numerical Measurement and Approximation of Probabilities} + +\Paragraph{1.} \First{The} possibility of numerical measurement, mentioned at +the close of \Chapref{III}., arises out of the Addition Theorem~(24.1). +In introducing the definitions and the axiom, which are +required in order to make the convention of numerical measurement +operative, we may appear, as in the case of the original +definitions of Addition and Multiplication, to be arguing in an +artificial way. This appearance is due, here as in \Chapref{XII}., +to our having given the names of addition and multiplication to +certain processes of compounding probabilities \emph{in advance of} +postulating that the processes in question have the properties +commonly associated with these names. As common sense is +hasty to impute the properties as soon as it hears the names, it +may overlook the necessity of formally introducing them. + +\Paragraph{2.} The definitions and the axiom which are needed in order +to give a meaning to numerical measurement are the following:--- +\begin{DPalign*}[m] +\lintertext{\indent XVII.} + & a/h + \left\{a/h + \left[a/h + (a/h + \ldots\ r \text{ terms})\right]\right\} + = r · a/h. + \rintertext{Def.}\\ +% +\lintertext{\rlap{\indent XVIII.}} + &\text{If $r · a/h = b/f$, then $a/h = \dfrac{1}{r} · b/f$.} + \rintertext{Def.}\\ +% +\lintertext{\indent XIX.} + &\text{If $b/f = q · c/g$, then $\dfrac{1}{r} · b/f = \dfrac{q}{r}\, c/g$.} + \rintertext{Def.} +\end{DPalign*} +Thus if $b/h = a/h + a/h + \ldots$ to $r$~terms, then the probability +$b/h$ is said to be $r$~times the probability~$a/h$; hence if $ab/h=0$ and +$a/h = b/h$, the probability $(a+b)/h$ is \emph{twice} the probability~$a/h$. +If $a$~and~$b$ are exhaustive as well as exclusive alternatives relatively +to~$h$, so that $(a+b)/h = 1$, since we take the relation of +certainty as our unit, then $a/h = b/h = \frac{1}{2}$. + +We also need the following axiom postulating the existence of +relations of probability corresponding to all proper fractions: +%% -----File: 170.png---Folio 159------- + +(vii.) If $q$~and~$r$ are any finite integers and $q < r$, there exists +a relation of probability which can be expressed, by means of the +convention of the foregoing definitions, as~$\dfrac{q}{r}$. + +\Paragraph{3.} From these axioms and definitions combined with those +of \Chapref{XII}., it is easy to show (certainty being represented +by unity and impossibility by zero) that we can manipulate +according to the ordinary laws of arithmetic the ``numbers'' +which by means of a special convention we have thus introduced +to represent probabilities. Of the kind of proofs necessary +for the complete demonstration of this the following is given as +an example: +\begin{DPgather*} +\lintertext{\indent(50)} + \text{If $a/f = \dfrac{1}{m}$ and $b/h = \dfrac{1}{n}$, $a/f + b/h = \dfrac{m + n}{mn}$.} +\end{DPgather*} + +Let the probability $\dfrac{1}{mn} = P$, which exists by~(vii.), +\begin{DPalign*} +\lintertext{then} + n · P &= \frac{1}{m} = a/f + \rintertext{by~(XIX.),}\\ +% +\lintertext{and} + m · P &= \frac{1}{n} = b/h, +\end{DPalign*} +\begin{DPalign*} +\therefore a/f + b/h + &= n · P + m · P, \text{ if this probability exists,}\\ + &= P + P \ldots \text{ to $n$~terms} + P + P\ldots \text{ to $m$~terms,}\\ + &= P + P \ldots \text{ to $m+n$~terms,}\\ + &= (m+n)P = \frac{m+n}{mn} + \rintertext{\PadTxt[r]{}{by~(XIX.).}} +\end{DPalign*} +This probability exists in virtue of~(vii.). + +\Paragraph{4.} Many probabilities---in fact all those which are equal to +the probability of some other argument which has the same +premiss and of which the conclusion is incompatible with that +of the original argument---are numerically measurable in the +sense that there is \emph{some} other probability with which they are +comparable in the manner described above. But they are not +numerically measurable in the most usual sense, unless the probability +with which they are thus comparable is the relation +of certainty. The conditions under which a probability $a/h$ is +numerically measurable and equal to~$\dfrac{q}{r}$ are easily seen. It +%% -----File: 171.png---Folio 160------- +\index{Probability, and relevant knowledge!comparison of}% +is necessary that there should exist probabilities $a_1/h_1$, $a_2/h_2 \ldots$, +$a_q/h_q \ldots a_r/h_r$, such that +\begin{gather*} +a_1/h_1 = a_2/h_2 = \ldots = a_q/h_q = \ldots = a_r/h_r, \\ +a/h = \textstyle\Sum_1^q a_s/h_s, \text{ and } + \textstyle\Sum_1^r a_s/h_s = 1. +\end{gather*} + +If $a/h = \dfrac{q_1}{r_1}$ and $b/h = \dfrac{q_2}{r_2}$, it follows from~(32) that $ab/h = \dfrac{q_1q_2}{r_1r_2}$ +only if $a/h$~and~$b/h$ are independent arguments. Unless, therefore, +we are dealing with independent arguments, we cannot +apply detailed mathematical reasoning even when the individual +probabilities are numerically measurable. The greater part of +mathematical probability, therefore, is concerned with arguments +which are \emph{both} independent \emph{and} numerically measurable. + +\Paragraph{5.} It is evident that the cases in which exact numerical +measurement is possible are a very limited class, generally +dependent on evidence which warrants a judgment of equiprobability +by an application of the Principle of Indifference. +\index{Principle of Indifference!measurement@{and measurement}}% +The fuller the evidence upon which we rely, the less likely is it to +be perfectly symmetrical in its bearing on the various alternatives, +and the more likely is it to contain some piece of relevant information +favouring one of them. In actual reasoning, therefore, +perfectly equal probabilities, and hence exact numerical measures, +will occur comparatively seldom. + +The sphere of inexact numerical comparison is not, however, +quite so limited. Many probabilities, which are incapable of +numerical measurement, can be placed nevertheless \emph{between} +numerical limits. And by taking particular non-numerical +probabilities as standards a great number of comparisons or +approximate measurements become possible. If we can place +a probability in an order of magnitude with some standard probability, +we can obtain its approximate measure by comparison. + +This method is frequently adopted in common discourse. +When we ask how probable something is, we often put our question +in the form---Is it more or less probable than so and so?---where +`so and so' is some comparable and better known probability. +We may thus obtain information in cases where it would +be impossible to ascribe any \emph{number} to the probability in question. +Darwin was giving a numerical limit to a non-numerical probability +%% -----File: 172.png---Folio 161------- +when he said of a conversation with Lyell that he thought +it no more likely that he should be right in nearly all points than +that he should toss up a penny and get heads twenty times +running.\footnote + {\textit{Life and Letters}, vol.~ii.\ p.~240.} +Similar cases and others also, where the probability +which is taken as the standard of comparison is itself non-numerical +and not, as in Darwin's instance, a numerical one, +\index{Darwin!Lyell@{and Lyell}}% +will readily occur to the reader. + +A specially important case of approximate comparison is that +of `practical certainty.' This differs from logical certainty since +its contradictory is not impossible, but we are in practice completely +satisfied with any probability which approaches such +a limit. The phrase has naturally not been used with complete +precision; but in its most useful sense it is essentially non-numerical---we +cannot measure practical certainty in terms of +logical certainty. We can only explain how great practical +certainty is by giving instances. We may say, for instance, that +it is measured by the probability of the sun's rising to-morrow. +The type which we shall be most likely to take will be that of a +well-verified induction. + +\Paragraph{6.} Most of such comparisons must be based on the principles +of \Chapref{V}\@. It is possible, however, to develop a systematic +method of approximation which may be occasionally useful. +The theorems given below are chiefly suggested by some work +of Boole's. His theorems were introduced for a different purpose, +\index{Boole!approximation@{and approximation}}% +and he does not seem to have realised this interesting +application of them; but analytically his problem is identical +with that of approximation.\footnote + {In Boole's \textit{Calculus} we are apt to be left with an equation of the second + or of an even higher degree from which to derive the probability of the conclusion; + and Boole introduced these methods in order to determine which of the + several roots of his equation should be taken as giving the true solution of the + problem in probability. In each case he shows that that root must be chosen + which lies between certain limits, and that only one root satisfies this condition. + The general theory to be applied in such cases is expounded by him in Chapter~XIX. + of \textit{The Laws of Thought}, which is entitled ``On Statistical Conditions.'' + But the solution given in that chapter is awkward and unsatisfactory, and he + subsequently published a much better method in the \textit{Philosophical Magazine} + for 1854 (4th~series, vol.~viii.)\ under the title ``On the Conditions by which the + Solutions of Questions in the Theory of Probabilities are limited.''} +This method of approximation +is also substantially the same analytically as that dealt with by +\index{Yule!approximation@{and approximation}}% +Mr.~Yule under the heading of \textit{Consistence}.\footnote + {\textit{Theory of Statistics}, chap.~ii.} +%% -----File: 173.png---Folio 162------- + +(51) $xy/h$ always lies between\footnote + {In this and the following theorems the term `between' includes the + limits.} +$x/h$ and $x/h + y/h - 1$ and +between $y/h$ and $x/h + y/h - 1$. +\begin{DPalign*} +\lintertext{For} +xy/h &= x/h - x\bar{y}/h + \rintertext{by (24.2),}\\ +% + &= x/h - \bar{y}/h · x/\bar{y}/h + \rintertext{by~X.} +\end{DPalign*} +\begin{DPgather*} +\lintertext{Now} +\text{$x/\bar{y}h$ lies between $0$~and~$1$} + \rintertext{by (2)~and~(3),} \\ +% +\text{$\therefore xy/h$ lies between $x/h$ and $x/h - \bar{y}/h$,}\\ +\text{\ie\ between $x/h$ and $x/h + y/h - 1$.} +\end{DPgather*} +As $xy/h \nless 0$, the above limits may be replaced by $x/h$ and~$0$, if +$x/h + y/h - 1 < 0$. + +We thus have limits for~$xy/h$, close enough sometimes to be +useful, which are available whether or not $x/h$ and~$y/h$ are \emph{independent} +arguments. For instance, if $y/h$ is nearly certain, $xy/h += x/h$ nearly, quite independently of whether or not $x$ and~$y$ are +independent. This is obvious; but it is useful to have a simple +and general formula for all such cases. + +(52) $x_1x_2 \ldots x_{n+1}/h$ is always greater than $\Sum_1^{n+1} x_r/h - n$. +\begin{DPalign*}[m] +\lintertext{For by~(51)} +x_1x_2 \ldots x_{n+1}/h + &> x_1x_2 \ldots x_n/h + x_{n+1}/h - 1\\ + &> x_1x_2 \ldots x_{n-1}/h + x_n/h + x_{n+1}/h - 2, +\end{DPalign*} +and so on. + +(53) $xy/h + \bar{x}\bar{y}/h$ is always less than $x/h - y/h + 1$, and less +than $y/h - x/h + 1$. +\begin{DPalign*}[m] +\lintertext{\rlap{For as in~(51)}} +xy/h &= x/h - x\bar{y}/h\\ +\lintertext{and} +\bar{x}\bar{y}/h &= \bar{y}/h - x\bar{y}/h,\\ +\therefore xy/h + \bar{x}\bar{y}/h &= x/h - y/h + 1 - 2x\bar{y}/h, +\end{DPalign*} +whence the required result. + +(54) $xy/h -\bar{x}\bar{y}/h = x/h + y/h - 1$. + +This proposition, which follows immediately from the above, +is really out of place here. But its close connection with conclusions +(51)~and~(53) is obvious. It is slightly unexpected, +perhaps, that the difference of the probabilities that both of two +events will occur and that neither of them will, is independent of +whether or not the events themselves are independent. + +\Paragraph{7.} It is not worth while to work out more of these results here. +Some less systematic approximations of the same kind are given +in the course of the solutions in \Chapref{XVII}\@. + +In seeking to compare the degree of one probability with that +of another we may desire to get rid of one of the terms, on account +%% -----File: 174.png---Folio 163------- +of its not being comparable with any of our standard probabilities. +Thus our object in general is to eliminate a given symbol of +quantity from a set of equations or inequations. If, for instance, +we are to obtain numerical limits within which our probability +must lie, we must eliminate from the result those probabilities +which are non-numerical. This is the general problem for +solution. + +(55)~A general method of solving these problems when we +can throw our equations into a linear shape so far as all symbols +of probability are concerned, is best shown in the following +example:--- +\begin{DPalign*} +\lintertext{\indent\PadTxt[l]{}{Suppose we have}} + \lambda + \nu &= a \rintertext{(i.)}\\ + \lambda + \sigma &= b \rintertext{(ii.)}\\ + \lambda + \nu + \sigma &= c \rintertext{(iii.)}\\ + \lambda + \mu + \nu + \rho &= d \rintertext{(iv.)}\\ + \lambda + \mu + \sigma + \tau &= e \rintertext{(v.)}\\ + \lambda + \mu + \nu + \rho + \sigma + \tau + \upsilon &= 1 \rintertext{(vi.)} +\end{DPalign*} +where $\lambda$, $\mu$, $\nu$, $\rho$, $\sigma$, $\tau$, $\upsilon$ represent probabilities which are to be +eliminated, and limits are to be found for~$c$ in terms of the +standard probabilities $a$,~$b$, $d$,~$e$, and~$1$. + +$\lambda$,~$\mu$,~etc., must all lie between $0$~and~$1$. + +From (i.)\ and~(iii.)\ $\sigma = c - a$; from (ii.)\ and~(iii.)\ $\nu = c - b$. + +From (i.),~(ii.), and~(iii.)\ $\lambda = a + b - c$, \\ +whence %[** TN: Not using condensed intertext] +\begin{gather*} + c - a \eqslantgtr 0,\quad c - b \eqslantgtr 0,\quad a + b - c \eqslantgtr 0,\\ +\intertext{substituting for $\sigma$,~$\nu$,~$\lambda$ in (iv.), (v.), and~(vi.)} +\mu + \rho = d - a,\quad +\mu + \tau = e - b,\quad +\mu + \rho + \tau + \upsilon = 1 - c, \\ +\intertext{whence} +\rho = d - a - \mu,\quad +\tau = e - b - \mu,\quad +\upsilon = 1 - c - d + a - e + b + \mu, \\ +\therefore d - a - \mu \eqslantgtr 0,\quad + e - b - \mu \eqslantgtr 0,\quad + 1 - c - d + a - e + b + \mu \eqslantgtr 0. \\ +\intertext{We have still to eliminate~$\mu$.} +\mu \eqslantgtr d - a,\quad \mu \eqslantgtr e - b, \\ +\mu \eqslantgtr c + d + e - a - b - 1, \\ +\text{$\therefore d - a \eqslantgtr c + d + e - a - b - 1$ and + $e - b \eqslantgtr c + d + e - a - b - 1$.} +\end{gather*} +Hence we have: + +Upper limits of~$c$:---$b + 1 - e$, $a + 1 - d$, $a + b$ (whichever is least), + +Lower limits of~$c$:---$a$, $b$ (whichever is greatest). + +This example, which is only slightly modified from one given +by Boole, represents the actual conditions of a well-known +problem in probability. +%% -----File: 175.png---Folio 164------- +\index{Calculus of Probability}% +\index{Causality!independence@{and independence}}% +\index{Independence, for knowledge!of events}% + + +\Chapter{XVI}{Observations on the Theorems of Chapter XIV. and their +Developments, including Testimony} + +\Paragraph{1.} \First{In} Definition~XIII. of \Chapref{XII}. a meaning was given to +the statement that $a_1/h$ and~$a_2/h$ are independent arguments. +In Theorem~(33) of \Chapref{XIV}. it was shown that, if $a_1/h$~and~$a_2/h$ +are independent, $a_1a_2/h = a_1/h · a_2/h$. Thus where on given +evidence there is independence between $a_1$~and~$a_2$, the probability +on this evidence of~$a_1a_2$ jointly is the product of the probabilities +of $a_1$~and~$a_2$ separately. It is difficult to apply mathematical +reasoning to the Calculus of Probabilities unless this condition +is fulfilled; and the fulfilment of the condition has often been +assumed too lightly. A good many of the most misleading +fallacies in the theory of Probability have been due to a use of +the Multiplication Theorem in its simplified form in cases where +this is illegitimate. + +\Paragraph{2.} These fallacies have been partly due to the absence of +a clear understanding as to what is meant by \emph{Independence.} +Students of Probability have thought of the independence of +events, rather than of the independence of arguments or propositions. +The one phraseology is, perhaps, as legitimate as the +other; but when we speak of the dependence of events, we are +led to believe that the question is one of direct causal dependence, +two events being dependent if the occurrence of one is a part +cause or a possible part cause of the occurrence of the other. In +this sense the result of tossing a coin is dependent on the existence +of bias in the coin or in the method of tossing it, but it is independent +of the actual results of other tosses; immunity from +smallpox is dependent on vaccination, but is independent of +statistical returns relating to immunity; while the testimonies +of two witnesses about the same occurrence are independent, +so long as there is no collusion between them. +%% -----File: 176.png---Folio 165------- +\index{Independence, for knowledge}% + +This sense, which it is not easy to define quite precisely, is +at any rate not the sense with which we are concerned when we +deal with independent probabilities. We are concerned, not with +direct causation of the kind described above, but with `dependence +for knowledge,' with the question whether the \emph{knowledge} of +one fact or event affords any rational ground for expecting the +existence of the other. The dependence for knowledge of two +events usually arises, no doubt, out of causal connection, or what +we term such, of \emph{some} kind. But two events are not independent +for knowledge merely because there is an absence of direct causal +connection between them; nor, on the other hand, are they +necessarily dependent because there is in fact a causal train which +brings them into an indirect connection. The question is whether +there is any \emph{known} probable connection, direct or indirect. A +knowledge of the results of other tossings of a coin may be hardly +less relevant than a knowledge of the bias of the coin; for a +knowledge of these results may be a ground for a probable knowledge +of the bias. There is a similar connection between the +statistics of immunity from smallpox and the causal relations +between vaccination and smallpox. The truthful testimonies +of two witnesses about the same occurrence have a common +cause, namely the occurrence, however independent (in the legal +sense of the absence of collusion) the witnesses may be. For the +purposes of probability two facts are only independent if the +existence of one is no \emph{indication} of anything which might be a +part cause of the other. + +\Paragraph{3.} While dependence and independence may be thus connected +with the conception of causality, it is not convenient to +found our definition of independence upon this connection. A +partial or possible cause involves ideas which are still obscure, and +I have preferred to define independence by reference to the conception +of relevance, which has been already discussed. Whether +there really are material external causal laws, how far causal +connection is distinct from logical connection, and other such +questions, are profoundly associated with the ultimate problems +of logic and probability and with many of the topics, especially +those of \Partref{III}., of this treatise. But I have nothing useful to +say about them. Nearly everything with which I deal can be +expressed in terms of logical relevance. And the relations between +logical relevance and material cause must be left doubtful. +%% -----File: 177.png---Folio 166------- +\index{Cournot, and frequency theory!independence@{and independence}}% + +\Paragraph{4.} It will be useful to give a few examples out of writers who, +as I conceive, have been led into mistakes through misapprehending +the significance of Independence. + +Cournot,\footnote + {For some account of Cournot, see Chapter~XXIV. §\;3.} +in his work on Probability, which after a long period +of neglect has come into high favour with a modern school of +thought in France, distinguishes between `subjective probability' +based on ignorance and `objective probability' based on the +calculation of `objective possibilities,' an `objective possibility' +being a chance event brought about by the combination or convergence +of phenomena belonging to \emph{independent} series. The +existence of objectively chance events depends on his doctrine +that, as there are series of phenomena causally dependent, so +there are others between the causal developments of which there +is independence. These objective possibilities of Cournot's, +whether they be real or fantastic, can have, however, small +importance for the theory of probability. For it is not known +to us what series of phenomena are thus independent. If we had +to wait until we knew phenomena to be independent in this sense +before we could use the simplified multiplication theorem, most +mathematical applications of probability would remain hypothetical. + +\Paragraph{5.} Cournot's `objective probability,' depending wholly on +objective fact, bears some resemblances to the conception in the +minds of those who adopt the frequency theory of probability. +The proper definition of independence on this theory has been +\index{Yule!independence@{and independence}}% +given most clearly by Mr.~Yule\footnote + {``Notes on the Theory of Association of Attributes in Statistics,'' \textit{Biometrika}, + vol.~ii.\ p.~125.} as follows: + +``Two attributes $A$~and~$B$ are usually defined to be independent, +within any given field of observation or `universe,' +when the chance of finding them together is the product of the +chances of finding either of them separately. The physical +meaning of the definition seems rather clearer in a different +form of statement, viz.\ if we define $A$~and~$B$ to be independent +\emph{when the proportion of~$A$'s amongst the~$B$'s of the given universe is +the same as in that universe at large}. If, for instance, the question +were put, `What is the test for independence of smallpox attack +and vaccination?' the natural reply would be, `The percentage +of vaccinated amongst the attacked should be the same as in +the general population.'\ldots'' +%% -----File: 178.png---Folio 167------- +\index{McColl, and symbolic probability!Boole@{and Boole}|inote}% +\index{Wilbraham, H., and Boole|inote}% + +This definition is consistent with the rest of the theory +to which it belongs, but is, at the same time, open to the +general objections to it.\footnote + {See \Chapref{VIII}\@.} +Mr.~Yule admits that $A$~and~$B$ may be +independent in the world at large but not in the world of~$C$'s. +The question therefore arises as to what world given evidence +specifies, and whether any step forward is possible when, as is +generally the case, we do not know for certain what the proportions +in a given world actually are. As in the case of Cournot's +independent series, it is in general impossible that we should +know whether $A$~and~$B$ are or are not independent in this sense. +The logical independence for knowledge which justifies our +reasoning in a certain way must be something different from +either of these objective forms of independence. + +\Paragraph{6.} I come now to Boole's treatment of this subject. The +\index{Boole!independence@{and independence}}% +central error in his system of probability arises out of his giving +two inconsistent definitions of `independence.'\footnote + {Boole's mistake was pointed out, accurately though somewhat obscurely, + by H.~Wilbraham in his review ``On the Theory of Chances developed in Professor + Boole's \textit{Laws of Thought}'' (\textit{Phil.\ Mag.}\ 4th~series, vol.~vii., 1854). Boole + failed to understand the point of Wilbraham's criticism, and replied hotly, + challenging him to impugn any individual results (``Reply to some Observations + published by Mr.~Wilbraham,'' \textit{Phil.\ Mag.}\ 4th~series, vol.~viii., 1854). He + returned to the same question in a paper entitled ``On a General Method in + the Theory of Probabilities,'' \textit{Phil.\ Mag.}\ 4th~series, vol.~viii., 1854, where he + endeavours to support his theory by an appeal to the Principle of Indifference. + McColl, in his ``Sixth Paper on Calculus of Equivalent Statements,'' saw + that Boole's fallacy turned on his definition of Independence; but I do + not think he understood, at least he does not explain, where precisely Boole's + mistake lay.} +He first wins +the reader's acquiescence by giving a perfectly correct definition: +``Two events are said to be independent when the +probability of the happening of either of them is unaffected by +our \emph{expectation} of the occurrence or failure of the other.''\footnote + {\textit{Laws of Thought}, p.~255. The italics in this quotation are mine.} +But +a moment later he interprets the term in quite a different sense; +for, according to Boole's second definition, we must regard the +events as independent unless we are told either that they \emph{must} +concur or that they \emph{cannot} concur. That is to say, they are independent +unless we know for certain that there is, in fact, an +invariable connection between them. ``The simple events, $x$,~$y$,~$z$, +will be said to be \emph{conditioned} when they are not free to occur in +every possible combination; in other words, when some compound +event depending upon them is precluded from occurring\ldots. +%% -----File: 179.png---Folio 168------- +\index{De Morgan!independence@{and independence}}% +Simple unconditioned events are by definition independent.''\footnote + {\textit{Op.~cit.}\ p.~258.} +In fact as long as $xz$ is \emph{possible}, $x$~and~$z$ are independent. This is +plainly inconsistent with Boole's first definition, with which he +makes no attempt to reconcile it. The consequences of his employing +the term independence in a double sense are far-reaching. +For he uses a method of reduction which is only valid when the +arguments to which it is applied are independent in the first +sense, and assumes that it is valid if they are independent in the +second sense. While his theorems are true if all the propositions +or events involved are independent in the first sense, they are not +true, as he supposes them to be, if the events are independent +only in the second sense. In some cases this mistake involves +him in results so paradoxical that they might have led him +to detect his fundamental error.\footnote + {There is an excellent instance of this, \textit{Laws of Thought}, p.~286. Boole + discusses the problem: Given the probability~$p$ of the disjunction `either~$Y$ + is true, or $X$~and~$Y$ are false,' required the probability of the conditional proposition, + `If $X$~is true, $Y$~is true.' The two propositions are formally equivalent; + but Boole, through the error pointed out above, arrives at the result $\dfrac{cp}{1 - p + cp}$, + where $c$~is the probability of `If either $Y$~is true, or $X$~and~$Y$ false, $X$~is true.' + His explanation of the paradox amounts to an assertion that, so long as two + propositions, which are formally equivalent when true, are only probable, they + are not necessarily equivalent.} +Boole was almost certainly +led into this error through supposing that the \textit{data} of a +problem can be of the form, ``Prob.~$x = p$,'' \ie~that it is +sufficient to state that the probability of a proposition is such +and such, without stating to what premisses this probability is +referred.\footnote + {In studying and criticising Boole's work on Probability, it is very important + to take into account the various articles which he contributed to the + \textit{Philosophical Magazine} during 1854, in which the methods of \textit{The Laws of + Thought} are considerably improved and modified. His last and most considered + contribution to Probability is his paper ``On the application of the Theory of + Probabilities to the question of the combination of testimonies or judgments,'' + to be found in the \textit{Edin.\ Phil.\ Trans.}\ vol.~xxi., 1857. This memoir contains a + simplification and general summary of the method originally proposed in \textit{The + Laws of Thought}, and should be regarded as superseding the exposition of that + book. In spite of the error already alluded to, which vitiates many of his + conclusions, the memoir is as full as are his other writings of genius and + originality.} + +It is interesting that De~Morgan should have given, +incidentally, a definition of independence almost identical +with Boole's second definition: ``Two events are independent +if the latter might have existed without the former, or the +%% -----File: 180.png---Folio 169------- +\index{Macfarlane, and independence|inote}% +former without the latter, for anything that we know to the +contrary''\footnote + {``Essay on Probabilities'' in the \textit{Cabinet Encyclopaedia}, p.~26. De~Morgan + is not very consistent with himself in his various distinct treatises on this + subject, and other definitions may be found elsewhere. Boole's second definition + of Independence is also adopted by Macfarlane, \textit{Algebra of Logic}, p.~21.} + +\Paragraph{7.} In many other cases errors have arisen, not through a +misapprehension of the meaning of independence, but merely +through careless assumptions of it, or through enunciating the +Theorem of Multiplication without its qualifying condition. +Mathematicians have been too eager to assume the legitimacy +of those complicated processes of multiplying probabilities, for +which the greater part of the mathematics of probability is +engaged in supplying simplifications and approximate solutions. +Even De~Morgan was careless enough in one of his writings\footnote + {``Theory of Probabilities'' in the \textit{Encyclopaedia Metropolitana}.} +to enunciate the Multiplication Theorem in the following form: +``The probability of the happening of two, three, or more events +is the product of the probabilities of their happening separately +(p.~398)\ldots. Knowing the probability of a compound event, +and that of one of its components, we find the probability +of the other by dividing the first by the second. This is a +mathematical result of the last too obvious to require further +proof (p.~401).'' + +An excellent and classic instance of the danger of wrongful +assumptions of independence is given by the problem of determining +the probability of throwing heads twice in two consecutive +tosses of a coin. The plain man generally assumes without +hesitation that the chance is~$(\frac{1}{2})^2$. For the \textit{à~priori} chance of +heads at the first toss is~$\frac{1}{2}$, and we might naturally suppose that +the two events are independent,---since the mere fact of heads +having appeared once can have no influence on the next toss. +But this is not the case unless we know for certain that the coin +is free from bias. If we do not know whether there is bias, or +which way the bias lies, then it is reasonable to put the probability +somewhat higher than~$(\frac{1}{2})^2$. The \emph{fact} of heads having appeared +at the first toss is not the cause of heads appearing at the second +also, but the \emph{knowledge}, that the coin has fallen heads already, +affects our forecast of its falling thus in the future, since heads in +the past may have been due to a cause which will favour heads +in the future. The possibility of bias in a coin, it may be noticed, +%% -----File: 181.png---Folio 170------- +\index{D'Alembert|inote}% +always favours `runs'; this possibility increases the probability +both of `runs' of heads and of `runs' of tails. + +This point is discussed at some length in \Chapref{XXIX}. and +further examples will be given there. In this chapter, therefore, +I will do no more than refer to an investigation by Laplace and +\index{Laplace!independence@{and independence}}% +to one real and one supposed fallacy of Independence of a type +with which we shall not be concerned in \Chapref{XXIX}\@. + +\Paragraph{8.}\Pagelabel{170} Laplace, in so far as he took account at all of the considerations +explained in §\;7, discussed them under the heading of \textit{Des +inégalités inconnues qui peuvent exister entre les chances que l'on +suppose égales}.\footnote + {\textit{Essai philosophique}, p.~49. See also ``Mémoire sur les Probabilités,'' \textit{Mém.\ + de l'Acad.}\ p.~228, and cp.\ D'Alembert, ``Sur le calcul des probabilités,'' + \textit{Opuscules mathématiques} (1780), vol.~vii.} +In the case, that is to say, of the coin with +unknown bias, he held that the true probability of heads even +at the first toss differed from~$\frac{1}{2}$ by an amount unknown. But +this is not the correct way of looking at the matter. In the +supposed circumstances the \emph{initial} chances for heads and tails +respectively at the first toss really are equal. What is not true +is that the initial probability of `heads twice' is equal to the +probability of `heads once' squared. + +Let us write `heads at first toss' $= h_1$; `heads at second toss' +$= h_2$. Then $h_1/h = h_2/h = \frac{1}{2}$, and $h_1h_2/h = h_2/h_1h · h_1/h$. Hence +$h_1h_2/h = \{h_1/h\}^2$ only if $h_2/h_1h = h_2/h$, \ie~if the knowledge that +heads has fallen at the first toss does not affect in the least the +probability of its falling at the second. In general, it is true that +$h_2/h_1h$ will not differ greatly from~$h_2/h$ (for relative to most hypotheses +heads at the first toss will \emph{not much} influence our expectation +of heads at the second), and $\frac{1}{4}$~will, therefore, give a good approximation +to the required probability. Laplace suggests an ingenious +method by which the divergence may be diminished. If we +throw two coins and define `heads' at any toss as the face thrown +by the second coin, he discusses the probability of `heads twice +running' with the first coin. The solution of this problem +involves, of course, particular assumptions, but they are of a kind +more likely to be realised in practice than the complete absence +of bias. As Laplace does not state them, and as his proof is +incomplete, it may be worth while to give a proof in detail. + +Let $h_1$,~$t_1$, $h_2$,~$t_2$ denote heads and tails respectively with +the first and second coins respectively at the first toss, and +$h_1'$,~$t_1'$, $h_2'$,~$t_2'$ the corresponding events at the second toss, then +%% -----File: 182.png---Folio 171------- +the probability (with the above convention) of `heads twice running,' +\ie~agreement between the two coins twice running, is +\[ +(h_2h_2' + t_2t_2')(h_1h_1' + t_1t_1')/h + = (h_2h_2' + t_2t_2')/(h_1h_1' + t_1t_1', h) · (h_1h_1' + t_1t_1')/h. +\] +Since $h_2h_2'/(h_1h_1' + t_1t_1', h) = t_2t_2'/(h_1h_1' + t_1t_1', h)$ by the Principle +of Indifference, and $h_2h_2't_2t_2'/h = 0$. +\[ +\therefore (h_2h_2' + t_2t_2')/(h_1h_1' + t_1t_1', h) + = 2 · h_2h_2'/(h_1h_1' + t_1t_1', h) + \text{ by~(24.1).} +\] +\begin{DPgather*} +\lintertext{Similarly} + (h_1h_1' + t_1t_1')/h = 2h_1h_1'/h. +\end{DPgather*} + +We may assume that $h_1/h_1'h = h_1/h$, \ie~that heads with one +coin is irrelevant to the probability of heads with the other; and +$h_1/h = h_1'/h = \frac{1}{2}$ by the Principle of Indifference, so that +\index{Principle of Indifference}% +\begin{align*} +(h_1h_1' + t_1t_1')/h = 2\left(\tfrac{1}{2}\right)^2 + &= \tfrac{1}{2}.\\ +\therefore (h_2h_2' + t_2t_2')(h_1h_1' + t_1t_1')/h + &= 2h_2h_2'/(h_1h_1' + t_1t_1', h) · \tfrac{1}{2}\\ + &= h_2h_2'/(h_1h_1' + t_1t_1', h)\\ + &= \tfrac{1}{2} h_2/(h_2', h_1h_1' + t_1t_1', h), +\end{align*} +since, $(h_1h_1' + t_1t_1')$ being irrelevant to $h'_2/h$, $h'_2/(h_1h_1' + t_1t_1',\,h) = +h_2'/h = \frac{1}{2}$. + +Now $h_2/(h_2', h_1h_1' + t_1t_1', h)$ is greater than~$\frac{1}{2}$, since the fact of +the coins having agreed once may be \emph{some} reason for supposing +they will agree again. But it is less than~$h_2/h_1h$: for we may +assume that $h_2/(h_2', h_1h_1' + t_1t_1', h)$ is less than $h_2/(h_2', h_1h_1', h)$, +and also that $h_2/(h_2', h_1h_1', h) = h_2/h_1h$, \ie~that heads twice +running with one coin does not increase the probability of heads +twice running with a different coin. Laplace's method of tossing, +therefore, yields with these assumptions, more or less legitimate +according to the content of~$h$, a probability nearer to~$\frac{1}{4}$ than is +$h_1h_2/h$. If $h_2/(h_2', h_1h_1' + t_1t_1', h) = \frac{1}{2}$, then the probability is +exactly~$\frac{1}{4}$. + +\Paragraph{9.} Two other examples will complete this rather discursive +commentary. It has been supposed that by the Principle of +Indifference the probability of the existence of iron upon Sirius +is~$\frac{1}{2}$, and that similarly the probability of the existence there of +any other element is also~$\frac{1}{2}$. The probability, therefore, that +not one of the $68$~terrestrial elements will be found on Sirius +is~$(\frac{1}{2})^{68}$, and that at least one will be found there is $1 - \left(\frac{1}{2}\right)^{68}$ or +approximately certain. This argument, or a similar one, has +been seriously advanced. It would seem to prove also, amongst +%% -----File: 183.png---Folio 172------- +\index{Kries, von!Principle of Indifference@{and Principle of Indifference}}% +\index{Maxwell|inote}% +\index{Nitsche, A.|inote}% +\index{Stumpf|inote}% +many other things, that at least one college exactly resembling +some college at either Oxford or Cambridge will almost certainly +be found on Sirius. The fallacy is partly due, as has been pointed +out by Von~Kries and others, to an illegitimate use of the Principle +of Indifference. The probability of iron on Sirius is \emph{not}~$\frac{1}{2}$. But +the result is also due to the fallacy of false independence. +It is assumed that the known existence of $67$~terrestrial +elements on Sirius would not increase the probability of the +sixty-eighth's being found there also, and that their known +absence would not decrease the sixty-eighth's probability.\footnote + {See Von Kries, \textit{Die Principien der Wahrscheinlichkeitsrechnung}, p.~10. + Stumpf (\textit{Über den Begriff der mathem.\ Wahrscheinlichkeit}, pp.~71--74) argues that + the fallacy results from not taking into account the fact that there might be as + many metals as atomic weights, and that therefore the chance of iron is~$\dfrac{1}{z}$, where + $z$~is the number of possible atomic weights. A.~Nitsche (\textit{Vierteljsch.\ f.~wissensch.\ + Philos.}, 1892) thinks that the real alternatives are~$0$, or only~$1$, or only~$2\ldots$ or + $68$~terrestrial elements on Sirius, and that these are equally probable, the chance + of each being~$\dfrac{1}{69}$.} + +\Paragraph{10.} The other example is that of Maxwell's classic mistake in +\index{Maxwell!theory of gases@{and theory of gases}}% +the theory of gases.\footnote + {I take the statement of this from Bertrand's \textit{Calcul des probabilités}, p.~30. +\index{Bertrand!Maxwell@{and Maxwell}|inote}% + Let me here quote a precocious passage on Probability regarded as a branch of + Logic, from a letter written by Maxwell in his nineteenth year (1850), before + he came up to Cambridge: ``They say that Understanding ought to work + by the rules of right reason. These rules are, or ought to be, contained in + Logic; but the actual science of logic is conversant at present only with things + either certain, impossible, or entirely doubtful, none of which (fortunately) + we have to reason on. Therefore the true logic for this world is the calculus + of Probabilities, which takes account of the magnitude of the probability + which is, or ought to be, in a reasonable man's mind'' (\textit{Life}, page~143).} +According to this theory molecules of gas +move with great velocity in every direction. Both the directions +and velocities are unknown, but the probability that a molecule +has a given velocity is a function of that velocity and is independent +of the direction. The maximum velocity and the mean +velocity vary with the temperature. Maxwell seeks to +determine, on these conditions alone, the probability that a +molecule has a given velocity. His argument is as follows: + +If $\phi(x)$ represents the probability that the component of +velocity parallel to the axis of~$X$ is~$x$, the probability that the +velocity has components $x$,~$y$,~$z$ parallel to the three axes is +$\phi(x)\phi(y)\phi(z)$. Thus if $F(v)$ represents the probability of a total +velocity~$v$, we have $\phi(x)\phi(y)\phi(z)=F(v)$, where $v^2 = x^2 + y^2 + z^2$. +It is not difficult to deduce from this (assuming that the +%% -----File: 184.png---Folio 173------- +\index{Kries, von!independence@{and independence}}% +\index{Poincaré, Henri!independence@{and independence}}% +functions are analytical) that $\phi(x)$~must be of the form +$Ge^{-k^2x^2}$. + +It is generally agreed at the present time that this result is +erroneous. But the nature of the error is, I think, quite different +from what it is commonly supposed to be. + +Bertrand,\index{Bertrand!independence@{and independence}}\footnote + {\textit{Calcul des probabilités}, p.~30.} +Poincaré,\footnote + {\textit{Calcul des probabilités} (2nd ed.), pp.~41--44.} +and Von~Kries,\footnote + {\textit{Wahrscheinlichkeitsrechnung}, p.~199.} +all cite this argument of +Maxwell's as an illustration of the fallacy of Independence; and +argue that $\phi(x)$,~$\phi(y)$, and~$\phi(z)$ cannot, as he assumes, represent +independent probabilities, if, as he also assumes, the probability +of a velocity is a function of that velocity. But it is not in this +way that the error in the result really arises. If we do not know +\emph{what} function of the velocity the probability of that velocity is, +a knowledge of the velocity parallel to the axes of $x$~and~$y$ tells +us nothing about the velocity parallel to the axis of~$z$. Maxwell +was, I think, quite right to hold that a mere assumption that the +probability of a velocity is \emph{some} function of that velocity, does +\emph{not} interfere with the mutual independence of statements as to +the velocity parallel to each of the three axes. Let us denote +the proposition, `the velocity parallel to the axis of~$X$ is~$x$' by +$X(x)$, the corresponding propositions relative to the axes of $Y$ +and~$Z$ by $Y(y)$ and~$Z(z)$, and the proposition `the total +velocity is~$v$' by~$V(v)$; and let $h$~represent our \textit{à~priori} data. +Then if $X(x)/h = \phi(x)$ it is a justifiable inference from the +Principle of Indifference that $Y(y)/h = \phi(y)$ and $Z(z)/h = \phi(z)$. +Maxwell infers from this that $X(x)Y(y)Z(z)/h = \phi(x)\phi(y)\phi(z)$. +That is to say, he assumes that $Y(y)/X(x) · h = Y(y)/h$ and +that $Z(z)/Y(y) · X(x) · h = Z(z)/h$. I do not agree with the +authorities cited above that this is illegitimate. So long as +we do not know what function of the total velocity the probability +of that velocity is, a knowledge of the velocities parallel +to the axes of $x$ and~$y$ has no bearing on the probability of a given +velocity parallel to the axis of~$z$. But Maxwell goes on to infer +that $X(x)Y(y)Z(z)/h = V(v)/h$ where $v^2 = x^2 + y^2 + z^2$. It is here, +and in a very elementary way, that the error creeps in. The +propositions $X(x)Y(y)Z(z)$ and~$V(v)$ are \emph{not} equivalent. The +latter follows from the former, but the former does not follow +from the latter. There is more than one set of values $x$,~$y$,~$z$, +%% -----File: 185.png---Folio 174------- +\index{Bayes, and Inverse Probability}% +\index{Bernoulli, Daniel, and Inverse Probability}% +\index{Price and Bayes|inote}% +which will yield the same value $v$. Thus the probability $V(v)/h$ +is much greater than the probability $X(x)Y(y)Z(z)/h$. As we do +not know the direction of the total velocity $v$, there are many +ways, not inconsistent with our \textit{data}, of resolving it into components +parallel to the axes. Indeed I think it is a legitimate +extension of the preceding argument to put $V(v)/h = \phi(v)$; for +there is no reason for thinking differently about the direction +$V$ from what we think about the direction $X$. + +A difficulty analogous to this occurs in discussing the problem +of the dispersion of bullets over a target---a subject round which, +on account of a curiosity which it seems to have raised in the +minds of many students of probability, a literature has grown up +of a bulk disproportionate to its importance. + +\Paragraph{11.} I now pass to the Principle of Inverse Probability, a +\index{Inverse Probability}% +theorem of great importance in the history of the subject. With +various arguments which have been based upon it I shall deal +in \Chapref{XXX}\@. But it will be convenient to discuss here the +history of the Principle itself and of attempts at proving it. + +It first makes its appearance somewhat late in the history of +the subject. Not until 1763, when Bayes's theorem was communicated +to the Royal Society,\footnote + {Published in the \textit{Phil.\ Trans.}\ vol.~liii., 1763, pp.~376--398. This Memoir + was communicated by Price after Bayes's death; there was a second Memoir + in the following year (vol.~liv.\ pp.~298--310), to which Price himself made some + contributions. See Todhunter's \textit{History}, pp.~299 \textit{et~seq}. Thomas Bayes was + a dissenting minister of Tunbridge Wells, who was a Fellow of the Royal Society + from 1741 until his death in 1761. A German edition of his contributions to + Probability has been edited by Timerding.} +was a rule for the determination +of inverse probabilities explicitly enunciated. It is true that +solutions to inductive problems requiring an implicit and more +or less fallacious use of the inverse principle had already been +propounded, notably by Daniel Bernoulli in his investigations +into the statistical evidence in favour of inoculation.\footnote + {``Essai d'une nouvelle analyse de la mortalité causée par la petite vérole, + et des avantages de l'inoculation pour la prévenir,'' \textit{Hist.\ de~l'Acad.}, Paris, 1760 + (published 1766). Bernoulli argued that the recorded results of inoculation + rendered it a probable cause of immunity. This is an inverse argument, though + Bayes's theorem is not used in the course of it. See also D.~Bernoulli's \textit{Memoir + on the Inclinations of the Planetary Orbits}.} +But the +appearance of Bayes's \textit{Memoir} marks the beginning of a new +stage of development. It was followed in 1767 by a contribution +\index{Michell!Inverse Probability@{and Inverse Probability}}% +from Michell\footnote + {Michell's argument owes more, perhaps, to Daniel Bernoulli than to + Bayes.} +to the \textit{Philosophical Transactions} on the distribution +%% -----File: 186.png---Folio 175------- +of the stars, to which further reference will be made in +\Chapref{XXV}\@. And in 1774 the rule was clearly, though not +quite accurately, enunciated by Laplace in his ``Mémoire sur +\index{Laplace!Inverse Probability@{and Inverse Probability}}% +la probabilité des causes par les évènemens'' (\textit{Mémoires +présentés à l'Académie des Sciences}, vol.~vi., 1774). He states +the principle as follows (p.~623): + +``Si\Pagelabel{175} un évènement peut être produit par un nombre~$n$ de +causes différentes, les probabilités de l'existence de ces causes +prises de l'évènement sont entre elles comme les probabilités de +l'évènement prises de ces causes; et la probabilité de l'existence +de chacune d'elles est égale à la probabilité de l'évènement prise +de cette cause, divisée par la somme de toutes les probabilités +de l'évènement prises de chacune de ces causes.'' + +He speaks as if he intended to prove this principle, but he only +\DPtypo{give}{gives} explanations and instances without proof. The principle is +not strictly true in the form in which he enunciates it, as will be +seen on reference to theorems~(38) of \Chapref{XIV}.; and the +omission of the necessary qualification has led to a number of +fallacious arguments, some of which will be considered in \Chapref{XXX}\@. + +\Paragraph{12.} The value and originality of Bayes's \textit{Memoir} are considerable, +and Laplace's method probably owes much more to +it than is generally recognised or than was acknowledged by +Laplace. The principle, often called by Bayes's name, does not +appear in his \textit{Memoir} in the shape given it by Laplace and +usually adopted since; but Bayes's enunciation is strictly correct +and his method of arriving at it shows its true logical connection +with more fundamental principles, whereas Laplace's enunciation +gives it the appearance of a \emph{new} principle specially introduced +for the solution of causal problems. The following passage\footnote + {Quoted by Todhunter, \textit{op.~cit.}\ p.~296. Todhunter underrates the importance +\index{Todhunter!Bayes@{and Bayes}}% + of this passage, which he finds unoriginal, yet obscure.} +gives, in my opinion, a right method of approaching the +problem: ``If there be two subsequent events, the probability +of the second~$\dfrac{b}{N}$ and the probability of both together~$\dfrac{P}{N}$, and, it +being first discovered that the second event has happened, from +hence I guess that the first event has also happened, the probability +I am in the right is~$\dfrac{P}{b}$.'' If the occurrence of the first event +%% -----File: 187.png---Folio 176------- +\index{Donkin, W. F.!Inverse Probability@{and Inverse Probability}}% +\index{Kries, von!Inverse Probability@{and Inverse Probability}}% +\index{McColl, and symbolic probability!Inverse Probability@{and Inverse Probability}}% +\index{Markoff, A. A.!Inverse Probability@{and Inverse Probability}}% +is denoted by~$a$ and of the second by~$b$, this corresponds to +$ab/h = a/bh · b/h$ and therefore $a/bh = \dfrac{ab/h}{b/h}$; for $ab/h = \dfrac{P}{N}$, $b/h = \dfrac{b}{N}$, +$a/bh = \dfrac{P}{b}$. The direct and indeed fundamental dependence of the +inverse principle on the rule for compound probabilities was not +appreciated by Laplace. + +\Paragraph{13.} A number of proofs of the theorem have been attempted +since Laplace's time, but most of them are not very satisfactory, +and are generally couched in such a form that they do no more +than recommend the plausibility of their thesis. Mr.~McColl\footnote + {``Sixth Paper on the Calculus of Equivalent Statements,'' \textit{Proc.\ Lond.\ + Math.\ Soc\DPtypo{}{.}}, 1897, vol.~xxviii.\ p.~567. See also p.~155 above.} +gave +a symbolic proof, closely resembling theorem~(38) when differences +of symbolism are allowed for; and a very similar proof +has also been given by A.~A. Markoff.\footnote + {\textit{Wahrscheinlichkeitsrechnung}, p.~178.} +I am not acquainted with +any other rigorous discussion of it. + +Von Kries\footnote + {\textit{Die Principien der Wahrscheinlichkeitsrechnung}, pp.~117--121. The above + account of Von Kries's argument is much condensed.} +presents the most interesting and careful example +of a type of proof which has been put forward in one shape or +another by a number of writers. We have initially, according to +this view, a certain number of hypothetical possibilities, all +equally probable, some favourable and some unfavourable to our +conclusion. Experience, or rather knowledge that the event +has happened, rules out a number of these alternatives, and we +are left with a field of possibilities narrower than that with which +we started. Only part of the original field or \textit{Spielraum} of +possibility is now admissible (\textit{zulässig}). Causes have \textit{à~posteriori} +probabilities which are proportional to the extent of their occurrence +in the now restricted field of possibility. + +There is much in this which seems to be true, but it hardly +amounts to a proof. The whole discussion is in reality an +appeal to intuition. For how do we know that the possibilities +admissible \textit{à~posteriori} are still, as they were assumed to be \textit{à~priori}, +equal possibilities? Von~Kries himself notices that there +is a difficulty; and I do not see how he is to avoid it, except by +the introduction of an axiom. + +This was in fact the course taken by Professor Donkin in 1851, +in an article which aroused some interest in the \textit{Philosophical +%% -----File: 188.png---Folio 177------- +\index{Markoff, A. A.|inote}% +Magazine} at the time, but which has since been forgotten. +Donkin's theory is, however, of considerable interest. He laid +down as one of the fundamental principles of probability the +following:\footnote + {``On certain Questions relating to the Theory of Probabilities,'' \textit{Phil.\ Mag\DPtypo{}{.}}\ + 4th~series, vol.~i., 1851.} + +``If there be any number of mutually exclusive hypotheses +$h_1h_2h_3 \ldots$ of which the probabilities relative to a particular state +of information are $p_1p_2p_3 \ldots$, and if new information be gained +which changes the probabilities of some of them, suppose of +$h_{m+1}$~and all that follow, without having otherwise any reference +to the rest, then the probabilities of these latter have the \emph{same +ratios} to one another, \emph{after} the new information, that they had +before.''\footnote + {It is interesting to notice that an axiom, practically equivalent to this, + has been laid down more lately by A.~A. Markoff (\textit{Wahrscheinlichkeitsrechnung}, + p.~8) under the title `Unabhängigkeitsaxiom.'} + +Donkin goes on to say that the most important case is where +the new information consists in the knowledge that some of the +hypotheses must be rejected, without any further information +as to those of the original set which are retained. This is the +proposition which Von~Kries requires. + +As it stands, the phrase ``without having otherwise any +reference to the rest'' obviously lacks precision. An interpretation, +however, \emph{can} be put upon it, with which the principle is +true. If, given the old information and the truth of one of the +hypotheses $h_1 \ldots h_m$ to the exclusion of the rest, the probability +of what is conveyed by the new information is the \emph{same} whichever +of the hypotheses $h_1 \ldots h_m$ has been taken, then Donkin's +principle is valid. For let $a$~be the old information, $a'$~the new, +and let $h_r/a = p_r$, $h_r/aa' = p_r'$; then +\[ +p_r' = h_r/aa' = \frac{h_ra'/a}{a/'a} = \frac{a'/h_ra · p_r}{a'/a}, +\] +$\therefore \dfrac{p_r'}{p_r} = p_s'/p_s$, etc., if $a'/h_ra = a'/h_sa$, which is the condition already +explained. + +\Paragraph{14.} Difficulties connected with the Inverse Principle have +arisen, however, not so much in attempts to \emph{prove} the principle +as in those to \emph{enunciate} it---though it may have been the lack +%% -----File: 189.png---Folio 178------- +\index{De Morgan!Inverse Probability@{and Inverse Probability}}% +\index{Whately and combination of misses}% +of a rigorous proof that has been responsible for the frequent +enunciation of an inaccurate principle. + +It will be noticed that in the formula~(38.2) the \textit{à~priori} +probabilities of the hypotheses $a_1$~and~$a_2$ drop out if $p_1 = p_2$, and +the results can then be expressed in a much simpler shape. This +is the shape in which the principle is enunciated by Laplace for +\index{Laplace!Inverse Probability@{and Inverse Probability}}% +the \emph{general} case,\footnote + {See the passage quoted above, \Pageref{175}.} +and represents the uninstructed view expressed +with great clearness by De~Morgan:\footnote + {``Essay on Probabilities,'' in the \textit{Cabinet Encyclopædia}, p.~27.} +``Causes are likely or unlikely, +just in the same proportion that it is likely or unlikely +that observed events should follow from them. The most +probable cause is that from which the observed event could most +easily have arisen.'' If this were true the principle of Inverse +Probability would certainly be a most powerful weapon of proof, +even equal, perhaps, to the heavy burdens which have been laid +on it. But the proof given in \Chapref{XIV}. makes plain the +necessity in general of taking into account the \textit{à~priori} probabilities +of the possible causes. Apart from formal proof this +necessity commends itself to careful reflection. If a cause is +very improbable in itself, the occurrence of an event, which +might very easily follow from it, is not necessarily, so long as +there are other possible causes, strong evidence in its favour. +Amongst the many writers who, forgetting the theoretic qualification, +have been led into actual error, are philosophers as diverse +\index{Jevons!Inverse Probability@{and Inverse Probability}}% +\index{Sigwart!Inverse Probability@{and inverse probability}}% +as Laplace, De~Morgan, Jevons, and Sigwart, Jevons\footnote + {\textit{Principles of Science}, vol.~i.\ p.~280.} +going +so far as to maintain that the fallacious principle he enunciates +is ``that which common sense leads us to adopt almost instinctively.'' + +\Paragraph{15.} The theory of the combination of premisses dealt with +\index{Combination of premisses}% +in §§\;7,~8 of \Chapref{XIV}. has not often been discussed, and the +history of it is meagre. Archbishop Whately\footnote + {\textit{Logic}, 8th~ed.\ p.~211: ``As in the case of two probable premisses, the + conclusion is not established except upon the supposition of their being \emph{both} + true, so in the case of two distinct and independent indications of the truth + of some proposition, unless \emph{both} of them fail, the proposition must be true: + we therefore multiply together the fractions indicating the probability of the + failure of each---the chances against it---and, the result being the total chances + against the establishment of the conclusion by these arguments, this fraction + being deducted from unity, the remainder gives the probability \emph{for} it. \Eg~a + certain book is conjectured to be by such and such an author, partly, 1st,~from + its resemblance in style to his known works; partly, 2nd,~from its being attributed + to him by some one likely to be pretty well informed. Let the probability + of the conclusion, as deduced from one of these arguments by itself, be supposed~$\frac{2}{5}$, + and in the other case~$\frac{3}{7}$; then the \emph{opposite} probabilities will be $\frac{3}{5}$ and~$\frac{4}{7}$, which + multiplied together give~$\frac{12}{35}$ as the probability against the conclusion\ldots.'' + + The Archbishop's error, in that a negative can always be turned into an + affirmative by a change of verbal expression, was first pointed out by a mere + diocesan, Bishop Terrot, in the \textit{Edin.\ Phil.\ Trans.}\ vol.~xxi. The mistake is well + explained by Boole in the same volume of the \textit{Edin.\ Phil.\ Trans.}: ``A confusion + may here be noted between the probability that a conclusion is proved, and the + probability in favour of a conclusion furnished by evidence which does not prove + it. In the proof and statement of his rule, Archbishop Whately adopts the + former view of the nature of the probabilities concerned in the data. In the + exemplification of it, he adopts the latter.''} +was led astray +%% -----File: 190.png---Folio 179------- +\index{De Morgan!combination@{and combination of premisses}}% +\index{Terrot, Bishop!Whately@{and Whately}|inote}% +\index{Terrot, Bishop!combination@{and combination of premisses}}% +by a superficial error, and De~Morgan, adopting the same mistaken +rule, pushed it to the point of absurdity.\footnote + {``Theory of Probabilities,'' \textit{Encyclopædia Metropolitana}, p.~400. He shows + by means of it that ``if any assertion appear neither likely nor unlikely in + itself, then any logical argument in favour of it, however weak the premisses, + makes it in some degree more likely than not---a theorem which will be readily + admitted on its own evidence.'' He then gives an example: ``\textit{à~priori} + vegetation on the planets is neither likely nor unlikely; suppose argument + from analogy makes it~$\frac{3}{10}$; then the total probability is $\frac{1}{2} + \frac{1}{2}·\frac{3}{10}$ or~$\frac{13}{20}$.'' De~Morgan + seems to accept without hesitation the conclusion to be derived from + this, that everything which is not impossible is as probable as not.} +Bishop Terrot\footnote + {``On the Possibility of combining two or more Probabilities of the same + Event, so as to form one definite Probability,'' \textit{Edin.\ Phil.\ Trans}., 1856, vol.~xxi.} +\index{Boole!Whately@{and Whately}}% +\index{Boole!combination@{and combination of premisses}}% +approached the question more critically. Boole's\footnote + {``On the Application of the Theory of Probabilities to the Question of the + Combination of Testimonies or Judgments,'' \textit{Edin.\ Phil.\ Trans}., 1857, vol.~xxi.} +last and +most considered contribution to the subject of probability dealt +with the same topic. I know of no discussion of it during the +past sixty years. + +Boole's treatment is full and detailed. He states the problem +as follows: ``Required the probability of an event~$z$, when two +circumstances $x$~and~$y$ are known to be present,---the probability +of the event~$z$, when we know only of the existence of the circumstances~$x$, +being~$p$, and the probability, when we only know of +the existence of~$y$, being~$q$.''\footnote + {\textit{Loc.~cit.}\ p.~631. Boole's principle (\textit{loc.~cit.}\ p.~620) that ``the mean strength + of any probabilities of an event which are founded upon different judgments + or observations is to be measured by that supposed probability of the event + \textit{à~priori} which those judgments or observations following thereupon would not + tend to alter,'' is not correct if it means more than that the mean strength of + $z/x$~and~$z/y$ is to be measured by~$z/xy$.} +His solution, however, is vitiated +by the fundamental error examined in §\;6 above. Two of his +conclusions may be mentioned for their plausibility, but neither +is valid. + +``If the causes in operation, or the testimonies borne,'' he +%% -----File: 191.png---Folio 180------- +\index{Cournot, and frequency theory!on testimony}% +\index{Testimony, theory of}% +argues, ``are, separately, such as to leave the mind in a state of +equipoise as respects the event whose probability is sought, +united they will but produce the same effect.'' If, that is to say, +$a/h_1 = \frac{1}{2}$ and $a/h_2 = \frac{1}{2}$, he concludes that $a/h_1h_2 = \frac{1}{2}$. The plausibility +of this is superficial. Consider, for example, the following +instance: $h_1 = $ $A$~is black and $B$~is black or white, $h_2 = $ $B$~is black +and $A$~is black or white, $a = $ both $A$~and~$B$ are black. Boole also +\index{Boole!testimony@{and testimony}}% +concluded without valid reason that $a/h_1h_2$ increases, the greater +the \textit{à~priori} improbability of the combination~$h_1h_2$. + +\Paragraph{16.}\Pagelabel{180} The theory of ``Testimony'' itself, the theory, that is to +say, of the combination of the evidence of witnesses, has occupied +so considerable a space in the traditional treatment of Probability +that it will be worth while to examine it briefly. It may, however, +be safely said that the principal conclusions on the subject set +out by Condorcet, Laplace, Poisson, Cournot, and Boole, are +\index{Condorcet!testimony@{and testimony}}% +\index{Laplace!testimony@{and testimony}}% +\index{Poisson!on testimony}% +demonstrably false. The interest of the discussion is chiefly due +to the memory of these distinguished failures. + +It seems to have been generally believed by these and other +logicians and mathematicians\footnote + {Perhaps M.~Bertrand should be registered as an honourable exception. + At least he points out a precisely analogous fallacy in an example where two + meteorologists prophesy the weather, \textit{Calcul des Probabilités}, p.~31.} +that the probability of two +witnesses speaking the truth, who are independent in the sense +that there is no collusion between them, is always the product +of the probabilities that each of them separately will speak the +truth.\footnote + {\Eg, + \begin{minipage}[t]{3in} + Boole, \textit{Laws of Thought}, p.~279. \\ + De~Morgan, \textit{Formal Logic}, p.~195. \\ + Condorcet, \textit{Essai}, p.~4. \\ + Lacroix, \textit{Traité}, p.~248. \\ + Cournot, \textit{Exposition}, p.~354. \\ + Poisson, \textit{Recherches}, p.~323. + \end{minipage} \\ +This list could be greatly extended.} +On this basis conclusions such as the following, for +example, are arrived at: + +$X$~and~$Y$ are independent witnesses (\ie\ there is no collusion +between them). The probability that $X$~will speak the truth is~$x$, +that $Y$~will speak the truth is~$y$. $X$~and~$Y$ agree in a particular +statement. The chance that this statement is true is +\[ +\frac{xy}{xy + (1 - x)(1 - y)}. +\] +For the chance that they both speak the truth is~$xy$, and the +chance that they both speak falsely is~$(1-x)(1-y)$. As, in this +%% -----File: 192.png---Folio 181------- +case, our hypothesis is that they agree, these two alternatives +are exhaustive; whence the above result, which may be found +in almost every discussion of the subject. + +The fallacy of such reasoning is easily exposed by a more +exact statement of the problem. For let $a_1$ stand for ``$X_1$~asserts~$a$,'' +and let $a/a_1h = x_1$, where~$h$, our general data, is by itself +irrelevant to~$a$, \ie, $x_1$~is the probability that a statement is true +of which we only know that $X_1$ has asserted it. Similarly let us +write $b/b_2h = x_2$ where $b_2$~stands for ``$X_2$~asserts~$b$.'' The above +argument then assumes that, if $X_1$ and~$X_2$ are witnesses who are +causally independent in the sense there is no collusion between +them direct or indirect, $ab/a_1b_2h = a/a_1h · b/b_2h = x_1x_2$. + +But $ab/a_1b_2h = a/a_1bb_2h · b/a_1b_2h$, and this is not equal to $x_1x_2$ +unless $a/a_1bb_2h = a/a_1h$ and $b/a_1b_2h = b/b_2h$. It is not a sufficient +condition for this, as seems usually to be supposed, that $X_1$~and~$X_2$ +should be witnesses causally independent of one another. It is +also necessary that $a$ and~$b$, \ie\ the propositions asserted by the +witnesses, should be irrelevant to one another and also each of +them irrelevant to the fact of the assertion of the other by a +witness. If a knowledge of~$a$ affects the probability either of~$b$ +or of~$b_1$, it is evident that the formula breaks down. In the one +extreme case, where the assertions of the two contradict one +another, $ab/a_1b_2h = 0$. In the other extreme, where the two agree +in the same assertion, \ie\ where $a \equiv b$, $a/a_1bb_2h = 1$ and not $= a/a_1h$. + +\Paragraph{17.} The special problem of the agreement of witnesses, who +make the same statement, can be best attacked as follows, a +certain amount of simplification being introduced. Let the +general data~$h$ of the problem include the hypothesis that $X_1$ and~$X_2$ +are each asked and reply to a question to which there is only +one correct answer. Let $a_i = $ ``$X_i$~asserts~$a$ in reply to the question,'' +and $m_i = $ ``$X_i$~gives the correct answer to the question.'' +Then +\[ +m_1/a_1h = x_1 \text{ and } m_2/a_2h = x_2, +\] +$x_1$ and~$x_2$ being, in the conventional language of this problem, +the ``credibilities'' of the witnesses. We have, since the witnesses +agree and since $a$~follows from~$m_ia_i$ and $m_i$~follows from~$aa_i$, +\begin{align*} +m_1/a_1a_2h &= m_1m_2/a_1a_2h = m_2/a_1a_2h;\\ +a/a_ih &= m_i/a_ih;\\ +a/a_im_ih &= 1;\quad m_i/aa_ih = 1. +\end{align*} +Also, since the witnesses are, in the ordinary sense, ``independent'' +%% -----File: 193.png---Folio 182------- +witnesses, $a_2/a_1ah = a_2/ah$ and $a_2/a_1\bar{a}h = a_2/\bar{a}h$; that is to say, the +probability of $X_2$'s asserting~$a$ is independent of the fact of $X_1$'s +having asserted~$a$, given we know that $a$~is, in fact, true or false, +as the case may be. + +The probability that, if the witnesses agree, their assertion is +true is +\begin{gather*} +a/a_1a_2h = m_1/a_1a_2h = \frac{m_1a_2/a_1h}{a_2/a_1h} \\ + = \frac{a_2/a_1m_1h . m_1/a_1 h}{a_2a/a_1h + a_2\bar{a}/a_1 h} + = \frac{a_2/a_1ah . x_1}{a_2/a_1ah . x_1 + a_2/a_1\bar{a}h . (1 - x_2)}. +\end{gather*} +If this is to be equal to $\dfrac{x_1x_2}{x_1x_2 + (1 - x_1)(1 - x_2)}$, we must have +\[ +\frac{a_2/a_1ah}{a_2/a_1\bar{a}h} = \frac{x_2}{1 - x_2}. +\] +\begin{DPalign*} +\lintertext{Now} +\frac{a_2/a_1ah}{a_2/a_1\bar{a}h} + &= \frac{a_2/ah}{a_2/\bar{a}h} \text{ by the hypothesis of ``independence''}\\ + &= \frac{aa_2/h}{\bar{a}a_2/h} . \frac{\bar{a}/h}{a/h} + = \frac{a/a_2h}{\bar{a}/a_2 h} . \frac{\bar{a}/h}{a/h}\\ + &= \frac{x_2}{1 - x_2} . \frac{\bar{a}/h}{a/h}. +\end{DPalign*} +This then is the assumption which has tacitly slipped into the +conventional formula,---that $a/h =\bar{a}/h = \frac{1}{2}$. It is assumed, that +is to say, that any proposition taken at random is as likely as +not to be true, so that \emph{any} answer to a given question is, \textit{à~priori}, +as likely as not to be correct. Thus the conventional formula +ought to be employed only in those cases where the answer +which the ``independent'' witnesses agree in giving is, \textit{à~priori} +and apart from their agreement, as likely as not. + +\Paragraph{18.} A somewhat similar confusion has led to the controversy +as to whether and in what manner the \textit{à~priori} improbability +of a statement modifies its credibility in the mouth of a witness +whose degree of reliability is known. The fallacy of attaching +the same weight to a testimony regardless of the character of +what is asserted, is pointed out, of course, by Hume in the \textit{Essay +\index{Hume!testimony@{and testimony}}% +on Miracles}, and his argument, that the great \textit{à~priori} improbability +of some assertions outweighs the force of testimony +otherwise reliable, depends on the avoidance of it. The correct +is also taken by Laplace in his \textit{Essai philosophique} (pp\DPtypo{}{.}~98--102), +\index{Laplace!testimony@{and testimony}}% +%% -----File: 194.png---Folio 183------- +\index{Diderot on testimony}% +\index{Johnson, W. E.!testimony@{and testimony}}% +where he argues that a witness is less to be believed +when he asserts an extraordinary fact, declaring the opposite +view (taken by Diderot in the article on ``Certitude'' in the +\textit{Encyclopédie}) to be inconceivable before ``le simple bon sens.'' + +The manner in which the resultant probability is affected +depends upon the precise meaning we attach to ``degree of reliability'' +or ``coefficient of credibility.'' If a witness's credibility +\index{Coefficient of Credibility}% +is represented by~$x$, do we mean that, if $a$~is the true answer, +the probability of his giving it is~$x$, or do we mean that if he +answers~$a$ the probability of $a$'s being true is~$x$? These two things +are not equivalent. + +Let $a_1$ stand for ``$a$~is asserted by the witness''; $h_1$~for our +evidence bearing on the witness's veracity; and $h_2$~for other +evidence bearing on the truth of~$a$. Let $a/h_1h_2$, \ie\ the \textit{à~priori} +probability of~$a$ apart from our knowledge of the fact that the +witness has asserted it, be represented by~$p$. + +Let $a/a_1h_1 = x_1$ and $a_1/ah_1 = x_2$; so that $x_1 = \dfrac{a/h_1}{a_1/h_1} · x_2$. In +general $a/h_1 \ne a_1/h_1$. Do we mean by the witness's credibility +$x_1$ or~$x_2$? + +We require $\DPtypo{a/a_1h_2h_2}{a/a_1h_1h_2}$. + +Let $a_1/\bar{a}h_1 = r$, \ie\ the probability, apart from our special +knowledge concerning~$a$, that, if $a$~is false, the witness will hit on +that particular falsehood. +\begin{align*} +a/a_1h_1h_2 + &= \frac{a_1/ah_1h_2 · a/h_1h_2}{a_1/h_1h_2} + = \frac{x_2p}{a_1a/h_1h_2 + a_1\bar{a}/h_1h_2}\\ + &= \frac{x_2p}{x_2p + a_1/\bar{a}h_1h_2 · (1 - p)} + = \frac{x_2p}{x_2p + r(1 - p)}; +\end{align*} +for $a_1/ah_1h_2 = a_1/ah_1$ and $a_1/\bar{a}h_1h_2 = a_1/\bar{a}h_1$, since, given \emph{certain} +knowledge concerning~$a$, $h_2$~is irrelevant to the probability of~$a_1$. + +\Paragraph{19.} Generally speaking, all problems, in regard to the combination +of testimonies or to the combination of evidence derived +from testimony with evidence derived from other sources, may +be treated as special instances of the general problem of the +combination of arguments. Beyond pointing out the above +plausible fallacies, there is little to add. Mr.~W.~E. Johnson, +however, has proposed a method of defining credibility, which +is sometimes valuable, because it regards the witness's credibility +not absolutely, but with reference to a given type of question, +%% -----File: 195.png---Folio 184------- +\index{Bicquilley and testimony|inote}% +\index{Craig and tradition}% +\index{De Morgan!tradition@{and tradition}|inote}% +\index{Lacroix|inote}% +\index{Lee and tradition|inote}% +\index{Peterson and tradition}% +so that it enables us to measure the force of the witness's testimony +under special circumstances. If $a$~represents the fact of $A$'s +testimony regarding~$x$, then we may define $A$'s credibility for~$x$ +as~$\DPtypo{a}{\alpha}$, where $\DPtypo{a}{\alpha}$~is given by the equation +\[ +x/ah = x/h + \DPtypo{a}{\alpha}\sqrt{x/h · \bar{x}/h}; +\] +so that $\DPtypo{a}{\alpha}\sqrt{x/h · \bar{x}/h}$ measures the amount by which $A$'s~assertion +of~$x$ increases its probability. + +\Paragraph{20.} One of the most ancient problems in probability is concerned +with the gradual diminution of the probability of a past +event, as the length of the tradition increases by which it is +established. Perhaps the most famous solution of it is that +propounded by Craig in his \textit{Theologiae Christianae Principia +Mathematica}, published in 1699.\footnote + {See Todhunter's \textit{History}, p.~54. It has been suggested that the anonymous +\index{Todhunter!Craig@{and Craig}}% + essay in the \textit{Phil.\ Trans.}\ for 1699 entitled ``A Calculation of the Credibility + of Human Testimony'' is due to Craig. In this it is argued that, if the + credibilities of a set of witnesses are $p_1 \ldots p_n$, then if they are + successive the resulting probability is the product $p_1p_2 \ldots p_n$; if they are + concurrent, it is: + \[ + 1 - (1 - p_1)(1 - p_2) \ldots (1 - p_n). + \] + This last result follows from the supposition that the first witness leaves an + amount of doubt represented by $1 - p_1$; of this the second removes the fraction~$p_2$, + and so on. See also Lacroix, \textit{Traité élémentaire}, p.~262. The above theory + was actually adopted by Bicquilley.} +He proves that suspicions of +any history vary in the duplicate ratio of the times taken from +the beginning of the history in a manner which has been described +as a kind of parody of Newton's \textit{Principia}. ``Craig,'' says +Todhunter, ``concluded that faith in the Gospel so far as it +depended on oral tradition expired about the year 880, and that +so far as it depended on written tradition it would expire in the +year 3150. Peterson by adopting a different law of diminution +concluded that faith would expire in 1789.''\footnote + {In the \textit{Budget of Paradoxes} De~Morgan quotes Lee, the Cambridge Orientalist, + to the effect that Mahometan writers, in reply to the argument that the Koran + has not the evidence derived from Christian miracles, contend that, as evidence + of Christian miracles is daily weaker, a time must at last arrive when it will + fail of affording assurance that they were miracles at all: whence the necessity + of another prophet and other miracles.} +About the same +time Locke raised the matter in chap.~xvi.\ bk.~iv.\ of the +\index{Locke!on tradition}% +\textit{Essay Concerning Human Understanding}: ``Traditional testimonies +the farther removed, the less their proof\ldots. No +Probability can rise higher than its first original.'' This is +evidently intended to combat the view that the long acceptance +by the human race of a reputed fact is an additional argument +%% -----File: 196.png---Folio 185------- +\index{Cayley, and tradition}% +\index{Macfarlane, and independence!tradition@{and tradition}}% +in its favour and that a long tradition increases rather than +diminishes the strength of an assertion. ``This is certain,'' says +Locke, ``that what in one age was affirmed upon slight grounds, +can never after come to be more valid in future ages, by being +often repeated.'' In this connection he calls attention to ``a +rule observed in the law of England, which is, that though the +attested copy of a record be good proof, yet the copy of a copy +never so well attested, and by never so credible witnesses, will +not be admitted as a proof in Judicature.'' If this is still a good +rule of law, it seems to indicate an excessive subservience to the +principle of the decay of evidence. + +But, although Locke affirms sound maxims, he gives no theory +that can afford a basis for calculation. Craig, however, was the +more typical professor of probability, and in attempting an +algebraic formula he was the first of a considerable family. The +last grand discussion of the problem took place in the columns +of the \textit{Educational Times}.\footnote + {Reprinted in \textit{Mathematics from the Educational Times}, vol.~xxvii.} +Macfarlane\footnote + {\textit{Algebra of Logic}, p.~151. Macfarlane attempts a solution of the general + problem without success. Its solution is not difficult, if enough unknowns are + introduced, but of very little interest.} +mentions that four +different solutions have been put forward by mathematicians +of the problem: ``$A$~says that $B$~says that a certain event took +place; required the probability that the event did take place, +$p_1$~and~$p_2$ being $A$'s~and~$B$'s respective probabilities of speaking +the truth.'' Of these solutions only Cayley's is correct. +%% -----File: 197.png---Folio 186------- + + +\Chapter{XVII}{Some Problems in Inverse Probability, including Averages} + +\Paragraph{1.} \First{The} present chapter deals with `problems'---that is to +say, with applications to particular abstract questions of some of +the fundamental theorems demonstrated in \Chapref{XIV}\@. It +is without philosophical interest and should probably be omitted +by most readers. I introduce it here in order to show the analytical +power of the method developed above and its advantage +in ease and especially in accuracy over other methods which +have been employed.\footnote + {Such examples as these might sometimes be set to test the wits of students. + The problems on Probability usually given are simply problems on mathematical + combinations. These, on the other hand, are really problems in logic.} +§\;2 is mainly based upon some problems +discussed by Boole. §§\;3--7 deal with the fundamental theory +connecting averages and laws of error. §§\;8--11 treat discursively +the Arithmetic Average, the Method of Least Squares, and +Weighting. + +\Paragraph{2.} In the following paragraph solutions are given of some +problems posed by Boole in chapter~xx.\ of his \textit{Laws of Thought}. +Boole's own method of solving them is constantly erroneous,\footnote + {For the reason given in §\;6 of \Chapref{XVI}\@. The solutions of problems + I.--VI., for example, in the \textit{Laws of Thought}, chap.~xx., are all erroneous.} +and the difficulty of his method is so great that I do not know +of any one but himself who has ever attempted to use it. The +term `cause' is frequently used in these examples where it might +have been better to use the term `hypothesis.' For by a possible +cause of an event no more is here meant than an antecedent +occurrence, the knowledge of which is relevant to our anticipation +of the event; it does not mean an antecedent from which the +event in question \emph{must} follow. + +(56) The \textit{à~priori} probabilities of two causes $A_1$~and~$A_2$ +are $c_1$~and~$c_2$ respectively. The probability that if the cause~$A_1$ +%% -----File: 198.png---Folio 187------- +\index{Cayley, and tradition!Challenge@{and Challenge Problem}}% +\index{Macfarlane, and independence!Challenge@{and `\textit{Challenge Problem}'}|inote}% +occur, an event~$E$ will accompany it (whether as a consequence +of~$A_1$ or not), is~$p_1$, and the probability that $E$ will accompany~$A_2$, +if $A_2$ present itself, is~$p_2$. Moreover, the event~$E$ cannot appear +in the absence of both the causes $A_1$~and~$A_2$. Required the probability +of the event~$E$. + +This problem is of great historical interest and has been called +Boole's `Challenge Problem.' Boole originally proposed it for +\index{Boole!Challenge@{and Challenge Problem}}% +solution to mathematicians in 1851 in the \textit{Cambridge and Dublin +Mathematical Journal}. A result was given by Cayley\footnote + {\textit{Phil.\ Mag}.\ 4th~series, vol.~vi.} +in the +\textit{Philosophical Magazine}, which Boole declared to be erroneous.\footnote + {Cayley's solution was defended against Boole by Dedekind (\textit{Crelle's Journal}, +\index{Dedekind and `\textit{Challenge Problem}'|inote}% + vol.~1.\ p.~268). The difference arises out of the extreme ambiguity as to the + meaning of the terms as employed by Cayley.} +He then entered the field with his own solution.\footnote + {``Solution of a Question in the Theory of Probabilities,'' \textit{Phil.\ Mag}.\ 4th~series, + vol.~vii., 1854. This solution is the same as that printed by Boole + shortly afterwards in the \textit{Laws of Thought}, pp.~321--326. In the \textit{Phil.\ Mag}.\ + Wilbraham gave as the solution $u = c_1p_1 + c_2p_2 - z$, where $z$~is necessarily less + than either $c_1p_1$ or~$c_2p_2$. This solution is correct so far as it goes, but is not + complete. The problem is also discussed by Macfarlane, \textit{Algebra of Logic}, + p.~154.} +``Several +attempts at its solution,'' he says, ``have been forwarded to me, +all of them by mathematicians of great eminence, all of them +admitting of particular verification, yet differing from each other +and from the truth.''\footnote + {In proposing the problem Boole had said: ``The motives which have + led me, after much consideration, to adopt, with reference to this question, a + course unusual in the present day, and not upon slight grounds to be revived, + are the following: First, I propose the question as a test of the sufficiency of + received methods. Secondly, I anticipate that its discussion will in some + measure add to our knowledge of an important branch of pure analysis.'' + When printing his own solution in the \textit{Laws of Thought}, he adds, that the + above ``led to some interesting private correspondence, but did not elicit a + solution.''} +After calculations of considerable length +and great difficulty he arrives at the conclusion that $u$ is the +probability of the event~$E$ where $u$ is that root of the equation +\[ +\frac{\left[1 - c_1(1 - p_1) - u\right]\left[1 - c_2(1 - p_2) - u\right]}{1 - u} + = \frac{(u - c_1p_1)(u - c_2p_2)}{c_1p_1 + c_2p_2 - u} +\] +which is not less than $c_1p_1$ and~$c_2p_2$ and not greater than +$1 - c_1(1 - p_1)$, $1 - c_2(1 - p_2)$ or $c_1p_1 + c_2p_2$. + +This solution can easily be seen to be wrong. For in the +case where $A_1$~and~$A_2$ cannot both occur, the solution is +$u = c_1p_1 + c_2p_2$; whereas Boole's equations do not reduce to +%% -----File: 199.png---Folio 188------- +\index{McColl, and symbolic probability!Challenge@{and `\textit{Challenge Problem}'}|inote}% +this simplified form. The mistake which Boole has made is +the one general to his system, referred to in \Chapref{XVI}., §\;6.\footnote + {Boole's error is pointed out and a correct solution given in Mr.~M\textsuperscript{c}Coll's + ``Sixth Article on the Calculus of Equivalent Statements'' (\textit{Proc.\ Lond.\ Math.\ + Soc.}\ vol.~xxviii.\ p.~562).} + +The correct solution, which is very simple, can be reached as +follows: + +Let $a_1$,~$a_2$,~$e$ assert the occurrences of the two causes and the +event respectively, and let $h$ be the \textit{data} of the problem. + +Then we have $a_1/h = c_1$, $a_2/h = c_2$, $e/a_1 h = p_1$, $e/a_2 h = p_2$: we +require~$e/h$. Let $e/h = u$, and let $a_1a_2/eh = z$. Since the event +cannot occur in the absence of both the causes, +\[ +e/\bar{a}_1 \bar{a}_2 h = 0. +\] +It follows from this that $\bar{a}_1 \bar{a}_2/eh = 0$, unless $e/h = 0$, +\begin{DPgather*} +\lintertext{\ie} +(a_1 + a_2)/eh = 1,\\ +% +\lintertext{whence} +a_1/eh + a_2/eh = 1 + a_1 a_2/eh + \rintertext{by~(24).}\\ +\lintertext{Now} +a_1/eh = \frac{c_1 p_1}{u} \text{ and } a_2/eh = \frac{c_2 p_2}{u},\\ +\therefore u = \frac{c_1 p_1 + c_2 p_2}{1 + z}, +\end{DPgather*} +where $z$ is the probability after the event that \emph{both} the causes were +present. + +If we write $e a_1 a_2/h = y$, +\begin{DPalign*} +y &= a_1 a_2/eh · e/h = uz\\ +\lintertext{so that} +u &= (c_1 p_1 + c_2 p_2) - y. +\end{DPalign*} + +Boole's solution fails by attempting to be independent of +$y$~or~$z$. + +(56.1)\DPtypo{.}{} Suppose that we wish to find limits for the solution +which are independent of $y$~and~$z$: then, since $y \eqslantless 0$, +$u \eqslantless c_1 p_1 + c_2 p_2$. + +Again +\[ +e/h = e\bar{a}_1/h + ea_1/h + \eqslantless \bar{a}_1/h + ea_1/h + \eqslantless 1 - c_2 + c_2 p_2 + \quad\text{by (24.2) and~(4).} +\] +Similarly $e/h \eqslantless -c_2 + c_2p_2$. From the same equations it appears +that $e/h \eqslantless c_1 p_1$ and~$c_2 p_2$. +%% -----File: 200.png---Folio 189------- +%[** TN: In orig, above ends mid-line, but next line not indented.] + +\noindent$\therefore u$ lies between +\[ +\text{the greatest of } +\smash[t]{\left\{ +\begin{aligned} +& c_1p_1\\ +& c_2p_2 +\end{aligned} +\right.} +\text{ and the least of } +\smash[t]{\left\{ +\begin{aligned} +&c_1p_1 + c_2p_2\\ +&1 - c_1(1 - p_1)\\ +&1 - c_2(1 - p_2). +\end{aligned} +\right.} +\] + +It will be seen that these numerical limits are the same as the +limits obtained by Boole for the roots of his equations. + +(56.2) Suppose that the \textit{à~priori} probabilities of the causes $c_1$~and~$c_2$ +are to be eliminated. The only limit we then have is +$u < p_1 + p_2$. + +(56.3) Suppose that one of the \textit{à~priori} probabilities~$c_2$ is to be +eliminated. We then have limits $c_1p_1 \eqslantless u \eqslantless 1 - c_1 + c_1p_1$. If, therefore, +$c_1$~is large, $u$~does not differ widely from~$c_1p_1$. + +(56.4) Suppose $p_2$ is to be eliminated. We then have +\begin{align*} +c_1p_1 \eqslantless u + &\eqslantless c_1p_1 + c_2\\ + &\eqslantless c_1p_1 + 1 - c_1. +\end{align*} + +If therefore $c_1$ is large or $c_2$ small, $u$~does not differ widely +from~$c_1p_1$. + +(56.5) If $a_1/a_2h = a_1/h$, \ie\ if our knowledge of each of the +causes is independent, we have a closer approximation. For +\begin{gather*} +y = ea_1a_2/h + = e/a_1a_2h · a_1/a_2h · a_2/h + = e/a_1a_2h · c_1c_2, \\ +% +\therefore u = c_1p_1 + c_2p_2 - c_1c_2 · e/a_1a_2h, \\ +% +\therefore u > c_1p_1 + c_2p_2 - c_1c_2. +\end{gather*} + +(57) We may now generalise~(56) and discuss the case of $n$~causes. +If an event can only happen as a consequence of one +\emph{or more} of certain causes $A_1$,~$A_2$, $\ldots~A_n$, and if $c_1$ is the \textit{à~priori} +probability of the cause $A_1$ and~$p_1$ the probability that, if the +cause~$A_1$ be known to exist, the event~$E$ will occur: required the +probability of~$E$. + +This is Boole's problem~VI. (\textit{Laws of Thought}, p.~336). As +the result of ten pages of mathematics, he finds the solution to be +the root lying between certain limits of an equation of the $n$\ordth~degree +which he cannot solve. I know no other discussion of the +problem. The solution is as follows: +\begin{DPgather*} +e/h = e\bar{a}_1/h + ea_1/h + = e\bar{a}_1/h + e/a_1h · a_1/h + = e\bar{a}_1/h + c_1p_1 + \rintertext{(i.)}\\ +% +e\bar{a}_1/h + = e\bar{a}_1\bar{a}_2/h + e\bar{a}_1/a_2h · a_2/h + = e\bar{a}_1\bar{a}_2/h + c_2 · e\bar{a}_1/a_2h, \\ +% +e\bar{a}_1/a_2h + = e/a_2h - ea_1/a_2h + = p_2 - \frac{1}{c_2} · ea_1a_2/h, +\end{DPgather*} +%% -----File: 201.png---Folio 190------- +\begin{gather*} +\therefore e/h + = e\bar{a}_1\bar{a}_2/h + c_1p_1 + c_2p_2 - ea_1a_2/h, \\ + e\bar{a}_1\bar{a}_2/h + = e\bar{a}_1\bar{a}_2\bar{a}_3/h + e\bar{a}_1\bar{a}_2a_3/h, +\end{gather*} +\begin{DPalign*} +\lintertext{and} +e\bar{a}_1\bar{a}_2a_3/h + &= e\bar{a}_1\bar{a}_2/a_3h · c_3 + = c_3\left\{e/a_3h - e\overline{\bar{a}_1\bar{a}_2}/a_3/h\right\} \\ + &= c_3p_3 - e\overline{\bar{a}_1\bar{a}_2}a_3/h, +\end{DPalign*} +\[ +\therefore e/h + = e\bar{a}_1\bar{a}_2\bar{a}_3/h + c_1p_1 + c_2p_2 + c_3p_3 + - e\bar{a}_1a_2/h - e\overline{\bar{a}_1\bar{a}_2}a_3/h. +\] +In general +\begin{align*} +e\bar{a}_1\bar{a}_2 \ldots \bar{a}_{r-1}/h + &= e\bar{a}_1\bar{a}_2 \ldots \bar{a}_{r-1}\bar{a}_r/h + + e\bar{a}_1\bar{a}_2 \ldots \bar{a}_{r-1}a_r/h \\ +% + &= e\bar{a}_1 \ldots \bar{a}_r/h + + e\bar{a}_1 \ldots \bar{a}_{r-1}/a_r h · c_r \\ +% + &= e\bar{a}_1 \ldots \bar{a}_r/h + + c_r \left\{e/a_rh - \overline{e\bar{a}_1 \ldots \bar{a}_{r-1}}/a_rh \right\} \\ +% + &= e\bar{a}_1 \ldots \bar{a}_r/h + c_rp_r + - e\overline{\bar{a}_1 \ldots \bar{a}_{r-1}}a_r/h. +\end{align*} +\[ +\therefore \text{ finally we have } e/h = e\bar{a}_1 \ldots \bar{a}_n/h + \Sum_1^n c_r p_r - \Sum_2^n e\overline{\bar{a} \ldots \bar{a}_{r-1}}a_r/h. +\] +But since the $n$~causes are supposed to be exhaustive +\begin{DPgather*} +e\bar{a}_1 \ldots \bar{a}_n/h = 0,\\ +% +\therefore e/h = \Sum_1^n c_r p_r - \Sum_2^n e\overline{\bar{a}_1 \ldots \bar{a}_{r-1}}a_r/h + \rintertext{(ii.).}\\ +% +\lintertext{Let} +e\overline{\bar{a}_1 \ldots \bar{a}_{r-1}}a_r/h = n_r;\\ +% +\lintertext{then} +e/h = \Sum_1^n c_r p_r - \Sum_2^n n_r + \rintertext{(iii.).} +\end{DPgather*} + +(57.1) If our knowledge of the several causes is independent, +if, that is to say, our knowledge of the existence of any one of +them is not relevant to the probability of the existence of any +other, so that $a_r/a_sh = a_r/h = c_r$, then +\begin{align*} +e\overline{\bar{a}_1 \ldots \bar{a}_{r-1}}a_r/h + &= e\overline{\bar{a}_1 \ldots \bar{a}_{r-1}}/a_rh · c_r\\ +% + &= c_r · e\overline{\bar{a}_1 \ldots \bar{a}_{r-1}}a_rh \{1 - \bar{a}_1 \ldots \bar{a}_{r-1}/a_rh\} \\ +% + &= c_r \left[1 - \Prod_1^{r-1}(1 - c_1) \ldots (1 - c_{r-1})\right]e + /\overline{\bar{a}_1 \ldots \bar{a}_{r-1}}a_rh. +\end{align*} +\begin{DPgather*} +\lintertext{Let} +e/\overline{\bar{a}_1 \ldots \bar{a}_{r-1}}a_rh = m_r, \\ +\lintertext{then} +e/h = \Sum_{r=1}^{r=n} c_r p_r + - \Sum_{r=2}^{r=n} c_r \left[1 - \Prod_{s=1}^{s=r-1}(1 - c_s)\right]m_r. +\end{DPgather*} + +These results do not look very promising as they stand, but +they lead to some useful approximations on the elimination of +$m_r$~and~$n_r$ and to some interesting special cases. +%% -----File: 202.png---Folio 191------- + +(57.2) From equation~(i.)\ it follows that $e/h \eqslantgtr c_1p_1$ and +$e/h \eqslantless 1 - c_1(1 - p_1)$; and from equation~(ii.)\ that $e/h \eqslantless \Sum_1^n c_r p_r$; + +$\therefore e/h$ lies between +\[ +\text{the greatest of } +\left\{ +\begin{aligned} +&c_1p_1 \\ +&\PadTo{c_1p_1}{\vdots} \\ +&c_np_n +\end{aligned} +\right. +\text{ and the least of } +\left\{ +\begin{aligned} +&\Sum_1^n c_rp_r \\ +&1 - c_1(1 - p_1) \\ +& \PadTo{1 - c_1(1 - p_1)}{\vdots} \\ +&1 - c_n(1 - p_n). +\end{aligned} +\right. +\] + +(57.3) Further, if the causes are independent it follows from~(57.1) +that +\[ +e/h \eqslantgtr \Sum_1^n c_r p_r + - \Sum_2^n c_r {\textstyle\left[1 - \Prod_1^{r-1}(1 - c_s)\right]}, +\] +so that $e/h$ lies between +\[ +\settowidth{\TmpLen}{greatest}%[** TN: Stacking text to get horizontal fit] +\parbox[c]{\TmpLen}{\centering the\\ greatest\\ of } +\left\{ +\begin{aligned} +&{\textstyle\Sum_1^n c_rp_r - \Sum_2^n c_r \bigl[1 - \Prod\limits_1^{r-1}(1 - c_s)\bigr]}\\ +&c_1p_1 \\ +&\PadTo{c_1p_1}{\vdots} \\ +&c_np_n +\end{aligned} +\right. +\settowidth{\TmpLen}{ and the}% +\parbox[c]{\TmpLen}{ and the\\ least of } +\left\{ +\begin{aligned} +&{\textstyle\Sum_1^n c_r p_r} \\ +&1 - c_1(1 - p_1)\\ +&\PadTo{1 - c_1(1 - p_1)}{\vdots} \\ +&1 - c_n(1 - p_n). +\end{aligned} +\right. +\] + +(57.4) Now consider the case in which $p_1 = p_2 = \ldots = p_n = 1$, +\ie\ in which any of the causes would be sufficient, and in which +the causes are independent. Then $m_r = 1$; so that +\begin{align*} +e/h &= \Sum_{r=1}^{r=n}c_r + - \Sum_{r=2}^{r=n}c_r \left[1 - \Prod_{s=1}^{s=r-1}(1 - c_s)\right] \\ + &= 1 - (1 - c_1)(1 - c_2) \ldots (1 - c_n). +\end{align*} + +(57.5) Let $c_1$, $c_2 \ldots c_n$ be small quantities so that their +squares and products may be neglected. +\begin{DPgather*} +\lintertext{\indent Then} +e/h = \Sum c_rp_r, +\end{DPgather*} +\ie\ the smaller the probabilities of the causes the more do they +approach the condition of being mutually exclusive.\footnote + {Boole arrives at this result, \textit{Laws of Thought}, p.~345, but I doubt his proof.} + +(57.6) The \textit{à~posteriori} probability of a particular cause~$a_r$ +after the event has been observed is +\begin{align*} +a_r/eh &= \frac{e/a_rh · a_r/h}{e/h} \\ + &= \frac{p_rc_r}{e/h}. +\end{align*} + +(This is Boole's problem~IX., p.~357). +%% -----File: 203.png---Folio 192------- + +(58) The probability of the occurrence of a certain natural +phenomenon under given circumstances is~$p$. There is also a +probability~$a$ of a permanent cause of the phenomenon, \ie\ of a +cause which would always produce the event under the circumstances +supposed. What is the probability that the phenomenon, +being observed $n$~times, will occur the~$n+1$\ordth? + +This is Boole's problem~X. (\textit{Laws of Thought}, p.~358). Boole +arrives by his own method at the same result as that given below. +It is necessary first of all to state the assumption somewhat +more precisely. If $x_r$~asserts the occurrence of the event at the +$r^\text{th}$~trial and $t$~the existence of the `permanent cause' we have +\begin{DPgather*} +x_r/h = p,\quad t/h = a,\quad x_r/th = 1, \\ +\lintertext{and we require} +x_{n+1}/x_1 \ldots x_nh = y_{n+1}. +\end{DPgather*} +It is also assumed that if there is \emph{no} permanent cause the probability +of~$x_s$ is not affected by the observations~$x_r$, etc.,~\ie\ +\begin{DPgather*} +x_s/x_r \ldots x_t\bar{t}h = x_s/\bar{t}h,\footnotemark \\ +% +x_s/\bar{t}h + = \frac{x_s\bar{t}/h}{\bar{t}/h} + = \frac{x_s/h - x_s t/h}{\bar{t}/h} + = \frac{p - a}{1 - a}, \\ +% +\begin{aligned} +x_r/x_1 \ldots x_{r-1}h + &= x_r t/x_1 \ldots x_{r-1}h + x_r\bar{t}/x_1 \ldots x_{r-1}h \\ + &= t/x_1 \ldots x_{r-1}h + x_r/\bar{t}x_1 \ldots x_{r-1}h · \bar{t}/x_1 \ldots x_{r-1}h \\ + &= \frac{x_1 \ldots x_{r-1}t/h}{x_1 \ldots x_{r-1}/h} + + \frac{p-a}{1-a} + · \frac{x_1 \ldots x_{r-1}/\bar{t}h · \bar{t}/h}{x_1 \ldots x_{r-1}/h} \\ +% + &= \frac{a}{y_1y_2 \ldots y_{r-1}} + + \frac{p-a}{1-a} + \frac{\left(\dfrac{p-a}{1-a}\right)^{r-1}(1-a)}{y_1y_2\ldots y_{r-1}}, +\end{aligned} \\ +\lintertext{\rlap{\ie}} +y_r = \frac{a + (p-a)\left(\dfrac{p-a}{1-a}\right)^{r-1}}{y_1y_2 \ldots y_{r-1}}. \\ +% +\lintertext{\rlap{Also}} +y_1 = p \text{ and } y_2 = \frac{a + (p-a)\dfrac{p-a}{1-a}}{y_1}, +\end{DPgather*} +\footnotetext{This assumption, which is tacitly introduced by Boole, is not generally +justifiable. I use it here, as my main purpose is to illustrate a method. The +same problem, \emph{without} this assumption, will be discussed in dealing with Pure +Induction.}% +%% -----File: 204.png---Folio 193------- +\begin{DPgather*} +\lintertext{so that} +y_{n+1} = \frac{a + (p-a)\left(\dfrac{p-a}{1-a}\right)^n} + {a + (p-a)\left(\dfrac{p-a}{1-a}\right)^{n-1}}. +\end{DPgather*} + +(58.1)~If $p=a$, $y_n=1$; for if an event can only occur as the +result of a permanent cause, a single occurrence makes future +occurrences certain under similar conditions. + +(58.2) +\[ +y_{n+1} - y_n = \frac{a(p-a)\left(\dfrac{p-a}{1-a}\right)^{n-2}\left(1 - \dfrac{p-a}{1-a}\right)} + {\left[a + (p-a)\left(\dfrac{p-a}{1-a}\right)^{n-1}\right]\left[a + (p-a)\left(\dfrac{p-1}{1-a}\right)^{n-2}\right]} +\] +(by easy algebra); + +%[** TN: Paragraph break in original] +and $p$ is always $>a$ and~$<1$. + +So that $(p-a)\left(\dfrac{p-a}{1-a}\right)^r$ is positive and decreases as $r$~increases, +\[ +\therefore y_{n+1} > y_n. +\] + +As $n$ increases $y_n=1-\epsilon$, where +\[ +\epsilon = (p-a)\left[1 - \left(\dfrac{p-a}{1-a}\right)\right]\frac{\left(\dfrac{p-a}{1-a}\right)^{n-2}}{a + (p-a)\left(\dfrac{p-a}{1-a}\right)^{n-2}}, +\] +so that for any positive value of~$\eta$ however small a value of~$n$ +can be found such that $\epsilon<\eta$ so long as $a$~is not zero. + +(58.3)~$t_n$\DPnote{** [sic] no comma} the \textit{à posteriori} probability of a permanent cause +after $n$~successful observations is +\begin{DPgather*} +t/x_1 \ldots x_nh + = \frac{x_1 \ldots x_n/th· t/h}{x_1 \ldots x_n/h} + = \frac{a}{y_1y_2 \ldots y_n},\\ +\lintertext{\ie} t_n + = \frac{a}{a + (p-a)\left(\dfrac{p-a}{1-a}\right)^n},\\ + t_n = 1-\epsilon'\text{, where } +\epsilon' + = \frac{(p-a)\left(\dfrac{p-a}{1-a}\right)^n} + {a + (p-a)\left(\dfrac{p-a}{1-a}\right)^n}. +\end{DPgather*} +%% -----File: 205.png---Folio 194------- +\index{Law of error|ifoll}% +\index{Means and laws of error|ifoll}% +So that $t_n$~approaches the limit unity as $n$~increases, so long as $a$ +is not zero. + +\Paragraph{3.} The following is a common type of statistical problem.\footnote + {The substance of §§\;3--7 has been printed in the \textit{Journal of the Royal + Statistical Society}, vol.~lxxiv.\ p.~323 (February 1911).} +We are given a series of measurements, or observations, or +estimates of the true value of a given quantity; and we wish to +determine what function of these measurements will yield us +the \emph{most probable} value of the quantity, on the basis of this evidence. +The problem is not determinate unless we have some +good ground for making an assumption as to how likely we are +in each case to make errors of given magnitudes. But such an +assumption, with or without justification, is frequently made. + +The functions of the original measurements which we commonly +employ, in order to yield us approximations to the most +probable value of the quantity measured, are the various kinds +of means or averages---the arithmetic mean, for example, or +the median. The relation, which we assume, between errors of +different magnitudes and the probabilities that we have made +errors of those magnitudes, is called a \emph{law of error}. Corresponding +to each law of error which we might assume, there is some function +of the measurements which represents the most probable value +of the quantity. The object of the following paragraphs is to +discover what laws of error, if we assume them, correspond to +each of the simple types of average, and to discover this by means +of a systematic method. + +\Paragraph{4.} Let us assume that the real value of the quantity is either +$b_1, \ldots b_r \ldots b_n$, and let $a_r$~represent the conclusion that the +value is, in fact,~$b_r$. Further let $x_r$~represent the evidence that +a measurement has been made of magnitude~$y_r$. + +If a measurement~$y_p$ has been made, what is the probability +that the real value is~$b_s$? The application of the theorem of +inverse probability yields the following result: +\[ +%[** TN: Omitting \quad-like space after · in orig.] +a_s/x_ph + = \frac{x_p/a_sh · a_s/h} + {\Sum^{r=n}_{r=1}x_p/a_rh · a_r/h} +\] +(the number of possible values of the quantity being~$n$), where +$h$~stands for any other relevant evidence which we may have, +in addition to the fact that a measurement~$x_p$ has been made. + +Next, let us suppose that a number of measurements $y_1 \ldots y_m$ +%% -----File: 206.png---Folio 195------- +\index{Independence, for knowledge!law of error@{and law of error}}% +have been made; what is now the probability that the real value +is~$b_s$? We require the value of $a_s/x_1x_2 \ldots x_mh$. As before, +\[ +%[** TN: Omitting \quad-like space after · in orig.] +a_s/x_1x_2 \ldots x_mh + = \frac{x_1 \ldots x_m/a_sh · a_s/h} + {\Sum^{r=x}_{r=1}x_1\ldots x_m/a_rh · a_r/h}. +\] + +At this point we must introduce the simplifying assumption +that, if we knew the real value of the quantity, the different +measurements of it would be \emph{independent}, in the sense that a +knowledge of what errors have actually been made in some of +the measurements would not affect in any way our estimate of +what errors are likely to be made in the others. We assume, +in fact, that $x_r/x_p \ldots x_sa_rh = x_r/a_rh$. This assumption is exceedingly +important. It is tantamount to the assumption that +our law of error is unchanged throughout the series of observations +in question. The general evidence~$h$, that is to say, which justifies +our assumption of the particular law of error which we do assume, +is of such a character that a knowledge of the actual errors made +in a number of measurements, not more numerous than those +in question, are absolutely or approximately irrelevant to the +question of what form of law we ought to assume. The law +of error which we assume will be based, presumably, on an +experience of the relative frequency with which errors of different +magnitudes have been made under analogous circumstances in +the past. The above assumption will \emph{not} be justified if the +additional experience, which a knowledge of the errors in the new +measurements would supply, is sufficiently comprehensive, relatively +to our former experience, to be capable of modifying our +assumption as to the shape of the law of error, or if it suggests +that the circumstances, in which the measurements are being +carried out, are not so closely analogous as was originally supposed. + +With this assumption, \ie\ that $x_1$,~etc., are independent of +one another relatively to evidence $a_r h$,~etc., it follows from the +ordinary rule for the multiplication of independent probabilities +that +\begin{DPgather*} +x_1\ldots x_m/a_sh = \Prod \limits^{q=m}_{q=1}x_q/a_s h. \\ +\lintertext{Hence} +a_s/x_1x_2 \ldots x_mh + = \frac{a_s/h · \Prod \limits^{q=m}_{q=1}x_q/a_sh} + {\Sum^{r=n}_{r=1} + \left[ \Prod \limits^{q=m}_{q=1} x_q/a_rh · a_r/h \right]}. +\end{DPgather*} +%% -----File: 207.png---Folio 196------- +\index{Gauss, and laws of error|inote}% + +The \emph{most probable} value of the quantity under measurement, +given the $m$~measurements $y_1$,~etc.---which is our \textit{quaesitum}---is +therefore that value which makes the above expression a maximum. +Since the denominator is the same for all values of~$b_s$, +we must find the value which makes the numerator a maximum. +Let us assume that $a_1/h = a_2/h = \ldots = a_n/h$. We assume, that +is to say, that we have no reason \textit{à~priori} (\ie\ before any measurements +have been made) for thinking any one of the possible +values of the quantity more likely than any other. We require, +therefore, the value of~$b_s$ which makes the expression $\Prod \limits^{q=m}_{q=1}x_q/a_s h$ +a maximum. Let us denote this value by~$y$. + +We can make no further progress without a further assumption. +Let us assume that $x_q/a_sh$---namely, the probability of a +measurement~$y_q$ assuming the real value to be~$b_s$---is an algebraic +function~$f$ of $y_q$ and~$b_s$, the same function for all values of $y_q$ and~$b_s$ +within the limits of the problem.\footnote + {Gauss, in obtaining the normal law of error, made, in effect, the more + special assumption that $x_q/a_sh$ is a function of $e_q$~only, where $e_q$ is the error and + $e_q = b_s - y_q$. We shall find in the sequel that all symmetrical laws of error, + such that positive and negative errors of the same absolute magnitude are + equally likely, satisfy this condition---the normal law, for example, and the + simplest median law. But other laws, such as those which lead to the geometric + mean, do not satisfy it.} +We assume, that is to say, +$x_q/a_sh = f(y_q, b_s)$, and we have to find the value of~$b_s$, namely~$y$, +which makes $\Prod \limits^{q=m}_{q=1}f(y_q, y)$ a maximum. Equating to zero the +differential coefficient of this expression with respect to~$y$, we +have $\Sum^{q=m}_{q=1}\dfrac{f'(y_q, y)}{f(y_q, y)} = 0$,\footnote + {Since none of the measurements actually made can be impossible, none of + the expressions $f(y_q, y)$ can vanish.} +where $f' = \dfrac{df}{dy}$. This equation may be +written for brevity in the form $\Sum \dfrac{f'_q}{f_q} = 0$. + +If we solve this equation for~$y$, the result gives us the value of +the quantity under observation, which is most probable relatively +to the measurements we have made. + +The act of differentiation assumes that the possible values of~$y$ +are so numerous and so uniformly distributed within the range +in question, that we may, without sensible error, regard them as +continuous. + +\Paragraph{5.} This completes the \textit{prolegomena} of the inquiry. We are +%% -----File: 208.png---Folio 197------- +\index{Arithmetic mean (or average)!laws of error@{and laws of error}}% +now in a position to discover what laws of error correspond to +given assumptions respecting the algebraic relation between the +measurements and the most probable value of the quantity, and +\textit{vice versa}. For the law of error determines the form of~$f(y_q,y)$. +\index{Law of error!arithmetic@{and arithmetic mean}}% +And the form $f(y_q,y)$ determines the algebraic relation $\Sum \dfrac{f'_q}{f_q} = 0$ +between the measurements and the most probable value. It +may be well to repeat that $f(y_q,y)$ denotes the probability to +us that an observer will make a measurement~$y_q$ in observing a +quantity whose true value we know to be~$y$. A law of error tells +us what this probability is for all possible values of $y_q$ and~$y$ +within the limits of the problem. + +(i.) If the most probable value of the quantity is equal to the +arithmetic mean of the measurements, what law of error does this +imply? +\setlength{\TmpLen}{3.5in}%[** TN: Hard-coded width.] +\begin{DPalign*}[m] +\Sum\frac{f'_q}{f_q} + &= \parbox[t]{\TmpLen}{$0$ + must be equivalent to $\Sum (y - y_q) = 0$, since the most + probable value~$y$ must equal $\dfrac{1}{m} \Sum^{q=m}_{q=1} y_q$.} \\ +% +\therefore \dfrac{f'_q}{f_q} + &= \parbox[t]{\TmpLen}{$\phi''(y)(y - y_q)$ + where $\phi''(y)$ is some function which + is not zero and is independent of~$y_q$.} \\ +% +\intertext{Integrating,} +\log f_q + &= \parbox[t]{\TmpLen}{$\int \phi''(y)(y - y_q)\, dy + \psi(y_q)$ + where $\psi(y_q)$ is some function + independent of~$y$.} \\ +% + &= \phi'(y)(y - y_q) - \phi(y) + \psi(y_q). \\ +\lintertext{\rlap{So that}} +f_q &= e^{\phi'(y)(y - y_q) - \phi(y) + \psi(y_q)}. +\end{DPalign*} + +Any law of error of this type, therefore, leads to the arithmetic +mean of the measurements as the most probable value of the +quantity measured. + +%[** TN: Reformatting/displaying next alignment] +If we put $\phi(y) = -k^2y^2$ and $\psi(y_q) = -k^2y_q^2 + \log A$, we obtain +\begin{align*} +f_q &= Ae^{-k^2(y - y_q)^2}, \text{ the form normally assumed}\DPtypo{.}{,} \\ + &= Ae^{-k^2z_q^2}, +\end{align*} +where $z_q$ is the absolute magnitude of the error in +the measurement~$y_q$. + +This is, clearly, only one amongst a number of possible solutions. +But with one additional assumption we can prove that +this is the only law of error which leads to the arithmetic mean. +%% -----File: 209.png---Folio 198------- +\index{Gauss, and laws of error}% +\index{M'Alister, Sir Donald, and laws of error}% +Let us assume that negative and positive errors of the same +absolute amount are equally likely. + +In this case $f_q$ must be of the form $Be^{\theta(y - y_q)^2}$, +\[ +\therefore \phi'(y)(y - y_q) - \phi(y) + \psi(y_q) = \theta(y - y_q)^2. +\] + +Differentiating with respect to~$y$, +\[ +\phi''(y) = 2\frac{d}{d(y - y_q)^2}\theta (y - y_q)^2. +\] + +But $\phi''(y)$ is, by hypothesis, independent of~$y_q$. + +$\therefore \dfrac{d}{d(y - y_q)^2} \theta(y - y_q)^2 = -k^2$ where $k$~is constant; integrating, +$\theta (y - y_q)^2 = -k^2(y - y_q)^2 + \log C$ and we have $f_q = Ae^{-k^2(y - y_q)^2}$ (where +$A = BC$). + +(ii.) What is the law of error, if the geometric mean of the +\index{Law of error!geometric@{and geometric mean}}% +measurements leads to the most probable value of the quantity? + +In this case $\Sum \dfrac{f'_q}{f_q}= 0$ must be equivalent to $\Prod \limits^{q=m}_{q=1} y_q = y^m$, \ie\ to +\[ +\Sum \log\frac{y_q}{y} = 0. +\] +Proceeding as before, we find that the law of error is +\[ +f_q = Ae^{\phi'(y) \log \frac{y_q}{y} + \int \frac{\phi'(y)}{y}\, dy + \psi(y_q)}. +\] +There is no solution of this which satisfies the condition that +negative and positive errors of the same absolute magnitude are +equally likely. For we must have +\begin{gather*} +\phi'(y) \log \frac{y_q}{y} + \int \frac{\phi'(y)}{y}\, dy + \psi(y_q) = \phi(y - y_q)^2 \\ +\text{or } \phi''(y) \log \frac{y_q}{y} = \frac{d}{dy} \phi(y - y_q)^2, +\end{gather*} +which is impossible. + +The simplest law of error, which leads to the geometric mean, +seems to be obtained by putting $\phi'(y) = -ky$, $\psi(y_q) = 0$. This +gives $f_q = A \left(\dfrac{y}{y_q}\right)^{ky} e^{-ky}$. + +A law of error, which leads to the geometric mean of the +observations as the most probable value of the quantity, has been +previously discussed by Sir~Donald McAlister (\textit{Proceedings of the +Royal Society}, vol.~xxix.\ (1879) p.~365). His investigation depends +upon the obvious fact that, if the geometric mean of the +%% -----File: 210.png---Folio 199------- +observations yields the most probable value of the quantity, the +arithmetic mean of the logarithms of the observations must yield +the most probable value of the logarithm of the quantity. Hence, +if we suppose that the logarithms of the observations obey the +normal law of error (which leads to their arithmetic mean as the +\index{Law of error!normal law}% +most probable value of the logarithms of the quantity), we can +by substitution find a law of error for the observations themselves +which must lead to the geometric mean of them as the most +probable value of the quantity itself. + +If, as before, the observations are denoted by $y_q$,~etc., and the +quantity by~$y$, let their logarithms be denoted by $l_q$,~etc., and by~$l$. +Then, if $l_q$,~etc., obey the normal law of error, $f(l_q, l) = Ae^{-k^2(l_q - l)^2}$. +Hence the law of error for $y_q$,~etc., is determined by +\begin{align*} +f(y_q, y) + &= Ae^{-k^2(\log y_q - \log y)^2}\\ + &= Ae^{-k^2(\log \frac{y_q}{y})^2}, +\end{align*} +and the most probable value of~$y$ must, clearly, be the geometric +mean of $y_q$,~etc. + +This is the law of error which was arrived at by Sir Donald +McAlister. It can easily be shown that it is a special case of the +generalised form which I have given above of all laws of error +leading to the geometric mean. For if we put $\psi(y_q) = -k^2(\log y_q)^2$, +and $\phi'(y) = 2k^2 \log y$, we have +\begin{align*} +f_q &= Ae^{2k^2 \log y \log \frac{y_q}{y} + \int 2k^2\, \frac{\log y}{y}\, dy - k^2(\log y_q)^2}\\ + &= Ae^{2k^2 \log y \log y_q - 2k^2 (\log y)^2 + k^2(\log y)^2 - k^2 (\log y_q)^2}\\ + &= Ae^{-k^2\left(\log \frac{y_q}{y}\right)^2}. +\end{align*} + +\index{Kapteyn, Prof.\ J. C.!law of error@{and law of error}}% +A similar result has been obtained by Professor J.~C. Kapteyn.\footnote + {\textit{Skew Frequency Curves}, p.~22, published by the Astronomical Laboratory +\index{Frequency curves}% + at Groningen (1903).} +But he is investigating frequency curves, not laws of error, and +this result is merely incidental to his main discussion. His +method, however, is not unlike a more generalised form of Sir +Donald McAlister's. In order to discover the frequency curve +of certain quantities~$y$, he supposes that there are certain other +quantities~$z$, functions of the quantities~$y$, which are given by +$z = F(y)$, and that the frequency curve of these quantities~$z$ is +\emph{normal}. By this device he is enabled in the investigation of a +type of skew frequency curve, which is likely to be met with +often, to utilise certain statistical constants corresponding to +%% -----File: 211.png---Folio 200------- +\index{Median and laws of error}% +those which have been already calculated for the normal +curve. + +In fact the main advantage both of Sir Donald McAlister's +law of error and of Professor Kapteyn's frequency curves lies in +\index{Law of error!median@{and median}}% +the possibility of adapting without much trouble to unsymmetrical +phenomena numerous expressions which have been already +calculated for the normal law of error and the normal curve of +frequency.\footnote + {It may be added that Professor Kapteyn's monograph brings forward + considerations which would be extremely valuable in determining the types of + phenomena to which geometric laws of error are likely to be applicable.} + +This method of proceeding from arithmetic to geometric laws +of error is clearly capable of generalisation. We have dealt with +the geometric law which can be derived from the normal arithmetic +law. Similarly if we start from the simplest geometric +law of error, namely, $f_q = A\left(\dfrac{y}{y_q}\right)^{k^2y} e^{-k^2y}$, we can easily find, by +writing $\log y = l$ and $\log y_q = l_q$, the corresponding arithmetic +law, namely, $f_q = Ae^{k^2el(l - l_q) - k^2el}$, which is obtained from the +generalised arithmetic law by putting $\phi(l) = k^2e^{l}$ and $\psi(l_q) = 0$. +And, in general, corresponding to the arithmetic law +\[ +f_q = Ae^{\phi'(y)(y - y_q) - \phi(y) + \psi(y_q)}, +\] +we have the geometric law +\[ +f_q = Ae^{\phi'_l(z) \log \frac{z_q}{z } + \int \frac{\phi_l(z)}{z}\, dz + \psi_l(z_q)}, +\] +where +\[ +y = \log z,\ +y_q = \log z_q,\ +\int \frac{\phi'_1(z)}{z}\, dz = \phi(\log z) \text{ and } \psi_l(z_q) = \psi(\log z_q). +\] + +(iii.) What law of error does the harmonic mean imply? + +In this case, $\Sum\dfrac{f'_q}{f_q} = 0$ must be equivalent to $\Sum\left(\dfrac{1}{y_q} - \dfrac{1}{y}\right) = 0$. + +Proceeding as before, we find that $f_q = Ae^{\phi'(y)\left[\frac{1}{y_q} - \frac{1}{y}\right] - \int \frac{\phi'(y)}{y^2}\, dy + \psi(y_q)}$. +A simple form of this is obtained by putting $\phi'(y) = -k^2y^2$ and +$\psi(y_q) = -k^2y_q$. Then $f_q = Ae^{\frac{k^2}{y_q} (y - y_q)^2} = Ae^{-k^2\frac{z_q^2}{y_q}}$. With this law, +positive and negative errors of the same absolute magnitude are +not equally likely. + +(iv.) If the most probable value of the quantity is equal to the +median of the measurements, what is the law of error? + +The median is usually defined as the measurement which +%% -----File: 212.png---Folio 201------- +\index{Fechner, and median}% +occupies the middle position when the measurements are ranged +in order of magnitude. If the number of measurements~$m$ is odd, +the most probable value of the quantity is the $\dfrac{m+1}{2}$th, and, if the +number is even, all values between the $\dfrac{m}{2}$th and the $\left(\dfrac{m}{2}+1\right)$th are +equally probable amongst themselves and more probable than +any other. For the present purpose, however, it is necessary to +make use of another property of the median, which was known +to Fechner (who first introduced the median into use) but which +seldom receives as much attention as it deserves. \emph{If~$y$ is the +median of a number of magnitudes, the sum of the absolute differences +\emph{(\ie\ \emph{the difference always reckoned positive})} between $y$ and each of +the magnitudes is a minimum.} The median~$y$ of $y_1y_2\ldots y_m$ is +found, that is to say, by making $\Sum^m_1 |y_q - y|$ a minimum where +$|y_q - y|$ is the difference always reckoned positive between $y_q$ +and~$y$. + +We can now return to the investigation of the law of error +corresponding to the median. + +Write $|y - y_q| = z_q$. Then since $\Sum^m_1 z_q$ is to be a minimum we +must have $\Sum^m_1 \dfrac{y - y_q}{z_q} = 0$. Whence, proceeding as before, we have +\[ +f_q = Ae^{\int \frac{y - y_q}{z_q}\, \phi''(y)\, dy + \psi(y_q).} +\] + +The simplest case of this is obtained by putting +\begin{DPalign*} +\phi''(y) &= -k^2, \\ +\psi(y_q) &= \frac{y-y_q}{z_q}\, k^2y_q, \\ +\lintertext{whence} +f_q &= Ae^{-k^2|y - y_q|} = Ae^{-k^2 z_q}. +\end{DPalign*} + +This satisfies the additional condition that positive and negative +errors of equal magnitude are equally likely. Thus in this +important respect the median is as satisfactory as the arithmetic +mean, and the law of error which leads to it is as simple. It also +resembles the normal law in that it is a function of the error \emph{only}, +and not of the magnitude of the measurement as well. + +The median law of error, $f_q = Ae^{-k^2 z_q}$, where $z_q$~is the absolute +amount of the error always reckoned positive, is of some historical +%% -----File: 213.png---Folio 202------- +\index{Least Squares and Venn!method of}% +interest, because it was the earliest law of error to be formulated. +\index{Law of error!normal law}% +The first attempt to bring the doctrine of averages into definite +relation with the theory of probability and with laws of error was +published by Laplace in 1774 in a memoir ``sur la probabilité des +\index{Laplace!doctrine@{and doctrine of averages}}% +causes par les événemens.''\DPnote{** TN: [sic], not événements}\footnote + {\textit{Mémoires présentés à l'Académie des Sciences}, vol.~vi.} +This memoir was not subsequently +incorporated in his \textit{Théorie analytique}, and does not represent his +more mature view. In the \textit{Théorie} he drops altogether the law +tentatively adopted in the memoir, and lays down the main lines +of investigation for the next hundred years by the introduction +of the \emph{normal} law of error. The popularity of the normal law, +with the arithmetic mean and the method of least squares as its +corollaries, has been very largely due to its overwhelming advantages, +in comparison with all other laws of error, for the purposes +of mathematical development and manipulation. And in +addition to these technical advantages, it is probably applicable +as a first approximation to a larger and more manageable group +of phenomena than any other single law. So powerful a hold +indeed did the normal law obtain on the minds of statisticians, +that until quite recent times only a few pioneers have seriously +considered the possibility of preferring in certain circumstances +other means to the arithmetic and other laws of error to the +normal. Laplace's earlier memoir fell, therefore, out of remembrance. +But it remains interesting, if only for the fact that a +law of error there makes its appearance for the first time. + +Laplace sets himself the problem in a somewhat simplified +form: ``Déterminer le milieu que l'on doit prendre entre trois +observations données d'un même phénomène.'' He begins by +assuming a law of error $z = \phi(y$), where $z$~is the probability of an +error~$y$; and finally, by means of a number of somewhat arbitrary +assumptions, arrives at the result $\phi (y) = \dfrac{m}{2} e^{-my}$. If this formula +is to follow from his arguments, $y$~must denote the \emph{absolute} error, +always taken positive. It is not unlikely that Laplace was led +to this result by considerations other than those by which he +attempts to justify it. + +Laplace, however, did not notice that his law of error led to +the median. For, instead of finding the most probable value, +which would have led him straight to it, he seeks the ``mean of +error''---the value, that is to say, which the true value is as likely +%% -----File: 214.png---Folio 203------- +\index{Mode, and law of error}% +to fall short of as to exceed. This value is, for the median law, +laborious to find and awkward in the result. Laplace works it +out correctly for the case where the observations are no more +than three. + +\Paragraph{6.} I do not think that it is possible to find by this method a +law of error which leads to the mode. But the following general +\index{Law of error!mode@{and mode}}% +formulae are easily obtained: + +(v.) If $\Sum \theta(y_q, y)$ is the law of relation between the measurements +and the most probable value of the quantity, then the law +of error $f_q(y_q, y)$ is given by $f_q = Ae^{\int \theta (y_qy)\phi''(y)\, dy + \psi(y_q)}$. Since $f_q$~lies +between $0$~and~$1$, $\int \theta (y_qy)\phi''(y)\, dy + \psi(y_q) + \log A$ must be negative +for all values of $y_q$~and~$y$ that are physically possible; and, since +the values of~$y_q$ are between them exhaustive, +\[ +\Sum Ae^{\int \theta (y_qy)\phi''(y)\, dy + \psi(y_q)} = 1, +\] +where the summation is for all terms that can be formed by giving~$y_q$ +every value \textit{à~priori} possible. + +(vi.) The most general form of the law of error, when it is +assumed that positive and negative errors of the same magnitude +are equally probable, is $Ae^{-k^2 f(y - y_q)^2}$, where the most probable +value of the quantity is given by the equation +\[ +\Sum (y - y_q) f'(y - y_q)^2 = 0,\text{ where } +f'(y - y_q)^2 = \frac{d}{d(y - y_q)^2} f(y - y_q)^2. +\] +The arithmetic mean is a special case of this obtained by putting +$f(y - y_q)^2 = (y - y_q)^2$; and the median is a special case obtained +by putting $f(y - y_q)^2 = +\sqrt{(y - y_q)^2}$. + +We can obtain other special cases by putting +\[ +f(y - y_q)^2 = (y - y_q)^4, +\] +when the law of error is $Ae^{-k^2(y - y_q)^2}$ and the most probable values +are the roots of $my^3 - 3y^2\Sum y_q + 3y\Sum y^2_q - \Sum y^3_q = 0$; and by putting +$f(y - y_q)^2 = \log(y - y_q)^2$, when the law of error is $\dfrac{A}{(y - y_q)} = 0$ and +the most probable values the roots of $\Sum \dfrac{1}{(y - y_q)} = 0$. In all these +cases the law is a function of the error only. + +\Paragraph{7.} These results may be summarised thus. We have +assumed: + +(a) That we have no reason, before making measurements, for +%% -----File: 215.png---Folio 204------- +\index{Independence, for knowledge!measurement@{and measurement}}% +supposing that the quantity we measure is more likely to have +any one of its possible values than any other. + +(b) That the errors are independent, in the sense that a +\emph{knowledge} of how great an error has been made in one case does +not affect our expectation of the probable magnitude of the error +in the next. + +(c) That the probability of a measurement of given magnitude, +when in addition to the \textit{à~priori} evidence the real value of the +quantity is supposed known, is an algebraic function of this +given magnitude of the measurement and of the real value of the +quantity. + +(d) That we may regard the series of possible values as continuous, +without sensible error. + +(e) That the \textit{à~priori} evidence permits us to assume a law of +error of the type specified in~(c); \ie\ that the algebraic function +referred to in~(c) is known to us \textit{à~priori}. + +Subject to these assumptions, we have reached the following +conclusions: + +(1) The most general form of the law of error is +\[ +f_q = Ae^{\int \phi''(y)\theta (y_qy)\, dy + \psi(y_q)}, +\] +leading to the equation $\Sum\theta (y_qy) = 0$, connecting the most probable +value and the actual measurements, where $y$~is the most probable +value and $y_q$,~etc., the measurements. + +(2) Assuming that positive and negative errors of the same +absolute magnitude are equally likely, the most general form is +$f_q = Ae^{-k^2 f(y - y_q)^2}$, leading to the equation $\Sum (y - y_q) f'(y - y_q)^2 = 0$, +where $f'z = \dfrac{d}{dz}fz$. Of the special cases to which this form gives +rise, the most interesting were + +(3) $f_q = Ae^{-k^2(y - y_q)^2} = Ae^{-k^2z_q^2}$, where $z_q = |y - y_q|$, leading to +the arithmetic mean of the measurements as the most probable +value of the quantity; and + +(4) $f_q = Ae^{-k^2z_q}$, leading to the median. + +(5) The most general form leading to the arithmetic mean is +$f_q = Ae^{\phi'(y - y_q) - \phi(y) + \psi(y_q)}$, with the special cases~(3), and + +(6) $f_q = Ae^{k^2e^y (y - y_q) - k^2 e^y}$. + +(7) The most general form leading to the geometric mean is +$f_q = Ae^{\phi'(y)\log \frac{y_q}{y} + \int \frac{\phi'(y)}{y}\, dy + \psi (y_q)}$, with the special cases: +%% -----File: 216.png---Folio 205------- +\index{Arithmetic mean (or average)}% +\index{Averages|ifoll}% +\index{Least Squares and Venn!method of}% + +(8) $f_q = A\left(\frac{y}{y_q}\right)^{k^2y} e^{-k^2y}$, and + +(9) $f_q = Ae^{-k^2\left(\log\frac{y_q}{y}\right)^2}$. + +(10) The most general form leading to the harmonic mean is +$f_q = Ae^{\phi'(y)\left[\frac{1}{y_q} - \frac{1}{y}\right] - \int\frac{\phi'(y}{y^2}\DPtypo{}{\,dy} + \psi(y_q)}$, with the special case + +(11) $f_q = Ae^{-k^2\frac{(y - y_q)^2}{y_q}} = Ae^{-k^2\frac{z_q^2}{y_q}}$. + +(12) The most general form leading to the median is +\[ +f_q = Ae^{\phi'(y)\frac{y - y_q}{z_q} + \psi(y_q)}, +\] +with the special case~(4). + +In each of these expressions, $f_q$~is the probability of a measurement~$y_q$, +given that the true value is~$y$. + +\Paragraph{8.} The doctrine of Means and the allied theory of Least +Squares comprise so extensive a subject-matter that they cannot +be adequately treated except in a volume primarily devoted to +them. As, however, they are one of the important practical +applications of the theory of probability, I am unwilling to pass +them by entirely; and the following discursive observations, +chiefly relating to the Normal Law of Error, will serve, taken in +\index{Law of error!normal law}% +conjunction with the paragraphs immediately preceding, to +illustrate the connection between the theories of this treatise +and the general treatment of averages. + +\Paragraph{9.} \textit{The Claims of the Arithmetic Average}.---By definition the +arithmetic average of a number of quantities is nothing more +than their arithmetic sum divided by their number. But the +utility of an average generally consists in our supposed right to +substitute, in certain cases, this single measure for the varying +measures of which it is a function. Sometimes this requires no +justification; the word ``average'' is in these cases used for +the sake of shortness, and merely to summarise a set of facts: +as, for instance, when we say that the birth-rate in England is +greater than the birth-rate in France. + +But there are other cases in which the average makes a more +substantial claim to add to our knowledge. After a number of +examiners of equal capacity have given varying marks to a +candidate for the same paper, it may be thought fair to allow +the candidate the average of the different marks allotted: and +in general if several estimates of a magnitude have been made, +%% -----File: 217.png---Folio 206------- +\index{Arithmetic mean (or average)!Laplace on}% +\index{Arithmetic mean (or average)!Gauss on}% +\index{De Witt and arithmetic averages}% +\index{Fresnel and simplicity}% +\index{Gauss, and laws of error!arithmetic mean@{and arithmetic mean}}% +\index{Least Squares and Venn}% +\index{Least Squares and Venn!method of}% +between the accuracy of which we have no reason to discriminate, +we often think it reasonable to act as if the true magnitude were +the average of the several measurements. Perhaps De~Witt, in +his report on Annuities to the States General in 1671,\footnote + {\textit{De vardye van de lif-renten na proportie van de \DPtypo{losrenten}{los-renten}}. The Hague, 1671.} +was the +first to use it scientifically. But as Leibniz points out: ``Our +\index{Leibniz!arithmetic average@{and arithmetic average}}% +peasants have made use of it for a long time according to their +natural mathematics. For example, when some inheritance or +land is to be sold, they form three bodies of appraisers; these +bodies are called \emph{Schurzen} in Low Saxon, and each body makes +an estimate of the property in question. Suppose, then, that +the first estimates its value to be $1000$ crowns, the second, $1400$, +the third, $1500$; the sum of these three estimates is taken, viz.\ +$3900$, and because they were three bodies, the third, \ie~$1300$, is +taken as the mean value asked for. This is the axiom: \textit{aequalibus +aequalia}, equal suppositions must have equal consideration.''\footnote + {\textit{Nouveaux Essais}. Engl.\ transl.\ p.~540.} + +But this is a very inadequate axiom. Equal suppositions +would have equal consideration, if the three estimates had been +multiplied together instead of being added. The truth is that +at all times the arithmetic mean has had \emph{simplicity} to recommend +it. It is always easier to add than to multiply. But simplicity +is a dangerous criterion: ``La nature,'' says Fresnel, ``ne s'est +pas embarassée des difficultés d'analyse, elle n'a évité que la +complication des moyens.'' + +With Laplace and Gauss there began a series of attempts to +\index{Laplace!arithmetic mean@{and arithmetic mean}}% +\emph{prove} the worth of the arithmetic mean. It was discovered that +its use involved the assumption of a particular type of law of +error for the \textit{à~priori} probabilities of given errors. It was also +found that the assumption of this law led on to a more complicated +rule, known as the Method of Least Squares for combining +the results of observations which contain more than one +doubtful quantity. In spite of a popular belief that, whilst the +Arithmetic Mean is intuitively obvious, the Method of Least +Squares depends upon doubtful and arbitrary assumptions, it +can be demonstrated that the two stand and fall together.\footnote + {Venn (\textit{Logic of Chance}, p.~40) thinks that the Normal Law of Error and +\index{Venn!Least Squares@{and Least Squares}|inote}% + the Method of Least Squares ``are not only totally distinct things, but they have + scarcely even any necessary connection with each other. The Law of Error + is the statement of a physical fact\ldots. The Method of Least Squares, on the + other hand, is not a law at all in the scientific sense of the term. It is simply + a rule or direction\ldots.''} +%% -----File: 218.png---Folio 207------- +\index{Ellis, Leslie!Least Squares@{and Least Squares}|inote}% +\index{Hagen, and error}% +\index{Maclaurin, Theorem of}% +\index{Mathematicians, and probability!laws of error@{and laws of error}}% + +The analytical theorems of Laplace and Gauss are complicated, +but the special assumptions upon which they are based are easily +stated.\footnote + {For an account of the three principal methods of arriving at the Method + of Least Squares and the Arithmetic Mean, see Ellis, \textit{Least Squares}. Gauss's + first method is in the \textit{Theoria Motus}, and his second in the \textit{Theoria Combinationis + Observationum}. Laplace's investigations are in chap.~iv.\ of the second + Book of the \textit{Théorie analytique}. Laplace's method was improved by Poisson +\index{Poisson!least errors@{and least errors}}% + in the \textit{Connaissance des temps} for 1827 and~1832.} +Gauss supposes (\textit{a})~that the probability of a given error +is a function of the error only and not also of the magnitude of +the observation, (\textit{b})~that the errors are so small that their cubes +and higher powers may be neglected. Assumption~(\textit{a}) is arbitrary,\footnote + {It does not follow, as G.~Hagen argues (\textit{Grundzüge der Wahrscheinlichkeitsrechnung}, + p.~29), that, because a larger error is less probable than a smaller, + \emph{therefore} the probability of a given error is a function of its magnitude + only.} +and Gauss did not state it explicitly. These two assumptions, +together with certain others, lead us to the result. For +let $\phi(z)$~be the law of error where $z$~is the error, and let us assume, +as it always is assumed in these proofs, that $\phi(z)$ can be expanded +by Maclaurin's Theorem. Then $\phi (x) = \phi (0) + z\phi'(0) + \dfrac{z^2}{2!}\phi''(0) + \dfrac{z^3}{3!}\phi'''(0) + \ldots$. +It is also supposed that positive and negative +errors are equally probable, \ie~$\phi(z) = \phi(-z)$, so that $\phi'(0)$ and~$\phi'''(0)$ +vanish. Since we may neglect~$z^4$ in comparison with~$z^2$, +$\phi{z} = \phi(0) + \frac{1}{2}z^2\phi''(0)$. But (neglecting~$z^4$ and higher powers) +$a + bz^2 = ae^{\frac{bz^2}{a}}$, so that $\phi(z) = ae^{\frac{bz^2}{a}}$. + +Gauss's proof looks much more complicated than this, but he +obtains the form $ae^{\frac{bz^2}{a}}$ by neglecting higher powers of~$z$, so that +this expression is really equivalent to $a + bz^2$. By this approximation +he has reduced all the possible laws to an equivalent +form.\footnote + {This is pointed out by Bertrand, \textit{Calcul des probabilités}, p.~267.} +It is true, therefore, that the normal law of error is, to +the second power of the error, equivalent to any law of error, +\emph{which is a function of the error only, and for which positive and +negative errors are equally probable}. Laplace also introduces +assumptions equivalent to these. + +While mathematicians have endeavoured to establish the +normal law of error and the arithmetic mean as a law of logic, +%% -----File: 219.png---Folio 208------- +\index{Fechner, and median!law of sensation@{and law of sensation}}% +\index{Pearson, Karl!arithmetic mean@{and arithmetic mean}}% +others have claimed for it the testimony of experience and have +deemed it a law of nature.\footnote + {This is, of course, a very common point of view indeed. Cf.~Bertrand, +\index{Bertrand!Law of Error@{and Law of Error}|inote}% + \textit{op.~cit.}\ p.~183: ``Malgré les objections précédentes, la formule de Gauss doit + être adoptée. L'observation la confirme: cela doit suffire dans les applications.''} + +That this cannot be so, is evident. For suppose that $x_1x_2 \ldots x_n$ +are a set of observations of an unknown quantity~$x$. Then, by +this principle, $x = \dfrac{1}{n}\Sum x_r$ gives the most probable value of~$x$. But +suppose we had wished to determine~$x^2$, our observations, assuming +that we can multiply correctly, would be $x_1^2$,~$x_2^2 \ldots x_n^2$, +and the most probable value of $x^2 = \dfrac{1}{n}\Sum x_r^2$. But $(\dfrac{1}{n}\Sum x_r)^2 = \dfrac{1}{n}\Sum x_r^2$. +And in general, $\dfrac{1}{n}\Sum f(x_r) \neq f(\dfrac{1}{n}\Sum x_r)$. Nor is this a consideration +which can safely be ignored in practice. For our ``observations'' +are often the result of some manipulation, and the particular +shape in which we get them is not necessarily fixed for us. It is +not easy to say what the \emph{direct} observation is. In particular if +any such law of sensation, as that enunciated by Fechner, is true +(\ie~that sensation varies as the logarithm of the stimulus), the +arithmetic mean must break down as a \emph{practical} rule in all cases +where human sensation is part of the instrument by means of +which the observations are recorded.\footnote + {This was noticed by Galton.} +\index{Galton!Fechner's law@{and Fechner's law}}% + +Apart, however, from theoretical refutations, statisticians now +recognise that the arithmetic mean and the normal law of error +can only be applied to certain special classes of phenomena. +\index{Quetelet!arithmetic mean@{and arithmetic mean}}% +Quetelet\footnote + {\DPtypo{E.g.}{\Eg}~\textit{Letters on the Theory of Probabilities}, p.~114.} +was, I think, the first to point this out. In England, +Galton drew attention to the fact many years ago, and Professor +Pearson\footnote + {On ``Errors of Judgment, etc.,'' \textit{Phil.\ Trans.}~A, vol.~cxcviii.\ pp.~235--299. + The following quotation is from his memoir \textit{On the General Theory of Skew + Correlation and \DPtypo{Nonlinear}{Non-linear} Regression}, where further references are given.} +has shown ``that the Gaussian-Laplace normal distribution +is very far from being a general law of frequency +distribution either for errors of observation or for the distribution +of deviations from type such as occur in organic populations\ldots. +It is not even approximately correct, for example, in the distribution +of barometric variations, of grades of fertility and incidence +of disease.'' +%% -----File: 220.png---Folio 209------- +\index{Ellis, Leslie!Least Squares@{and Least Squares}}% +\index{Least Squares and Venn!method of}% +\index{Merriman, Mansfield, and Least Squares}% + +The Arithmetic Mean occupies, therefore, no unique position; +and it is worth while, from the point of view of probability, to +discuss the properties of other possible means and laws of error, +as, for example, on the lines indicated in the earlier part of this +chapter. + +\Paragraph{10.} \textit{The Method of Least Squares.}---The problem, to which this +method is applied, is no more than the application of the same +considerations, as those which we have just been discussing, to +cases where the relation between the observed measurements and +the quantity whose most probable value we require, involves +more than one unknown. + +Owing to the surprising character of its conclusions, if they +could be accepted as universally valid, and to the obscurity of +the mathematical fabric that has been reared on and about it, +this method has been surrounded by an unnecessary air of +mystery. It is true that in recent times scepticism has grown +at the expense of mystery. It is also true that just views have +been held by individuals for sixty years past, notably by Leslie +Ellis. But the old mistakes are not always corrected in the +current text-books, and even so useful and generally used a +treatise on Least Squares, as Professor Mansfield Merriman's, +opens with a series of very fallacious statements. + +The controversial side of the Method of Least Squares is +purely logical; in the later developments there is much elaborate +mathematics of whose correctness no one is in doubt. What it +is important to state with the utmost possible clearness is the +precise assumptions on which the mathematics is based; when +these assumptions have been set forth, it remains to determine +their applicability in particular cases. + +In dealing with averages we supposed ourselves to be presented +with a number of direct observations of some quantity +which it is desired to determine. But it is obvious that direct +observations will be in many cases either impracticable or inconvenient; +and our natural course will be to measure certain +other quantities which we know to bear fixed and invariable +relations to the unknowns we wish to determine. In surveying, +for instance, or in astronomy, we constantly prefer to take +measurements of angles or distances in which we are not interested +for their own sakes, but which bear known geometrical relationships +to the set of ultimate unknowns. +%% -----File: 221.png---Folio 210------- +\index{Astronomers and Least Squares}% +\index{Boscovitch and Least Squares}% +\index{Euler and Least Squares}% +\index{Gauss, and laws of error!Least Squares@{and Least Squares}}% +\index{Lambert and Least Squares}% +\index{Legendre and Least Squares}% +\index{Mayer and Least Squares}% +\index{Simpson and Least Squares}% + +If we wish to determine the most probable values of a set of +unknowns $x_1, x_2, \ldots x_r$, instead of obtaining a number of +sets of direct observations of each, we may obtain a number of +equations of observation of the following type: +\[ +\begin{array}{*{4}{r@{}}l} +a_1x_1 &{}+ a_2x_2 &{}+ \ldots &{}+ a_rx_r &{}= V_1,\\ +b_1x_1 &{}+ b_2x_2 &{}+ \ldots &{}+ b_rx_r &{}= V_2,\\ +\hdotsfor[10]{5}\\ +k_1x_1 &{}+ k_2x_2 &{}+ \ldots &{}+ k_rx_r &{}= V_n, +\end{array} +\] +where $V_1$,~etc., are the quantities \emph{directly} observed, and the $a'$s, +$b'$s,~etc., are supposed known~($n>r$). + +We have in such a case $n$~equations to determine $r$~unknowns, +and since the observations are likely to be inexact, there may be +no precise solution whatever. In these circumstances we wish to +know the most probable set of values of the~$x$'s warranted by +these observations. + +The problem is precisely similar in kind to that dealt with +by averages and differs only in the degree of its complexity. It +is the problem of finding the most probable solution of such a set +of discrepant equations of observation that the Method of Least +Squares claims to solve. + +By 1750 the astronomers were obtaining such equations of +observation in the course of their investigations, and the question +arose as to the proper manner of their solution. Boscovich in +Italy, Mayer and Lambert in Germany, Laplace in France, Euler +\index{Laplace!Least Squares@{and Least Squares}}% +in Russia, and Simpson in England proposed different methods +of solution. Simpson, in 1757, was the first to introduce, by way +of simplification, the assumption or axiom that positive and +negative errors are equally probable.\footnote + {See Merriman's \textit{Method of Least Squares}, p.~181, for an historical sketch, + from which the above is taken. In 1877 Merriman published in the \textit{Transactions + of the Connecticut Academy} a list of writings relating to the Method of + Least Squares and the theory of accidental errors of observation, which comprised + 408~titles---classified as 313~memoirs, 72~books, 23~parts of books.} +The Method of Least +Squares was first definitely stated by Legendre in 1805, who +proposed it as an advantageous method of adjusting observations. +This was soon followed by the `proofs' of Laplace and Gauss. +But it is easily shown that these proofs involve the normal law +of error $y = ke^{-k^2x^2}$, and the theory of Least Squares simply +develops the mathematical results of applying to equations of +observation, which involve more than one unknown, that law +%% -----File: 222.png---Folio 211------- +\index{Index numbers}% +of error which leads to the Arithmetic Mean in the case of a single +unknown. + +\Paragraph{11.} \textit{The Weighting of Averages.}---It is necessary to recur to +\index{Averages!weighting of}% +\index{Weighting of averages}% +the distinction made at the beginning of §\;9 between the two +types to which our average, or, as it is generally termed in social +inquiries, our index number, may belong. The average or index +number may simply summarise a set of facts and give us the +actual value of a composite quantity, as, for example, the index +number of the cost of living. In such cases the composite +quantity, in which we are interested, need not contain precisely +the same number of units of each of the elementary quantities of +which it is composed, so that the `weights,' which denote the +numbers of each elementary quantity appropriate to the composite +quantity, are part of the definition of the composite +quantity, and can no more be dispensed with than the magnitudes +of the elementary quantities themselves. Nor in such cases is +the rejection of discordant observations permissible; if, that is +to say, some of the elementary quantities are subject to much +wider variation, or to variations of a different type than the +majority, that is no reason for rejecting them. + +On the other hand, the individual items, out of which the +average is composed, may each be \emph{indications} or approximate +estimates of some \emph{one single} quantity; and the average, instead +of representing the measure of a composite quantity, may be +selected as furnishing the most probable value of the single +quantity, given, as evidence of its magnitude, the values of the +various terms which make up the average. + +If this is the character of our average, the problem of weighting +depends upon what we know about the individual observations +or samples or indications, out of which our average is to be built +up. The units in question may be \emph{known} to differ in respects +relevant to the probable value of the \textit{quaesitum}. Thus there +may be reasons, quite apart from the actual results of the individual +observations or samples, for trusting some of them more +than others. Our knowledge may indicate to us, in fact, that +the constants of the laws of error appropriate to the several +instances, even if the type of the law can be assumed to be +constant, should be varied according to the data we possess about +each. It may also indicate to us that the condition of \emph{independence} +between the instances, which the method of averages +%% -----File: 223.png---Folio 212------- +\index{Independence, for knowledge!averages@{and averages}}% +\index{Jevons!index numbers@{and index numbers}}% +presumes, is imperfectly satisfied, and consequently that our +mode of combining the instances in an average must be modified +accordingly. + +Some modern statisticians, who, really influenced perhaps by +practical considerations, have been inclined to deprecate the +importance of weighting on theoretical grounds, have not always +been quite clear what kind of average they supposed themselves +to be dealing with. In particular, discussions of the question of +weighting in connection with index numbers of the value of +money have suffered from this confusion. It has not been clear +whether such index numbers really represent measures of a +composite quantity or whether they are probable estimates of +the value of a single quantity formed by combining a number of +independent approximations towards the value of this quantity. +The original Jevonian conception of an index number of the +value of money was decidedly of the latter type. Modern work +on the subject has been increasingly dominated by the other +conception. A discussion of where the truth lies would lead me +too far into the field of a subject-matter alien to that of this +treatise. + +Theoretical arguments against weighting have sometimes +been based on the fact that to weight the items of the average +in an irrelevant manner, or, as it is generally expressed, in a +random manner, is not likely, provided the variations between +the weights are small compared with the variations between the +items, to affect the result very much. But why should any one +wish to weight an average ``at random''? Such observations +overlook the real meaning and significance of weights. They are +probably inspired by the fact that a superficial treatment of +statistics would sometimes lead to the introduction of weights +which are irrelevant. In drawing a conclusion, for example, +from the vital statistics of various towns, the figures of population +for the different towns may or may not be relevant to our conclusion. +It depends on the character of the argument. If they +are relevant, it may be right to employ them as weights. If they +are irrelevant, it must be wrong and unnecessary to do so. The +fact that what is a more important article of consumption than +pins \emph{may}, on certain assumptions, be irrelevant to the usefulness +of variations in the price of each article as indications of variation +in the value of money. With other assumptions, it may be +%% -----File: 224.png---Folio 213------- +\index{Discordant observations, rejection of}% +extremely relevant. Or again, we may know that observations +with a particular instrument tend to be too large and must, +therefore, be weighted down. It is contrary both to theory and +to common sense to suppose that the possession of information +as to the relative reliability of different statistics is not useful. +There is no place, therefore, in my judgment, for a \emph{generalised} +argument as to the propriety or impropriety of weighting an +average. + +It should be added that, where we seek to build up an index +number of a conception, which is quantitative but is not itself +numerically measurable in any defined or unambiguous sense, by +combining a number of numerical quantities, which, while they +do not measure our \textit{quaesitum} are nevertheless indications of its +quantitative variations and tend to fluctuate in the same sense, +as, for example, by means of what are sometimes called \emph{economic +barometers} of the state of business, or the prosperity of the country +or the like, some very confusing questions can arise both as to +what sort of a thing our resulting index really is, and as to the +mode of compilation appropriate to it. + +These confusing questions always arise when, instead of +measuring a quantity directly, we seek an index to fluctuations +in its magnitude by combining in an average the fluctuations of +a series of magnitudes, which are, each of them in a different way, +to some extent (but only to some extent), correlated with fluctuations +in our \textit{quaesitum}. I must not burden this book with a +discussion of the problems of Index Numbers. But I venture to +think that they would be sooner cleared up if the natures and +purposes of differing index numbers were more sharply distinguished---those, +namely, which are simply descriptive of a composite +commodity, those which seek to combine results differing from +one another in a way analogous to the variations of an instrument +of precision, and those which combine results, not of the \textit{quaesitum} +itself, but of various other quantities, variations in which are +partly due to variations in the \textit{quaesitum}, but which we well +know to be also due to other distinguishable influences. Index +numbers of the third type are often treated by methods and +arguments only appropriate to those of the second type. + +\Paragraph{12.} \textit{The Rejection of Discordant Observations}.---This differs +from the problem just discussed, because we have supposed so +far that our system of weighting is determined by data which we +%% -----File: 225.png---Folio 214------- +\index{Hagen, and error!discordant observations@{and discordant observations}|inote}% +\index{Independence, for knowledge!discordant observations@{and discordant observations}}% +possess prior to and apart from our knowledge of the actual +magnitude of the items of our average. The principle of the +rejection of discordant observations comes in when it is argued +that, if one or more of our observations show great discrepancies +from the results of the greater number, these ought to be partly +or entirely neglected in striking the average, even if there is no +reason, except their discrepancy from the rest, for attributing +less weight to them than to the others. By some this practice +has been thought to be in accordance with the dictates of common +sense; by others it is denounced as savouring even of forgery.\footnote + {\Eg\ G.~Hagen's \textit{Grundzüge der Wahrscheinlichkeitsrechnung}, p.~63: ``Die + Täuschung, die man durch Verschweigen von Messungen begeht, lässt sich + eben so wenig entschuldigen, als wenn man Messungen fälschen oder fingiren + wollte.''} + +This controversy, like so many others in Probability, is due +to a failure to understand the meaning of `independence.' The +mathematics of the orthodox theory of Averages and Least +\index{Averages!discordant observations@{and discordant observations}}% +Squares depend, as we have seen, upon the assumption that the +observations are `independent'; but this has sometimes been +interpreted to mean a \emph{physical} independence. In point of fact, +the theory requires that the observations shall be independent, +in the sense that a \emph{knowledge} of the result of some does not affect +the probability that the others, when known, involve given +errors. + +Clearly there may be initial data in relation to which this +supposition is entirely or approximately accurate. But in many +cases the assumption will be inadmissible. A knowledge of the +results of a number of observations may lead us to modify our +opinion as to the relative reliabilities of others. + +The question, whether or not discordant observations should +be specially weighted down, turns, therefore, upon the nature of +the preliminary data by which we have been guided in initially +adopting a particular law of error as appropriate to the observations. +If the observations are, relevant to these data, strictly +`independent,' in the sense required for probability, then rejection +is not permissible. But if this condition is not fulfilled, a bias +against discordant observations may be well justified. +%% -----File: 226.png---Folio 215------- + + +\Part{III}{Induction and Analogy} +%% -----File: 227.png---Folio 216------- +%[Blank Page] +%% -----File: 228.png---Folio 217------- +\index{Logic, academic!Induction@{and Induction}}% + + +\Chapter{XVIII}{Introduction} + +\begin{Quote} +Nothing so like as eggs; yet no one, on account of this apparent similarity, +expects the same taste and relish in all of them. 'Tis only after a long course +of uniform experiments in any kind, that we attain a firm reliance and security +with regard to a particular event. Now where is that process of reasoning, +which from one instance draws a conclusion, so different from that which it +infers from a hundred instances, that are no way different from that single +instance? This question I propose as much for the sake of information, as +with any intention of raising difficulties. I cannot find, I cannot imagine any +such reasoning. But I keep my mind still open to instruction, if any one will +vouchsafe to bestow it on me.---\textsc{Hume}.\footnote + {\textit{Philosophical Essays concerning Human Understanding.}} +\end{Quote} + +\Paragraph{1.} \First{I have} described Probability as comprising that part of +logic which deals with arguments which are rational but not +conclusive. By far the most important types of such arguments +are those which are based on the methods of Induction and +\index{Induction!Logic@{and Logic}}% +Analogy. Almost all empirical science rests on these. And the +decisions dictated by experience in the ordinary conduct of life +generally depend on them. To the analysis and logical justification +of these methods the following chapters are directed. + +Inductive processes have formed, of course, at all times a +vital, habitual part of the mind's machinery. Whenever we learn +by experience, we are using them. But in the logic of the schools +they have taken their proper place slowly. No clear or satisfactory +account of them is to be found anywhere. Within and +yet beyond the scope of formal logic, on the line, apparently, +between mental and natural philosophy, Induction has been +admitted into the organon of scientific proof, without much help +from the logicians, no one quite knows when. + +\Paragraph{2.} What are its distinguishing characteristics? What are +the qualities which in ordinary discourse seem to afford strength +to an inductive argument? +%% -----File: 229.png---Folio 218------- +\index{Analogy, principle of!induction@{and induction}}% + +I shall try to answer these questions before I proceed to +the more fundamental problem---What ground have we for regarding +such arguments as rational? + +Let the reader remember, therefore, that in the first of the +succeeding chapters my main purpose is no more than to state +in precise language what elements are commonly regarded as +adding weight to an empirical or inductive argument. This +requires some patience and a good deal of definition and special +terminology. But I do not think that the work is controversial. +At any rate, I am satisfied myself that the analysis of \Chapref{XIX}. +is fairly adequate. + +In the next section, Chapters \Chapref[]{XX}.~and~\Chapref[]{XXI}., I continue in +part the same task, but also try to elucidate what sort of assumptions, +\emph{if} we could adopt them, lie behind and are required by the +methods just analysed. In \Chapref{XXII}. the nature of these +assumptions is discussed further, and their possible justification +is debated. + +\Paragraph{3.} The passage quoted from Hume at the head of this chapter +\index{Hume!Induction@{and Induction}}% +is a good introduction to our subject. Nothing so \emph{like} as eggs, +and after a \emph{long} course of uniform experiments we can expect +with a firm reliance and security the same taste and relish in all +of them. The eggs must be like eggs, and we must have tasted +many of them. This argument is based partly upon \emph{Analogy} +and partly upon what may be termed \emph{Pure Induction}. We argue +\index{Induction!pure}% +from Analogy in so far as we depend upon the \emph{likeness} of the eggs, +and from Pure Induction when we trust the \emph{number} of the experiments. + +It will be useful to call arguments \emph{inductive} which depend +in any way on the methods of Analogy and Pure Induction. But +I do not mean to suggest by the use of the term \emph{inductive} that these +methods are necessarily confined to the objects of phenomenal +experience and to what are sometimes called empirical questions; +or to preclude from the outset the possibility of their use in +abstract and metaphysical inquiries. While the term \emph{inductive} +will be employed in this general sense, the expression \emph{Pure +Induction} must be kept for that part of the argument which +arises out of the repetition of instances. + +\Paragraph{4.} Hume's account, however, is incomplete. His argument +could have been improved. His experiments should not have +been too uniform, and ought to have differed from one another +%% -----File: 230.png---Folio 219------- +\index{Analogy, principle of!negative}% +as much as possible in all respects save that of the likeness of the +eggs. He should have tried eggs in the town and in the country, +in January and in June. He might then have discovered that +eggs could be good or bad, however like they looked. + +This principle of varying those of the characteristics of the +instances, which we regard in the conditions of our generalisation +as non-essential, may be termed \emph{Negative Analogy}. + +It will be argued later on that an increase in the \emph{number} of +experiments is \emph{only} valuable in so far as, by increasing, or possibly +increasing, the variety found amongst the non-essential characteristics +\index{Variety!induction@{and induction}}% +of the instances, it strengthens the Negative Analogy. +If Hume's experiments had been \emph{absolutely} uniform, he would +have been right to raise doubts about the conclusion. There is +no process of reasoning, which from one instance draws a conclusion +different from that which it infers from a hundred instances, +if the latter are known to be in \emph{no} way different from +the former. Hume has unconsciously misrepresented the typical +inductive argument. + +When our control of the experiments is fairly complete, and +the conditions in which they take place are well known, there is +not much room for assistance from Pure Induction. If the +Negative Analogies are known, there is no need to count the +instances. But where our control is incomplete, and we do not +know accurately in what ways the instances differ from one +another, then an increase in the mere number of the instances +helps the argument. For unless we know for certain that the +instances are perfectly uniform, each new instance \emph{may} possibly +add to the Negative Analogy. + +Hume might also have weakened his argument. He expects +no more than the same taste and relish from his eggs. He +attempts no conclusion as to whether his stomach will always +draw from them the same nourishment. He has conserved the +force of his generalisation by keeping it narrow. + +\Paragraph{5.} In an inductive argument, therefore, we start with a +number of instances similar in some respects~$AB$, dissimilar in +others~$C$. We pick out one or more respects~$A$ in which the +instances are similar, and argue that some of the other respects~$B$ +in which they are also similar are likely to be associated with +the characteristics~$A$ in other unexamined cases. The more +comprehensive the essential characteristics~$A$, the greater the +%% -----File: 231.png---Folio 220------- +\index{Analogy, principle of!positive}% +\index{Mill, and inductive correlations}% +variety amongst the non-essential characteristics~$C$, and the less +comprehensive the characteristics~$B$ which we seek to associate +with~$A$, the stronger is the likelihood or probability of the generalisation +we seek to establish. + +These are the three ultimate logical elements on which the +probability of an empirical argument depends,---the Positive +and the Negative Analogies and the scope of the generalisation. + +\Paragraph{6.} Amongst the generalisations arising out of empirical +argument we can distinguish two separate types. The first of +these may be termed \emph{universal induction}. Although such inductions +\index{Induction!universal}% +are themselves susceptible of any degree of probability, +they affirm \emph{invariable} relations. The generalisations which they +assert, that is to say, claim universality, and are upset if a +single exception to them can be discovered. Only in the more +exact sciences, however, do we aim at establishing universal +inductions. In the majority of cases we are content with that +other kind of induction which leads up to laws upon which +we can generally depend, but which does not claim, however +adequately established, to assert a law of more than probable +connection.\footnote + {What Mill calls `approximate generalisations.'} +This second type may be termed \emph{Inductive Correlation}. +\index{Inductive correlation}% +If, for instance, we base upon the data, that this and that +and those swans are white, the conclusion that \emph{all} swans are white, +we are endeavouring to establish a universal induction. But if +we base upon the data that this and those swans are white and +that swan is black, the conclusion that \emph{most} swans are white, +or that the probability of a swan's being white is such and such, +then we are establishing an inductive correlation. + +Of these two types, the former---universal induction---presents +both the simpler and the more fundamental problem. In +this part of my treatise I shall confine myself to it almost entirely. +In \Partref{V}., on the Foundations of Statistical Inference, I shall +discuss, so far as I can, the logical basis of inductive correlation. + +\Paragraph{7.} The fundamental connection between Inductive Method +and Probability deserves all the emphasis I can give it. Many +writers, it is true, have recognised that the conclusions which we +reach by inductive argument are probable and inconclusive. +Jevons, for instance, endeavoured to justify inductive processes +by means of the principles of inverse probability. And it is true +also that much of the work of Laplace and his followers was +\index{Laplace!Induction@{and Induction}}% +%% -----File: 232.png---Folio 221------- +\index{Evidence, and measurement of Probability!Induction@{and Induction}}% +directed to the solution of essentially inductive problems. But +it has been seldom apprehended clearly, either by these writers +or by others, that the validity of every induction, strictly interpreted, +\index{Induction!validity of}% +depends, not on a matter of fact, but on the existence of +a relation of probability. An inductive argument affirms, not +that a certain matter of fact \emph{is} so, but that \emph{relative to certain +evidence} there is a probability in its favour. The validity of the +induction, relative to the original evidence, is not upset, therefore, +if, as a fact, the truth turns out to be otherwise. + +The clear apprehension of this truth profoundly modifies +our attitude towards the solution of the inductive problem. The +validity of the inductive method does \emph{not} depend on the success +of its predictions. Its repeated failure in the past may, of course, +supply us with new evidence, the inclusion of which will modify +the force of subsequent inductions. But the force of the old +induction \emph{relative to the old evidence} is untouched. The evidence +with which our experience has supplied us in the past may have +proved misleading, but this is entirely irrelevant to the +question of what conclusion we ought reasonably to have +drawn from the evidence then before us. The validity and +reasonable nature of inductive generalisation is, therefore, a +question of logic and not of experience, of formal and not of +material laws. The actual constitution of the phenomenal +universe determines the character of our evidence; but it cannot +determine what conclusions \emph{given} evidence \emph{rationally} supports. +%% -----File: 233.png---Folio 222------- +\index{Analogy, principle of!induction@{and induction}}% +\index{definition of}% +\index{Jevons!Induction@{and Induction}}% +\index{Propositional function!induction@{and induction}}% + + +\Chapter{XIX}{The Nature of Argument by Analogy} + +\begin{Quote} +All kinds of reasoning from causes or effects are founded on two particulars, +viz.\ the constant conjunction of any two objects in all past experience, and the +resemblance of a present object to any of them. Without some degree of +\index{Hume!analogy@{and analogy}}% +resemblance, as well as union, 'tis impossible there can be any reasoning.---\textsc{Hume}.\footnote + {\textit{A Treatise of Human Nature.}} +\end{Quote} + +\Paragraph{1.} \First{Hume} rightly maintains that some degree of resemblance +must always exist between the various instances upon which a +generalisation is based. For they must have this, at least, in +common, that they are instances of the proposition which +generalises them. Some element of analogy must, therefore, +lie at the base of every inductive argument. In this chapter I +shall try to explain with precision the meaning of Analogy, and +to analyse the reasons, for which, rightly or wrongly, we usually +regard analogies as strong or weak, without considering at present +whether it is possible to find a \emph{good} reason for our instinctive +principle that likeness breeds the expectation of likeness. + +\Paragraph{2.} There are a few technical terms to be defined. We mean +by a \emph{generalisation} a statement that all of a certain definable class +of propositions are true. It is convenient to specify this class +in the following way. If $f(x)$~is true for all those values of~$x$ for +which $\phi(x)$~is true, then we have a generalisation about $\phi$~and~$f$ +which we may write $g(\phi, f)$. If, for example, we are dealing with +the generalisation, ``all swans are white,'' this is equivalent to +the statement ``\,`$x$~is white' is true for all those values of~$x$ for +which `$x$~is a swan' is true.'' The proposition $\phi(a) · f(a)$ is an +\emph{instance} of the generalisation~$g(\phi, f)$. + +By thus defining a generalisation in terms of propositional +functions, it becomes possible to deal with all kinds of generalisations +%% -----File: 234.png---Folio 223------- +\index{Analogy, principle of!negative}% +\index{Analogy, principle of!positive}% +\index{Analogy, principle of!generalisation@{and generalisation}}% +in a uniform way; and also to bring generalisation into +convenient connection with our definition of Analogy. + +If some one thing is true about both of two objects, if, that is +to say, they both satisfy the same propositional function, then to +this extent there is an \emph{analogy} between them. Every generalisation +$g(\phi, f)$, therefore, asserts that one analogy is always accompanied +by another, namely, that between all objects having the +analogy~$\phi$ there is also the analogy~$f$. The set of propositional +functions, which are satisfied by both of the two objects, constitute +the \emph{positive analogy}. The analogies, which would be +disclosed by complete knowledge, may be termed the \emph{total positive +analogy}; those which are relative to partial knowledge, the +\emph{known positive analogy}. + +As the positive analogy measures the resemblances, so the +negative analogy measures the differences between the two objects. +The set of functions, such that each is satisfied by one and not +by the other of the objects, constitutes the \emph{negative analogy}. +We have, as before, the distinction between the \emph{total negative +analogy} and the \emph{known negative analogy}. + +This set of definitions is soon extended to the cases in which +the number of instances exceeds two. The functions which are +true of \emph{all} of the instances constitute the positive analogy of the +set of instances, and those which are true of \emph{some only}, and are +false of others, constitute the negative analogy. It is clear that +a function, which represents positive analogy for a group of +instances taken out of the set, may be a negative analogy for the +set as a whole. Analogies of this kind, which are positive for +a sub-class of the instances, but negative for the whole class, we +may term \emph{sub-analogies}. By this it is meant that there are +\index{Sub-analogies}% +resemblances which are common to some of the instances, but +not to all. + +A simple notation, in accordance with these definitions, will +be useful. If there is a positive analogy~$\phi$ between a set of instances +$a_1$,~$\ldots$~$a_n$, whether or not this is the total analogy +between them, let us write this--- +\[ +\underset{a_1 \ldots a_n}{A}(\phi).\footnotemark +\] +\footnotetext{Hence $\underset{a_1 \ldots a_n}{A}(\phi) \equiv \phi(a_1) · \phi(a_2) \ldots \phi(a_n) \equiv \Prod \limits^{x=a_n}_{x=a_1} \phi(x)$.}% +%% -----File: 235.png---Folio 224------- + +And if there is a negative analogy~$\phi'$, let us write this--- +\[ +\underset{a_1 \ldots a_n}{\bar{A}}(\phi').\footnotemark +\] +\footnotetext{Hence $\underset{a_1 \ldots a_n}{\bar{A}}(\phi') \equiv \Sum^{x=a_s}_{x=a_r}\phi'(x) · \Sum^{x=a_s'}_{x=a_r'} \overline{\phi'(x)}$.}% + +Thus $\underset{a_1 \ldots a_n}{A}(\phi)$ expresses the fact that there is a set of +characteristics~$\phi$ which are common to all the instances, and +$\underset{a_1 \ldots a_n}{\bar{A}}(\phi')$ that there is a set of characteristics $\phi'$ which is +true of at least one of the instances and false of at least one. + +\Paragraph{3.} In the typical argument from analogy we wish to generalise +from one part to another of the total analogy which experience +has shown to exist between certain selected instances. In all the +cases where one characteristic~$\phi$ has been found to exist, another +characteristic~$f$ has been found to be associated with it. We argue +from this that any instance, which is known to share the first +analogy~$\phi$, is likely to share also the second analogy~$f$. We have +found in certain cases, that is to say, that both $\phi$~and~$f$ are true +of them; and we wish to assert~$f$ as true of other cases in which +we have only observed~$\phi$. We seek to establish the generalisation +$g(\phi, f)$, on the ground that $\phi$~and~$f$ constitute between them an +observed positive analogy in a given set of experiences. + +But while the argument is of this character, the grounds, upon +which we attribute more or less weight to it, are often rather +complex; and we must discuss them, therefore, in a systematic +manner. + +\Paragraph{4.} According to the view suggested in the last chapter, the +value of such an argument depends partly upon the nature of the +conclusion which we seek to draw, partly upon the evidence +which supports it. If Hume had expected the same degree of +\index{Hume!analogy@{and analogy}}% +nourishment as well as the same taste and relish from all of the +eggs, he would have drawn a conclusion of weaker probability. +Let us consider, then, this dependence of the probability upon the +\emph{scope} of the generalisation $g(\phi,f)$,---upon the comprehensiveness, +that is to say, of the condition~$\phi$ and the conclusion~$f$ respectively. + +The more comprehensive the condition~$\phi$ and the less comprehensive +the conclusion~$f$, the greater \textit{à~priori} probability do +we attribute to the generalisation~$g$. With every increase in~$\phi$ +this probability increases, and with every increase in~$f$ it will +diminish. +%% -----File: 236.png---Folio 225------- + +The condition $\phi (\equiv \phi_1\phi_2)$ is more comprehensive than the +condition~$\phi_1$, relative to the general evidence~$h$, if $\phi_2$ is a condition +independent of~$\phi_1$ relative to~$h$, $\phi_2$~being independent of~$\phi_1$, if +$g(\phi_1, \phi_2)/h\neq 1$, \ie~if, relative to~$h$, the satisfaction of~$\phi_2$ is not +inferrible from that of~$\phi_1$. + +Similarly the conclusion $f(\equiv f_1 f_2)$ is more comprehensive than +the conclusion~$f_1$, relative to the general evidence~$h$, if $f_2$~is a conclusion +independent of~$f_1$, relative to~$h$, \ie~if $g(f_1, f_2)/h \neq 1$. + +If $\phi \equiv \phi_1\phi_2$ and $f \equiv f_1 f_2$, where $\phi_1$~and~$\phi_2$ are independent and +$f_1$~and~$f_2$ are independent relative to~$h$, we have--- +\begin{DPalign*} +g(\phi_1, f)/h + &= g(\phi_1\phi_2, f) · g(\phi_1\bar{\phi}_2, f)/h \\ + &\eqslantless g(\phi, f)/h, \\ +\lintertext{and} +g(\phi, f)/h + &= g(\phi, f_1f_2)/h \\ + &= g(\phi f_1, f_2)/h · g(\phi, f_1)/h \\ + &\leq g(\phi, f_1)/h, \\ +\lintertext{so that} +g(\phi, f_1)/h + &\geq g(\phi, f)/h \geq g(\phi_1, f)/h. +\end{DPalign*} + +This proves the statement made above. It will be noticed +that we cannot necessarily compare the \textit{à~priori} probabilities +of two generalisations in respect of more and less, unless the condition +of the first is included in the condition of the second, and +the conclusion of the second is included in that of the first. + +We see, therefore, that some generalisations stand \emph{initially} +in a stronger position than others. In order to attain a given +degree of probability, generalisations require, according to their +scope, different amounts of favourable evidence to support them. + +\Paragraph{5.} Let us now pass from the character of the generalisation +\textit{à~priori} to the evidence by which we support it. Since, whenever +the conclusion~$f$ is complex, \ie~resolvable into the form +$f_1f_2$ where $g(f_1, f_2)/h \neq 1$, we can express the probability of the +generalisation $g(\phi, f)$ as the product of the probabilities of the +two generalisations $g(\phi f_1, f_2)$ and~$g(\phi, f)$, we may assume in what +follows, that the conclusion~$f$ is simple and not capable of further +analysis, without diminishing the generality of our argument. + +We will begin with the simplest case, namely, that which +arises in the following conditions. First, let us assume that our +knowledge of the examined instances is complete, so that we know +of every statement, which is about the examined instances, +whether it is true or false of each.\footnote + {If $\psi(a)$ is a proposition and $\psi(a)=h · \theta(a)$, where $h$~is a proposition not + involving~$a$, then we must regard~$\theta(a)$, not~$\psi(a)$\DPtypo{}{,} as the statement \emph{about}~$a$.} +Second, let us assume that +%% -----File: 237.png---Folio 226------- +\index{Uniformity of Nature, Law of}% +all the instances which are known to satisfy the condition~$\phi$, +are also known to satisfy the conclusion~$f$ of the generalisation. +And third let us assume that there is nothing which is true of +\emph{all} the examined instances and yet not included either in~$\phi$ or +in~$f$, \ie\ that the positive analogy between the instances is +exactly \DPchg{co-extensive}{coextensive} with the analogy~$\phi f$ which is covered by the +generalisation. + +Such evidence as this constitutes what we may term a perfect +analogy. The argument in favour of the generalisation cannot +be further improved by a knowledge of additional instances. +Since the positive analogy between the instances is exactly +coextensive with the analogy covered by the generalisation, and +since our knowledge of the examined instances is complete, there +is no need to take account of the negative analogy. + +An analogy of this kind, however, is not likely to have much +practical utility; for if the analogy covered by the generalisation, +covers the \emph{whole} of the positive analogy between the instances +it is difficult to see to what \emph{other} instances the generalisation can +be applicable. Any instance, about which everything is true +which is true of all of a set of instances, must be identical with +one of them. Indeed, an argument from perfect analogy can +only have practical utility, if, as will be argued later on, there are +some distinctions between instances which are \emph{irrelevant} for the +purposes of analogy, and if, in a perfect analogy, the positive +analogy, of which we must take account, need cover only those +distinctions which are relevant. In this case a generalisation +based on perfect analogy might cover instances numerically +distinct from those of the original set. + +The law of the Uniformity of Nature appears to me to amount +to an assertion that an analogy which is perfect, except that mere +differences of position in time and space are treated as irrelevant, +\index{Space!uniformity@{and uniformity}}% +\index{Time!uniformity@{and uniformity}}% +is a valid basis for a generalisation, two total causes being regarded +as the \emph{same} if they only differ in their positions in time +or space. This, I think, is the whole of the importance which +this law has for the theory of inductive argument. It involves +the assertion of a generalised judgment of irrelevance, namely, +of the irrelevance of mere position in time and space to generalisations +which have no reference to particular positions in time +and space. It is in respect of such position in time or space that +`nature' is supposed `uniform.' The significance of the law +%% -----File: 238.png---Folio 227------- +and the nature of its justification, if any, are further discussed +in \Chapref{XXII}\@. + +\Paragraph{6.} Let us now pass to the type which is next in order of +simplicity. We will relax the first condition and no longer assume +that the \emph{whole} of the positive analogy between the instances is +covered by the generalisation, though retaining the assumption +that our knowledge of the examined instances is complete. We +know, that is to say, that there are some respects in which the +examined instances are all alike, and yet which are not covered +by the generalisation. If $\phi_1$~is the part of the positive analogy +between the instances which is \emph{not} covered by the generalisation, +then the probability of this type of argument from analogy can +be written--- +\[ +g(\phi, f) \Big/ \underset{a_1 \ldots a_n}{A}(\phi \phi_1 f). +\] + +The value of this probability turns on the comprehensiveness +of~$\phi_1$. There are some characteristics~$\phi_1$ common to all the +instances, which the generalisation treats as unessential, but +the less comprehensive these are the better. $\phi_1$~stands for the +characteristics in which all the instances resemble one another +outside those covered by the generalisation. To reduce these +resemblances between the instances is the same thing as to +increase the differences between them. And hence any increase +in the Negative Analogy involves a reduction in the comprehensiveness +of~$\phi_1$. When, however, our knowledge of the +instances is complete, it is not necessary to make separate +mention of the negative analogy $\underset{a_1 \ldots a_n}{\bar{A}}(\phi')$ in the above formula. +For $\phi'$~simply includes all those functions about the instances, +which are not included in~$\phi \phi_1 f$, and of which the contradictories +are not included in them; so that in stating $\underset{a_1 \ldots a_n}{A}(\phi\phi_1f)$, we +state by implication $\underset{a_1 \ldots a_n}{\bar{A}}(\phi')$ also. + +The whole process of strengthening the argument in favour +of the generalisation $g(\phi, f)$ by the accumulation of further experience +appears to me to consist in making the argument +approximate as nearly as possible to the conditions of a perfect +analogy, by steadily reducing the comprehensiveness of those +resemblances~$\phi_1$ between the instances which our generalisation +disregards. Thus the advantage of additional instances, derived +%% -----File: 239.png---Folio 228------- +from experience, arises not out of their number as such, but out +of their tendency to limit and reduce the comprehensiveness of~$\phi_1$, +or, in other words, out of their tendency to increase the negative +analogy~$\phi'$, since $\phi_1 \phi'$~comprise between them whatever is not +covered by~$\phi f$. The more numerous the instances, the less comprehensive +are their superfluous resemblances likely to be. But +a single additional instance which greatly reduced~$\phi_1$ would increase +the probability of the argument more than a large number +of instances which affected~$\phi_1$ less. + +\Paragraph{7.} The nature of the argument examined so far is, then, that +the instances all have some characteristics in common which +we have ignored in framing our generalisation; but it is still +assumed that our knowledge about the examined instances is +complete. We will next dispense with this latter assumption, and +deal with the case in which our knowledge of the characteristics +of the examined instances themselves is or may be incomplete. + +It is now necessary to take explicit account of the known +negative analogy. For when the known positive analogy falls +short of the total positive analogy, it is not possible to infer the +negative analogy from it. Differences may be known between the +instances which cannot be inferred from the known positive +analogy. The probability of the argument must, therefore, be +written--- +\[ +g(\phi, f)\Big/\underset{a_1 \ldots a_n}{A}(\phi \phi_1 f) + \underset{a_1 \ldots a_n}{\bar{A}}(\phi'), +\] +where $\phi \phi_1 f$~stands for the characteristics in which all $n$~instances +$a_1 \ldots a_n$ are \emph{known} to be alike, and $\phi'$~stands for the characteristics +in which they are \emph{known} to differ. + +This argument is strengthened by any additional instance or +by any additional knowledge about the former instances which +diminishes the known superfluous resemblances~$\phi_1$ or increases the +negative analogy~$\phi'$. The object of the accumulation of further +experience is still the same as before, namely, to make the form +of the argument approximate more and more closely to that of +perfect analogy. Now, however, that our knowledge of the +instances is no longer assumed to be complete, we must take +account of the mere \emph{number}~$n$ of the instances, as well as of our +specific knowledge in regard to them; for the more numerous +the instances are, the greater the opportunity for the \emph{total} +negative analogy to exceed the \emph{known} negative analogy. But +%% -----File: 240.png---Folio 229------- +the more complete our knowledge of the instances, the less +attention need we pay to their mere number, and the more +imperfect our knowledge the greater the stress which must be +laid upon the argument from number. This part of the argument +will be discussed in detail in the following chapter on +Pure Induction. + +\Paragraph{8.} When our knowledge of the instances is incomplete, there +may exist analogies which are known to be true of some of the +instances and are not known to be false of any. These sub-analogies +\index{Sub-analogies}% +(see §\;2) are not so dangerous as the positive analogies~$\phi_1$, +which are known to be true of \emph{all} the instances, but their existence +is, evidently, an element of weakness, which we must endeavour +to eliminate by the growth of knowledge and the multiplication +of instances. A sub-analogy of this kind between the instances +$a_r \ldots a_s$ may be written $\underset{a_r \ldots a_s}{A}(\psi_k)$; and the formula, if it +is to take account of all the relevant information, ought, therefore, +to be written--- +\[ +g(\phi,f)\Big/\underset{a_1 \ldots a_n}{A}(\phi\phi_1f) + \underset{a_1 \ldots a_n}{\bar{A}}(\phi') + \Prod\left\{\underset{a_r \ldots a_s}{A}(\psi_k)\right\}, +\] +where the terms of $\Prod\left\{\underset{a_r \ldots a_s}{A}(\psi_k)\right\}$ stand for the various sub-analogies +between sub-classes of the instances, which are not +included in~$\phi \phi_1 f$ or in~$\phi'$. + +\Paragraph{9.} There is now another complexity to be introduced. We +must dispense with the assumption that the whole of the analogy +covered by the generalisation is known to exist in all the instances. +For there may be some instances within our experience, about +which our knowledge is incomplete, but which show \emph{part} of the +analogy required by the generalisation and nothing which contradicts +it; and such instances afford some support to the +generalisation. Suppose that ${}_b\phi$~and~${}_bf$ are \emph{part} of $\phi$~and~$f$ respectively, +then we may have a set of instances $b_1 \ldots b_m$ which +show the following analogies: +\[ +\underset{b_1 \ldots b_m}{A}({}_b\phi\, {}_b\phi_{1} \,{}_bf) +\underset{b_1 \ldots b_m}{\bar{A}}({}_b\phi') +\Prod\left\{\underset{b_r \ldots b_s}{A}({}_b\psi_k)\right\}, +\] +where ${}_b\phi_1$~is the analogy not covered by the generalisation, and +so on, as before. +%% -----File: 241.png---Folio 230------- + +The formula, therefore, is now as follows: +\[ +g(\phi,f)\Big/ +\Prod_{a,b\ldots}\left\{ + \underset{a_1 \ldots a_n}{A}({}_a\phi\, {}_a\phi_{1}\, {}_af) + \underset{a_1 \ldots a_n}{\bar{A}}({}_a\phi')\right\} + \Prod\left\{\underset{a_r\, b_s\ldots}{A}(\psi_k)\right\}. +\] +In this expression ${}_a\phi$,~${}_a f$ are the whole \emph{or} part of $\phi$,~$f$; the product +$\Prod\limits_{a,b\ldots}$ is composed of the positive and negative analogies for each +of the sets of instances $a_1 \ldots a_n$, $b_1 \ldots b_m$,~etc.; and the +product~$\Prod$ contains the various sub-analogies of different sub-classes +of all the instances $a_1 \ldots a_n$, $b_1 \ldots b_m$,~etc., regarded as +\emph{one} set.\footnote + {Even if we want to distinguish between the sub-analogies of the $a$~set and + the sub-analogies of the $b$~set, this information can be gathered from the product~$\Prod$.} + +\Paragraph{10.} This completes our classification of the \emph{positive} evidence +which supports a generalisation; but the probability may also +be affected by a consideration of the negative evidence. We +have taken account so far of that part of the evidence only which +shows the whole or part of the analogy we require, and we have +neglected those instances of which~$\phi$, the condition of the generalisation, +or~$f$, its conclusion, or part of~$\phi$ or of~$f$ is \emph{known to be false}. +Suppose that there are instances of which $\phi$~is true and $f$~false, it +is clear that the generalisation is ruined. But cases in which we +know \emph{part} of~$\phi$ to be true and $f$~to be false, and are ignorant as +to the truth or falsity of the rest of~$\phi$, weaken it to some extent. +We must take account, therefore, of analogies +\[ +\underset{a_1' \ldots a_n''}{A}({}_{a'}\phi\, {}_{a'}\bar f), +\] +%[**TN: Unclear. A formatter claims to have checked this against a hard copy.] +where~${}_{a'}\phi$, part of~$\phi$, is true of all the set, and~${}_{a'}f$, part of~$f$, is +false of all the set, while the truth or falsity of some part of $\phi$~and~$f$ +is unknown. The negative evidence, however, can strengthen +as well as weaken the evidence. We deem instances favourably +relevant in which $\phi$~and~$f$ are both false together.\footnote + {I am disposed to think that we need not pay attention to instances for + which part of~$\phi$ is known to be false, and part of~$f$ to be true. But the + question is a little perplexing.} + +Our final formula, therefore, must include terms, similar to +those in the formula which concludes §\;9, not only for sets of +instances which show analogies~${}_a\phi\, {}_a f$ where ${}_a\phi$~and~${}_a f$ are parts +of $\phi$~and~$f$, but also for sets which show analogies ${}_a\bar\phi\, {}_a f$, +%% -----File: 242.png---Folio 231------- +or analogies~${}_a\bar{\phi}\, {}_a\bar{f}$, where ${}_a\phi$~and~${}_af$ are the whole or part of $\phi$~and~$f$, +and \DPtypo{$\bar{\phi}\,\bar{f}$}{$\bar{\phi}$, $\bar{f}$} are the contradictories of $\phi$~and~$f$.\footnote + {Where the conclusion~$f$ is simple and not complex (see §\;5), some of these + complications cannot, of course, arise.} + +It should be added, perhaps, that the theoretical classification +of most empirical arguments in daily use is complicated by +the account which we reasonably take of generalisations previously +established. We often take account indirectly, therefore, +of evidence which supports in some degree other generalisations +than that which we are concerned to establish or refute at the +moment, but the probability of which is relevant to the problem +under investigation. + +\Paragraph{11.} The argument will be rendered unnecessarily complex, +without much benefit to its theoretical interest, if we deal with +the most general case of all. What follows, therefore, will deal +with the formula of the third degree of generality, namely--- +\[ +g(\phi,f)\Big/\underset{a_1 \ldots a_n}{A}(\phi\phi_1f) + \underset{a_1 \ldots a_n}{\bar{A}}(\phi') + \Prod\left\{\underset{a_r \ldots a_s}{A}(\psi_k)\right\}, +\] +in which no \emph{partial} instances occur, \ie\ no instances in which part +only of the analogy, required by the generalisation, is known to +exist. In this third degree of generality, it will be remembered, +our knowledge of the characteristics of the instances is incomplete, +there is more analogy between the instances than is +covered by the generalisation, and there are some sub-analogies +to be reckoned with. In the above formula the incompleteness +of our knowledge is implicitly recognised in that $\phi \phi_1 f \phi'$ are +not between them entirely comprehensive. It is also supposed +that all the evidence we have is positive, no knowledge is +assumed, that is to say, of instances characterised by the conjunctions +${}_a\bar{\phi}\, {}_af$,~${}_a\phi\ {}_a\bar f$, or~${}_a\bar{\phi}\, {}_a\bar f$, where ${}_a\phi$~and~${}_a f$ are part of $\phi$~and~$f$. + +An argument, therefore, from experience, in which, on the +basis of examined instances, we establish a generalisation applicable +beyond these instances, can be strengthened, if we restrict our +attention to the simpler type of case, by the following means: + +(1) By reducing the resemblances~$\phi_1$ known to be common to +all the instances, but ignored as unessential by the generalisation. + +(2) By increasing the differences~$\phi'$ known to exist between +the instances. +%% -----File: 243.png---Folio 232------- + +(3) By diminishing the sub-analogies or unessential resemblances~$\psi_k$ +known to be common to some of the instances and not +known to be false of any. + +These results can generally be obtained in two ways, either by +increasing the number of our instances or by increasing our knowledge +of those we have. + +The reasons why these methods seem to common sense to +strengthen the argument are fairly obvious. The object of~(1) is to +avoid the possibility that $\phi_1$~as well as~$\phi$ is a necessary condition +of~$f$. The object of~(2) is to avoid the possibility that there may +be some resemblances additional to~$\phi$, common to all the instances, +which have escaped our notice. The object of~(3) is to get rid +of indications that the total value of~$\phi_1$ may be greater than the +known value. When $\phi\phi_1 f$ is the \emph{total} positive analogy between +the instances, so that the known value of~$\phi_1$ is its total value, it +is~(1) which is fundamental; and we need take account of (2)~and~(3) +only when our knowledge of the instances is incomplete. +But when our knowledge of the instances is incomplete, so that +$\phi_1$~falls short of its total value and we cannot infer~$\phi'$ from it, +it is better to regard~(2) as fundamental; in any case every +reduction of~$\phi_1$ must increase~$\phi'$. + +\Paragraph{12.} I have now attempted to analyse the various ways in +which common practice seems to assume that considerations +of Analogy can yield us presumptive evidence in favour of a +generalisation. + +It has been my object, in making a classification of empirical +arguments, not so much to put my results in forms closely similar +to those in which problems of generalisation commonly present +themselves to scientific investigators, as to inquire whether +ultimate uniformities of method can be found beneath the +innumerable modes, superficially differing from another, in +which we do in fact argue. + +I have not yet attempted to justify this way of arguing. +After turning aside to discuss in more detail the method of Pure +Induction, I shall make this attempt; or rather I shall try to see +\emph{what sort} of assumptions are capable of justifying empirical +reasoning of this kind. +%% -----File: 244.png---Folio 233------- + + +\Chapter{XX}{The Value of Multiplication of Instances, or Pure +Induction} + +\Paragraph{1.} \First{It} has often been thought that the essence of inductive argument +lies in the multiplication of instances. ``Where is that +\index{Multiplication!of instances|ifoll}% +process of reasoning,'' Hume inquired, ``which from one instance +\index{Hume!Induction@{and Induction}}% +draws a conclusion, so different from that which it infers from +a hundred instances, that are no way different from that single +instance?'' I repeat that by emphasising the number of the instances +Hume obscured the real object of the method. If it +were strictly true that the hundred instances are \emph{no} way different +from the single instance, Hume would be right to wonder in what +manner they can strengthen the argument. The object of increasing +the number of instances arises out of the fact that we +are nearly always aware of \emph{some} difference between the instances, +and that even where the known difference is insignificant we may +suspect, especially when our knowledge of the instances is very +incomplete, that there may be more. Every new instance \emph{may} +diminish the unessential resemblances between the instances and +by introducing a new difference increase the Negative Analogy. +\index{Analogy, principle of!negative}% +For this reason, and for this reason only, new instances are +valuable. + +If our premisses comprise the body of memory and tradition +which has been originally derived from direct experience, and +the conclusion which we seek to establish is the Newtonian theory +of the Solar System, our argument is one of Pure Induction, in +so far as we support the Newtonian theory by pointing to the +great number of consequences which it has in common with the +facts of experience. The predictions of the Nautical Almanack +are a consequence of the Newtonian theory, and these predictions +are verified many thousand times a day. But even here the +%% -----File: 245.png---Folio 234------- +force of the argument largely depends, not on the mere number +of these predictions, but on the knowledge that the circumstances +in which they are fulfilled differ widely from one another in a +vast number of important respects. The \emph{variety} of the circumstances, +\index{Variety}% +in which the Newtonian generalisation is fulfilled, rather +than the number of them, is what seems to impress our reasonable +faculties. + +\Paragraph{2.} I hold, then, that our object is always to increase the +Negative Analogy, or, which is the same thing, to diminish the +characteristics common to all the examined instances and yet not +taken account of by our generalisation. Our method, however, +maybe one which certainly achieves this object, or it may be one +which possibly achieves it. The former of these, which is obviously +the more satisfactory, may consist either in increasing our +definite knowledge respecting instances examined already, or in +finding additional instances respecting which definite knowledge +is obtainable. The second of them consists in finding additional +instances of the generalisation, about which, however, our definite +knowledge may be meagre; such further instances, if our +knowledge about them were more complete, would either increase +or leave unchanged the Negative Analogy; in the former case +they would strengthen the argument and in the latter case they +would not weaken it; and they must, therefore, be allowed some +weight. The two methods are not entirely distinct, because +new instances, about which we have some knowledge but not +much, may be known to increase the Negative Analogy a little +by the first method, and suspected of increasing it further by the +second. + +It is characteristic of advanced scientific method to depend +on the former, and of the crude unregulated induction of ordinary +experience to depend on the latter. It is when our definite +knowledge about the instances is limited, that we must pay +attention to their number rather than to the specific differences +between them, and must fall back on what I term Pure Induction. + +In this chapter I investigate the conditions and the manner +in which the mere repetition of instances can add to the force +of the argument. The chief value of the chapter, in my judgment, +is negative, and consists in showing that a line of advance, +which might have seemed promising, turns out to be a blind +alley, and that we are thrown back on known Analogy. Pure +%% -----File: 246.png---Folio 235------- +Induction will not give us any very substantial assistance in +getting to the bottom of the general inductive problem. + +\Paragraph{3.} The problem of generalisation\footnote + {In the most general sense we can regard any proposition as the generalisation + of all the propositions which follow from it. For if $h$~is any proposition, + and we put $\phi(x)\equiv \text{`$x$~can be inferred from~$h$'}$ and, $f(x)\equiv x$, then $g(\phi, f)\equiv h$. Since + Pure Induction consists in finding as many instances of a generalisation as + possible, it is, in the widest sense, the process of strengthening the probability + of any proposition by adducing numerous instances of known truths which + follow from it. The argument is one of Pure Induction, therefore, in so far as + the probability of a conclusion is based upon the number of independent consequences + which the conclusion and the premisses have in common.} +by Pure Induction can be +stated in the following symbolic form: + +Let $h$~represent the general \textit{à~priori data} of the investigation; +let $g$~represent the generalisation which we seek to establish; +let $x_1x_2\ldots x_n$ represent instances of~$g$. + +Then $x_1/gh=1$, $x_2/gh=1 \ldots x_n/gh=1$; given~$g$, that is to +say, the truth of each of its instances follows. The problem is +to determine the probability $g/hx_1x_2 \ldots x_n$, \ie~the probability +of the generalisation when $n$~instances of it are given. Our +analysis will be simplified, and nothing of fundamental importance +will be lost, if we introduce the assumption that there is nothing +in our \textit{à~priori data} which leads us to distinguish between the +\textit{à~priori} likelihood of the different instances; we assume, that is +to say, that there is no reason \textit{à~priori} for expecting the occurrence +of any one instance with greater reliance than any other,~\ie +\begin{DPgather*}[m] +x_1/h = x_2/h = \ldots = x_n/h. \\ +\lintertext{\rlap{Write}} +g/hx_1x_2 \ldots x_n = p_n \\ +\lintertext{\rlap{and}} +x_{n+1}/hx_1x_2 \ldots x_n = y_{n+1}; \\ +\intertext{then} +\begin{aligned} +\frac{p_n}{p_{n-1}} = \frac{g/hx_1\ldots x_n}{g/hx_1 \ldots x_{n-1}} + &= \frac{gx_n/hx_1 \ldots x_{n-1}} + {g/hx_1 \ldots x_{n-1} · x_n/hx_1 \ldots x_{n-1}}\\ +% + &= \frac{x_n/ghx_1 \ldots x_{n-1}}{x_n/hx_1 \ldots x_{n-1}} \\ +% + &= \frac{1}{y_n}. +\end{aligned} +\end{DPgather*} +$\therefore \dfrac{p_n}{p_{n-1}} = \dfrac{1}{y_n}$, and hence $p_n = \dfrac{1}{y_1y_2\ldots y_n} · p_0$, where $p_0 = g/h$, \ie~$p_0$ +is the \textit{à~priori} probability of the generalisation. +%% -----File: 247.png---Folio 236------- + +It follows, therefore, that $p_n > p_{n-1}$ so long as $y_n > 1$. + +Further, +\begin{align*} +x_1x_2 \ldots x_n/h + &= x_n/hx_1x_2 \ldots x_{n-1} · x_1x_2 \ldots x_{n-1}/h \\ + &= y_n · x_1x_2 \ldots x_{n-1}/h \\ + &= y_n y_{n-1} \ldots y_1. \\ +\therefore p_n + &= \frac{p_0}{y_1y_2 \ldots y_n} + = \frac{p_0}{x_1x_2 \ldots x_n/h} \\ + &= \frac{p_0}{x_1x_2 \ldots x_ng/h + x_1x_2 \ldots x_n\bar{g}/h} \\ + &= \frac{p_0}{g/h + x_1x_2 \ldots x_n/\bar{g}h · \bar{g}/h} \\ + &= \frac{p_0}{p_0 + x_1x_2 \ldots x_n/\bar{g}h(1 - p_0)} +\end{align*} + +This approaches unity as a limit, if $x_1x_2 \ldots x_n/\bar{g}h · \dfrac{1}{p_0}$ +approaches zero as a limit, when $n$~increases. + +\Paragraph{4.} We may now stop to consider how much this argument has +proved. We have shown that if each of the instances necessarily +follows from the generalisation, then each additional instance +increases the probability of the generalisation, so long as the new +instance could not have been predicted with certainty from a +knowledge of the former instances.\footnote + {Since $p_n > p_{n-1}$ so long as $y_n \neq 1$.} +This condition is the same +as that which came to light when we were discussing Analogy. +If the new instance were identical with one of the former instances, +a knowledge of the latter would enable us to predict it. +If it differs or may differ in analogy, then the condition required +above is satisfied. + +The common notion, that each successive verification of a +doubtful principle strengthens it, is formally proved, therefore, +without any appeal to conceptions of law or of causality. \emph{But +we have not proved} that this probability approaches certainty as +a limit, or even that our conclusion becomes more likely than not, +as the number of verifications or instances is indefinitely increased. + +\Paragraph{5.} What are the conditions which must be satisfied in order +that the rate, at which the probability of the generalisation +increases, may be such that it will approach certainty as a +%% -----File: 248.png---Folio 237------- +\index{Probability, and relevant knowledge!finite}% +limit when the number of independent instances of it are indefinitely +increased? We have already shown, as a basis for +this investigation, that $p_n$~approaches the limit of certainty for +a generalisation~$g$, if, as $n$~increases, $x_1x_2 \ldots x_n/\bar{g}h$ becomes +small compared with~$p_0$, \ie~if the \textit{à~priori} probability of so many +instances, assuming the falsehood of the generalisation, is small +compared with the generalisation's \textit{à~priori} probability. It +follows, therefore, that the probability of an induction tends +towards certainty as a limit, when the number of instances is +increased, provided that +\[ +x_r/x_1 x_2 \ldots x_{r-1} \bar{g}h < 1 - \epsilon +\] +for all values of~$r$, and $p_0>\eta$, where $\epsilon$~and~$\eta$ are finite probabilities, +separated, that is to say, from impossibility by a value +of some finite amount, however small. These conditions appear +simple, but the meaning of a `finite probability' requires a +word of explanation.\footnote + {The proof of these conditions, which is obvious, is as follows: + \[ + x_1 x_2\ldots x_n / \bar{g}h = x_{n-1}\bar{g}h · x_1 x_2 \ldots x_{n-1} / \bar{g}h < (1-\epsilon)^n, + \] + where $\epsilon$~is finite and $p_0 > \eta$ where $\eta$~is finite. There is always, under these + conditions, some finite value of~$n$ such that both $(1 - \epsilon)^n$ and~$\dfrac{(1 - \epsilon)^n}{\eta}$ are less + than any given finite quantity, however small.} + +I argued in \Chapref{III}. that not all probabilities have an +exact numerical value, and that, in the case of some, one can say +no more about their relation to certainty and impossibility than +that they fall short of the former and exceed the latter. There +is one class of probabilities, however, which I called the numerical +class, the ratio of each of whose members to certainty can be +expressed by some number less than unity; and we can sometimes +compare a non-numerical probability in respect of more and less +with one of these numerical probabilities. This enables us to +give a definition of `finite probability' which is capable of application +to non-numerical as well as to numerical probabilities. I +define a `finite probability' as one which \emph{exceeds} some numerical +probability, the ratio of which to certainty can be expressed by +a finite number.\footnote + {Hence a series of probabilities $p_1p_2 \ldots p_r$ approaches a limit~$L$, if, given + any positive finite number~$\epsilon$ however small, a positive integer~$n$ can always be + found such that for all values of~$r$ greater than~$n$ the difference between $L$~and~$p_r$ + is less than~$\epsilon · \gamma$, where $\gamma$~is the measure of certainty.} +The principal method, in which a probability +can be proved finite by a process of argument, arises either when +%% -----File: 249.png---Folio 238------- +its conclusion can be shown to be one of a finite number of alternatives, +which are between them exhaustive or, at any rate, have +a finite probability, and to which the Principle of Indifference +is applicable; or (more usually), when its conclusion is \emph{more} +probable than some hypothesis which satisfies this first condition. + +\Paragraph{6.} The conditions, which we have now established in order +that the probability of a pure induction may tend towards +certainty as the number of instances is increased, are (1)~that +$x_r/x_1x_2 \ldots x_{r-1} \bar{g}h$ falls short of certainty by a finite amount +for all values of~$r$, and (2)~that~$p_0$, the \textit{à~priori} probability of our +generalisation, exceeds impossibility by a finite amount. It is +easy to see that we can show by an exactly similar argument that +the following more general conditions are equally satisfactory: + +(1) That $x_r/x_1x_2 \ldots x_{r-1} \bar{g}h$ falls short of certainty by a finite +amount for all values of~$r$ beyond a specified value~$s$. + +(2) That $p_s$, the probability of the generalisation relative to +a knowledge of these first $s$~instances, exceeds impossibility by +a finite amount. + +In other words Pure Induction can be usefully employed to +strengthen an argument if, after a certain number of instances +have been examined, we have, from some other source, a finite +probability in favour of the generalisation, and, assuming the +generalisation is false, a finite uncertainty as to its conclusion +being satisfied by the next hitherto unexamined instance which +satisfies its premiss. To take an example, Pure Induction can +be used to support the generalisation that the sun will rise every +morning for the next million years, provided that with the experience +we have actually had there are finite probabilities, +however small, \emph{derived from some other source}, first, in favour of +the generalisation, and, second, in favour of the sun's \emph{not} rising +to-morrow assuming the generalisation to be false. Given these +finite probabilities, obtained otherwise, however small, then the +probability can be strengthened and can tend to increase towards +certainty by the mere multiplication of instances provided +that these instances are so far distinct that they are not +inferrible one from another. + +\Paragraph{7.} Those supposed proofs of the Inductive Principle, which +are based openly or implicitly on an argument in inverse probability, +are all vitiated by unjustifiable assumptions relating +to the magnitude of the \textit{à~priori} probability~$p_0$. Jevons, for +\index{Jevons!Induction@{and Induction}}% +%% -----File: 250.png---Folio 239------- +\index{Metaphysics and certainty}% +instance, avowedly assumes that we may, in the absence of special +information, suppose any unexamined hypothesis to be as likely +as not. It is difficult to see how such a belief, if even its most +immediate implications had been properly apprehended, could +have remained plausible to a mind of so sound a practical judgment +as his. The arguments against it and the contradictions +to which it leads have been dealt with in \Chapref{IV}\@. The +demonstration of Laplace, which depends upon the Rule of +\index{Laplace!Induction@{and Induction}}% +Succession, will be discussed in \Chapref{XXX}\@. + +\Paragraph{8.} The prior probability, which must always be found, before +the method of pure induction can be usefully employed to support +a substantial argument, is derived, I think, in most ordinary +cases---with what justification it remains to discuss---from considerations +of Analogy. But the conditions of valid induction +as they have been enunciated above, are quite independent of +analogy, and might be applicable to other types of argument. +In certain cases we might feel justified in assuming \emph{directly} that +the necessary conditions are satisfied. + +Our belief, for instance, in the validity of a logical scheme is +based partly upon inductive grounds---on the \emph{number} of conclusions, +each seemingly true on its own account, which can be +derived from the axioms---and partly on a degree of self-evidence +in the axioms themselves sufficient to give them the initial +probability upon which induction can build. We depend upon +the initial presumption that, if a proposition appears to us to +be true, this is by itself, in the absence of opposing evidence, +\emph{some reason} for its \emph{being} as well as appearing true. We cannot +deny that what appears true is sometimes false, but, unless we +can assume some substantial relation of probability between +the appearance and the reality of truth, the possibility of +even probable knowledge is at an end. + +The conception of our having \emph{some} reason, though not a +conclusive one, for certain beliefs, arising out of direct inspection, +may prove important to the theory of epistemology. The old +metaphysics has been greatly hindered by reason of its having +always demanded demonstrative certainty. Much of the cogency +of Hume's criticism arises out of the assumption of methods +\index{Hume}% +of certainty on the part of those systems against which it was +directed. The earlier realists were hampered by their not perceiving +that lesser claims in the beginning might yield them +%% -----File: 251.png---Folio 240------- +\index{Moore, G. E.|inote}% +what they wanted in the end. And transcendental philosophy +has partly arisen, I believe, through the belief that there is no +knowledge on these matters short of certain knowledge, being +combined with the belief that such certain knowledge of metaphysical +questions is beyond the power of ordinary methods. + +When we allow that probable knowledge is, nevertheless, real, +a new method of argument can be introduced into metaphysical +discussions. The demonstrative method can be laid on one side, +and we may attempt to advance the argument by taking account +of circumstances which seem to give \emph{some} reason for preferring +one alternative to another. Great progress may follow if the +nature and reality of objects of perception,\footnote + {A paper by Mr.~G.~E. Moore entitled, ``The Nature and Reality of Objects + of Perception,'' which was published in the \textit{Proceedings of the Aristotelian Society + for 1906}, seems to me to apply for the first time a method somewhat resembling + that which is described above.} +for instance, can be +usefully investigated by methods not altogether dissimilar from +those employed in science and with the prospect of obtaining as +high a degree of certainty as that which belongs to some scientific +conclusions; and it may conceivably be shown that a belief in +the conclusions of science, enunciated in any reasonable manner +however restricted, involves a preference for some metaphysical +conclusions over others. + +\Paragraph{9.} Apart from analysis, careful reflection would hardly lead +us to expect that a conclusion which is based on no other than +grounds of pure induction, defined as I have defined them as +consisting of repetition of instances merely, could attain in this +way to a high degree of probability. To this extent we ought +all of us to agree with Hume. We have found that the suggestions +of common sense are supported by more precise methods. +Moreover, we constantly distinguish between arguments, which +we call inductive, upon other grounds than the number of instances +upon which they are based; and under certain conditions +we regard as crucial an insignificant number of experiments. The +method of pure induction may be a useful means of strengthening +a probability based on some other ground. In the case, however, +of most scientific arguments, which would commonly be called +inductive, the probability that we are right, when we make +predictions on the basis of past experience, depends not so +much on the number of past experiences upon which we rely, +as on the degree in which the circumstances of these experiences +%% -----File: 252.png---Folio 241------- +resemble the known circumstances in which the prediction is +to take effect. Scientific method, indeed, is mainly devoted to +discovering means of so heightening the known analogy that +we may dispense as far as possible with the methods of pure +induction. + +When, therefore, our previous knowledge is considerable +and the analogy is good, the purely inductive part of the argument +may take a very subsidiary place. But when our knowledge +of the instances is slight, we may have to depend upon pure +induction a good deal. In an advanced science it is a last resort,---the +least satisfactory of the methods. But sometimes it must +be our first resort, the method upon which we must depend in +the dawn of knowledge and in fundamental inquiries where +we must presuppose nothing. +%% -----File: 253.png---Folio 242------- +\index{Fermat, formula of}% + + +\Chapter{XXI}{The Nature of Inductive Argument Continued} + +\Paragraph{1.} \First{In} the enunciation, given in the two preceding chapters, of the +Principles of Analogy and Pure Induction there has been no +reference to experience or causality or law. So far, the argument +has been perfectly formal and might relate to a set of propositions +of any type. But these methods are most commonly +employed in physical arguments where material objects or +experiences are the terms of the generalisation. We must consider, +therefore, whether there is any good ground, as some +logicians seem to have supposed, for restricting them to this +kind of inquiry. + +I am inclined to think that, whether reasonably or not, we +naturally apply them to all kinds of argument alike, including +formal arguments as, for example, about numbers. When we +are told that Fermat's formula for a prime, namely, $2^{2^a}+1$~for +all values of~$a$, has been verified in every case in which verification +is not excessively laborious---namely, for $a=1$, $2$,~$3$, +and~$4$, we feel that this is \emph{some} reason for accepting it, or, at +least, that it raises a sufficient presumption to justify a +further examination of the formula.\footnote + {This formula has, in fact, been disproved in recent times, \eg~$2^{2^5}+1= + 4,294,967,297 = 641 × 6,700,417$. Thus it is no longer so good an illustration + as it would have been a hundred years ago.} +Yet there can be no reference +here to the uniformity of nature or physical causation. If +inductive methods are limited to natural objects, there can no +more be an appreciable ground for thinking that $2^{2^a}+1$ is a true +formula for primes, because empirical methods show that it +yields primes up to $a=4$, or \emph{even if they showed that it yielded +primes for every number up to a million million}, than there is +to think that any formula which I may choose to write down +%% -----File: 254.png---Folio 243------- +at random is a true source of primes. To maintain that there is +no appreciable ground in such a case is paradoxical. If, on the +other hand, a partial verification does raise some just appreciable +presumption in the formula's favour, then we must include +numbers, at any rate, as well as material objects amongst the +proper subjects of the inductive method. The conclusion of +the previous chapter indicates, however, that, if arguments of +this kind have force, it can only be in virtue of there being +some finite \textit{à~priori} probability for the formula based on other +than inductive grounds. + +\index{Jevons!Induction@{and Induction}}% +There are some illustrations in Jevons's \textit{Principles of Science},\footnote + {Pp.~229--231 (one volume edition). Jevons uses these illustrations, not + for the purpose to which I am here putting them, but to demonstrate the fallibility + of empirical laws.} +which are relevant to this discussion. We find it to be true of +the following six numbers: +\[ +5,\ 15,\ 35,\ 45,\ 65,\ 95 +\] +that they all end in five, and are all divisible by five without remainder. +Would this fact, by itself, raise any kind of presumption +that all numbers ending in five are divisible by five without +remainder? Let us also consider the six numbers, +\[ +7,\ 17,\ 37,\ 47,\ 67,\ 97. +\] +They all end in seven and also agree in being primes. Would +this raise a presumption in favour of the generalisation that all +numbers are prime, which end in seven? We might be prejudiced +in favour of the first argument, because it would lead us to a +true conclusion; but we ought not to be prejudiced against the +second because it would lead us to a false one; for the validity +of empirical arguments as the foundation of a probability cannot +be affected by the actual truth or falsity of their conclusions. +If, on the evidence, the analogy is similar and equal, and if the +scope of the generalisation and its conclusion is similar, then the +value of the two arguments must be equal also. + +Whether or not the use of empirical argument appears plausible +to us in these particular examples, it is certainly true that many +mathematical theorems have actually been discovered by such +methods. Generalisations have been suggested nearly as often, +perhaps, in the logical and mathematical sciences, as in the +%% -----File: 255.png---Folio 244------- +\index{Jevons|inote}% +\index{Newton, and induction}% +physical, by the recognition of particular instances, even where +formal proof has been forthcoming subsequently. Yet if the +suggestions of analogy have no appreciable probability in the +formal sciences, and should be permitted only in the material, it +must be unreasonable for us to pursue them. If no finite probability +exists that a formula, for which we have empirical verification, +is in fact universally true, Newton was acting fortunately, +but not reasonably, when he hit on the Binomial Theorem by +methods of empiricism.\footnote + {See Jevons, \textit{loc.\ cit.}\ p.~231.} + +\Paragraph{2.} I am inclined to believe, therefore, that, if we trust the +promptings of common sense, we have the same kind of ground +for trusting analogy in mathematics that we have in physics, +and that we ought to be able to apply any justification of the +method, which suits the latter case, to the former also. This +does not mean that the \textit{à~priori} probabilities, from some other +source than induction, which the inductive method requires as +its foundation, may not be sought and found differently in the +two types of inquiry. A reason why it has been thought +that analogy ought to be confined to natural laws may be, +perhaps, that in most of those cases, in which we could +support a mathematical theorem by a very strong analogy, the +existence of a formal proof has done away with the necessity +for the limping methods of empiricism; and because in most +mathematical investigations, while in our earliest thoughts +we are not ashamed to consult analogy, our later work will be +more profitably spent in searching for a formal proof than in +establishing analogies which must, at the best, be relatively weak. +As the modern scientist discards, as a rule, the method of pure +induction, in favour of experimental analogy, where, if he +takes account of his previous knowledge, one or two cases may +prove immensely significant; so the modern mathematician +prefers the resources of his analysis, which may yield him +certainty, to the doubtful promises of empiricism. + +\Paragraph{3.} The main reason, however, why it has often been held that +we ought to limit inductive methods to the content of the particular +material universe in which we live, is, most probably, the +fact that we can easily imagine a universe so constructed that +such methods would be useless. This suggests that analogy and +induction, while they happen to be useful to us in this world, +%% -----File: 256.png---Folio 245------- +\index{Frazer, Sir J.}% +\index{Logic, academic!Induction@{and Induction}}% +\index{Primitive people and rational belief}% +cannot be universal principles of logic, on the same footing, for +instance, as the syllogism. + +In one sense this opinion may be well founded. I do not deny +or affirm at present that it may be necessary to confine inductive +methods to arguments about certain kinds of objects or certain +kinds of experiences. It may be true that in every useful argument +from analogy our premisses must contain fundamental +assumptions, obtained directly and not inductively, which some +possible experiences might preclude. Moreover, the success of +induction in the past can certainly affect its probable usefulness +for the future. We may discover something about the nature +of the universe---we may even discover it by means of induction +itself---the knowledge of which has the effect of destroying the +further utility of induction. I shall argue later on that the +confidence with which we ourselves use the method does in +fact depend upon the nature of our past experience. + +But this empirical attitude towards induction may, on the +other hand, arise out of either one of two possible confusions. +It may confuse, first, the reasonable character of arguments +with their practical usefulness. The usefulness of induction +depends, no doubt, upon the actual content of experience. If +there were no repetition of detail in the universe, induction +would have no utility. If there were only a single object in the +universe, the laws of addition would have no utility. But the +processes of induction and addition would remain reasonable. +It may confuse, secondly, the validity of attributing probability +to the conclusion of an argument with the question of the actual +truth of the conclusion. Induction tells us that, on the basis of +certain evidence, a certain conclusion is reasonable, \emph{not} that it is +true. If the sun does not rise to-morrow, if Queen Anne still +lives, this will not prove that it was foolish or unreasonable of us +to have believed the contrary. + +\Paragraph{4.} It will be worth while to say a little more in this connection +about the not infrequent failure to distinguish the rational from +the true. The excessive ridicule, which this mistake has visited +on the supposed irrationality of barbarous and primitive peoples, +affords some good examples. ``Reflection and enquiry should +satisfy us,'' says Dr.~Frazer in the \textit{Golden Bough}, ``that to our +predecessors we are indebted for much of what we thought most +our own, and that their errors were not wilful extravagances +%% -----File: 257.png---Folio 246------- +\index{Pythagoras and `\textit{seven}'}% +or the ravings of insanity, but simply hypotheses, justifiable as +such at the time when they were propounded, but which a fuller +experience has proved to be inadequate\ldots. Therefore, in +reviewing the opinions and practices of ruder ages and races we +shall do well to look with leniency upon their errors as inevitable +slips made in the search for truth\ldots.'' The first introduction of +iron ploughshares into Poland, he tells in another passage, having +been followed by a succession of bad harvests, the farmers attributed +the badness of the crops to the iron ploughshares, and discarded +them for the old wooden ones. The method of reasoning +of the farmers is not different from that of science, and may, +surely, have had for them some appreciable probability in its +favour. ``It is a curious superstition,'' says a recent pioneer in +Borneo, ``this of the Dusuns, to attribute anything---whether +good or bad, lucky or unlucky---that happens to them to something +novel which has arrived in their country. For instance, +my living in Kindram has caused the intensely hot weather we +have experienced of late.''\footnote + {\textit{Golden Bough}, p.~174.} +What is this curious superstition +but the Method of Difference? +\index{Method of Difference}% + +The following passage from Jevons's \textit{Principles of Science} well +\index{Jevons!analogy@{and analogy}}% +illustrates the tendency, to which he himself yielded, to depreciate +the favourite analogies of one age, because the experience of +their successors has confuted them. Between things which are +the same in number, he points out, there is a certain resemblance, +namely in number; and in the infancy of science men could not +be persuaded that there was not a deeper resemblance implied +in that of number. ``Seven days are mentioned in Genesis; +infants acquire their teeth at the end of seven months; they +change them at the end of seven years; seven feet was the limit +of man's height; every seventh year was a climacteric or critical +year, at which a change of disposition took place. In natural +science there were not only the seven planets, and the seven +metals, but also the seven primitive colours, and the seven tones +of music. So deep a hold did this doctrine take that we still have +its results in many customs, not only in the seven days of the +week, but the seven years' apprenticeship, puberty at fourteen +years, the second climacteric, and legal majority at twenty-one +years, the third climacteric.'' Religious systems from Pythagoras +to Comte have sought to derive strength from the virtue of seven. +\index{Comte!seven@{and `\textit{seven}'}}% +%% -----File: 258.png---Folio 247------- +\index{Hudson, W. H., and animism|inote}% +\index{Newton, and induction!ans seven@{and `\textit{seven}'}}% +``And even in scientific matters the loftiest intellects have occasionally +yielded, as when Newton was misled by the analogy +between the seven tones of music and the seven colours of his +spectrum\ldots. Even the genius of Huyghens did not prevent +\index{Huyghens!six@{and `\textit{six}'}}% +him from inferring that but one satellite could belong to Saturn, +because, with those of Jupiter and the earth, it completed the +perfect number of six.'' But is it certain that Newton and +Huyghens were only reasonable when their theories were true, +and that their mistakes were the fruit of a disordered fancy? +Or that the savages, from whom we have inherited the most +fundamental inductions of our knowledge, were always superstitious +when they believed what we now know to be +preposterous? + +It is important to understand that the common sense of the +race has been impressed by very weak analogies and has attributed +to them an appreciable probability, and that a logical +theory, which is to justify common sense, need not be afraid of +including these marginal cases. Even our belief in the real +existence of other people, which we all hold to be well established, +may require for its justification the combination of +experience with a just appreciable \textit{à~priori} possibility for +Animism generally.\footnote + {``This is animism, or that sense of something in Nature which to the + enlightened or civilised man is not there, and in the civilised man's child, if it + be admitted that he has it at all, is but a faint survival of a phase of the + primitive mind. And by animism I do not mean the theory of a soul in + nature, but the tendency or impulse or instinct, in which all myth originates, + to \emph{animate} all things; the projection of ourselves into nature; the sense and + apprehension of an intelligence like our own, but more powerful in all visible + things'' (Hudson, \textit{Far Away and Long Ago}, pp.~224--5). This `tendency or + impulse or instinct,' refined by reason and enlarged by experience, may be + required, in the shape of an intuitive \textit{à~priori} probability, if some of those + universal conclusions of common sense, which the most sceptical do not kick + away, are to be supported with rational foundations.} +If we actually possess evidence which +renders some conclusion absurd, it is very difficult for us to +appreciate the relation of this conclusion to data which are +different and less complete; but it is essential that we should +realise arguments from analogy as \emph{relative to premisses}, if we are +to approach the logical theory of Induction without prejudice. + +\Paragraph{5.} While we depreciate the former probability of beliefs +which we no longer hold, we tend, I think, to exaggerate the +present degree of certainty of what we still believe. The preceding +paragraph is not intended to deny that savages often greatly +%% -----File: 259.png---Folio 248------- +\index{Grimsehl|inote}% +\index{Relativity, of knowledge!doctrine of, and the Law of Uniformity|inote}% +\index{Uniformity of Nature, Law of}% +\index{Universal Causation, Law of}% +overestimate the value of their crude inductions, and are to this +extent irrational. It is not easy to distinguish between a belief's +being the most reasonable of those which it is open to us to +believe, and its being more probable than not. In the same way +we, perhaps, put an excessive confidence in those conclusions---the +existence of other people, for instance, the law of gravity, or +to-morrow's sunrise---of which, in comparison with many other +beliefs, we are very well assured. We may sometimes confuse +the practical certainty, attaching to the class of beliefs upon which +it is rational to act with the utmost confidence, with the more +wholly objective certainty of logic. We might rashly assert, for +instance, that to-morrow's sunrise is as likely to us as failure, +and the special virtue of the number seven as unlikely, even to +Pythagoras, as success, in an attempt to throw heads a hundred +times in succession with an unbiassed coin.\footnote + {Yet if every inhabitant of the world, Grimsehl has calculated, were to toss + a coin every second, day and night, this latter event would only occur once on + the average in every twenty billion years.} + +\Paragraph{6.} As it has often been held upon various grounds, with +reason or without, that the validity of Induction and Analogy +depends in some way upon the character of the actual world, +logicians have sought for material laws upon which these methods +can be founded. The Laws of Universal Causation and the +Uniformity of Nature, namely, that all events have \emph{some} cause +and that the same total cause always produces the same effect, +are those which commonly do service. But these principles +merely assert that there are \emph{some} data from which events posterior +to them in time could be inferred. They do not seem to yield us +much assistance in solving the inductive problem proper, or in +determining how we can infer with probability from \emph{partial} data. +It has been suggested in \Chapref{XIX}\DPtypo{}{.} that the Principle +of the Uniformity of Nature amounts to an assertion that an +argument from perfect analogy (defined as I have defined it) is +valid when applied to events only differing in their positions in +time or space.\footnote + {Is this interpretation of the Principle of the Uniformity of Nature affected + by the Doctrine of Relativity?} +It has also been pointed out that ordinary inductive +arguments appear to be strengthened by any evidence +which makes them approximate more closely in character to a +perfect analogy. But this, I think, is the whole extent to which +this principle, even if its truth could be assumed, would help us. +%% -----File: 260.png---Folio 249------- +\index{Atomic Uniformity}% +\index{Principle of superposition of small!effects}% +States of the universe, identical in every particular, may never +recur, and, even if identical states were to recur, we should not +know it. + +The kind of fundamental assumption about the character of +material laws, on which scientists appear commonly to act, +seems to me to be much less simple than the bare principle of +Uniformity. They appear to assume something much more like +what mathematicians call the principle of the superposition of +small effects, or, as I prefer to call it, in this connection, the +\emph{atomic} character of natural law. The system of the material +universe must consist, if this kind of assumption is warranted, +of bodies which we may term (without any implication as to +their size being conveyed thereby) \emph{legal atoms}, such that each of +them exercises its own separate, independent, and invariable +effect, a change of the total state being compounded of a number +of separate changes each of which is solely due to a separate +portion of the preceding state. We do not have an invariable +relation between particular bodies, but nevertheless each has on +the others its own separate and invariable effect, which does not +change with changing circumstances, although, of course, the +total effect may be changed to almost any extent if all the other +accompanying causes are different. Each atom can, according +to this theory, be treated as a separate cause and does +not enter into different organic combinations in each of which +it is regulated by different laws. + +Perhaps it has not always been realised that this atomic +uniformity is in no way implied by the principle of the +Uniformity of Nature. Yet there might well be quite different +laws for wholes of different degrees of complexity, and laws of +connection between complexes which could not be stated in +terms of laws connecting individual parts. In this case +natural law would be organic and not, as it is generally +supposed, atomic. If every configuration of the Universe were +subject to a separate and independent law, or if very small +differences between bodies---in their shape or size, for instance,---led +to their obeying quite different laws, prediction would be +impossible and the inductive method useless. Yet nature might +still be uniform, causation sovereign, and laws timeless and +absolute. + +The scientist wishes, in fact, to assume that the occurrence +%% -----File: 261.png---Folio 250------- +of a phenomenon which has appeared as part of a more complex +phenomenon, may be \emph{some} reason for expecting it to be associated +on another occasion with part of the same complex. Yet if +different wholes were subject to different laws \textit{quâ}~wholes and +not simply on account of and in proportion to the differences of +their parts, knowledge of a part could not lead, it would seem, +even to presumptive or probable knowledge as to its association +with other parts. Given, on the other hand, a number of legally +atomic units and the laws connecting them, it would be possible +to deduce their effects \textit{pro~tanto} without an exhaustive knowledge +of all the coexisting circumstances. + +We do habitually assume, I think, that the size of the atomic +unit is for mental events an individual consciousness, and for +material events an object small in relation to our perceptions. +These considerations do not show us a way by which we can +justify Induction. But they help to elucidate the kind of assumptions +which we do actually make, and may serve as an introduction +to what follows. +%% -----File: 262.png---Folio 251------- +\index{Necessary connection, law of}% + + +\Chapter{XXII}{The Justification of these Methods} + +\Paragraph{1.} \First{The} general line of thought to be followed in this chapter may +be indicated, briefly, at the outset. + +A system of facts or propositions, as we ordinarily conceive +it, may comprise an indefinite number of members. But the +ultimate constituents or indefinables of the system, which all +the members of it are about, are less in number than these +members themselves. Further, there are certain laws of necessary +connection between the members, by which it is meant (I do not +stop to consider whether \emph{more} than this is meant) that the truth +or falsity of every member can be inferred from a knowledge of +the laws of necessary connection together with a knowledge of the +truth or falsity of some (but not all) of the members. + +The ultimate constituents together with the laws of necessary +connection make up what I shall term the \emph{independent variety} +of the system. The more numerous the ultimate constituents +and the necessary laws, the greater is the system's independent +variety. It is not necessary for my present purpose, which is +merely to bring before the reader's mind the sort of conception +which is in mine, that I should attempt a complete definition +of what I mean by a system. + +Now it is characteristic of a system, as distinguished from +a collection of heterogeneous and independent facts or propositions, +that the number of its premisses, or, in other words, the +amount of independent variety in it, should be less than the +number of its members. But it is not an obviously essential +characteristic of a system that its premisses or its independent +variety should be actually finite. We must distinguish, +therefore, between systems which may be termed finite and +infinite respectively, the terms \emph{finite} and \emph{infinite} referring not to +%% -----File: 263.png---Folio 252------- +the number of members in the system but to the amount of independent +variety in it. + +The purpose of the discussion, which occupies the greater +part of this chapter, is to maintain that, if the premisses of our +argument permit us to assume that the facts or propositions, +with which the argument is concerned, belong to a \emph{finite} system, +then probable knowledge can be validly obtained by means of +an inductive argument. I now proceed to approach the question +from a slightly different standpoint, the controlling idea, however, +being that which is outlined above. + +\Paragraph{2.} What is our actual course of procedure in an inductive +argument? We have before us, let us suppose, a set of $n$~instances +which have $r$~known qualities, $a_1a_2\ldots a_r$ in common, +these $r$~qualities constituting the known positive analogy. From +these qualities three (say) are picked out, namely, $a_1$,~$a_2$,~$a_3$ and +we inquire with what probability \emph{all} objects having these three +qualities have also certain other qualities which we have picked +out, namely, $a_{r-1}$,~$a_r$. We wish to determine, that is to say, +whether the qualities $a_{r-1}$,~$a_r$ are \emph{bound up} with the qualities +$a_1$,~$a_2$,~$a_3$. In thus approaching this question we seem to +suppose that the qualities of an object are bound together in +a limited number of \emph{groups}, a sub-class of each group being an +infallible symptom of the coexistence of certain other members +of it also. + +Three possibilities are open, any of which would prove +destructive to our generalisation. It may be the case (1)~that +$a_{r-1}$~or~$a_r$ is independent of all the other qualities of the instances---they +may not overlap, that is to say, with any other groups; +or (2)~that $a_1a_2a_3$~do not belong to the same groups as~$a_{r-1}a_r$; +or (3)~that $a_1a_2a_3$, while they belong to the same group as~$a_{r-1}a_r$, +are not sufficient to specify this group uniquely---they belong, +that is to say, to other groups also which do not include $a_{r-1}$~and~$a_r$. +The precautions we take are directed towards reducing the +likelihood, so far as we can, of each of these possibilities. We +distrust the generalisation if the terms typified by~$a_{r-1}a_r$ are +numerous and comprehensive, because this increases the likelihood +that some at least of them fall under heading~(1), and also +because it increases the likelihood of~(3). We trust it if the +terms typified by~$a_1a_2a_3$ are numerous and comprehensive, +because this decreases the likelihood both of~(2) and of~(3). If +%% -----File: 264.png---Folio 253------- +we find a new instance which agrees with the former instances in +$a_1a_2a_3a_{r-1}a_r$ but not in~$a_4$, we welcome it, because this disposes of +the possibility that it is~$a_4$, alone or in combination, that is bound +up with~$a_{r-1}a_r$. We desire to increase our knowledge of the +properties, lest there be some positive analogy which is escaping us, +and when our knowledge is incomplete we multiply instances, +which we do not know to increase the negative analogy for +certain, in the hope that they may do so. + +If we sum up the various methods of Analogy, we find, I +think, that they are all capable of arising out of an underlying +assumption, that if we find two sets of qualities in coexistence +there is a finite probability that they belong to the same group, +and a finite probability also that the first set specifies this group +uniquely. Starting from this assumption, the object of the +methods is to increase the finite probability and make it large. +Whether or not anything of this sort is explicitly present to our +minds when we reason scientifically, it seems clear to me that we +do act exactly as we should act, if this were the assumption from +which we set out. + +In most cases, of course, the field is greatly simplified from +the first by the use of our pre-existing knowledge. Of the +properties before us we generally have good reason, derived +from prior analogies, for supposing some to belong to the same +group and others to belong to different groups. But this does +not affect the theoretical problem confronting us. + +\Paragraph{3.} What kind of ground could justify us in assuming the +existence of these finite probabilities which we seem to require? +If we are to obtain them, not directly, but by means of argument, +we must somehow base them upon a finite number of exhaustive +alternatives. + +The following line of argument seems to me to represent, on +the whole, the kind of assumption which is obscurely present to +our minds. We suppose, I think, that the almost innumerable +apparent properties of any given object all arise out of a finite +number of generator properties, which we may call~$\phi_1\phi_2\phi_3\ldots$. +\index{Generator properties}% +Some arise out of $\phi_1$~alone, some out of~$\phi_1$ in conjunction with~$\phi_2$, +and so on. The properties which arise out of $\phi_1$~alone form one +group; those which arise out of~$\phi_1\phi_2$ in conjunction form another +group, and so on. Since the number of generator properties is +finite, the number of groups also is finite. If a set of apparent +%% -----File: 265.png---Folio 254------- +properties arise (say) out of three generator properties~$\phi_1\phi_2\phi_3$, +\index{Generator properties!plurality of}% +then this set of properties may he said to specify the group~$\phi_1\phi_2\phi_3$. +Since the total number of apparent properties is assumed +to be greater than that of the generator properties, and since the +number of groups is finite, it follows that, if two sets of apparent +properties are taken, there is, in the absence of evidence to the +contrary, a finite probability that the second set will belong +to the group specified by the first set. + +There is, however, the possibility of a plurality of generators. +The first set of apparent properties may specify more than one +group,---there is more than one group of generators, that is to +say, which are competent to produce it; and some only of these +groups may contain the second set of properties. Let us, for +the moment, rule out this possibility. + +When we argue from an analogy, and the instances have +two groups of characters in common, namely $\phi$~and~$f$, either $f$~belongs +to the group~$\phi$ or it arises out of generators partly distinct +from those out of which $\phi$~arises. For the reason already explained +there is a finite probability that $f$~and~$\phi$ belong to the +same group. If this is the case, \ie\ if the generalisation~$g(\phi f)$ +is valid, then $f$~will certainly be true of all other cases in which +$\phi$~is true; if this is not the case, then $f$~will not always be true +when $\phi$~is true. We have, therefore, the preliminary conditions +necessary for the application of pure induction. If $x_r$,~etc., are +the instances, +\begin{DPgather*} +g/h = p_0, \text{ where $p_0$ is finite},\\ +x_r/gh = 1, \text{ etc.}, \\ +\lintertext{and} +x_r/x_1x_2\ldots x_{r-1}\bar{g}h = 1 - \epsilon, \text{ where $\epsilon$ is finite}. +\end{DPgather*} +And hence, by the argument of \Chapref{XX}., the probability of a +generalisation, based on such evidence as this, is capable, under +suitable conditions, of tending towards certainty as a limit, when +the number of instances is increased. + +If $\phi$~is complex and includes a number of characters which +are not always found together, it must include a number of +separate generator properties and specify a large group; hence +the initial probability that $f$~belongs to this group is relatively +large. If, on the other hand, $f$~is complex, there will be, for the +same reasons \textit{mutatis mutandis}, a relatively smaller initial probability +than otherwise that $f$~belongs to any other given group. +%% -----File: 266.png---Folio 255------- +\index{Uniformity of Nature, Law of}% + +When the argument is mainly by analogy, we endeavour to +obtain evidence which makes the initial probability~$p_0$ relatively +high; when the analogy is weak and the argument depends for +its strength upon pure induction, $p_0$~is small and~$p_m$, which is +based upon numerous instances, depends for its magnitude upon +their number. But an argument from induction must always +involve some element of analogy, and, on the other hand, few +arguments from analogy can afford to ignore altogether the +strengthening influence of pure induction. + +\Paragraph{4.} Let us consider the manner in which the methods of +analogy increase the initial likelihood that two characters belong +to the same group. The numerous characters of an object which +are known to us may be represented by $a_1a_2\ldots a_n$. We select +two sets of these, $a_r$~and~$a_s$, and seek to determine whether $a_s$~always +belongs to the group specified by~$a_r$. Our previous knowledge +will enable us, in general, to rule out many of the object's +characters as being irrelevant to the groups specified by $a_r$~and~$a_s$, +although this will not be possible in the most fundamental inquiries. +We may also know that certain characters are always +associated with $a_r$~or with~$a_s$. But there will be left a residuum +of whose connection with $a_r$~or~$a_s$ we are ignorant. These +characters, whose relevance is in doubt, may be represented by +$a_{r+1}\ldots a_{s-1}$. If the analogy is perfect, these characters are +eliminated altogether. Otherwise, the argument is weakened +in proportion to the comprehensiveness of these doubtful characters. +For it may be the case that some of $a_{r+1}\ldots a_{s-1}$ are +necessary as well as~$a_r$, in order to specify all the generators +which are required to produce~$a_s$. + +\Paragraph{5.} We may possibly be justified in neglecting certain of the +characters $a_{r+1}\ldots a_{s-1}$ by \emph{direct} judgments of irrelevance. +\index{Irrelevance}% +There are certain properties of objects which we rule out from +the beginning as wholly or largely independent and irrelevant to +all, or to some, other properties. The principal judgments of +this kind, and those alone about which we seem to feel much +confidence, are concerned with absolute position in time and +\index{Time}% +space, this class of judgments of irrelevance being summed up, +\index{Space}% +I have suggested, in the Principle of the Uniformity of Nature. +We judge that \emph{mere} position in time and space cannot possibly +affect, as a determining cause, any other characters; and this +belief appears so strong and certain, although it is hard to see +%% -----File: 267.png---Folio 256------- +\index{Generator properties!plurality of}% +how it can be based on experience, that the judgment by which +we arrive at it seems perhaps to be direct. A further type of +instance in which some philosophers seem to have trusted direct +judgments of relevance in these matters arises out of the relation +between mind and matter. They have believed that no mental +event can possibly be a \emph{necessary} condition for the occurrence of +a material event. + +The Principle of the Uniformity of Nature, as I interpret it, +supplies the answer, if it is correct, to the criticism that the +instances, on which generalisations are based, are all alike in +being past, and that any generalisation, which is applicable to +the future, must be based, for this reason, upon imperfect analogy. +We judge directly that the resemblance between instances, which +consists in their being past, is in itself irrelevant, and does not +supply a valid ground for impugning a generalisation. + +But these judgments of irrelevance are not free from difficulty, +and we must be suspicious of using them. When I say that position +is irrelevant, I do not mean to deny that a generalisation, the +premiss of which specifies position, may be true, and that the +same generalisation without this limitation might be false. But +this is because the generalisation is incompletely stated; it +happens that objects so specified have the required characters, +and hence their position supplies a sufficient criterion. Position +may be relevant as a sufficient condition but never as a \emph{necessary} +condition, and the inclusion of it can only affect the truth of a +generalisation when we have left out some other essential condition. +A generalisation which is true of one instance must be +true of another which \emph{only} differs from the former by reason of +its position in time or space. + +\Paragraph{6.} Excluding, therefore, the possibility of a plurality of +generators, we can justify the method of perfect analogy, and +other inductive methods in so far as they can be made to +approximate to this, by means of the assumption that the +objects in the field, over which our generalisations extend, do +not have an infinite number of independent qualities; that, in +other words, their characteristics, however numerous, cohere +together in groups of invariable connection, which are finite +in number. This does not limit the number of entities which +are only \emph{numerically} distinct. In the language used at the +beginning of this chapter, the use of inductive methods can be +%% -----File: 268.png---Folio 257------- +\index{Broad, C.~D.|inote}% +\index{Generator properties!plurality of}% +justified if they are applied to what we have reason to suppose +a finite system.\footnote + {Mr.~C.~D. Broad, in two articles ``On the Relation between Induction and + Probability'' (\textit{Mind}, 1918 and~1920), has been following a similar line of + thought.} + +\Paragraph{7.} Let us now take account of a possible plurality of +generators. I mean by this the possibility that a given character +can arise in more than one way, can belong to more than +one distinct group, and can arise out of more than one generator. +$\phi$~might, for instance, be sometimes due to a generator~$\DPtypo{a}{\alpha}_1$, and +$\DPtypo{a}{\alpha}_1$~might invariably produce~$f$. But we could not generalise +from $\phi$~to~$f$, if $\phi$~might be due in other cases to a different +generator~$\DPtypo{a}{\alpha}_2$ which would \emph{not} be competent to produce~$f$. + +If we were dealing with inductive correlation, where we do +\index{Inductive correlation}% +not claim universality for our conclusions, it would be sufficient +for us to assume that the number of distinct generators, to which +a given property~$\phi$ can be due, is always finite. To obtain validity +for universal generalisations it seems necessary to make the more +comprehensive and less plausible assumption that a finite probability +always exists that there is \emph{not}, in any given case, a plurality +of causes. With this assumption we have a valid argument from +pure induction on the same lines, nearly, as before. + +\Paragraph{8.} We have thus two distinct difficulties to deal with, and we +require for the solution of each a separate assumption. The +point may be illustrated by an example in which only one of the +difficulties is present. There are few arguments from analogy of +which we are better assured than the existence of other people. +We feel indeed so well assured of their existence that it has been +thought sometimes that our knowledge of them must be in some +way direct. But analogy does not seem to me unequal to the +proof. We have numerous experiences in our own person of +acts which are associated with states of consciousness, and we +infer that similar acts in others are likely to be associated with +similar states of consciousness. But this argument from analogy +is superior in one respect to nearly all other empirical arguments, +and this superiority may possibly explain the great confidence +which we feel in it. We do seem in this case to have +direct knowledge, such as we have in no other case, that our +states of consciousness are, sometimes at least, causally connected +with some of our acts. We do not, as in other cases, +%% -----File: 269.png---Folio 258------- +\index{Analogy, principle of!logical foundation of}% +\index{Inductive correlation}% +merely observe invariable sequence or coexistence between consciousness +and act; and we do believe it to be vastly improbable +in the case of some at least of our own physical acts that they +could have occurred without a mental act to support them. +Thus, we seem to have a special assurance of a kind not usually +available for believing that there is \emph{sometimes} a necessary connection +between the conclusion and the condition of the +generalisation; we doubt it only from the possibility of a +plurality of causes. + +The objection to this argument on the ground that the analogy +is always imperfect, in that all the observed connections of +consciousness and act are alike in being \emph{mine}, seems to me to be +invalid on the same ground as that on which I have put on one +side objections to future generalisations, which are based on the +fact that the instances which support them are all alike in being +\emph{past}. If direct judgments of irrelevance are ever permissible, +there seems some ground for admitting one here. + +\Paragraph{9.} As a logical foundation for Analogy, therefore, we seem to +need some such assumption as that the amount of variety in the +\index{Variety!limitation of}% +universe is limited in such a way that there is no one object so +complex that its qualities fall into an infinite number of independent +groups (\ie~groups which might exist independently +as well as in conjunction); or rather that none of the objects +about which we generalise are as complex as this; or at least +that, though some objects may be infinitely complex, we sometimes +have a finite probability that an object about which we +seek to generalise is not infinitely complex. + +To meet a possible plurality of causes some further assumption +is necessary. If we were content with Inductive Correlations +and sought to prove merely that there was a probability in favour +of \emph{any} instance of the generalisation in question, without inquiring +whether there was a probability in favour of \emph{every} instance, +it would be sufficient to suppose that, while there may be more +than one sufficient cause of a character, there is not an infinite +number of distinct causes competent to produce it. And this +involves no new assumption; for if the aggregate variety of the +system is finite, the possible plurality of causes must also be finite. +If, however, our generalisation is to be universal, so that it breaks +down if there is a single exception to it, we must obtain, by some +means or other, a finite probability that the set of characters, +%% -----File: 270.png---Folio 259------- +\index{Measurement of Probability!induction@{and induction}}% +which condition the generalisation, are \emph{not} the possible effect of +more than one distinct set of fundamental properties. I do not +know upon what ground we could establish a finite probability +to this effect. The necessity for this seemingly arbitrary hypothesis +strongly suggests that our conclusions should be in the +form of inductive correlations, rather than of universal generalisations. +Perhaps our generalisations should always run: `It is +probable that any given~$\phi$ is~$f$,' rather than, `It is probable that +all~$\phi$ are~$f$.' Certainly, what we commonly seem to hold with conviction +is the belief that the sun will rise \emph{to-morrow}, rather than +the belief that the sun will \emph{always} rise so long as the conditions +explicitly known to us are fulfilled. This will be matter for +further discussion in \Partref{V}., when Inductive Correlation is +specifically dealt with. + +\Paragraph{10.} There is a vagueness, it may be noticed, in the number of +instances, which would be required on the above assumptions +to establish a given numerical degree of probability, which +corresponds to the vagueness in the degree of probability which +we do actually attach to inductive conclusions. We assume +that the necessary number of instances is finite, but we do not +know what the number is. We know that the probability of a +well-established induction is great, but, when we are asked to +name its degree, we cannot. Common sense tells us that some +inductive arguments are stronger than others, and that some +are very strong. But how much stronger or how strong we +cannot express. The probability of an induction is only +numerically definite when we are able to make definite assumptions +about the number of independent equiprobable influences +at work. Otherwise, it is non-numerical, though bearing relations +of greater and less to numerical probabilities according to the +approximate limits within which our assumption as to the possible +number of these causes lies. + +\Paragraph{11.} Up to this point I have supposed, for the sake of simplicity, +that it is necessary to make our assumptions as to the limitation +of independent variety in an absolute form, to assume, that is to +say, the finiteness of the system, to which the argument is applied, +\emph{for certain}. But we need not in fact go so far as this. + +If our conclusion is~$C$ and our empirical evidence is~$E$, then, +in order to justify inductive methods, our premisses must include, +in addition to~$E$, a general hypothesis~$H$ such that~$C/H$, the +%% -----File: 271.png---Folio 260------- +\textit{à~priori} probability of our conclusion, has a finite value. The +effect of~$E$ is to increase the probability of~$C$ above its initial +\textit{à~priori} value, $C/HE$~being greater than~$C/H$. But the method +of strengthening~$C/H$ by the addition of evidence~$E$ is valid quite +apart from the particular content of~$H$. If, therefore, we have +another general hypothesis~$H'$ and other evidence~$E'$, such that +$H/H'$ has a finite value, we can, without being guilty of a circular +argument, use evidence~$E'$ by the same method as before to +strengthen the probability~$H/H'$. If we call~$H$, namely, the +absolute assertion of the finiteness of the system under consideration, +the \emph{inductive hypothesis}, and the process of strengthening~$C/H$ +\index{Inductive hypothesis}% +by the addition~$E$ the \emph{inductive method}, it is not circular to +\index{Inductive method}% +use the inductive method to strengthen the inductive hypothesis +itself, relative to some more primitive and less far-reaching assumption. +If, therefore, we have any reason~($H'$) for attributing +\textit{à~priori} a finite probability to the Inductive Hypothesis~($H$), then +the actual conformity of experience \textit{à~posteriori} with expectations +based on the assumption of~$H$ can be utilised by the inductive +method to attribute an enhanced value to the probability of~$H$. +To this extent, therefore, we can support the Inductive Hypothesis +by experience. In dealing with any particular question we can +take the Inductive Hypothesis, not at its \textit{à~priori} value, but at +the value to which experience in general has raised it. What +we require \textit{à~priori}, therefore, is not the certainty of the Inductive +Hypothesis, but a finite probability in its favour.\footnote + {I have implicitly assumed in the above argument that if $H'$~supports~$H$, it + strengthens an argument which $H$ would strengthen. This is not \emph{necessarily} + the case for the reasons given on pp.\ \Pageref[]{68}~and~\Pageref[]{147}. In these passages the + necessary conditions for the above are elucidated. I am, therefore, assuming + that in the case now in question these conditions actually are fulfilled.} + +Our assumption, in its most limited form, then, amounts to +this, that we have a finite \textit{à~priori} probability in favour of +the Inductive Hypothesis as to there being some limitation +of independent variety (to express shortly what I have already +\index{Variety!limitation of}% +explained in detail) in the objects of our generalisation. Our +experience might have been such as to diminish this probability +\textit{à~posteriori}. It has, in fact, been such as to increase it. It is +because there has been so much repetition and uniformity in our +experience that we place great confidence in it. To this extent +the popular opinion that Induction depends upon experience for +its validity is justified and does not involve a circular argument. +%% -----File: 272.png---Folio 261------- + +\Paragraph{12.} I think that this assumption is adequate to its purpose +and would justify our ordinary methods of procedure in inductive +argument. It was suggested in the previous chapter that our +theory of Analogy ought to be as applicable to mathematical +as to material generalisations, if it is to justify common sense. +The above assumptions of the limitation of independent variety +sufficiently satisfy this condition. There is nothing in these +assumptions which gives them a peculiar reference to material +objects. We believe, in fact, that all the properties of numbers +can be derived from a \emph{limited} number of laws, and that the same +set of laws governs all numbers. To apply empirical methods to +such things as numbers renders it necessary, it is true, to make +an assumption about the nature of numbers. But it is the same +kind of assumption as we have to make about material objects, +and has just about as much, or as little, plausibility. There is +no new difficulty. + +The assumption, also, that the system of Nature is finite is +in accordance with the analysis of the underlying assumption of +scientists, given at the close of the previous chapter. The +hypothesis of atomic uniformity, as I have called it, while not +formally equivalent to the hypothesis of the limitation of independent +variety, amounts to very much the same thing. If the +fundamental laws of connection changed altogether with variations, +for instance, in the shape or size of bodies, or if the laws +governing the behaviour of a complex had no relation whatever +to the laws governing the behaviour of its parts when belonging +to other complexes, there could hardly be a limitation of independent +variety in the sense in which this has been defined. And, +on the other hand, a limitation of independent variety seems +necessarily to carry with it some degree of atomic uniformity. +The underlying conception as to the character of the System of +Nature is in each case the same. + +\Paragraph{13.} We have now reached the last and most difficult stage of +the discussion. The logical part of our inquiry is complete, and +it has left us, as it is its business to leave us, with a question of +epistemology. Such is the premiss or assumption which our +\index{Epistemology!inductive hypothesis@{and inductive hypothesis}}% +logical processes need to work upon. What right have we to +make it? It is no sufficient answer in philosophy to plead that +the assumption is after all a very little one. + +I do not believe that any conclusive or perfectly satisfactory +%% -----File: 273.png---Folio 262------- +answer to this question can be given, so long as our knowledge +\index{Knowledge!direct and indirect}% +of the subject of epistemology is in so disordered and undeveloped +a condition as it is in at present. No proper answer has yet been +given to the inquiry---of what sorts of things are we capable of +direct knowledge? The logician, therefore, is in a weak position, +when he leaves his own subject and attempts to solve a particular +instance of this general problem. He needs guidance as to what +\emph{kind} of reason we could have for such an assumption as the use +of inductive argument appears to require. + +On the one hand, the assumption may be absolutely \textit{à~priori} +in the sense that it would be equally applicable to all possible +objects. On the other hand, it may be seen to be applicable to +some classes of objects only. In this Case it can only arise out +of some degree of particular knowledge as to the nature of the +objects in question, and is to this extent dependent on experience. +But if it is experience which in this sense enables us to know the +assumption as true of certain amongst the objects of experience, +it must enable us to know it in some manner which we may term +direct and not as the result of an inference. + +Now an assumption, that \emph{all} systems of fact are finite (in the +sense in which I have defined this term), cannot, it seems perfectly +plain, be regarded as having absolute, universal validity in the +sense that such an assumption is self-evidently applicable to every +kind of object and to all possible experiences. It is not, therefore, +in quite the same position as a self-evident \emph{logical} axiom, and does +not appeal to the mind in the same way. The most which can +be maintained is that this assumption is true of \emph{some} systems of +fact, and, further, that there are some objects about which, as +soon as we understand their nature, the mind is able to apprehend +directly that the assumption in question \emph{is} true. + +In \Chapref{II}. §\;7, I wrote: ``By some mental process of +which it is difficult to give an account, we are able to pass from +direct acquaintance with things to a knowledge of propositions +about the things of which we have sensations or understand the +meaning.'' Knowledge, so obtained, I termed direct knowledge. +From a sensation of yellow and from an understanding of the +meaning of `yellow' and of `colour,' we could, I suggested, +have direct knowledge of the fact or proposition `yellow is a +colour;' we might also know that colour cannot exist without +extension, or that two colours cannot be perceived at the same +%% -----File: 274.png---Folio 263------- +\index{Causality}% +\index{Proposition, characterisation of!synthetic}% +\index{Uniformity of Nature, Law of}% +time in the same place. Other philosophers might use terms +differently and express themselves otherwise; but the substance +of what I was there trying to say is not very disputable. But +when we come to the question as to what kinds of propositions +we can come to know in this manner, we enter upon an unexplored +field where no certain opinion is discoverable. + +In the case of logical terms, it seems to be generally agreed +that if we understand their meaning we can know directly propositions +about them which go far beyond a mere expression of +this meaning;---propositions of the kind which some philosophers +have termed \emph{synthetic}. In the case of non-logical or +empirical entities, it seems sometimes to be assumed that our +direct knowledge must be confined to what may be regarded as +an expression or description of the meaning or sensation apprehended +by us. If this view is correct the Inductive Hypothesis +is not the kind of thing about which we can have direct knowledge +as a result of our acquaintance with objects. + +I suggest, however, that this view is incorrect, and that we +are capable of direct knowledge about empirical entities which +goes beyond a mere expression of our understanding or sensation +of them. It may be useful to give the reader two examples, more +familiar than the Inductive Hypothesis, where, as it appears to +me, such knowledge is commonly assumed. The first is that of the +causal irrelevance of mere position in time and space, commonly +called the Uniformity of Nature. We do believe, and yet have +no adequate inductive reason whatever for believing, that mere +position in time and space cannot make any difference. This +belief arises directly, I think, out of our acquaintance with +the objects of experience and our understanding of the concepts +of `time' and `space.' The second is that of the Law of +Causation. We believe that every object in time has a `necessary' +connection\footnote + {I do not propose to define the meaning of this.} +with some set of objects at a previous time. +This belief also, I think, arises in the same way. It is to be +noticed that neither of these beliefs clearly arises, in spite of the +directness which may be claimed for them, out of any one single +experience. In a way analogous to these, the validity of assuming +the Inductive Hypothesis, as applied to a particular class of +objects, appears to me to be justified. + +Our justification for using inductive methods in an argument +%% -----File: 275.png---Folio 264------- +about numbers arises out of our perceiving directly, when we +understand the meaning of a number, that they are of the required +character.\footnote + {Since numbers are logical entities, it may be thought less unorthodox to + make such an assumption in their case.} +And when we perceive the nature of our +phenomenal experiences, we have a direct assurance that in their +case also the assumption is legitimate. We are capable, that +is to say, of direct synthetic knowledge about the nature +of the objects of our experience. On the other hand, there +may be some kinds of objects, about which we have no such +assurance and to which inductive methods are not reasonably +applicable. It may be the case that some metaphysical questions +are of this character and that those philosophers have been right +who have refused to apply empirical methods to them. + +\Paragraph{14.} I do not pretend that I have given any perfectly adequate +reason for accepting the theory I have expounded, or any such +theory. The Inductive Hypothesis stands in a peculiar position +\index{Inductive hypothesis}% +in that it seems to be neither a self-evident logical axiom nor an +object of direct acquaintance; and yet it is just as difficult, as +though the inductive hypothesis were either of these, to remove +from the organon of thought the inductive method which can +only be based on it or on something like it. + +As long as the theory of knowledge is so imperfectly +understood as now, and leaves us so uncertain about the grounds +of many of our firmest convictions, it would be absurd to +confess to a special scepticism about this one. I do not think +that the foregoing argument has disclosed a reason for such +scepticism. We need not lay aside the belief that this conviction +gets its invincible certainty from some valid principle darkly +present to our minds, even though it still eludes the peering +eyes of philosophy. +%% -----File: 276.png---Folio 265------- +\index{Bacon|ifoll}% +\index{Ellis, Leslie!Bacon@{and Bacon}|inote}% +\index{Mill, and inductive correlations!induction@{and induction}|ifoll}% +\index{Newton, and induction!Bacon@{and Bacon}}% +\index{Spedding and Ellis and Bacon|inote}% + + +\Chapter{XXIII}{Some Historical Notes on Induction} + +\Paragraph{1.} \First{The} number of books, which deal with inductive\footnote + {See note at the end of this chapter on ``The Use of the Term \emph{Induction}.''} +theory, is +extraordinarily small. It is usual to associate the subject with +the names of Bacon, Hume, and Mill. In spite of the modern +\index{Hume!Induction@{and Induction}}% +tendency to depreciate the first and the last of these, they are the +principal names, I think, with which the history of induction +ought to be associated. The next place is held by Laplace and +\index{Laplace!Induction@{and Induction}}% +Jevons. Amongst contemporary logicians there is an almost +\index{Jevons!Induction@{and Induction}}% +complete absence of constructive theory, and they content +themselves for the most part with the easy task of criticising +Mill, or with the more difficult one of following him. + +That the inductive theories of Bacon and of Mill are full of +errors and even of absurdities, is, of course, a commonplace of +criticism. But when we ignore details, it becomes clear that they +were really attempting to disentangle the essential issues. We +depreciate them partly, perhaps, as a reaction from the view once +held that they helped the progress of scientific discovery. For +it is not plausible to suppose that Newton owed anything to Bacon, +or Darwin to Mill. But with the logical problem their minds +\index{Darwin!Mill@{and Mill}}% +were truly occupied, and in the history of logical theory they +should always be important. + +It is true, nevertheless, that the advancement of science was +the main object which Bacon himself, though not Mill, believed +that his philosophy would promote. The \textit{Great Instauration} was +intended to promulgate an actual method of discovery entirely +different from any which had been previously known.\footnote + {He speaks of himself as being ``in hac re plane protopirus, et vestigia + nullius sequutus''; and in the \textit{Praefatio Generalis} he compares his method to + the mariner's compass, until the discovery of which no wide sea could be + crossed (see Spedding and Ellis, vol.~i.\ p.~24).} +It did +%% -----File: 277.png---Folio 266------- +\index{Ellis, Leslie!Bacon@{and Bacon}|inote}% +\index{Macaulay and Bacon}% +\index{Spedding and Ellis and Bacon|inote}% +not do this, and against such pretensions Macaulay's well-known +essay was not unjustly directed. Mill, however, expressly disclaimed +in his preface any other object than to classify and +generalise the practices ``conformed to by accurate thinkers in +their scientific inquiries.'' Whereas Bacon offered rules and +demonstrations, hitherto unknown, with which any man could +solve all the problems of science by taking pains, Mill admitted +that ``in the existing state of the cultivation of the sciences, +there would be a very strong presumption against any one +who should imagine that he had effected a revolution in the +theory of the investigation of truth, or added any fundamentally +new process to the practice of it.'' + +\Paragraph{2.} The theories of both seem to me to have been injured, +though in different degrees, by a failure to keep quite distinct +the three objects: (1)~of helping the scientist, (2)~of explaining +and analysing his practice, and (3)~of justifying it. Bacon was +really interested in the second as well as in the first, and was +led to some of his methods by reflecting upon what distinguished +good arguments from bad in actual investigations. To logicians +his methods were as new as he claimed, but they had their +origin, nevertheless, in the commonest inferences of science and +daily life. But his main preoccupation was with the first, which +did injury to his treatment of the third. He himself became +aware as the work progressed that, in his anxiety to provide +an infallible mode of discovery, he had put forth more than he +would ever be able to justify.\footnote + {This view is taken in the edition of James Spedding and Leslie Ellis. + Their introductions to Bacon's philosophical works seem to me to be very greatly + superior to the accounts to be found elsewhere. They make intelligible, what + seems, according to other commentaries, fanciful and without sense or reason.} +His own mind grew doubtful, +and the most critical parts of the description of the new method +were never written. No one who has reflected much upon Induction +need find it difficult to understand the progress and +development of Bacon's thoughts. To the philosopher who first +distinguished some of the complexities of empirical proof in a +generalised, and not merely a particular, form, the prospects of +systematising these methods must have seemed extraordinarily +hopeful. The first investigator could not have anticipated that +Induction, in spite of its apparent certainty, would prove so +elusive to analysis. + +Mill also was led, in a not dissimilar way, to attempt a too +%% -----File: 278.png---Folio 267------- +\index{Mill, and inductive correlations!plurality of causes@{and plurality of causes}|inote}% +\index{Plurality of causes and Mill}% +simple treatment, and, in seeking for ease and certainty, to +\index{Certainty!Bacon@{and Bacon}}% +treat far too lightly the problem of justifying what he had +claimed. Mill shirks, almost openly, the difficulties; and scarcely +attempts to disguise from himself or his readers that he grounds +induction upon a circular argument. + +\Paragraph{3.} Some of the most characteristic errors both of Bacon and +of Mill arise, I think, out of a misapprehension, which it has been +a principal object of this book to correct. Both believed, without +hesitation it seems, that induction is capable of establishing a +conclusion which is absolutely certain, and that an argument +is invalid if the generalisation, which it supports, admits of +exceptions in fact. ``Absolute certainty,'' says Leslie Ellis,\footnote + {\textit{Op.~cit.}\ vol.~i.\ p.~23.} ``is +one of the distinguishing characters of the Baconian induction.'' +It was, in this respect, mainly that it improved upon the older +induction \textit{per enumerationem simplicem}. ``The induction which +the logicians speak of,'' Bacon argues in the \textit{Advancement of +Learning}, ``is utterly vicious and incompetent\ldots. For to conclude +upon an enumeration of particulars, without instance +contradictory, is no conclusion but a conjecture.'' The conclusions +of the new method, unlike those of the old, are not liable to be +upset by further experience. In the attempt to justify these +claims and to obtain demonstrative methods, it was necessary +to introduce assumptions for which there was no warrant. + +Precisely similar claims were made by Mill, although there +are passages in which he abates them,\footnote + {When he deals with Plurality of Causes, for instance.} +for his own rules of procedure. +An induction has no validity, according to him as +according to Bacon, unless it is absolutely certain. The following +passage\footnote + {Bk.~iii.\ chap.~iii.~3 (the italics are mine).} +is significant of the spirit in which the subject +was approached by him: ``Let us compare a few cases +of incorrect inductions with others which are acknowledged +to be legitimate. Some, we know, which were believed for +centuries to be correct, were nevertheless incorrect. \emph{That all +swans are white, cannot have been a good induction, since the +conclusion has turned out erroneous.} The experience, however, on +which the conclusion rested was genuine.'' Mill has not justly +apprehended the relativity of all inductive arguments to the +evidence, nor the element of uncertainty which is present, more +%% -----File: 279.png---Folio 268------- +\index{Analogy, principle of!Bacon@{and Bacon}}% +\index{Mill, and inductive correlations!probability@{and probability}|inote}% +or less, in all the generalisations which they support.\footnote + {This misapprehension may be connected with Mill's complete failure to + grasp with any kind of thoroughness the nature and importance of the theory of + probability. The treatment of this topic in the \textit{System of Logic} is exceedingly + bad. His understanding of the subject was, indeed, markedly inferior to the + best thought of his own time.} +Mill's +methods would yield certainty, if they were correct, just as +Bacon's would. It is the necessity, to which Mill had subjected +himself, of obtaining certainty that occasions their want of +reality. Bacon and Mill both assume that experiment can +shape and analyse the evidence in a manner and to an extent +which is not in fact possible. In the aims and expectations with +which they attempt to solve the inductive problem, there is on +fundamental points an unexpectedly close resemblance \DPtypo{beween}{between} +them. + +\Paragraph{4.} Turning from these general criticisms to points of greater +detail, we find that the line of thought pursued by Mill was +essentially the same as that which had been pursued by Bacon, +and, also, that the argument of the preceding chapters is, in +spite of some real differences, a development of the same fundamental +ideas which underlie, as it seems to me, the theories of +Mill and Bacon alike. + +We have seen that all empirical arguments require an initial +probability derived from analogy, and that this initial probability +may be raised towards certainty by means of pure induction +or the multiplication of instances. In some arguments we depend +mainly upon analogy, and the initial probability obtained by +means of it (with the assistance, as a rule, of previous knowledge) +is so large that numerous instances are not required. In other +arguments pure induction predominates. As science advances +and the body of pre-existing knowledge is increased, we depend +increasingly upon analogy; and only at the earlier stages of our +investigations is it necessary to rely, for the greater part of our +support, upon the multiplication of instances. Bacon's great +achievement, in the history of logical theory, lay in his being the +first logician to recognise the importance of methodical analogy +to scientific argument and the dependence upon it of most well-established +conclusions. The \textit{Novum Organum} is mainly concerned +with explaining methodical ways of increasing what I +have termed the Positive and Negative Analogies, and of avoiding +false Analogies. The use of exclusions and rejections, to which +%% -----File: 280.png---Folio 269------- +\index{Ellis, Leslie!Bacon@{and Bacon}|inote}% +\index{Mill, and inductive correlations!pure induction@{and pure induction}}% +Bacon attached supreme importance, and which he held to constitute +\index{Bacon!tables of}% +the essential superiority of his method over those which +preceded it, entirely consists in the determination of what characters +(or natures as he would call them) belong to the positive +and negative analogies respectively. The first two tables with +which the investigation begins are, first, the table \textit{essentiae et +praesentiae}, which contains all known instances in which the +given nature is present, and, second, the table \textit{declinationis sive +absentiae in proximo}, which contains instances corresponding in +each case to those of the first table, but in which, notwithstanding +this correspondence, the given nature is absent.\footnote + {Ellis, vol.~i.\ p.~33.} +The doctrine +of prerogative instances is concerned no less plainly with the +methodical determination of Analogy. And the doctrine of +idols is expounded for the avoidance of \emph{false} analogies, standing, +he says, in the same relation to the interpretation of Nature, as +the doctrine of fallacies to ordinary logic.\footnote + {Ellis, vol.~i.\ p.~89.} +Bacon's error lay +in supposing that, because these methods were new to logic, they +were therefore new to practice. He exaggerated also their precision +and their certainty; and he underestimated the importance +of pure induction. But there was, at bottom, nothing about +his rules impracticable or fantastic, or indeed unusual. + +\Paragraph{5.} Almost the whole of the preceding paragraph is equally +applicable to Mill. He agreed with Bacon in depreciating the +part played in scientific inquiry by pure induction, and in +emphasising the importance of analogy to all systematic investigators. +But he saw further than Bacon in allowing for the +Plurality of Causes, and in admitting that an element of pure +induction was therefore made necessary. ``The Plurality of +Causes,'' he says,\footnote + {Book~iv.\ chap.~x.~2.} +``is the only reason why mere number of instances +is of any importance in inductive inquiry. The tendency +of unscientific inquirers is to rely too much on number, without +analysing the instances\ldots. Most people hold their conclusions +with a degree of assurance proportioned to the mere \emph{mass} of the +experience on which they appear to rest; not considering that +by the addition of instances to instances, all of the same kind, +that is, differing from one another only in points already recognised +as immaterial, nothing whatever is added to the evidence of +%% -----File: 281.png---Folio 270------- +\index{Mill, and inductive correlations!methods of}% +\index{Uniformity of Nature, Law of!Mill@{and Mill}}% +the conclusion. A single instance eliminating some antecedent +which existed in all the other cases, is of more value than the +greatest multitude of instances which are reckoned by their +number alone.'' Mill did not see, however, that our knowledge +of the instances is seldom complete, and that new instances, which +are not known to differ from the former in material respects, may +add, nevertheless, to the negative analogy, and that the multiplication +of them may, for this reason, strengthen the evidence. +It is easy to see that his methods of Agreement and Difference +closely resemble Bacon's, and aim, like Bacon's, at the determination +\index{Bacon!limited variety@{and limited variety}}% +of the Positive and Negative Analogies. By allowing +for Plurality of Causes Mill advanced beyond Bacon. But he +was pursuing the same line of thought which alike led to Bacon's +rules and has been developed in the chapters of this book. +Like Bacon, however, he exaggerated the precision with which +his canons of inquiry could be used in practice. + +\Paragraph{6.} No more need be said respecting method and analysis. +But in both writers the exposition of method is closely intermingled +with attempts to justify it. There is nothing in Bacon +which at all corresponds to Mill's appeals to Causation or to the +Uniformity of Nature, and, when they seek for the ground of +induction, there is much that is peculiar to each writer. It is +my purpose, however, to consider in this place the details common +to both, which seem to me to be important and which exemplify +the only line of investigation which seems likely to be fruitful; +and I shall pursue no further, therefore, their numerous points +of difference. + +The attempt, which I have made to justify the initial probability +which Analogy seems to supply, primarily depends upon +a certain limitation of independent variety and upon the derivation +of all the properties of any given object from a limited +number of primary characters. In the same way I have supposed +that the number of primary characters which are capable of +producing a given property is also limited. And I have argued +that it is not easy to see how a finite probability is to be obtained +unless we have in each case some such limitation in the number +of the ultimate alternatives. + +It was in a manner which bears fundamental resemblances +to this that Bacon endeavoured to demonstrate the cogency of +his method. He considers, he says, ``the simple forms or difference +%% -----File: 282.png---Folio 271------- +\index{Mill, and inductive correlations!limited variety@{and limited variety}}% +of things which are few in number, and the degrees and +co-ordinations whereof make all this variety,'' And in \textit{Valerius +Terminus} he argues ``that every particular that worketh any +effect is a thing compounded more or less of diverse single natures, +more manifest and more obscure, and that it appeareth not to +which of the natures the effect is to be ascribed.''\footnote + {Quoted by Ellis, vol.~i.\ p.~41.\index{Ellis, Leslie!Bacon@{and Bacon}|inote}} +It is indeed +essential to the method of exclusions that the matter to which it +is applied should be somehow resolvable into a finite number of +elements. But this assumption is not peculiar, I think, to +Bacon's method, and is involved, in some form or other, in every +argument from Analogy. In making it Bacon was initiating, +perhaps obscurely, the modern conception of a finite number of +laws of nature out of the combinations of which the almost boundless +variety of experience ultimately arises. Bacon's error was +double and lay in supposing, first, that these distinct elements +lie upon the surface and consist in visible characters, and second, +that their natures are, or easily can be, known to us, although +the part of the \textit{Instauration}, in which the manner of conceiving +simple natures was to be explained, he never wrote. These +beliefs falsely simplified the problem as he saw it, and led him +to exaggerate the ease, certainty, and fruitfulness of the new +method. But the view that it is possible to reduce all the +phenomena of the universe to combinations of a limited number +of simple elements---which is, according to Ellis,\footnote + {Vol.~i.\ p.~28.} +the central +point of Bacon's whole system---was a real contribution to philosophy. + +\Paragraph{7.} The assumption that every event can be analysed into a +limited number of ultimate elements, is never, so far as I am +aware, explicitly avowed by Mill. But he makes it in almost +every chapter, and it underlies, throughout, his mode of procedure. +His methods and arguments would fail immediately, if we were +to suppose that phenomena of infinite complexity, due to an +infinite number of independent elements, were in question, or +if an infinite plurality of causes had to be allowed for. + +In distinguishing, therefore, analogy from pure induction, +and in justifying it by the assumption of a \emph{limited} complexity in +the problems which we investigate, I am, I think, pursuing, with +numerous differences, the line of thought which Bacon first +%% -----File: 283.png---Folio 272------- +\index{Analogy, principle of!Leibniz@{and Leibniz}}% +\index{Couturat|inote}% +pursued and which Mill popularised. The method of treatment +is dissimilar, but the subject-matter and the underlying beliefs +are, in each case, the same. + +\Paragraph{8.} Between Bacon and Mill came Hume. Hume's sceptical +\index{Hume!Induction@{and Induction}}% +criticisms are usually associated with causality; but argument +by induction---inference from past particulars to future generalisations---was +the real object of his attack. Hume showed, not that +inductive methods were false, but that their validity had never +been established and that all possible lines of proof seemed +equally unpromising. The \emph{full} force of Hume's attack and the +nature of the difficulties which it brought to light were never +appreciated by Mill, and he makes no adequate attempt to +deal with them. Hume's statement of the case against induction +has never been improved upon; and the successive attempts +of philosophers, led by Kant, to discover a transcendental solution +\index{Kant!Hume@{and Hume}}% +have prevented them from meeting the hostile arguments on +their own ground and from finding a solution along lines which +might, conceivably, have satisfied Hume himself. + +\Paragraph{9.} It would not be just here to pass by entirely the name +of the great Leibniz, who, wiser in correspondence and fragmentary +\index{Leibniz!Induction@{and Induction}}% +projects than in completed discourses, has left to us +sufficient indications that his private reflections on this subject +were much in advance of his contemporaries'. He distinguished +three degrees of conviction amongst opinions, logical certainty +\index{Certainty!Leibniz@{and Leibniz}}% +(or, as we should say, propositions known to be formally true), +physical certainty which is only logical probability, of which a +well-established induction, as that man is a biped, is the type, +and physical probability (or, as we should say, an inductive +correlation), as for example that the south is a rainy quarter.\footnote + {Couturat, \textit{Opuscules et fragments inédits de Leibniz}, p.~232.} +He condemned generalisations based on mere repetition of +instances, which he declared to be without logical value, and he +insisted on the importance of \emph{Analogy} as the basis of a valid +induction.\footnote + {Couturat, \textit{La Logique de Leibniz d'après des documents inédits}, pp\DPtypo{}{.}~262,~267.} +He regarded a hypothesis as more probable in +proportion to its \emph{simplicity} and its \emph{power}, that is to say, to the +number of the phenomena it would explain and the fewness of +the assumptions it involved. In particular a power of accurate +prediction and of explaining phenomena or experiments previously +%% -----File: 284.png---Folio 273------- +\index{Analogy, principle of!Jevons@{and Jevons}}% +untried is a just ground of secure confidence, of which +he cites as a nearly perfect example the key to a cryptogram.\footnote + {Letter to Conring, 19th~March 1678.} + +\Paragraph{10.} Whewell and Jevons furnished logicians with a storehouse +\index{Jevons!Induction@{and Induction}}% +of examples derived from the practice of scientists. +Jevons, partly anticipated by Laplace, made an important +\index{Laplace!Induction@{and Induction}}% +advance when he emphasised the close relation between +Induction and Probability. Combining insight and error, he +spoilt brilliant suggestions by erratic and atrocious arguments. +His application of Inverse Probability to the inductive problem +is crude and fallacious, but the idea which underlies it is +substantially good. He, too, made explicit the element of +Analogy, which Mill, though he constantly employed it, had +seldom called by its right name. There are few books, so +superficial in argument yet suggesting so much truth, as Jevons's +\textit{Principles of Science}. + +\Paragraph{11.} Modern text-books on Logic all contain their chapters on +Induction, but contribute little to the subject. Their recognition +of Mill's inadequacy renders their exposition, which, in spite +of criticisms, is generally along his lines, nerveless and confused. +Where Mill is clear and offers a solution, they, confusedly +criticising, must withhold one. The best of them, Sigwart and +\index{Sigwart!induction@{and induction}}% +Venn, contain criticism and discussion which is interesting, but +\index{Venn!induction@{and induction}}% +constructive theory is lacking. Hitherto Hume has been master, +only to be refuted in the manner of Diogenes or Dr.~Johnson. +%% -----File: 285.png---Folio 274------- + + +%[** TN: Omitting italic smallcaps "Induction"] +\Notes{i}{Notes on Part III}{On the Use of the Term Induction} +\index{Induction@{`\textit{Induction}'}}% + +\Paragraph{1.} \First{Induction} is in origin a translation of the Aristotelian \textgreek{>epagwg'h}. +This term was used by Aristotle in two quite distinct senses---first, +\index{Aristotle!induction@{and induction}}% +and principally, for the process by which the observation of particular +instances, in which an abstract notion is exemplified, enables us to +realise and comprehend the abstraction itself; secondly, for the type +of argument in which we generalise after the complete enumeration +and assertion of \emph{all} the particulars which the generalisation embraces. +From this second sense it was sometimes extended to cases in which +we generalise after an \emph{incomplete} enumeration. In post-Aristotelian +writers the induction \textit{per enumerationem simplicem} approximates to +induction in Aristotle's second sense, as the number of instances is +increased. To Bacon, therefore, ``the induction of which the logicians +speak'' meant a method of argument by multiplication of instances. +He himself deliberately extended the use of the term so as to cover +all the systematic processes of empirical generalisation. But he +also used it, in a manner closely corresponding to Aristotle's \emph{first} use, +for the process of forming scientific conceptions and correct notions +of ``simple natures.''\footnote + {See Ellis's edition of Bacon's \textit{Works}, vol.~i.\ p.~37. On the first occasion + \index{Ellis, Leslie!Bacon@{and Bacon}|inote}% + on which Induction is mentioned in the \textit{Novum Organum}, it is used in this + secondary sense.} + +\Paragraph{2.} The modern use of the term is derived from Bacon's. Mill +defines it as ``the operation of discovering and proving general +propositions.'' His philosophical system required that he should +define it as widely as this; but the term has really been used, both +by him and by other logicians, in a narrower sense, so as to cover +those methods of proving general propositions, which we call empirical, +and so as to exclude generalisations, such as those of mathematics, +which have been proved formally. Jevons was led, partly by the +\index{Jevons!Induction@{and Induction}}% +linguistic resemblance, partly because in the one case we proceed +from the particular to the general and in the other from the general +to the particular, to define Induction as the inverse process of +Deduction. In contemporary logic Mill's use prevails; but there +%% -----File: 286.png---Folio 275------- +\index{Cause@{`\textit{Cause}'}}% +\index{Cournot, and frequency theory!causality@{and causality}}% +is, at the same time, a suggestion---arising from earlier usage, and +because Bacon and Mill never quite freed themselves from it---of +argument by mere multiplication of instances. I have thought it +best, therefore, to use the term \emph{pure induction} to describe arguments +which are based upon the \emph{number} of instances, and to use \emph{induction} +itself for all those types of arguments which combine, in one form or +another, pure induction with analogy.\\ + + +%[** TN: Omitting italic smallcaps "Cause"] +\NoteSec{ii}{On the Use of the Term Cause} + +\Pagelabel{275}% +\Paragraph{1.} Throughout the preceding argument, as well as in \Partref{II}., +I have been able to avoid the metaphysical difficulties which surround +the true meaning of \emph{cause}. It was not necessary that I should +inquire whether I meant by \emph{causal} connection an invariable connection +in fact merely, or whether some more intimate relation was +involved. It has also been convenient to speak of causal relations +between objects which do not strictly stand in the position of cause +and effect, and even to speak of \emph{a probable cause}, where there is no +implication of necessity and where the antecedents will sometimes +lead to particular consequents and sometimes will not. In making +this use of the term, I have followed a practice not uncommon amongst +writers on probability, who constantly use the term \emph{cause}, where +\emph{hypothesis} might seem more appropriate.\footnote + {Cf.\ Czuber, \textit{Wahrscheinlichkeitsrechnung}, p.~139. In dealing with Inverse +\index{Czuber!cause@{and `\textit{cause}'}|inote}% + Probability Czuber explains that he means by \emph{possible cause} the various \textit{Bedingungskomplexe} + from which the cause \emph{can} result.} + +One is led, almost inevitably, to use `cause' more widely than +`sufficient cause' or than `necessary cause,' because, the necessary +causation of particulars by particulars being rarely apparent to us, +the strict sense of the term has little utility. Those antecedent +circumstances, which we are usually content to accept as causes, are +only so in strictness under a favourable conjunction of innumerable +other influences. + +\Paragraph{2.} As our knowledge is partial, there is constantly, in our use +of the term \emph{cause}, some reference implied or expressed to a limited +body of knowledge. It is clear that, whether or not, as Cournot\footnote + {See \Chapref{XXIV}. §\;3.} +maintains, there are such things as independent series in the order +of causation, there is often a sense in which we may hold that there +is a closer intimacy between some series than between others. This +intimacy is relative, I think, to particular information, which is +actually known to us, or which is within our reach. It will be useful, +therefore, to give precise definitions of these wider senses in which +it is often convenient to use the expression \emph{cause}. +%% -----File: 287.png---Folio 276------- +\index{Causality}% +\index{Kries, von!knowledge@{and knowledge}}% +\index{Proposition, characterisation of!existential}% +\index{Uniformity of Nature, Law of}% + +We must first distinguish between assertions of law and assertions +of fact, or, in the terminology of Von~Kries,\footnote + {\textit{Die Principien der Wahrscheinlichkeitsrechnung}, p.~86.} +between nomologic and +ontologic knowledge. It may be convenient in dealing with some +\index{Knowledge!homologic and ontologic}% +questions to frame this distinction with reference to the special +circumstances. But the distinction generally applicable is between +propositions which contain no reference to \emph{particular} moments of +time, and existential propositions which cannot be stated without +reference to specific points in the time series. The Principle of the +Uniformity of Nature amounts to the assertion that natural laws +are all, in this sense, timeless. We may, therefore, divide our \textit{data} +into two portions $k$~and~$l$, such that $k$~denotes our formal and +nomologic evidence, consisting of propositions whose predication +does not involve a particular time reference, and $l$~denotes the +existential or ontologic propositions. + +\Paragraph{3.} Let us now suppose that we are investigating two existential +propositions $a$~and~$b$, which refer two events $A$~and~$B$ to particular +moments of time, and that $A$~is referred to moments which are all +prior to those at which $B$~occurred. What various meanings can we +give to the assertion that $A$~and~$B$ are \emph{causally connected}? + +(i.) If $b/ak=1$, $A$~is a sufficient cause of~$B$. In this case $A$~is a +cause of~$B$ in the strictest sense. $b$~can be \emph{inferred} from~$a$, and no +additional knowledge consistent with~$k$ can invalidate this. + +(ii.) If $b/\bar{a}k=0$, $A$~is a necessary cause of~$B$. + +(iii.) If $k$~includes all the laws of the existent universe, then $A$~is +\emph{not} a sufficient cause of~$B$ unless $b/ak=1$. The Law of Causation, +therefore, which states that every existent has to some other previous +existent the relation of effect to sufficient cause, is equivalent to the +proposition that, if $k$~is the body of natural law, then, if $b$~is true, +there is always another true proposition~$a$, which asserts existences +prior to~$B$, such that $b/ak=1$. No use has been made so far of our +existential knowledge~$l$, which is irrelevant to the definitions preceding. + +(iv.) If $b/akl=1$ and~$b/kl\neq 1$, $A$~is a sufficient cause of~$B$ under +conditions~$l$. + +(v.) If $b/\bar{a}kl=0$ and~$b/kl\neq 0$, $A$~is a necessary cause of~$B$ under +conditions~$l$. + +(vi.) If there is any existential proposition~$h$ such that $b/ahk=1$ +and~$b/hk\neq 1$, $A$~is, relative to~$k$, a possible sufficient cause of~$B$. + +(vii.) If there is an existential proposition~$h$ such that $b/\bar{a}hk=0$ +and~$b/hk\neq 0$, $A$~is, relative to~$k$, a possible necessary cause of~$B$. + +(viii.) If $b/ahkl=1$, $b/hk\neq 1$, and~$h/akl\neq 0$, $A$~is, relative to~$k$, +a possible sufficient cause of~$B$ under conditions~$l$. + +(ix.) If $b/\bar{a}hkl=0$, $b/hkl\neq0$, $h/\bar{a}kl=0$, and~$h/akl\neq 0$, $A$~is, +relative to~$k$, a possible necessary cause of~$B$ under conditions~$l$. +%% -----File: 288.png---Folio 277------- +Thus an event is a possible necessary cause of another, relative to +given nomologic data, if circumstances can arise, not inconsistent +with our existential data, in which the first event will be indispensable +if the second is to occur. + +(x.) Two events are \emph{causally independent} if no part of either is, +relative to our nomologic data, a possible cause of any part of the +other under the conditions of our existential knowledge. The greater +the scope of our existential knowledge, the greater is the likelihood +of our being able to pronounce events causally dependent or independent. + +\Paragraph{4.} These definitions preserve the distinction between `causally +independent' and `independent for probability,'---the distinction +between \textit{causa essendi} and \textit{causa cognoscendi}. If $b/ahkl\neq b/\bar{a}hkl$, +where $a$~and~$b$ may be any propositions whatever and are not limited +as they were in the causal definitions, we have `dependence for +probability,' and $a$~is a \textit{causa cognoscendi} for~$b$, relative to data~$kl$. +If $a$~and~$b$ are causally dependent, according to definition~(x.), $b$~is a +possible \textit{causa essendi}, relative to data~$kl$. + +But, after all, the essential relation is that of `independence for +probability.' We wish to know whether knowledge of one fact +throws light \emph{of any kind} upon the likelihood of another. The theory +of causality is only important because it is thought that by means of +its assumptions light \emph{can} be thrown by the experience of one phenomenon +upon the expectation of another. +%% -----File: 289.png---Folio 278------- +\index{Psychical Research|ifoll}% +%[Blank Page] +%% -----File: 290.png---Folio 279------- + + +\Part[Philosophical Applications]{IV}{Some Philosophical Applications of +Probability} +%% -----File: 291.png---Folio 280------- +%[Blank Page] +%% -----File: 292.png---Folio 281------- +\index{Chance, objective}% +\index{Probability, and relevant knowledge!objective relation of}% + + +\Chapter{XXIV}{The Meanings of Objective Chance, and of Randomness} +\index{Randomness}% + +\Paragraph{1.} \First{Many} important differences of opinion in the treatment of +Probability have been due to confusion or vagueness as to +what is meant by Randomness and by Objective Chance, as +distinguished from what, for the purposes of this chapter, may be +termed Subjective Probability. It is agreed that there is a sort +of Probability which depends upon knowledge and ignorance, and +\index{Knowledge!ignorance@{and ignorance}}% +is relative, in some manner, to the mind of the subject; but it is +supposed that there is also a more objective Probability which +is not thus dependent, or less completely so, though precisely +what this conception stands for is not plain. The relation of +Randomness to the other concepts is also obscure. The problem +of clearing up these distinctions is of importance if we are to +criticise certain schools of opinion intelligently, as well as to the +treatment of the foundations of Statistical Inference which is to +be attempted in \Partref{V}. + +There are at least three distinct issues to be kept apart. There +is the antithesis between knowledge and ignorance, between +events, that is to say, which we have some reason to expect, and +events which we have no reason to expect, which gives rise to +the theory of subjective probability and subjective chance; and, +connected with this, the distinction between `random' selection +and `biassed' selection. There are next objective probability and +objective chance, which are as yet obscure, but which are commonly +held to arise out of the antithesis between `cause' and +`chance,' between events, that is to say, which are causally connected +and events which are not causally connected. And there +is, lastly, the antithesis between chance and design, between +`blind causes' and `final causes,' where we oppose a `chance' +%% -----File: 293.png---Folio 282------- +\index{Cournot, and frequency theory!chance@{and chance}}% +\index{Spinoza|inote}% +event to one, part of whose cause is a volition following on a +conscious desire for the event.\footnote + {This is discussed in \Chapref{XXV}. §\;4.} + +\Paragraph{2.} The method of this treatise has been to regard subjective +probability as fundamental and to treat all other relevant conceptions +as derivative from this. That there is such a thing as +probability in this sense has been admitted by all sensible philosophers +since the middle of the eighteenth century at least.\footnote + {D'Alembert, collecting (largely from Hume, many passages being translated +\index{D'Alembert!chance@{and chance}}% +\index{Hume!chance@{and chance}}% + almost \textit{verbatim}) in the \textit{Encyclopédie méthodique} the most up-to-date + commonplaces of the subject, found it natural to write: ``Il n'y a point de + hasard à proprement parler; mais il y a son équivalent: l'ignorance, où nous + sommes des vraies causes des événemens, a sur notre esprit l'influence qu'on + suppose au hasard.'' Compare also the sentences from Spinoza quoted on + p.~117 above.} +But +there is also, many writers have supposed, something else which +may be fitly described as objective probability; and there is, +besides, a long tradition in favour of the view that it is this (whatever +it may be) which is logically and philosophically important, +subjective probability being a vague and mainly psychological +conception about which there is very little to be said. + +The distinction exists already in Hume: ``Probability is of +two kinds, either when the object is really in itself uncertain, +and to be determined by chance; or when, though the object be +already certain, yet 'tis uncertain to our judgment, which finds +a number of proofs on each side of the question.''\footnote + {\textit{A Treatise of Human Nature}, Book~ii.\ part~iii.\ section~ix.} +But the +distinction is not elucidated, and one can only infer from other +passages that Hume did not intend to imply in this passage the +existence of objective chance in a sense contradictory to a determinist +theory of the Universe. In Condorcet all is confused; and +\index{Condorcet!chance@{and chance}}% +in Laplace nearly all. In the nineteenth century the distinction +\index{Laplace!chance@{and chance}}% +begins to grow explicit in the writings of Cournot. ``Les explications +que j'ai données\ldots,'' he writes in the preface to his +\textit{Exposition}, ``sur le double sens du mot de probabilité, qui +tantôt se rapporte à une certaine mesure de nos connaissances, et +tantôt à une mesure de la possibilité des choses, indépendamment +de la connaissance que nous en avons: ces explications, dis-je, +me semblent propres à resoudre les difficultés qui ont rendu +jusqu'ici suspecte à de bons esprits toute la théorie de la probabilité +mathématique.'' It will be worth while to pause for a +moment to consider the ideas of Cournot. +%% -----File: 294.png---Folio 283------- +\index{Chance, objective!Couturat on}% +\index{Cournot, and frequency theory!chance@{and chance}}% +\index{Definitions!de la Placette, Jean, and chance}% +\index{Independence, for knowledge!chance@{and chance}}% +\index{Series of probabilities!independent}% + +\Paragraph{3.} Cournot, while admitting that there is such a thing as subjective +chance, was concerned to dispute the opinion that chance +is \emph{merely} the offspring of ignorance, saying that in this case +``le calcul des chances'' is merely ``un calcul des illusions.'' +The chance, upon which ``le calcul des chances'' is based, is +something different, and depends, according to him, on the combination +or convergence of phenomena belonging to \emph{independent} +series. By ``independent series'' he means series of phenomena +which develop as parallel or successive series without any causal +interdependence or link of solidarity whatever.\footnote + {``Le mot hasard,'' Cournot writes in his \textit{Essai sur les fondemenis de nos + connaissances}, ``n'indique pas une cause substantielle, mais une idée: cette idée + est celle de la combinaison entre plusieurs séries de causes ou de faits qui se + développent chacun dans sa série propre, indépendamment les uns des autres.'' + This is very like the definition given by Jean de~la~Placette in his \textit{Traité des jeux + de hasard}, to which Cournot refers: ``Pour moi, je suis persuadé que le hasard + renferme quelque chose de réal et de positif, savoir un concours de deux ou + plusieurs événements contingents, chacun desquels a ses causes, mais en sorte + que leur concours n'en a aucune que l'on connaisse.''} +No one, he +says by way of example, seriously believes that in striking the +ground with his foot he puts out the navigator in the Antipodes, +or disturbs the system of Jupiter's satellites. Separate trains of +events, that is to say, have been set going by distinct initial acts of +creation, so to speak.\footnote + {\textit{Essai sur les fondements de nos connaissances}, i.~134: ``La nature ne se + gouverne pas par une loi unique\ldots\ ses lois ne sont pas toutes dérivées les + unes des autres, on dérivées toutes d'une loi supérieure par une nécessité purement + logique\ldots\ nous devons les concevoir au contraire comme ayant pu + étre décrétées séparément d'une infinité de manières.''} +Every event is causally connected with +previous events belonging to its own series, but it cannot be +modified by contact with events belonging to a different series. +A `chance' event is a complex due to the concurrence in time +or place of events belonging to causally independent series. + +This theory, as it stands, is evidently unsatisfactory. Even +if there are series of phenomena which are independent in Cournot's +sense, it is not clear how we can know which they are, or how we +can set up a calculus which presumes an acquaintance with them. +Just as it is likely that we are all cousins if we go back far enough, +so there may be, after all, remote relationships between ourselves +and Jupiter. A remote connection or a reaction quantitatively +small is a matter of degree and not by any means the same thing +as absolute independence. Nevertheless Cournot has contributed +something, I think, to the stock of our ideas. He has +%% -----File: 295.png---Folio 284------- +\index{Chance, objective!Poincaré on}% +\index{Chance, objective!Condorcet on}% +\index{Darbon, A., and Cournot}% +\index{Kries, von!Cournot@{and Cournot}|inote}% +\index{Poincaré, Henri!chance@{and chance}}% +hinted at, even if he has not disentangled, one of the elements +in a common conception of chance; and of the notion, which he +seems to have in his mind, we must in due course take account.\footnote + {Cournot's work on Probability has been highly praised by authorities as + diverse and distinguished as Boole and Von~Kries, and has been made the +\index{Boole!Cournot@{and Cournot}|inote}% + foundation of a school by some recent French philosophers (see the special + number of the \textit{Revue de métaphysique et de morale}, devoted to Cournot and published + in~1905, and the bibliography at the end of the present volume \textit{passim}). + The best account with which I am acquainted, of Cournot's theory of probability, + is to be found in A.~Darbon's \textit{Le Concept du hasard}. Cournot's philosophy of + the subject is developed, not so much in his \textit{Exposition de la théorie des chances}, + as in later works, especially in his \textit{Essai sur les fondements de nos connaissances}. + Cournot never touched any subject without contributing something to it, but, + on the whole, his work on Probability is, in my opinion, disappointing. No + doubt his \textit{Exposition} is superior to other French text-books of the period, of + which there is so large a variety, and his work, both here and elsewhere, is not + without illuminating ideas: but the philosophical treatment is so confused and + indefinite that it is difficult to make much of it beyond the one specific point + treated above.} + +\Paragraph{4.} In the writings of Condorcet, I have said above, all is confused. +\index{Condorcet!chance@{and chance}}% +But in Bertrand's criticism of him a relevant distinction, +\index{Bertrand!chance@{and chance}}% +though not elucidated, is brought before the mind. ``The +motives for believing,'' wrote Condorcet, ``that, from ten million +white balls mixed with one black, it will not be the black ball +which I shall draw at the first attempt is \emph{of the same kind} as the +motive for believing that the sun will not fail to rise to-morrow.'' +``The assimilation of the two cases,'' Bertrand writes in criticism +of the above,\footnote + {\textit{Calcul des probabilités}, p.~xix.} +``is not legitimate: one of the probabilities is +objective, the other subjective. The probability of drawing +the black ball at the first attempt is~$\frac{1}{10,000,000}$, neither more nor +less. Whoever evaluates it otherwise makes a mistake. The +probability that the sun will rise varies from one mind to another. +A scientist might hold on the basis of a false theory, without being +utterly irrational, that the sun will soon be extinguished; he +would be within his rights, just as Condorcet is within his; both +would exceed their rights in accusing of error those who think +differently.'' Before commenting on this distinction, let us have +before us also some interesting passages by Poincaré. + +\Paragraph{5.} We certainly do not use the term `chance,' Poincaré points +out, as the ancients used it, in opposition to determinism. For +us therefore the natural interpretation of `chance' is subjective,---`Chance +is only the measure of our ignorance. Fortuitous +phenomena are, by definition, those, of the laws of which we are +%% -----File: 296.png---Folio 285------- +ignorant.'' But Poincaré immediately adds: ``Is this definition +very satisfactory? When the first Chaldaean shepherds followed +with their eyes the movements of the stars, they did not yet +know the laws of astronomy, but would they have dreamed of +saying that the stars move by chance? If a modern physicist +is studying a new phenomenon, and if he discovers its law on +Tuesday, would he have said on Monday that the phenomenon +was fortuitous?''\footnote + {\textit{Calcul des probabilités} (2nd~edition), p.~2. This passage also appears in an + article in the \textit{Revue du mois} for 1907 and in the author's \textit{Science et méthode}, of + the English translation of which I have made use above,---at the cost of doing + incomplete justice to Poincaré's most admirable style.} + +There is also another type of case in which ``chance must be +something more than the name we give to our ignorance.'' Among +the phenomena, of the causes of which we are ignorant, there are +some, such as those dealt with by the manager of a life insurance +\index{Insurance}% +company, about which the calculus of probabilities can give real +information. Surely it cannot be thanks to our ignorance, +Poincaré urges, that we are able to arrive at valuable conclusions. +If it were, it would be necessary to answer an inquirer thus: +``You ask me to predict the phenomena that will be produced. +If I had the misfortune to know the laws of these phenomena, I +could not succeed except by inextricable calculations, and I should +have to give up the attempt to answer you; but since I am +fortunate enough to be ignorant of them, I will give you an answer +at once. And, what is more extraordinary still, my answer will +be right.'' The ignorance of the manager of the life insurance +company as to the prospects of life of his individual policy-holders +does not prevent his being able to pay dividends to his +shareholders. + +Both these distinctions seem to be real ones, and Poincaré +proceeds to examine further instances in which we seem to +distinguish \emph{objectively} between events according as they are or +are not due to `chance.' He takes the case of a cone balanced +upon its tip; we know for certain that it will fall, but not on +which side---chance will determine. ``A very small cause which +escapes our notice determines a considerable effect that we cannot +fail to see, and then we say that that effect is due to chance.'' +The weather, and the distribution of the minor planets on the +Zodiac, are analogous instances. And what we term `games of +chance' afford, it has always been recognised, an almost perfect +%% -----File: 297.png---Folio 286------- +\index{Chance, objective|ifoll}% +example. ``It may happen that small differences in the initial +conditions produce very great ones in the final phenomena. A +small error in the former will produce an enormous error in the +latter. Prediction becomes impossible, and we have the fortuitous +phenomenon.'' ``The greatest chance is the birth of a great +man. It is only by chance that the meeting occurs of two genital +cells of different sex that contain precisely, each on its side, the +mysterious elements, the mutual reaction of which is destined +to produce genius\ldots. How little it would have taken to make +the spermatozoid which carried them deviate from its course. +It would have been enough to deflect it a hundredth part of an +inch, and Napoleon would not have been born and the destinies +of a continent changed. No example can give a better comprehension +of the true character of chance.'' + +Poincaré calls attention next to another class of events, which +we commonly assign to `chance,' the distinguishing characteristic +of which seems to be that their causes are very numerous and +complex,---the motions of molecules of gas, the distribution of +drops of rain, the shuffling of a pack of cards, or the errors of +observation. Thirdly there is the type, usually connected with +one of the first two, and specially emphasised, as we have seen +above, by Cournot, in which something comes about through +the concurrence of events which we regard as belonging to distinct +causal trains,---a man is walking along the street and is killed by +the fall of a tile. + +\Paragraph{6.} When we attribute such events, as those illustrated by +Poincaré, to \emph{chance}, we certainly do not mean merely to assert +that we do not know how they arose or that we had no special +reason for anticipating them \textit{à~priori}. So far from this being the +case, we mean to make a definite assertion as to the kind of way +in which they arose;---though exactly what we mean to assert +about them it is extremely difficult to say. + +Now a careful examination of all the cases in which various +writers claim to detect the presence of `objective chance' confirms +the view that `subjective chance,' which is concerned with +knowledge and ignorance, is fundamental, and that so-called +`objective chance,' however important it may turn out to be +from the practical or scientific point of view, is really a special +kind of `subjective chance' and a derivative type of the latter. +For none of the adherents of `objective chance' wish to question +%% -----File: 298.png---Folio 287------- +\index{Chance, objective!definition of}% +the determinist character of natural order; and the possibility +of this objective chance of theirs seems always to depend on the +possibility that a particular kind of knowledge either is ours or +is within our powers and capacity. Let me try to distinguish as +exactly as I can the criterion of objective chance. + +\Paragraph{7.} When we say that an event has happened by chance, we +do not mean that previous to its occurrence the event was, on +the available evidence, very improbable; this may or may not +have been the case. We say, for example, that if a coin falls heads +it is `by chance,' whereas its falling heads is not at all improbable. +The term `by chance' has reference rather to the state of our +information about the concurrence of the event considered and +the event premised. The fall of the coin is a chance event if +our knowledge of the circumstances of the throw is \emph{irrelevant} +to our expectation of the possible alternative results. If the +number of alternatives is very large, then the occurrence of +the event is not only subject to chance but is also very improbable. +In general two events may be said to have a chance +connection, in the subjective sense, when knowledge of the +first is irrelevant to our expectation of the second, and produces +no additional presumption for or against it; when, that is to +say, the probabilities of the propositions asserting them are +\emph{independent} in the sense defined in \Chapref{XII}. §\;8. + +The above definition deals with chance in the widest sense. +What is the \textit{differentia} of the narrower group of cases to which +it is desired to apply the term `\emph{objective} chance'? The occurrence +of an event may be said to be subject to objective chance, +I think, when it is not only a chance event in the above sense, +but when we also have good reason to suppose that the addition +of further knowledge of a given kind, if it were procurable, would +not affect its chance character. We must consider, that is to say, +the probability which is relative not to \emph{actual} knowledge but to +the \emph{whole} of a \emph{certain kind} of knowledge. We may be able to +infer from our evidence that, even with certain kinds of +additions to our knowledge, the connections between the events +would still be subject to chance in the sense just defined, and +we may be able to infer this without actually having the additional +information in question. If, however complete our +knowledge of certain kinds of things might be, there would still +exist independence between the propositions, the conjunction +%% -----File: 299.png---Folio 288------- +\index{Casual@{`\textit{Casual}'}}% +of which we are investigating, then we may say there is an +\emph{objective} sense in which the actual conjunction of these propositions +is due to chance. + +\Paragraph{8.} This is, I think, the right line of inquiry. It remains to +decide, \emph{what} kinds of information must be irrelevant to the +connection, in order that the presence of objective chance may +be established. + +When we attribute a coincidence to objective chance, we +mean not only that we do not actually know a law of connection, +but, speaking roughly, that there is no law of connection to be +known. And when we say that the occurrence of one alternative +rather than another is due to chance, we mean not only +that we know no principle by which to choose between the +alternatives, but also that no such principle is knowable. This +use of the term closely corresponds to what Venn means by the +\index{Venn!chance@{and chance}}% +term `casual': ``We call a coincidence casual, I apprehend, +when we mean to imply that no knowledge of one of the two +\index{Knowledge!homologic and ontologic}% +elements, which we can suppose to be practically attainable, +would enable us to expect the other.''\footnote + {\textit{Logic of Chance}, p.~245.} + +To make this more precise, we must revive our distinction,\footnote + {See \Partref{III}. Note~(ii.)\ §\;2, \Pageref{275}.} +between nomologic knowledge and ontologic knowledge, +between knowledge of laws and knowledge of facts or existence. +Given certain facts~$f(a)$ about~$a$ and certain laws of connection,~$L$, +we can infer certainly or probably other facts~$\phi(a)$ about~$a$. If +a \emph{complete} knowledge of laws of connection together with~$f(a)$ +yields no appreciable probability for preferring~$\phi(a)$ to other +alternatives, then I suggest that an actual connection between $\phi$~and~$f$ +in a particular instance may be said to be due to chance in +a sense which usage justifies us in calling objective. We do +not, in fact, when we speak of objective chance, always use it +in so strict a sense as this, but this is, I think, the underlying +conception to which current usage approximates. Current +usage diverges from this sense mainly for two reasons. We +speak of objective chance if in the above conditions our +grounds for preference, though appreciable, are very small; and +we are not insistent to assert the rule of chance if a comparatively +\emph{slight} addition to our ontologic knowledge would render the +probability or the grounds for preference appreciable. +%% -----File: 300.png---Folio 289------- +\index{Chance, objective!Poincaré on}% +\index{Poincaré, Henri!chance@{and chance}}% + +To sum up the above, an event is due to objective chance if +in order to predict it, or to prefer it to alternatives, at present +\DPchg{equi-probable}{equiprobable}, with any high degree of probability, it would be +necessary to know a great many more facts of existence about +it than we actually do know, and if the addition of a wide +knowledge of general principles would be little use. +\index{Knowledge!chance@{and chance}}% + +It must be added that we make a distinction between facts of +existence which are highly variable from case to case and those +which are constant or nearly constant over a certain field of +observation or experience. Within the limits of this field we +regard the \emph{permanent} facts of existence as being, from the standpoint +of chance, in nearly the same position as laws. A connection +is not due to chance, therefore, if a knowledge of the permanent +facts of existence could lead to their prediction. + +To sum up again therefore,---if within a given field of observation +or experience a knowledge of those facts of existence which are +permanent or invariable within that field, together with a knowledge +of all the relevant fundamental causal laws or general +principles, and of a \emph{few} other facts of existence, would not +permit us, given~$f(a)$, to attribute an appreciable probability to~$\phi(a)$ +(or an appreciable probability to the alternative~$\phi_1(a)$ +rather than~$\phi_2(a)$); then the conjunction of~$\phi(a)$ (or of~$\phi_1(a)$ +rather than $\phi_2(a)$ with~$f(a)$) is due to objective chance. + +\Paragraph{9.} If we return to the examples of Poincaré, the above definition +appears to conform satisfactorily with the usages of common +sense. It is when an \emph{exact} knowledge of fact, as distinguished +from principle, is required for even approximate prediction that +the expression `objective chance' seems applicable. But +neither our definition nor usage is precise as to the \emph{amount} of +knowledge of fact which must be required for prediction, in +order that, in the absence of it, the event may be regarded as +subject to objective chance. + +It may be added that the expression `chance' can be used +with reference to general statements as well as to particular facts. +We say, for example, that it is a matter of chance if a man dies +on his birthday, meaning that, as a general principle and in the +absence of special information bearing on a particular case, there +is no presumption whatever in favour of his dying on his birthday +rather than on any other day. If as a general rule there were celebrations +on such a day such as would be not unlikely to accelerate +%% -----File: 301.png---Folio 290------- +\index{Frequency theory!randomness@{and randomness}}% +death, we should say that a man's dying on his birthday was not +altogether a matter of chance. If we knew no such general rule +but did not know enough about birthdays to be assured that there +was no such rule, we could not call the chance `objective'; we +could only speak of it thus, if on the evidence before us there was a +strong presumption against the existence of any such general rule. + +\Paragraph{10.} The philosophical and scientific importance of objective +chance as defined above cannot be made plain, until Part V., on +the Foundations of Statistical Inference, has been reached. There +it will appear in more than one connection, but chiefly in connection +with the application of Bernoulli's formula. In cases where +the use of this formula is valid, important inferences can be drawn; +and it will he shown that, when the conditions for objective chance +are approximately satisfied, it is probable that the conditions +for the application of Bernoulli's formula will be approximately +satisfied also. + +\Paragraph{11.} The term \emph{random} has been used, it is well recognised, in +\index{Venn!random@{and `\textit{random}'}}% +several distinct senses. Venn\footnote + {\textit{Logic of Chance}, chap.~v., ``The Conception \emph{Randomness} and its Scientific +\index{Randomness}% + Treatment.''} +and other adherents of the +`frequency' theory have given to it a precise meaning, but one +which has avowedly very little relation to popular usage. A +\index{Peirce!randomness@{and randomness}}% +random sample, says Peirce,\footnote + {``A Theory of Probable Inference'' (published in Johns Hopkins \textit{Studies in + Logic}), p.~152.} +is one ``taken according to a precept +or method, which, being applied over and over again indefinitely, +would in the long run result in the drawing of any one set of instances +as often as any other set of the same number.'' The +same fundamental idea has been expressed with greater precision +by Professor Edgeworth in connection with his investigations +\index{Edgeworth!randomness@{and randomness}}% +into the law of error.\footnote + {``Law of Error,'' \textit{Camb.\ Phil.\ Trans.}, 1904, p.~128.} +It is a fatal objection, in my opinion, to +this mode of defining randomness, that in general we can only +know whether or not we have a random sample when our knowledge +is nearly complete. Its divergence from ordinary usage is +well illustrated by the fact that there would be perfect randomness +in the distribution of stars in the heavens, as Venn explicitly points +out, if they were disposed in an exact and symmetrical pattern.\footnote + {But it may be added that this seems inconsistent with Venn's conception + of randomness as that of aggregate order and \emph{individual irregularity}; nor is it + concordant with Venn's typically random diagram (p.~118). His usage, therefore, + is sometimes nearer than his definition to the popular usage.} +%% -----File: 302.png---Folio 291------- +\index{Probability, and relevant knowledge!randomness@{and randomness}}% + +I do not believe, therefore, that this kind of definition is a +useful one. The term must be defined with reference to probability, +not to what will happen ``in the long run''; though +there may be two senses of it, corresponding to subjective and +objective probability respectively. + +The most important phrase in which the term is used is that +of `a random selection' or `taken at random.' When we apply +this term to a particular member of a series or collection of +objects, we may mean one of two things. We may mean that +our knowledge of the method of choosing the particular member +is such that \textit{à~priori} the member chosen is as likely to be any +one member of the series as any other. We may also mean, +not that we have \emph{no} knowledge as to which particular member +is in question, but that such knowledge as we have respecting +the particular member, as distinguished from other members of +the series, is irrelevant to the question as to whether or not +this member has the characteristic under examination. In the +first case the particular member is a random member of the +series for \emph{all} characteristics; in the second case it is a random +member for some only. As the second case is the more general, +we had better take that for the purpose of defining `random +selection.' + +The point will be brought out further if we discuss the +more difficult use of the term. What exactly do we mean by +the statement: ``Any number, taken at random, is equally +likely to be odd or even''? According to the frequency theory, +this simply means that there are as many odd numbers as there +are even. Taking it in a sense corresponding to subjective +chance (and to the explanations given above), I propose as +a definition the following: $a$~is taken at random from the +class~$S$ for the purposes of the propositional function $S(x)·\phi(x)$, +\index{Propositional function!randomness@{and randomness}}% +relative to evidence~$h$, if `$x$~is~$a$' is irrelevant to the probability +$\phi(x)/S(x)· h$. Thus `the number of the inhabitants of France is +odd' is, relative to my knowledge, a random instance of the +propositional function `$x$~is an odd number,' since `$a$~is the +number of the inhabitants of France' is irrelevant to the probability +of `$a$~is odd'.\footnote + {In the above $S(x)$ stands for `$x$~is a number', $\phi(x)$~stands for `$x$~is odd,' + $a$~stands for `the number of inhabitants of France.'} +Thus to say that a number taken at +random is as likely to be odd as even, means that there is a +%% -----File: 303.png---Folio 292------- +\index{Selection, random}% +probability~$\frac{1}{2}$ that any instance taken at random of the +generalisation `all numbers are odd' (or of the corresponding +generalisation `all numbers are even') is true; an instance being +taken at random in respect of evenness or oddness, if our +knowledge about it satisfies the conditions defined above. +Whether or not a given instance is taken at random, depends, +therefore, upon what generalisation is in question. + +\Paragraph{12.} We may or may not have reason to believe that, if we take +a series of random selections, the proportionate number of +occurrences of one particular type of result will very probably +lie within certain limits. For reasons to be explained in \Chapref{XXIX}., +random selection relative to such information may +conveniently be termed `random selection under Bernoullian +conditions.' It is this kind of random selection which is scientifically +and statistically important. But, as this corresponds to +`objective chance,' it is convenient to have a wider definition +of `random selection' unqualified, corresponding to `subjective +chance'; and it is this wider definition which is given above. + +The term opposite to `random selection' in ordinary usage +is `biassed selection.' When I use this phrase without qualification +I shall use it as the opposite of `random selection' in the +wider unqualified sense. +%% -----File: 304.png---Folio 293------- +\index{Bernoulli, Daniel, and Inverse Probability!planets@{and planets}|inote}% +\index{Chance, objective!planets@{and planets}}% +\index{Planets, movements of}% +\index{Probability, and relevant knowledge!planetary orbits@{and planetary orbits}}% + + +\Chapter{XXV}{Some Problems arising out of the Discussion of Chance} + +\Paragraph{1.} \First{There} are two classical problems in which attempts have been +made to attribute certain astronomical phenomena to a specific +cause, rather than to objective chance in some such sense as has +been defined in the preceding chapter. + +The first of these is concerned with the inclinations to the +ecliptic of the orbits of the planets of the solar system. This +problem has a long history, but it will be sufficient to take De +Morgan's statement of it.\footnote + {Article on \textit{Probabilities} in \textit{Encyclopaedia Metropolitana}, p.~412, §\;46. De~Morgan + takes this without acknowledgment from Laplace, \textit{Théorie analytique +\index{Laplace!planets@{and planets}|inote}% + des probabilités} (1st~edition), pp.~257, 258. Laplace also allows for the fact + that all the planets move in the same sense as the earth. He concludes: ``On + verra que l'existence d'une cause commune qui a dirigé tous ces mouvemens + dans le sens de la rotation du soleil, et sur des plans peu inclinés à celui de son + équateur, est indiquée avec une probabilité bien supérieure à celle du plus + grand nombre des faits historiques sur lesquels on ne se permet aucun doute.'' + Laplace had in his turn borrowed the example, also without acknowledgment, + from Daniel Bernoulli. See also D'Alembert, \textit{Opuscules mathématiques}, vol.~iv., +\index{D'Alembert!planets@{and planets}}% + 1768, pp.~89 and 292.} +If we suppose that each of the orbits +might have \emph{any} inclination, we obtain a vast number of combinations +of which only a small number are such that their sum is as +small or smaller than the sum of those of the actual system. +But the very existence of ourselves and our world can be shown +to imply that one of this small number has been selected, and +De~Morgan derives from this an enormous presumption that +\index{De Morgan!planets@{and planets}}% +``there was a necessary cause in the formation of the solar system +for the inclinations being what they are.'' + +The answer to this was pointed out by D'Alembert\footnote + {\textit{Op.\ cit.}\ p.~292. ``Il y a certainement d'infini contre un à parier que les + Planètes ne devraient pas se trouver dans le même plan; ce n'est pas une raison + pour en conclure que cette disposition, si elle avoit lieu, auroit nécessairement + d'autre cause que le hasard; car il y auroit de même l'infini contre un à parier + que les Planètes pourroient n'avoir pas une certaine disposition déterminée à +volonté\ldots.'' + + D'Alembert is employing the instance for his own purposes, in order to build + up an \textit{ad~hominem} argument in favour of his theory concerning `runs' against + D.~Bernoulli (see also \Pageref{317}).} +in criticising +%% -----File: 305.png---Folio 294------- +\index{Bernoulli, Daniel, and Inverse Probability!planets@{and planets}}% +\index{Boole|inote}% +\index{Forbes, J. D.|inote}% +\index{Herschell and binary stars}% +\index{Probability, and relevant knowledge!binary stars@{and binary stars}}% +\index{Stars, binary}% +\index{Todhunter|inote}% +\index{Venn|inote}% +Daniel Bernoulli. De~Morgan could have reached a similar +result \emph{whatever} the configuration might have happened to be. +\emph{Any} arbitrary disposition over the celestial sphere is vastly +improbable \textit{à~priori}, that is to say in the absence of known laws +tending to favour particular arrangements. It does not follow +from this, as De~Morgan argues, that any actual disposition +possesses \textit{à posteriori} a peculiar significance. + +\Paragraph{2.} The second of these problems is known as Michell's problem +\index{Michell!binary stars@{and binary stars}}% +of binary stars. Michell's Memoir was published in the \textit{Philosophical +Transactions} for 1767.\footnote + {See also Todhunter's \textit{History}, pp.~332--4; Venn, \textit{Logic of Chance}, p.~260; + Forbes, ``On the Alleged Evidence for a Physical Connexion between Stars + forming Binary or Multiple Groups, deduced from the Doctrine of Chances,'' + \textit{Phil.\ Mag}., 1850, and Boole, ``On the Theory of Probabilities and in particular + on Michell's Problem of the Distribution of the Fixed Stars,'' \textit{Phil.\ + Mag.}, 1851.} +It deals with the question as to +whether stars which are optically double, \ie\ which are so situated +as to appear close together to an observer on the earth---are also +physically so ``either by an original act of the Creator, or in consequence +of some general law, such perhaps as gravity.'' He +argues that if the stars ``were scattered by mere chance as it +might happen~\ldots\ it is manifest~\ldots\ that every star being +as likely to be in any one situation as another, the probability that +any one particular star should happen to be within a certain +distance (as, for example, one degree) of any other given star +would be represented~\ldots\ by a fraction whose numerator would +be to its denominator as a circle of one degree radius to a circle +whose radius is the diameter of a great circle~\ldots\ that is, about +$1$~in~$13131$.'' From this beginning he derives an immense presumption +against the scattering of the several contiguous stars +that may be observed ``by mere chance as it might happen.'' +And he goes on to argue that, if there are causal laws directly +tending to produce the observed proximities, we may reasonably +suppose that the proximities are actual, and not merely optical +and apparent. The fact that Michell's induction was confirmed +by the later investigations of Herschell adds interest to the +speculation. But apart from this the argument is evidently +%% -----File: 306.png---Folio 295------- +\index{Chance, objective}% +\index{Chance, objective!binary stars@{and binary stars}}% +subtler than in the first example. Michell argues that there are +more stars optically contiguous, than would be likely if there +were no special cause acting towards this end, and further that, +if such a cause is in operation, it must be \emph{real}, and not merely +optical, contiguity that results from it. + +Let us analyse the argument more closely. By ``mere chance +as it might happen'' Michell cannot be supposed to mean ``uncaused.'' +He is thinking of objective chance in the sense in +which I have defined this in the preceding chapter. We +speak of a chance occurrence when it is brought about by the +coincidence of forces and circumstances so numerous and complex +that knowledge sufficient for its prediction is of a kind altogether +out of our reach. Michell uses the term vaguely but means, I +think, something of this kind: An event is due to \emph{mere chance} +when it can only occur if a large number of independent\footnote + {See §\;3 of \hyperref[note:ii]{Note~(ii.)}\ to \Partref{III}\@.} +conditions +are fulfilled simultaneously. The alternatives which +Michell is discussing are therefore these: Are binary stars merely +due to the interaction of a vast variety of stellar laws and positions +or are they the result of a few fundamental tendencies, +which might be the subject of knowledge and which would lead +us to expect such stars in relative profusion? + +The existence of numerous binary stars may give a real +inductive argument in favour of their arising out of the interaction +of a relatively small number of independent causes. But +it is not possible to arrive at such precise results as Michell's. +If there is some finite probability \textit{à~priori} that binary stars, +when they arise, do arise in this way, then, since the frequent +coincidence of a given set of independent causes relatively few +in number is more likely than that of a set relatively numerous, +the observation of binary stars will raise this probability \textit{à posteriori} +to an extent which depends upon the relative profusion +in which such stars appear. If, in short, the first of the two +alternatives proposed above is assumed, there is no greater +presumption for a distribution, covering a part of the heavens, +in which binary stars appear, than for any other distribution; +if the second is assumed, there \emph{is} a greater presumption. The +observation of numerous distributions in which binary stars +appear \emph{increases}, therefore, by the inverse principle, any \textit{à~priori} +probability which may exist in favour of the second hypothesis. +%% -----File: 307.png---Folio 296------- +\index{Probability, and relevant knowledge!star drifts@{and star drifts}}% +But more than this the argument cannot justify. That Michell's +argument is, as it stands, no more valid than De~Morgan's, +becomes plain when we notice that he would still have a high +probability for his conclusion even if \emph{only one} binary star had +been observed. The valuable part of the argument must clearly +turn upon the observation of \emph{numerous} binary stars. + +Let us now turn to Michell's second step. He argues that, +if binary stars arise out of the interaction of a small number of +independent forces, they must be physically and not merely +optically double. The force of this argument seems to depend +upon our possessing previous knowledge as to the nature of the +principal natural laws, and upon an assumption, arising out of +this, that there are not likely to be forces tending to arrange +stars, in reality at great distances from one another, so as to +\emph{appear} double from this particular planet. But Michell, in +arguing thus, was neglecting the possibility that the optical +connection between the stars might be due to the observer and +his means of observation. It was not impossible that there should +be a law, connected with the transmission of light for example, +which would cause stars to appear to an observer to be much +nearer together than they really are. + +While, therefore, a relative profusion of binary stars constitutes +evidence favourably relevant to Michell's conclusion, the argument +is more complex and much less conclusive than he seems to +have supposed. This is a criticism which is applicable to many +such arguments. The simplicity of the evidence, which arises +out of the lack of much relevant information, is liable, unless we +are careful, to lead us into deceptive calculations and into assertions +of high numerical probabilities, upon which we should never +venture in cases where the evidence is full and complicated, but +where, in fact, the conclusion is established far more strongly. +The enormously high probability in favour of his conclusion, to +which Michell's calculations led him, should itself have caused +him to suspect the accuracy of the reasoning by which he +reached it. + +\Paragraph{3.} Some more recent problems of this type seem, however, so +far as I am acquainted with them, to follow safer lines of argument. +The most important are concerned with the existence +of star drifts. It seems to me not at all impossible to possess +\index{Star drifts}% +\textit{data} on which a valid argument can be constructed from the +%% -----File: 308.png---Folio 297------- +\index{Cause, final}% +\index{Pearson, Karl!stars@{and stars}}% +\index{Probability, and relevant knowledge!final causes@{and final causes}}% +observation of optically apparent star drifts to the probability +of a real uniformity of motion amongst certain sets of stars +relatively to others. + +Another problem, somewhat analogous to the preceding, has +been recently discussed by Professor Karl Pearson.\footnote + {``On the Improbability of a Random Distribution of the Stars in Space,'' + \textit{Proceedings of Royal Society}, series~A, vol.~84, pp.~47--70, 1910.} +The title +might prove a little misleading, perhaps, until the explanation +has been reached of the sense in which the term `random' is +used in it. But Professor Pearson uses the term in a perfectly +precise sense. He defines a random distribution as one in which +spherical shells of equal volume about the sun as centre contain +the same number of stars.\footnote + {It is, therefore, independent of direction, and the distribution is random + even if the stars are massed in particular quarters of the heavens. The definition + is, therefore, exceedingly arbitrary.} +He argues that the observed facts +render probable the following disjunction: Either the distribution +of stars is not random in the sense defined above, \emph{or} there is +a correlation between their distance and their brilliancy, such as +might be produced, for example, by the absorption of light in its +transmission through space, \emph{or} the space within which they all +lie is limited in volume and not spherical in form.\footnote + {This should run more correctly, I think, ``not a sphere \emph{with the sun as +centre}.''} +But it is +useless to employ the term \emph{random} in this sense in such inquiries +as Michell's. For there is no reason to suppose that a non-*random +distribution is more likely than a random distribution +to depend upon the interaction of a small number of independent +forces, and there might even exist a presumption the other way. +This arbitrary interpretation of randomness does not help us to +\index{Randomness!Pearson's use of}% +the solution of any interesting problem. + +\Paragraph{4.} The discussion of \emph{final} causes and of the argument from +design has suffered confusion from its supposed connection with +theology. But the logical problem is plain and can be determined +upon formal and abstract considerations. The argument is in all +cases simply this---an event has occurred and has been observed +which would be very improbable \textit{à~priori} if we did not know that +it had actually happened; on the other hand, the event is of such +a character that it might have been not unreasonably predicted +if we had assumed the existence of a conscious agent whose +motives are of a certain kind and whose powers are sufficient. +%% -----File: 309.png---Folio 298------- +\index{Coover, J.|inote}% +\index{Society for Psychical Research|inote}% + +Symbolically: Let $h$ be our original \textit{data}, $a$~the occurrence +of the event, $b$~the existence of the supposed conscious agent. +Then $a/h$~is assumed very small in comparison with~$a/bh$; and +we require~$b/ah$, the probability, that is to say, of~$b$ after $a$~is +known. The inverse principle of probability already demonstrated +shows that $b/ah = a/bh·\dfrac{b/h}{a/h}$, and $b/ah$ is therefore not +determinate in terms of $a/bh$~and~$a/h$ alone. Thus we cannot +measure the probability of the conscious agent's existence \emph{after} +the event, unless we can measure its probability \emph{before} the event. +And it is our ignorance of this, as a rule, that we are endeavouring +to remedy. The argument tells us that the existence of the +hypothetical agent is more likely after the event than before +it; but, as in the case of the general inductive problem dealt +with in \Partref{III}., unless there is an \emph{appreciable} probability first, +there cannot be an appreciable probability afterwards. No +conclusion, therefore, which is worth having, can be based on the +argument from design \emph{alone}; like induction, this type of argument +can only strengthen the probability of conclusions, for +which there is something to be said on \emph{other} grounds. We cannot +say, for example, that the human eye is due to design more +probably than not, unless we have some reason, apart from the +nature of its construction, for suspecting conscious workmanship. +But the necessary \textit{à~priori} probability, derived from some other +source, may sometimes be forthcoming. The man who upon a +desert island picks up a watch, or who sees the symbol \textit{John +Smith} traced upon the sand, can use with reason the argument +from design. For he has other grounds for supposing that +beings, capable of designing such objects, do exist, and that +their presence on the island, now or formerly, is appreciably +possible. + +\Paragraph{5.} The most important problems at the present day, in which +arguments of this kind are employed, are those which arise in +connection with psychical research.\footnote + {The probability that a remarkable success in naming playing cards is due + to psychic agency, was discussed by Professor Edgeworth in \textit{Metretike}. This +\index{Edgeworth!Psychical Research@{and Psychical Research}|inote}% + was, I think, the first application of probabilities to these questions. See also + \textit{Proceedings of the Society for Psychical Research}, Parts VIII.~and~X.; Professor + Edgeworth's article on \textit{Psychical Research and Statistical Method}, Stat.\ Journ.\ + vol.~lxxxii.\ (1919) p.~222; and \textit{Experiments in Psychical Research at Leland + Stanford Junior University}, by J.~Coover.} +The analysis of the `cross-correspondences,' +%% -----File: 310.png---Folio 299------- +\index{Hypothetical entities}% +\index{Logic, academic!initial probability@{and initial probability}}% +\index{Physics and initial probability}% +which have played so large a part in recent +discussions, presents many points of difficulty which are not +dissimilar to those which arise in other scientific inquiries of +great complexity in which our initial knowledge is small. An +important part of the \emph{logical} problem, therefore, is to distinguish +the \emph{peculiarity} of psychical problems and to discover what special +evidence they demand beyond what is required when we deal with +other questions. There is a certain tendency, I think, arising out +of the belief that psychical problems are in some way peculiar, +to raise sceptical doubts against them, which are equally valid +against \emph{all} scientific proofs. Without entering into any questions +of detail, let us endeavour to separate those difficulties which +seem peculiar to psychical research from those which, however +great, are not different from the difficulties which confront +students of heredity, for instance, and which are not less likely +than these to yield ultimately to the patience and the insight of +investigators. + +For this purpose it is necessary to recur, briefly, to the analysis +of \Partref{III}\@. It was argued there that the methods of empirical +proof, by which we strengthen the probability of our conclusions, +are not at all dissimilar, when we apply them to the discovery +of formal truth, and when we apply them to the discovery of the +laws which relate material objects, and that they may possibly +prove useful even in the case of metaphysics; but that the +\emph{initial} probability which we strengthen by these means is differently +obtained in each class of problem. In logic it arises out +of the postulate that apparent self-evidence invests what seems +self-evident with \emph{some} degree of probability; and in physical +science, out of the postulate that there is a limitation to the +amount of \emph{independent} variety amongst the qualities of material +objects. But both in logic and in physical science we may wish +to consider hypotheses which it is not possible to invest with any +\textit{à~priori} probability and which we entertain solely on account of +the known truth of many of their consequences. An axiom +which has no self-evidence, but which it seems necessary to combine +with other axioms which are self-evident in order to deduce +\index{Axioms!non-self-evident}% +the generally accepted body of formal truth, stands in this +category. A scientific entity, such as the ether or the electron, +whose qualities have \emph{never} been observed but whose existence we +postulate for purposes of explanation, stands in it also. If the +%% -----File: 311.png---Folio 300------- +\index{Probability, and relevant knowledge!spirits@{and spirits}}% +\index{Probability, and relevant knowledge!telepathy@{and telepathy}}% +\index{Spirits, probability of}% +\index{Telepathy, probability of}% +analysis of \Partref{III}. is correct, we can never attribute a finite +probability\footnote + {I am \emph{assuming} that there is \emph{no} argument, arising either from self-evidence + or analogy, in addition to the argument arising from the truth of their consequences, + in favour of the truth of such axioms or the existence of such objects; + but I daresay that this may not certainly be the case. The reader may be reminded + also that, when I deny a finite probability this is not the same thing as + to affirm that the probability is infinitely small. I mean simply that it is not + greater than some numerically measurable probability.} +to the truth of such axioms or to the existence of +such scientific entities, however many of their consequences +we find to be true. They may be \emph{convenient} hypotheses, because, +if we confine ourselves to \emph{certain classes} of their consequences, +we are not likely to be led into error; but they stand, nevertheless, +in a position altogether different from that of such generalisations +as we have reason to invest with an initial probability. + +Let us now apply these distinctions to the problems of psychical +research. In the case of some of them we can obtain the initial +probability, I think, by the same kind of postulates as in physical +science, and our conclusions need not be open to a greater degree +of doubt than these. In the case of others we cannot; and these +must remain, unless some method is open to us peculiar to +psychical research, as tentative unproved hypotheses in the +same category as the ether. + +The best example of the first class is afforded by telepathy. +We know that the consciousnesses which, if our hypothesis is +correct, act upon one another, do exist; and I see no \emph{logical} difference +between the problem of establishing a law of telepathy and +that of establishing the law of gravitation. There is at present a +\emph{practical} difference on account of the much narrower scope of our +knowledge, in the case of telepathy, of cognate matters. We can, +therefore, be much less certain; but there seems no reason why +we should necessarily remain less certain after more evidence +has been accumulated. It is important to remember that, in +the case of telepathy, we are merely discovering a relation between +objects \emph{which we already know to exist}. + +The best example of the other class is afforded by attempts +to attribute psychic phenomena to the agency of `spirits' other +than human beings. Such arguments are weakened at present +by the fact that no phenomena are known, so far as I am aware, +which cannot be explained, though improbably in some cases, +in other ways. But even if phenomena were to be observed of +%% -----File: 312.png---Folio 301------- +\index{James, W., and spirits}% +\index{Probability, and relevant knowledge!spirits@{and spirits}}% +which no known agency could afford even an improbable explanation, +the hypothesis of `spirits' would still lie in the same +logical limbo as the hypothesis of the `ether,' in which they +might be supposed not inappropriately to move. + +Such an hypothesis as the existence of `spirits' could only +become substantial if some peculiar method of knowledge were +within our power which would yield us the initial probability +which is demanded. That such a method exists, it is not infrequently +claimed. If we can directly perceive these `spirits,' +as many of those who are described in James's \textit{Varieties of +Religious Experience} think they can, the problem is, logically, +altogether changed. We have, in fact, very much the same kind +of reason, though it may be with less probability, that we have +for believing in the existence of other people. The preceding +paragraph applies only to attempts at proving the existence of +`spirits' from such evidence as is discussed by the Society for +Psychical Research. + +In between these two extremes comes a class of cases, with +regard to which it is extremely difficult to come to a decision---that +of attempts to attribute psychic phenomena to the conscious +agency of the dead. I wish to discuss here, not the nature of the +existing evidence, but the question whether it is possible for +\emph{any} evidence to be convincing. In this case the object whose +existence we are endeavouring to demonstrate resembles \emph{in +many respects} objects which we know to exist. The question +of epistemology, which is before us, is this: Is it necessary, in +order that we may have an initial probability, that the object of +our hypothesis should resemble in every relevant particular +some \emph{one} object which we know to exist, or is it sufficient that we +should know instances of all its supposed qualities, though never +in combination? It is clear that \emph{some} qualities may be irrelevant---position +in time and space, for example---and that `every +\index{Space!irrelevance of}% +\index{Time!irrelevance of}% +relevant particular' need not include these. But can the initial +probability exist if our hypothesis assumes qualities, which have +plainly some degree of relevance, in \emph{new} combinations? If we +have no knowledge of consciousness existing apart from a living +body, can \emph{indirect} evidence of whatever character afford us any +probability of such a thing? Could any evidence, for example, +persuade us that a tree felt the emotion of amusement, even if +it laughed repeatedly when we made jokes? Yet the analogy +%% -----File: 313.png---Folio 302------- +\index{Calculus of Probability!Psychical Research@{and Psychical Research}}% +\index{Measurement of Probability!psychical research@{and psychical research}}% +\index{Occurrences, remarkable}% +\index{Remarkableness}% +which we demand seems to be a matter of degree; for it does not +seem unreasonable to attribute consciousness to dogs, although +this constitutes a combination of qualities unlike in many respects +to any which we \emph{know} to exist. + +This discussion, however, is wandering from the subject of +probability to that of epistemology, and it will not be solved until +\index{Epistemology}% +we possess a more comprehensive account of this latter subject +than we have at present. I wish only to distinguish between those +cases in which we obtain the initial probability in the same +manner as in physical science from those in which we must get +it, if at all, in some other way. The distinctions I have made +are sufficiently summarised by a recapitulation of the following +comparisons: We compared the proof of telepathy to the proof +of gravitation, the proof of non-human `spirits' to the proof +of the ether, and, much less closely, the proof of the consciousness +of the dead to the proof of the consciousness of trees, or, perhaps, +of dogs. + +Before passing to the next of the rather miscellaneous topics +of this chapter, it may be worth while to add that we should be +very chary of applying to problems of psychical research the +\emph{calculus} of probabilities. The alternatives seldom satisfy the +conditions for the application of the Principle of Indifference, +\index{Principle of Indifference!Psychical Research@{and Psychical Research}}% +and the initial probabilities are not capable of being measured +numerically. If, therefore, we endeavour to \emph{calculate} the probability +that some phenomenon is due to `abnormal' causes, +our mathematics will be apt to lead us into unjustifiable +conclusions. + +\Paragraph{6.} Uninstructed common sense seems to be specially unreliable +in dealing with what are termed `remarkable occurrences.' +Unless a `remarkable occurrence' is simply one which produces +on us a particular psychological effect, that of surprise, we can +only define it as an event which \emph{before} its occurrence is very improbable +on the available evidence. But it will often occur---whenever, +in fact, our \textit{data} leave open the possibility of a large +number of alternatives and show no preference for any of them---that +\emph{every} possibility is exceedingly improbable \textit{à~priori}. It +follows, therefore, that what actually occurs does not derive any +peculiar significance merely from the fact of its being `remarkable' +in the above sense. Something further is required before we +can build with success. Yet Michell's argument and the argument +\index{Michell}% +%% -----File: 314.png---Folio 303------- +\index{Calculus of Probability}% +from design derive a good deal of their plausibility, I think, +from the `remarkable' character of the actual constitution +whether of the heavens or of the universe, in forgetfulness of the +fact that it is impossible to propound \emph{any} constitution which +would if it existed be other than `remarkable.' It is supposed +that a remarkable occurrence is specially in need of an explanation, +and that any \emph{sufficient} explanation has a high probability +in its favour. That an explanation is particularly required, +possesses a measure of truth; for it is likely that our original +\textit{data} were much lacking in completeness, and the occurrence of +the extraordinary event brings to light this deficiency. But +that we are not justified in adopting with confidence any sufficient +explanation, has been shown already. + +Such arguments, however, get a part of their plausibility from +a quite different source. There is a general supposition that some +kinds of occurrences are more likely than others to be \emph{susceptible} +of an explanation \emph{by us}; and, therefore, any explanation which +deals with such cases falls in prepared soil. Results which, +judging from ourselves, conscious agents would be likely to produce +fall into this category. Results which would be probable, +supposing a direct and predominant causal dependence between +the elements whose concomitance is remarked, belong to it also. +There is, in fact, a sort of argument from analogy as to whether +certain sorts of phenomena are or are not likely to be due to +`chance.' This may explain, for example, why the particular +concurrence of atoms that go to compose the human eye, why a +series of correct guesses in naming playing cards, why special +symmetry or special asymmetry amongst the stars, seem to +require explanation in no ordinary degree. \emph{Prior} to an explanation +these particular concurrences or series or distributions are +no more improbable than any other. But the causes of such +conjunctions as these are more likely to be discoverable by the +human mind than are the causes of others, and the attempt to +explain them deserves, therefore, to be more carefully considered. +This supposition, derived by analogy or induction from those +cases in which we believe the causes to be known to us, has, perhaps, +some weight. But the direct application of the Calculus +of Probabilities can do no more in these cases than suggest matter +for investigation. The fact that a man has made a long series +of correct guesses in cases where he is cut off from the ordinary +%% -----File: 315.png---Folio 304------- +channels of communication, is a fact worthy of investigation, +because it is more likely to be susceptible of a \emph{simple} causal explanation, +which may have many applications, than a case in +which false and true guesses follow one another with no apparent +regularity. + +\Paragraph{7.} In the case of empirical laws, such as Bode's law, which have +\index{Bode's Law}% +no more than a very slight connection with the general body of +scientific knowledge, it is sometimes thought that the law is more +probable if it is proposed \emph{before} the examination of some or all of +the available instances than if it is proposed after their examination. +Supposing, for example, that Bode's law is accurately +true for seven planets, it is held that the law would be more +probable if it was suggested after the examination of six and +was confirmed by the subsequent discovery of the seventh, than +it would be if it had not been propounded until after all seven +had been observed. The arguments in favour of such a conclusion +\index{Peirce}% +are well put by Peirce:\footnote + {C.~S. Peirce, \textit{A Theory of Probable Inference}, pp.~162--167; published in +Johns Hopkins \textit{Studies in Logic}, 1883.} +``All the qualities of objects may be +conceived to result from variations of a number of continuous +variables; hence any lot of objects possesses some character in +common, not possessed by any other.'' Hence if the common +character is not predesignate we can conclude nothing. Cases +must not be used to prove a generalisation which has only been +suggested by the cases themselves. He takes the first five poets +from a biographical dictionary with their ages at death: +\begin{center} +\setlength{\TmpLen}{1.5in} +\begin{tabular}{p{\TmpLen}@{}r<{\quad}|>{\quad}p{\TmpLen}@{}r} +Aagard \dotfill & 48 & Abunowas \dotfill & 48\\ +Abeille \dotfill & 76 & Accords \dotfill & 45\\ +Abulola \dotfill & 84 & & \\ +\end{tabular} +\end{center} + +\hspace*{16pt}%[** TN: Must match Subpars item width] +``These five ages have the following characters in common: +\begin{Subpars} +\item[``1.] The difference of the two digits composing the number, +divided by three, leaves a remainder of \emph{one}. + +\item[``2.] The first digit raised to the power indicated by the second, +and then divided by three, leaves a remainder of \emph{one}. + +\item[``3.] The sum of the prime factors of each age, including \emph{one} as +a prime factor, is divisible by \emph{three}.'' +\end{Subpars} + +He compares a generalisation regarding the ages of poets based +%% -----File: 316.png---Folio 305------- +\index{Playfair, Dr.\ Lyon}% +\index{Prediction, value of}% +on this evidence to Dr.~Lyon Playfair's argument about the +specific gravities of the three allotropic forms of carbon: +\begin{center} +\setlength{\TmpLen}{2in} +\begin{tabular}{p{\TmpLen}@{}l} +Diamond \dotfill & $3.48 = \sqrt[2]{12}$ \\ +Graphite \dotfill & $2.29 = \sqrt[3]{12}$ \\ +Charcoal \dotfill & $1.88 = \sqrt[4]{12}$ +\end{tabular} +\end{center} +approximately, the atomic weight of carbon being~$12$. Dr.~Playfair +thinks that the above renders it probable that the specific +gravities of the allotropic forms of other elements would, if we +knew them, be found to equal the different roots of their atomic +weight. + +The weakness of these arguments, however, has a different +explanation. These inductions are very improbable, because they +are out of relation to the rest of our knowledge and are based on +a very small number of instances. The apparent absurdity, +moreover, of the inductive law of Poets' Ages is increased by the +fact that we take account of the knowledge we actually possess +that the ages of poets are not in fact connected by any such law. +If we knew nothing whatever about poets' ages except what is +stated above, the induction would be as valid as any other which +is based on a very weak analogy and a very small number of +instances and is unsupported by indirect evidence. + +The peculiar virtue of prediction or predesignation is altogether +imaginary. The number of instances examined and the analogy +between them are the essential points, and the question as to +whether a particular hypothesis happens to be propounded before +or after their examination is quite irrelevant. If all our inductions +had to be thought of before we examined the cases to +which we apply them, we should, doubtless, make fewer inductions; +but there is no reason to think that the few we should make +would be any better than the many from which we should be +precluded. The plausibility of the argument is derived from a +different source. If an hypothesis is proposed \textit{à~priori}, this +commonly means that there is some ground for it, arising out of +our previous knowledge, \emph{apart from} the purely inductive ground, +and if such is the case the hypothesis is clearly stronger than one +which reposes on inductive grounds only. But if it is a mere +guess, the lucky fact of its preceding some or all of the cases which +verify it adds nothing whatever to its value. It is the union of +%% -----File: 317.png---Folio 306------- +\index{Statistics, and prediction}% +prior knowledge, with the inductive grounds which arise out of +the immediate instances, that lends weight to an hypothesis, and +not the occasion on which the hypothesis is first proposed. It is +sometimes said, to give another example, that the daily fulfilment +of the predictions of the \textit{Nautical Almanack} constitutes the most +cogent proof of the laws of dynamics. But here the essence of +the verification lies in the variety of cases which can be brought +accurately under our notice by means of the \textit{Almanack}, and in +the fact that they have all been obtained on a uniform principle, +\emph{not} in the fact that the verification is preceded by a prediction. + +The same point arises not uncommonly in statistical inquiries. +If a theory is first proposed and is then confirmed by the examination +of statistics, we are inclined to attach more weight to it than +to a theory which is constructed in order to suit the statistics. +But the fact that the theory which precedes the statistics is more +likely than the other to be supported by general considerations---for +it has not, presumably, been adopted for no reason at all---constitutes +the only valid ground for this preference. If it does +\emph{not} receive more support than the other from general considerations, +then the circumstances of its origin are no argument in its +favour. The opposite view, which the unreliability of some +statisticians has brought into existence,---that it is a positive +advantage to approach statistical evidence \emph{without} preconceptions +based on general grounds, because the temptation to `cook' +the evidence will prove otherwise to be irresistible,---has no +\emph{logical} basis and need only be considered when the impartiality of +an investigator is in doubt. +%% -----File: 318.png---Folio 307------- +\index{Belief, rational}% +\index{Conduct and Probability}% +\index{Ethics|ifoll}% +\index{Probability, and relevant knowledge!ethics@{and ethics}}% + + +\Chapter{XXVI}{The Application of Probability to Conduct} + +\Paragraph{1.} \First{Given} as our basis what knowledge we actually have, the +probable, I have said, is that which it is rational for us to believe. +This is not a definition. For it is not rational for us to believe +that the probable is true; it is only rational to have a probable +belief in it or to believe it in preference to alternative beliefs. To +believe one thing \emph{in preference} to another, as distinct from believing +the first true or more probable and the second false or less probable, +must have reference to action and must be a loose way of expressing +the propriety of \emph{acting} on one hypothesis rather than +on another. We might put it, therefore, that the probable is +the hypothesis on which it is rational for us to act. It is, however, +not so simple as this, for the obvious reason that of two hypotheses +it may be rational to act on the less probable if it leads to the +greater good. We cannot say more at present than that the +probability of a hypothesis is one of the things to be determined +and taken account of before acting on it. + +\Paragraph{2.} I do not know of passages in the ancient philosophers which +explicitly point out the dependence of the duty of pursuing +goods on the reasonable or probable expectation of attaining +them relative to the agent's knowledge. This means only that +analysis had not disentangled the various elements in rational +action, not that common sense neglected them. Herodotus +\index{Herodotus}% +puts the point quite plainly. ``There is nothing more profitable +for a man,'' he says, ``than to take good counsel with himself; +for even if the event turns out contrary to one's hope, still one's +decision was right, even though fortune has made it of no effect: +whereas if a man acts contrary to good counsel, although by luck +he gets what he had no right to expect, his decision was not any +the less foolish.''\footnote + {Herod.\ vii.~10.} +%% -----File: 319.png---Folio 308------- +\index{Jesuits}% +\index{Taylor, Jeremy|inote}% + +\Paragraph{3.} The first contact of theories of probability with modern +ethics appears in the Jesuit doctrine of probabilism. According +\index{Probabilism}% +to this doctrine one is justified in doing an action for which there +is \emph{any} probability, however small, of its results being the best +possible. Thus, if any priest is willing to permit an action, that +fact affords some probability in its favour, and one will not be +damned for performing it, however many other priests denounce +it.\footnote + {Compare with this doctrine the following curious passage from Jeremy + Taylor:---``We being the persons that are to be persuaded, we must see that + we be persuaded reasonably. And it is unreasonable to assent to a lesser + evidence when a greater and clearer is propounded: but of that every man for + himself is to take cognisance, if he be able to judge; if he be not, he is not + bound under the tie of necessity to know anything of it. That that is + necessary shall be certainly conveyed to him: God, that best can, will certainly + take care for that; for if he does not, it becomes to be not necessary; or if it + should still remain necessary, and he be damned for not knowing it, and yet to + know it be not in his power, then who can help it! There can be no further + care in this business.''} +It may be suspected, however, that the object of this +doctrine was not so much duty as safety. The priest who permitted +you so to act assumed thereby the responsibility. The +correct application of probability to conduct naturally escaped +the authors of a juridical ethics, which was more interested in +the fixing of responsibility for definite acts, and in the various +specified means by which responsibility might be disposed of, +than in the greatest possible sum-total of resultant good. + +A more correct doctrine was brought to light by the efforts of +the philosophers of the Port Royal to expose the fallacies of probabilism. +``In order to judge,'' they say, ``of what we ought to +do in order to obtain a good and to avoid an evil, it is necessary +to consider not only the good and evil in themselves, but also +the probability of their happening and not happening, and to +regard geometrically the proportion which all these things have, +taken together.''\footnote + {\textit{The Port Royal Logic} (1662), Eng.\ Trans.\ p.~367.} +\index{Port Royal logic!probabilism@{and probabilism}}% +Locke perceived the same point, although +\index{Locke}% +not so clearly.\footnote + {\textit{Essay concerning Human Understanding}, book~ii.\ chap.~xxi.\ §\;66.} +By Leibniz this theory is advanced more +\index{Leibniz}% +explicitly; in such judgments, he says, ``as in other estimates +disparate and heterogeneous and, so to speak, of more than one +dimension, the greatness of that which is discussed is in reason +composed of both estimates (\ie\ of goodness and of probability), +and is like a rectangle, in which there are two considerations, +viz.\ that of length and that of breadth\ldots. Thus we should +%% -----File: 320.png---Folio 309------- +\index{Butler, Bishop}% +\index{Moore, G. E.}% +still need the art of thinking and that of estimating probabilities, +besides the knowledge of the value of goods and evils, in order +properly to employ the art of consequences.''\footnote + {\textit{Nouveaux Essais}, book~ii.\ chap.~xxi.} + +In his preface to the \textit{Analogy} Butler insists on ``the absolute +and formal obligation'' under which even a low probability, +if it is the greatest, may lay us: ``To us probability is the very +guide of life.'' + +\Paragraph{4.} With the development of a utilitarian ethics largely concerned +with the summing up of consequences, the place of probability +in ethical theory has become much more explicit. But +although the general outlines of the problem are now clear, there +are some elements of confusion not yet dispersed. I will deal with +some of them. + +In his \textit{Principia Ethica} (p.~152) Dr.~Moore argues that ``the +first difficulty in the way of establishing a probability that one +course of action will give a better total result than another, lies +in the fact that we have to take account of the effects of both +throughout an infinite future\ldots. We can certainly only pretend +to calculate the effects of actions within what may be called an +`immediate future.'\ldots\ We must, therefore, certainly have +some reason to believe that no consequences of our action in a +further future will generally be such as to reverse the balance of +good that is probable in the future which we can foresee. This +large postulate must be made, if we are ever to assert that the +results of one action will be even probably better than those of +another. Our utter ignorance of the far future gives us no justification +for saying that it is even probably right to choose the +greater good within the region over which a probable forecast +may extend.'' + +This argument seems to me to be invalid and to depend on +a wrong philosophical interpretation of probability. Mr.~Moore's +reasoning endeavours to show that there is not even a \emph{probability} +by showing that there is not a \emph{certainty}. We must not, of course, +have reason to believe that remote consequences will \emph{generally} +be such as to reverse the balance of immediate good. But we +need not be certain that the opposite is the case. If good is +additive, if we have reason to think that of two actions one produces +more good than the other in the near future, and if we have +no means of discriminating between their results in the distant +%% -----File: 321.png---Folio 310------- +\index{Butler, Bishop}% +\index{Goodness, organic nature of}% +future, then by what seems a legitimate application of the +Principle of Indifference we may suppose that there is a probability +\index{Principle of Indifference!ethics@{and ethics}}% +in favour of the former action. Mr.~Moore's argument +must be derived from the empirical or frequency theory of +probability, according to which we must know for certain what +will happen \emph{generally} (whatever that may mean) before we can +assert a probability. + +The results of our endeavours are very uncertain, but we have +a genuine probability, even when the evidence upon which it is +founded is slight. The matter is truly stated by Bishop Butler: +``From our short views it is greatly uncertain whether this +endeavour will, in particular instances, produce an overbalance +of happiness upon the whole; since so many and distant things +must come into the account. And that which makes it our duty +is that there is some appearance that it will, and no positive +appearance to balance this, on the contrary side\ldots.''\footnote + {This passage is from the \emph{Analogy}. The Bishop adds: ``\ldots\ and also + that such benevolent endeavour is a cultivation of that most excellent of all + virtuous principles, the active principle of benevolence.''} + +The difficulties which exist are not chiefly due, I think, to our +ignorance of the remote future. The possibility of our knowing +that one thing rather than another is our duty depends upon the +assumption that a greater goodness in any part makes, in the +absence of evidence to the contrary, a greater goodness in the +whole more probable than would the lesser goodness of the part. +We assume that the goodness of a part is \emph{favourably} relevant to +the goodness of the whole. Without this assumption we have no +reason, not even a probable one, for preferring one action to any +other on the whole. If we suppose that goodness is always +\emph{organic}, whether the whole is composed of simultaneous or +successive parts, such an assumption is not easily justified. The +case is parallel to the question, whether physical law is organic or +atomic, discussed in \Chapref{XXI}. §\;6. + +Nevertheless we can admit that goodness is partly organic +and still allow ourselves to draw probable conclusions. For the +alternatives, that \emph{either} the goodness of the whole universe +throughout time is organic \emph{or} the goodness of the universe is the +arithmetic sum of the goodnesses of infinitely numerous and +infinitely divided parts, are not exhaustive. We may suppose +that the goodness of conscious persons is organic for each distinct +%% -----File: 322.png---Folio 311------- +\index{Couturat|inote}% +\index{Law|inote}% +\index{Measurement of Probability}% +\index{Measurement of Probability!ethics@{and ethics}}% +and individual personality. Or we may suppose that, when +conscious units are in conscious relationship, then the whole +which we must treat as organic includes both units. These are +only examples. We must suppose, in general, that the units +whose goodness we must regard as organic and indivisible are +not always larger than those the goodness of which we can +perceive and judge directly. + +\Paragraph{5.} The difficulties, however, which are most fundamental +from the standpoint of the student of probability, are of a different +kind. Normal ethical theory at the present day, if there can be +said to be any such, makes two assumptions: first, that degrees +of goodness are numerically measurable and arithmetically +additive, and second, that degrees of probability also are numerically +measurable. This theory goes on to maintain that what +we ought to add together, when, in order to decide between two +courses of action, we sum up the results of each, are the `mathematical +expectations' of the several results. `Mathematical +expectation' is a technical expression originally derived from the +scientific study of gambling and games of chance, and stands for +the product of the possible gain with the probability of attaining +it.\footnote + {Priority in the conception of mathematical expectation can, I think, be +\index{Mathematical Expectation}% + claimed by Leibniz, \textit{De incerti aestimatione,} 1678 (Couturat, \textit{Logigue de Leibniz}, + p.~248). In a letter to Placcius, 1687 (Dutens, vi.~i.~36 and Couturat, \textit{op.~cit.}\ + p.~246) Leibniz proposed an application of the same principle to jurisprudence, + by virtue of which, if two litigants lay claim to a sum of money, + and if the claim of the one is twice as probable as that of the other, the sum + should be divided between them in that proportion. The doctrine, seems + sensible, but I am not aware that it has ever been acted on\DPtypo{}{.}} +In order to obtain, therefore, a measure of what ought to +be our preference in regard to various alternative courses of action, +we must sum for each course of action a series of terms made +up of the amounts of good which may attach to each of its +possible consequences, each multiplied by its appropriate probability. + +The first assumption, that quantities of goodness are duly +subject to the laws of arithmetic, appears to me to be open to a +certain amount of doubt. But it would take me too far from +my proper subject to discuss it here, and I shall allow, for the +purposes of further argument, that in some sense and to some +extent this assumption can be justified. The second assumption, +however, that degrees of probability are wholly subject to the +laws of arithmetic, runs directly counter to the view which has +%% -----File: 323.png---Folio 312------- +\index{Intuition \textit{versus} experience!ethical judgment@{and ethical judgment}}% +\index{Weight, of evidence}% +been advocated in Part~I.~of this treatise. Lastly, if both these +points be waived, the doctrine that the `mathematical expectations' +of alternative courses of action are the proper measures of +our degrees of preference is open to doubt on two grounds---first, +because it ignores what I have termed in \Partref{I}. the `weights' +of the arguments, namely, the amount of evidence upon which +each probability is founded; and second, because it ignores the +element of `risk' and assumes that an even chance of heaven +or hell is precisely as much to be desired as the certain attainment +of a state of mediocrity. Putting on one side the first of +these grounds of doubt, I will treat each of the others in turn. + +\Paragraph{6.} In \Chapref{III}. of \Partref{I}. I have argued that only in a +strictly limited class of cases are degrees of probability numerically +measurable. It follows from this that the `mathematical +expectations' of goods or advantages are not always numerically +measurable; and hence, that even if a meaning can be given to +the sum of a series of non-numerical `mathematical expectations,' +not every pair of such sums are numerically comparable in respect +of more and less. Thus even if we know the degree of advantage +which might be obtained from each of a series of alternative +courses of actions and know also the probability in each case of +obtaining the advantage in question, it is not always possible by +a mere process of arithmetic to determine which of the alternatives +ought to be chosen. If, therefore, the question of right action is +under all circumstances a determinate problem, it must be in +virtue of an intuitive judgment directed to the situation as a +whole, and not in virtue of an arithmetical deduction derived +from a series of separate judgments directed to the individual +alternatives each treated in isolation. + +We must accept the conclusion that, if one good is greater +than another, but the probability of attaining the first less than +that of attaining the second, the question of which it is our duty +to pursue may be indeterminate, unless we suppose it to be +within our power to make direct quantitative judgments of probability +and goodness jointly. It may be remarked, further, +that the difficulty exists, whether the numerical indeterminateness +of the probability is intrinsic or whether its numerical value +is, as it is according to the Frequency Theory and most other +theories, simply unknown. + +\Paragraph{7.} The second difficulty, to which attention is called above, +%% -----File: 324.png---Folio 313------- +\index{Bernoulli, Jac.!weight of evidence}% +is the neglect of the `weights' of arguments in the conception +of `mathematical expectation.' In \Chapref{VI}. of \Partref{I}. the +significance of `weight' has been discussed. In the present +connection the question comes to this---if two probabilities are +equal in degree, ought we, in choosing our course of action, to +prefer that one which is based on a greater body of knowledge? + +The question appears to me to be highly perplexing, and it is +difficult to say much that is useful about it. But the degree of +completeness of the information upon which a probability is +based does seem to be relevant, as well as the actual magnitude +of the probability, in making practical decisions. Bernoulli's +maxim,\footnote + {\textit{Ars Conjectandi}, p.~215: ``Non sufficit expendere unum alterumve argumentum, + sed conquirenda sunt omnia, quae in cognitionem nostram venire + possunt, atqne ullo modo ad probationem rei facere videntur.''} +that in reckoning a probability we must take into account +all the information which we have, even when reinforced by +\index{Locke!weight@{and weight of evidence}}% +Locke's maxim that we must get all the information we can,\footnote + {\textit{Essay concerning Human Understanding}, book~ii.\ chap.~xxi.\ §\;67: ``He + that judges without informing himself to the utmost that he is capable, cannot + acquit himself of \emph{judging amiss}.''} +does not seem completely to meet the case. If, for one alternative, +the available information is necessarily small, that does not seem +to be a consideration which ought to be left out of account +altogether. + +\Paragraph{8.} The last difficulty concerns the question whether, the +former difficulties being waived, the `mathematical expectation' +of different courses of action accurately measures what our +preferences ought to be---whether, that is to say, the undesirability +of a given course of action increases in direct proportion +to any increase in the uncertainty of its attaining its object, or +whether some allowance ought to be made for `risk,' its undesirability +\index{Risk!ethics@{and ethics}}% +increasing more than in proportion to its uncertainty. + +In fact the meaning of the judgment, that we ought to act in +such a way as to produce most probably the greatest sum of +goodness, is not perfectly plain. Does this mean that we +ought so to act as to make the sum of the goodnesses of each of +the possible consequences of our action multiplied by its probability +a maximum? Those who rely on the conception of +`mathematical expectation' must hold that this is an indisputable +proposition. The justifications for this view most commonly +advanced resemble that given by Condorcet in his ``Réflexions +\index{Condorcet!ethics@{and ethics}}% +%% -----File: 325.png---Folio 314------- +sur la règle générale, qui prescrit de prendre pour valeur d'un +événement incertain, la probabilité de cet événement multipliée +par la valeur de l'événement en lui-même,''\footnote + {\textit{Hist.\ de~l'Acad}., Paris, 1781.} +where he argues +from Bernoulli's theorem that such a rule will lead to satisfactory +\index{Bernoulli's Theorem}% +results if a very large number of trials be made. As, however, +it will be shown in \Chapref{XXIX}. of \Partref{V}. that Bernoulli's +theorem is not applicable in by any means every case, this +argument is inadequate as a general justification. + +In the history of the subject, nevertheless, the theory of +`mathematical expectation' has been very seldom disputed. +As D'Alembert has been almost alone in casting serious doubts +\index{D'Alembert!mathematical expectation@{and mathematical expectation}}% +upon it (though he only brought himself into disrepute by doing +so), it will be worth while to quote the main passage in which he +declares his scepticism: ``Il me sembloit'' (in reading Bernoulli's +\textit{Ars Conjectandi}) ``que cette matière avoit besoin d'être traitée +d'une manière plus claire; je voyois bien que l'espérance étoit +plus grande, \Primo~que la somme espérée étoit plus grande, \Secundo~que +la probabilité de gagner l'étoit aussi. Mais je ne voyois pas avec +la même évidence, et je ne le vois pas encore, \Primo~que la probabilité +soit estimée exactement par les méthodes usitées; \Secundo~que quand +elle le seroit, l'espérance doive être proportionnelle à cette probabilité +simple, plutôt qu'à une puissance ou même à une fonction +de cette probabilité; \Tertio~que quand il y a plusieurs combinaisons +qui donnent différens avantages ou différens risques (qu'on +regarde comme des avantages négatifs) il faille se contenter +d'\emph{ajouter} simplement ensemble toutes les espérances pour avoir +l'espérance totale.''\footnote + {\textit{Opuscules mathématiques}, vol.~iv., 1768 (extraits de lettres), pp.~284,~285. + See also p.~88 of the same volume.} + +In extreme cases it seems difficult to deny some force to +D'Alembert's objection; and it was with reference to extreme +cases that he himself raised it. Is it certain that a larger good, +which is extremely improbable, is precisely equivalent ethically +to a smaller good which is proportionately more probable? We +may doubt whether the moral value of speculative and cautious +action respectively can be weighed against one another in a +simple arithmetical way, just as we have already doubted whether +a good whose probability can only be determined on a slight +basis of evidence can be compared by means merely of the +%% -----File: 326.png---Folio 315------- +\index{Weight, of evidence!ethics@{and ethics}}% +magnitude of this probability with another good whose likelihood +is based on completer knowledge. + +There seems, at any rate, a good deal to be said for the conclusion +that, other things being equal, that course of action is +preferable which involves least risk, and about the results of +\index{Risk}% +which we have the most complete knowledge. In marginal cases, +therefore, the coefficients of weight and risk as well as that +of probability are relevant to our conclusion. It seems natural +to suppose that they should exert some influence in other cases +also, the only difficulty in this being the lack of any principle for +the calculation of the degree of their influence. A high weight +and the absence of risk increase \textit{pro tanto} the desirability of the +action to which they refer, but we cannot measure the amount +of the increase. + +The `risk' may be defined in some such way as follows. If +$A$~is the amount of good which may result, $p$~its probability +$(p+q=1)$, and $E$~the value of the `mathematical expectation,' +\index{Mathematical Expectation}% +so that $E=pA$, then the `risk' is~$R$, where $R = p(A-E) = +p(1-p)A = pqA = qE$. This may be put in another way: $E$~measures +the net immediate sacrifice which should be made in the +hope of obtaining~$A$; $q$~is the probability that this sacrifice will +be made in vain; so that $qE$~is the `risk.'\footnote + {The theory of \textit{Risiko} is briefly dealt with by Czuber, \textit{Wahrscheinlichkeitsrechnung}, +\index{Czuber!risk@{and risk}|inote}% + vol.~i.\ pp.~219 \textit{et seq}. If $R$~measures the first insurance, this leads to a + \textit{Risiko} of the second order, $R_1 = qR = q^2E$. This again may be insured against, + and by a sufficient number of such reinsurances the risk can be completely + shifted: + \[ + E + R_1 + R_2 + \ldots + = E(1 + q + q^2 + \ldots) = \frac{E}{1-q} = \frac{E}{p} = A. + \]} +The ordinary theory +supposes that the ethical value of an expectation is a function +of~$E$ only and is entirely independent of~$R$. + +We could, if we liked, define a conventional coefficient~$c$ of +weight and risk, such as $c=\dfrac{2pw}{(1+q)(1+w)}$, where $w$~measures the +`weight,' which is equal to unity when $p=1$ and $w=1$, and +to zero when $p=0$ or $w=0$, and has an intermediate value +in other cases.\footnote + {If $pA = p'A'$, $w>w'$, and $q=q'$, then $cA>c'A'$; if $pA=p'A'$, $w=w'$, and + $q<q'$, then $cA>c'A'$; if $pA=p'A'$, $w>w'$, and $q<q'$, then $cA>c'A'$; but if + $pA=p'A'$, $w=w'$, and $q>q'$, we cannot in general compare $cA$ and $c'A'$.} +But if doubts as to the sufficiency of the +conception of `mathematical expectation' be sustained, it is not +likely that the solution will lie, as D'Alembert suggests, and as +has been exemplified above, in the discovery of some more +%% -----File: 327.png---Folio 316------- +\index{Apprehension, direct, and ethical judgment}% +\index{Bernoulli, Daniel, and Inverse Probability!Petersburg Paradox@{and Petersburg Paradox}}% +\index{D'Alembert!ethics@{and ethics}}% +\index{Mathematicians, and probability!ethics@{and ethics}}% +complicated function of the probability wherewith to compound +the proposed good. The judgment of goodness and the judgment +of probability both involve somewhere an element of direct +apprehension, and both are quantitative. We have raised a +doubt as to whether the magnitude of the `oughtness' of an +action can be in all cases directly determined by simply multiplying +together the magnitudes obtained in the two direct judgments; +and a new direct judgment may be required, respecting +the magnitude of the `oughtness' of an action under given +circumstances, which need not bear any simple and necessary +relation to the two former. + +The hope, which sustained many investigators in the course +of the nineteenth century, of gradually bringing the moral sciences +under the sway of mathematical reasoning, steadily recedes---if +we mean, as they meant, by mathematics the introduction of +precise numerical methods. The old assumptions, that all +quantity is numerical and that all quantitative characteristics +are additive, can be no longer sustained. Mathematical reasoning +now appears as an aid in its symbolic rather than in its numerical +character. I, at any rate, have not the same lively hope as +Condorcet, or even as Edgeworth, ``éclairer les Sciences morales +\index{Condorcet!ethics@{and ethics}}% +\index{Edgeworth!ethics@{and ethics}}% +et politiques par le flambeau de l'Algèbre.'' In the present case, +even if we are able to range goods in order of magnitude, and also +their probabilities in order of magnitude, yet it does not follow +that we can range the products composed of each good and its +corresponding probability in this order. + +\Paragraph{9.} Discussions of the doctrine of Mathematical Expectation, +\index{Mathematical Expectation}% +apart from its directly ethical bearing, have chiefly centred +\index{Petersburg Paradox}% +round the classic Petersburg Paradox,\Pagelabel{316}\footnote + {For the history of this paradox see Todhunter. The name is due, he says, +\index{Todhunter!Petersburg Paradox@{and Petersburg Paradox}}% + to its having first appeared in a memoir by Daniel Bernoulli in the \textit{Commentarii} + of the Petersburg Academy.} +which has been treated by +almost all the more notable writers, and has been explained by +them in a great variety of ways. The Petersburg Paradox arises +out of a game in which Peter engages to pay Paul one shilling +if a head appears at the first toss of a coin, two shillings if it does +not appear until the second, and, in general, $2^{r-1}$~shillings if no +head appears until the $r$\ordth~toss.\DPnote{** TN: Ordinal not ital in orig.} What is the value of Paul's +expectation, and what sum must he hand over to Peter before +the game commences, if the conditions are to be fair? +%% -----File: 328.png---Folio 317------- +\index{Bernoulli, Daniel, and Inverse Probability!Petersburg Paradox@{and Petersburg Paradox}}% + +The mathematical answer is $\Sum_{1}^{n}(\frac{1}{2})^r2^{r-1}$, if the number of tosses +is not in any case to exceed $n$~in all, and $\Sum_{1}^{\infty}(\frac{1}{2})^r2^{r-1}$ if this restriction +is removed. That is to say, Paul should pay $\dfrac{n}{2}$~shillings in the +first case, and an infinite sum in the second. Nothing, it is said, +could be more paradoxical, and no sane Paul would engage on +these terms even with an honest Peter. + +Many of the solutions which have been offered will occur at +once to the reader. The conditions of the game \emph{imply} contradiction, +say Poisson and Condorcet; Peter has undertaken +\index{Condorcet}% +\index{Poisson!Petersburg Paradox@{and Petersburg Paradox}}% +engagements which he cannot fulfil; if the appearance of heads +is deferred even to the $100$th~toss, he will owe a mass of silver +greater in bulk than the sun. But this is no answer. Peter has +promised much and a belief in his solvency will strain our imagination; +but it is imaginable. And in any case, as Bertrand points +\index{Bertrand!Petersburg Paradox@{and Petersburg Paradox}}% +out, we may suppose the stakes to be, not shillings, but grains of +sand or molecules of hydrogen. + +D'Alembert's principal explanations are, first, that true expectation +\index{D'Alembert!Petersburg Paradox@{and Petersburg Paradox}}% +is not necessarily the product of probability and +profit (a view which has been discussed above), and second, that +very long runs are not only very improbable, but do not occur +at all. + +The\Pagelabel{317} next type of solution is due, in the first instance, to Daniel +Bernoulli, and turns on the fact that no one but a miser regards +the desirability of different sums of money as directly proportional +to their amount; as Buffon says, ``L'avare est comme le +\index{Buffon}% +mathématicien: tous deux estiment l'argent par sa quantité +numérique.'' Daniel Bernoulli deduced a formula from the +assumption that the importance of an increment is inversely +proportional to the size of the fortune to which it is added. +Thus, if $x$~is the `physical' fortune and $y$~the `moral' fortune, +\[ +dy = k\, \frac{dx}{x}, +\] +or $y = k\log\dfrac{x}{a}$, where $k$~and~$a$ are constants. + +On the basis of this formula of Bernoulli's a considerable +%% -----File: 329.png---Folio 318------- +\index{Cramer and Petersburg Paradox}% +\index{Petersburg Paradox!psychology of}% +\index{Todhunter|inote}% +theory has been built up both by Bernoulli\footnote + {``Specimen Theoriae Novae de Mensura Sortis,'' \textit{Comm.\ Acad.\ Petrop.}\ + vol.~v.\ for 1730 and~1731, pp.~175--192 (published 1738). See Todhunter, + pp.~213 \textit{et~seq}.} +himself and by +\index{Laplace}% +Laplace.\footnote + {\textit{Théorie analytique}, chap.~x.\ ``De l'espérance morale,'' pp.~432--445.} +It leads easily to the further formula--- +\[ +x = (a+x_1)p_1(a+x_2)p_2\ldots, +\] +where $a$ is the initial `physical' fortune, $p_1$,~etc., the probabilities +of obtaining increments $x_1$,~etc., to~$a$, and $x$~the `physical' fortune +whose present possession would yield the same `moral' fortune +as does the expectation of the various increments $x_1$,~etc. By +means of this formula Bernoulli shows that a man whose fortune +is £$1000$ may reasonably pay a £$6$~stake in order to play the +Petersburg game with £$1$~units. Bernoulli also mentions two +solutions proposed by Cramer. In the first all sums greater +than~$2^{24}$ ($16,777,116$) are regarded as `morally' equal; this +leads to~£$13$ as the fair stake. According to the other formula +the pleasure derivable from a sum of money varies as the square +root of the sum; this leads to £$2:9$s.\ as the fair stake. But +little object is served by following out these arbitrary hypotheses. + +As a solution of the Petersburg problem this line of thought +is only partially successful: if increases of `physical' fortune +beyond a certain finite limit can be regarded as `morally' +negligible, Peter's claim for an infinite initial stake from Paul is, +it is true, no longer equitable, but with any reasonable law of +diminution for successive increments Paul's stake will still remain +paradoxically large. Daniel Bernoulli's suggestion is, however, +of considerable historical interest as being the first explicit +attempt to take account of the important conception known to +modern economists as the diminishing marginal utility of money,---a +\index{Marginal utility}% +conception on which many important arguments are founded +relating to taxation and the ideal distribution of wealth. + +Each of the above solutions probably contains a part of the +psychological explanation. We are unwilling to be Paul, partly +because we do not believe Peter will pay us if we have good +fortune in the tossing, partly because we do not know what we +should do with so much money or sand or hydrogen if we won it, +partly because we do not believe we ever should win it, and +partly because we do not think it would be a rational act to risk +%% -----File: 330.png---Folio 319------- +\index{Bernoulli's Theorem|inote}% +\index{Bradley|inote}% +\index{Laurent and gambling}% +an infinite sum or even a very large finite sum for an infinitely +larger one, whose attainment is infinitely unlikely. + +When we have made the proper hypotheses and have eliminated +these elements of psychological doubt, the theoretic dispersal +of what element of paradox remains must be brought about, I +think, by a development of the theory of risk. It is primarily +\index{Risk!Petersburg Paradox@{and Petersburg Paradox}}% +the great \emph{risk} of the wager which deters us. Even in the case +where the number of tosses is in no case to exceed a finite number, +the risk~$R$, as already defined, may be very great, and the relative +risk~$\dfrac{R}{E}$ will be almost unity. Where there is no limit to the +number of tosses, the risk is infinite. A relative risk, which +approaches unity, may, it has been already suggested, be a factor +which must be taken into account in ethical calculation. + +\Paragraph{10.} In establishing the doctrine, that all private gambling +\index{Gambling}% +must be with certainty a losing game, precisely contrary arguments +are employed to those which do service in the Petersburg +problem. The argument that ``you must lose if only you go on +long enough'' is well known. It is succinctly put by Laurent:\footnote + {\textit{Calcul des probabilités}, p.~129.} +Two players $A$~and~$B$ have $a$~and~$b$ francs respectively. $f(a)$~is +the chance that $A$~will be ruined. Thus $f(a)=\dfrac{b}{a+b}$,\footnote + {This would possibly follow from the theorem of Daniel Bernoulli. The + reasoning by which Laurent obtains it seems to be the result of a mistake.} +so that +the poorer a gambler is, relatively to his opponent, the more +likely he is to be ruined. But further, if $b=\infty$, $f(a)=1$, \ie~ruin +is certain. The infinitely rich gambler is the public. It is against +the public that the professional gambler plays, and his ruin is +therefore certain. + +Might not Poisson and Condorcet reply, The conditions of +\index{Condorcet!gambling@{and gambling}}% +\index{Poisson!gambling@{and gambling}}% +the game \emph{imply} contradiction, for no gambler plays, as this argument +supposes, for ever?\footnote + {Cf.\ also Mr.~Bradley, \textit{Logic}, p.~217.} +At the end of any \emph{finite} quantity of +play, the player, even if he is not the public, \emph{may} finish with +winnings of any finite size. The gambler is in a worse position if +his capital is smaller than his opponents'---at poker, for instance, +or on the Stock Exchange. This is clear. But our desire for +moral improvement outstrips our logic if we tell him that he +\emph{must} lose. Besides it is paradoxical to say that everybody +%% -----File: 331.png---Folio 320------- +individually must lose and that everybody collectively must win. +For every individual gambler who loses there is an individual +gambler or syndicate of gamblers who win. The true moral is +this, that poor men should not gamble and that millionaires +should do nothing else. But millionaires gain nothing by gambling +with one another, and until the poor man departs from the +path of prudence the millionaire does not find his opportunity. +If it be replied that in fact most millionaires are men originally +poor who departed from the path of prudence, it must be +admitted that the poor man is not doomed with certainty. +Thus the philosopher must draw what comfort he can from the +conclusion with which his theory furnishes him, that millionaires +are often fortunate fools who have thriven on unfortunate +ones.\footnote + {From the social point of view, however, this moral against gambling may + be drawn---that those who start with the largest initial fortunes are most likely + to win, and that a given increment to the wealth of these benefits them, on the + assumption of a diminishing marginal utility of money, less than it injures those + from whom it is taken.} + +\Paragraph{11.} In conclusion we may discuss a little further the conception +of `moral' risk, raised in §\;8 and at the end of §\;9. Bernoulli's +\index{Risk!moral@{`\textit{moral}'}}% +formula crystallises the undoubted truth that the value of a sum +of money to a man varies according to the amount he already +possesses. But does the value of an amount of goodness also +vary in this way? May it not be true that the addition of a given +good to a man who already enjoys much good is less good than +its bestowal on a man who has little? If this is the case, it +follows that a smaller but relatively certain good is better than +a greater but proportionately more uncertain good. + +In order to assert this, we have only to accept a particular +theory of organic goodness, applications of which are common +enough in the mouths of political philosophers. It is at the root +of all principles of equality, which do not arise out of an assumed +diminishing marginal utility of money. It is behind the numerous +arguments that an equal distribution of benefits is better than a +very unequal distribution. If this is the case, it follows that, the +sum of the goods of all parts of a community taken together +being fixed, the organic good of the whole is greater the more +equally the benefits are divided amongst the individuals. If the +doctrine is to be accepted, moral risks, like financial risks, must +not be undertaken unless they promise a profit actuarially. +%% -----File: 332.png---Folio 321------- +\index{Butler, Bishop!risk@{and risk}}% + +There is a great deal which could be said concerning such a +doctrine, but it would lead too far from what is relevant to the +study of Probability. One or two instances of its use, however, +may be taken from the literature of Probability. In his essay, +``Sur l'application du calcul des probabilités à l'inoculation de +la petite vérole,''\footnote + {\textit{Opuscules mathématiques}, vol.~ii.} +D'Alembert points out that the community +\index{D'Alembert}% +would gain on the average if, by sacrificing the lives of one in five +of its citizens, it could ensure the health of the rest, but he argues +that no legislator could have the right to order such a sacrifice. +Galton, in his \textit{Probability, the Foundation of Eugenics}, employed +\index{Galton}% +an argument which depends essentially on the same point. +Suppose that the members of a certain class cause an average +detriment~$M$ to society, and that the mischiefs done by the +several individuals differ more or less from~$M$ by amounts whose +average is~$D$, so that $D$~is the average amount of the individual +deviations, all regarded as positive, from~$M$; then, Galton argued, +the smaller $D$~is, the stronger is the justification for taking such +drastic measures against the propagation of the class as would +be consonant to the feelings, if it were known that each individual +member caused a detriment~$M$. The use of such arguments +seems to involve a qualification of the simple ethical doctrine +that right action should make the sum of the benefits of the +several individual consequences, each multiplied by its probability, +a maximum. + +On the other hand, the opposite view is taken in the \textit{Port Royal +Logic} and by Butler, when they argue that everything ought to +be sacrificed for the hope of heaven, even if its attainment be +thought infinitely improbable, since ``the smallest degree of +facility for the attainment of salvation is of higher value than +all the blessings of the world put together.''\footnote + {\textit{Port Royal Logic} (Eng.\ trans.), p.~369: ``It belongs to infinite things alone, +\index{Port Royal logic}% + as eternity and salvation, that they cannot be equalled by any temporal advantage; + and thus we ought never to place them in the balance with any of the + things of the world. This is why the smallest degree of facility for the attainment + of salvation is of higher value than all the blessings of the world put + together\ldots.''} +The argument is, +that we ought to follow a course of conduct which may with the +slightest probability lead to an infinite good, until it is logically +disproved that such a result of our action is impossible. The +Emperor who embraced the Roman Catholic religion, not because +%% -----File: 333.png---Folio 322------- +\index{Bernoulli, Jac.!second axiom of}% +\index{Probability, and relevant knowledge!truth@{and truth}}% +\index{Truth and probability}% +he believed it, but because it offered insurance against a disaster +whose future occurrence, however improbable, he could not +certainly disprove, may not have considered, however, whether +the product of an infinitesimal probability and an infinite good +might not lead to a finite or infinitesimal result. In any case the +argument does not enable us to choose between different courses +of conduct, unless we have reason to suppose that one path is +\emph{more} likely than another to lead to infinite good. + +\Paragraph{12.} In estimating the risk, `moral' or `physical,' it must be +\index{Risk!moral@{`\textit{moral}'}}% +\index{Risk!physical@{`\textit{physical}'}}% +remembered that we cannot necessarily apply to individual +cases results drawn from the observation of a long series resembling +them in some particular. I am thinking of such arguments +as Buffon's when he names $\frac{1}{10,000}$ as the limit, beyond +\index{Buffon}% +which probability is negligible, on the ground that, being the +chance that a man of fifty-six \emph{taken at random} will die within a +day, it is practically disregarded by a man of fifty-six \emph{who knows +his health to be good}. ``If a public lottery,'' Gibbon truly pointed +\index{Gibbon}% +out, ``were drawn for the choice of an immediate victim, and if +our name were inscribed on one of the ten thousand tickets, +should we be perfectly easy?'' + +Bernoulli's second axiom,\footnote + {See \Pageref{76}.} +that in reckoning a probability +we must take everything into account, is easily forgotten in these +cases of statistical probabilities. The statistical result is so +attractive in its definiteness that it leads us to forget the more +vague though more important considerations which may be, in a +given particular case, within our knowledge. To a stranger the +probability that I shall send a letter to the post unstamped may +be derived from the statistics of the Post Office; for me those +figures would have but the slightest bearing upon the question. + +\Paragraph{13.} It has been pointed out already that no knowledge of +probabilities, less in degree than certainty, helps us to know what +conclusions are true, and that there is no direct relation between +the truth of a proposition and its probability. Probability begins +and ends with probability. That a scientific investigation +pursued on account of its probability will generally lead to truth, +rather than falsehood, is at the best only probable. The proposition +that a course of action guided by the most probable +considerations will generally lead to success, is not certainly true +and has nothing to recommend it but its probability. +%% -----File: 334.png---Folio 323------- + +The importance of probability can only be derived from the +judgment that it is \emph{rational} to be guided by it in action; and a +practical dependence on it can only be justified by a judgment +that in action we \emph{ought} to act to take some account of it. It is +for this reason that probability is to us the ``guide of life,'' since +to us, as Locke says, ``in the greatest part of our concernment, +\index{Locke}% +God has afforded only the Twilight, as I may so say, of Probability, +suitable, I presume, to that state of Mediocrity and +Probationership He has been pleased to place us in here.'' +%% -----File: 335.png---Folio 324------- +%[Blank Page] +%% -----File: 336.png---Folio 325------- + + +\Part[Statistical Inference]{V}{The Foundations of Statistical +Inference} +%% -----File: 337.png---Folio 326------- +%[Blank Page] +%% -----File: 338.png---Folio 327------- +\index{Statistical inference|ifoll}% +\index{Statistics, and prediction!descriptive and inductive}% + + +\Chapter{XXVII}{The Nature of Statistical Inference} +\index{Inference!statistical|ifoll}% + +\Paragraph{1.} \First{The} Theory of Statistics, as it is now understood,\footnote + {See Yule, \textit{Introduction to Statistics}, pp.~1--5, for a very interesting account +\index{Yule!statistics@{and `\textit{statistics}'}}% + of the evolution of the meaning of the term \emph{statistics}.} +can be +divided into two parts which are for many purposes better kept +distinct. The first function of the theory is purely \emph{descriptive}. +It devises numerical and diagrammatic methods by which certain +salient characteristics of large groups of phenomena can be briefly +described; and it provides formulae by the aid of which we can +measure or summarise the variations in some particular character +which we have observed over a long series of events or instances. +The second function of the theory is \emph{inductive}. It seeks to extend +its description of certain characteristics of observed events to +the corresponding characteristics of other events which have not +been observed. This part of the subject may be called the +Theory of Statistical Inference; and it is this which is closely +bound up with the theory of probability. + +\Paragraph{2.} The union of these two distinct theories in a single science +is natural. If, as is generally the case, the development of +some inductive conclusion which shall go beyond the actually +observed instances is our ultimate object, we naturally choose +those modes of description, while we are engaged in our preliminary +investigation, which are most capable of extension +beyond the particular instances which they primarily describe. +But this union is also the occasion of a great deal of confusion. The +statistician, who is mainly interested in the technical methods of +his science, is less concerned to discover the precise conditions in +which a description can be legitimately extended by induction. +\index{Induction!statistics@{and statistics}|ifoll}% +He slips somewhat easily from one to the other, and having +found a complete and satisfactory mode of description he +%% -----File: 339.png---Folio 328------- +\index{Frequency curves!statistics@{and statistics}}% +\index{definition of!from statistics}% +may take less pains over the transitional argument, which is +to permit him to use this description for the purposes of +generalisation. + +One or two examples will show how easy it is to slip from +description into generalisation. Suppose that we have a series +of similar objects one of the characteristics of which is under +observation;---a number of persons, for example, whose age at +death has been recorded. We note the proportion who die at +each age, and plot a diagram which displays these facts graphically. +We then determine by some method of curve fitting a +mathematical frequency curve which passes with close approximation +through the points of our diagram. If we are given the +equation to this curve, the number of persons who are comprised +in the statistical series, and the degree of approximation (whether +to the nearest year or month) with which the actual age has been +recorded, we have a very complete and succinct account of one +particular characteristic of what may constitute a very large +mass of individual records. In providing this comprehensive +description the statistician has fulfilled his first function. But in +determining the accuracy with which this frequency curve can be +employed to determine the probability of death at a given age +in the population at large, he must pay attention to a new class +of considerations and must display a different kind of capacity. +He must take account of whatever extraneous knowledge may be +available regarding the sample of the population which came +under observation, and of the mode and conditions of the observations +themselves. Much of this may be of a vague kind, and most +of it will be necessarily incapable of exact, numerical, or statistical +treatment. He is faced, in fact, with the normal problems of +inductive science, \emph{one} of the data, which must be taken into +account, being given in a convenient and manageable form by +the methods of descriptive statistics. + +Or suppose, again, that we are given, over a series of years, +the marriage rate and the output of the harvest in a certain area +of population. We wish to determine whether there is any +apparent degree of correspondence between the variations of the +two within this field of observation. It is technically difficult to +measure such degree of correspondence as may appear to exist +between the variations in two series, the terms of which are in +some manner associated in couples,---by coincidence, in this case, +%% -----File: 340.png---Folio 329------- +\index{Error, probable}% +\index{Whitehead, and frequency theory!invalid inference@{and invalid inference}|inote}% +of time and place. By the method of correlation tables and +\index{Correlation}% +correlation coefficients the descriptive statistician is able to effect +this object, and to present the inductive scientist with a highly +significant part of his data in a compact and instructive form. +But the statistician has not, in calculating these coefficients of +observed correlation, covered the whole ground of which the inductive +scientist must take cognisance. He has recorded the +results of the observations in circumstances where they cannot +be recorded so clearly without the aid of technical methods; but +the precise nature of the conditions in which the observations +took place and the numerous other considerations of one sort or +another, of which we must take account when we wish to +generalise, are not usually susceptible of numerical or statistical +expression. + +The truth of this is obvious; yet, not unnaturally, the more +complicated and technical the preliminary statistical investigations +become, the more prone inquirers are to mistake the statistical +description for an inductive generalisation.\footnote + {Cf.\ Whitehead, \textit{Introduction to Mathematics}, p.~27: ``There is no more + common error than to assume that, because prolonged and accurate mathematical + calculations have been made, the application of the result to some fact + of nature is absolutely certain.''} +This tendency, +which has existed in some degree, as, I think, the whole history of +the subject shows, from the eighteenth century down to the +present time, has been further encouraged by the terminology in +ordinary use. For several statistical coefficients are given the +same name when they are used for purely descriptive purposes, +as when corresponding coefficients are used to measure the force +or the precision of an induction. The term `probable error,' +for example, is used \emph{both} for the purpose of supplementing +and improving a statistical description, \emph{and} for the +purpose of indicating the precision of some generalisation. +The term `correlation' itself is used \emph{both} to describe an +observed characteristic of particular phenomena \emph{and} in the +enunciation of an inductive law which relates to phenomena +in general. + +\Paragraph{3.} I have been at pains to enforce this contrast between +statistical description and statistical induction, because the +chapters which follow are to be entirely about the latter, whereas +nearly all statistical treatises are mainly concerned with the +former. My object will be to analyse, so far as I can, the logical +%% -----File: 341.png---Folio 330------- +\index{Frequency, statistical}% +\index{Great Numbers, Law of}% +basis of statistical modes of argument. This involves a double +task. To mark down those which are invalid amongst arguments +having the support of authority is relatively easy. +The other branch of our investigation, namely, to analyse +the ground of validity in the case of those arguments the +force of which all of us do in fact admit, presents the same +kind of fundamental difficulties as we met with in the case +of Induction. + +\Paragraph{4.} The arguments with which we have to deal fall into three +main classes: + +(i.) Given the probability relative to certain evidence of each +of a series of events, what are the probabilities, relative to the +same evidence, of various proportionate frequencies of occurrence +for the events over the whole series? Or more briefly, how often +may we expect an event to happen over a series of occasions, given +its probability on each occasion? + +(ii.) Given the frequency with which an event has occurred +on a series of occasions, with what probability may we expect it +on a further occasion? + +(iii.) Given the frequency with which an event has occurred +on a series of occasions, with what frequency may we probably +expect it on a further series of occasions? + +In the first type of argument we seek to infer an unknown +statistical frequency from an \textit{à~priori} probability. In the second +type we are engaged on the inverse operation, and seek to base +the calculation of a probability on an observed statistical frequency. +In the third type we seek to pass from an observed +statistical frequency, not merely to the probability of an individual +occurrence, but to the probable values of other unknown statistical +frequencies. + +Each of these types of argument can be further complicated +by being applied not simply to the occurrence of a simple event +but to the concurrence under given conditions of two or more +events. When this two or more dimensional classification replaces +the one dimensional, the theory becomes what is sometimes +termed Correlation, as distinguished from simple Statistical +\index{Correlation!statistical frequency@{and statistical frequency}}% +Frequency. + +\Paragraph{5.} In \Chapref{XXVIII}. I touch briefly on the observed +phenomena which have given rise to the so-called Law of +Great Numbers, and the discovery of which first set statistical +%% -----File: 342.png---Folio 331------- +investigation going. In \Chapref{XXIX}. the first type of argument, +as classified above, is analysed, and the conditions which +are required for its validity are stated. The crucial problem +of attacking the second and third types of argument is the +subject of my concluding chapters. +%% -----File: 343.png---Folio 332------- +\index{Halley and mortality statistics}% + + +\Chapter{XXVIII}{The Law of Great Numbers} + +\begin{Quote} +Natura quidem suas habet consuetudines, natas ex reditu causarum, sed non +nisi \textgreek{<ws `ep`i t`o pol'u}. Novi morbi inundant subiande humanum genus, quodsi +ergo de mortibus quotcunque experimenta feceris, non ideo naturae rerum limites +posuisti, ut pro futuro variare non possit.---\textsc{Leibniz} \textit{in a letter to Bernoulli, +December}~3, 1703. +\end{Quote} + +\Paragraph{1.} \First{It} has always been known that, while some sets of events +\emph{invariably} happen together, other sets \emph{generally} happen together. +That experience shows one thing, while not always a sign of +another, to be a usual or probable sign of it, must have been one +of the earliest and most primitive forms of knowledge. If a dog +is \emph{generally} given scraps at table, that is sufficient for him to judge +it reasonable to be there. But this kind of knowledge was slow +to be made precise. Numerous experiments must be carefully +recorded before we can know at all accurately \emph{how} usual the +association is. It would take a dog a long time to find out that +he was given scraps except on fast days, and that there was the +same number of these in every year. + +The necessary kind of knowledge began to be accumulated +during the seventeenth and eighteenth centuries by the early +statisticians. Halley and others began to construct mortality +tables; the proportion of the births of each sex were tabulated; +and so forth. These investigations brought to light a new fact +which had not been suspected previously---namely, that in certain +cases of partial association the degree of association, \ie~the proportion +of instances in which it existed, shows a very surprising +regularity, and that this regularity becomes more marked the +greater the number of the instances under consideration. It was +found, for example, not merely that boys and girls are born on +the whole in about equal proportions, but that the proportion, +%% -----File: 344.png---Folio 333------- +\index{Bernoulli, Jac.!regular frequency@{and regular frequency}}% +\index{Bortkiewicz, von, and great numbers|inote}% +\index{Great Numbers, Law of|ifoll}% +\index{Lotteries|inote}% +\index{Quetelet|inote}% +\index{Süssmilch and regular frequencies}% +which is not one of complete equality, tends everywhere, when +the number of recorded instances becomes large, to approximate +towards a certain definite figure. + +During the eighteenth century matters were not pushed much +further than this, that in certain cases, of which comparatively +few were known, there was this surprising regularity, increasing +in degree as the instances became more numerous. Bernoulli, +however, took the first step towards giving it a theoretical basis +by showing that, if the \textit{à~priori} probability is known throughout, +then (subject to certain conditions which he himself did not make +clear) \emph{in the long run} a certain determinate frequency of occurrence +is to be expected. Süssmilch (\textit{Die göttliche Ordnung in den +Veränderungen des menschlichen Geschlechts}, 1741) discovered a +theological interest in these regularities. Such ideas had become +sufficiently familiar for Gibbon to characterise the results of +\index{Gibbon}% +probability as ``so true in general, so fallacious in particular.'' +Kant found in them (as many later writers have done) some +\index{Kant}% +bearing on the problem of Free Will.\footnote + {In \textit{Idee zu einer allgemeinen Geschichte in weltbürgerlicher Absicht}, 1784. For + a discussion of this passage and for the connection between Kant and Süssmilch, + see Lottin's \textit{Quetelet}, pp.~367,~368.} + +But with the nineteenth century came bolder theoretical +methods and a wider knowledge of facts. After proving his +\index{Bernoulli's Theorem}% +extension of Bernoulli's Theorem,\footnote + {See \Pageref{345}.} +Poisson applied it to the +\index{Poisson!great numbers@{and great numbers}}% +observed facts, and gave to the principle underlying these +regularities the title of the \textit{Law of Great Numbers}. ``Les choses +de toutes natures,'' he wrote,\footnote + {\textit{Recherches}, pp.~7--12. Von Bortkiewicz (\textit{Kritische Betrachtungen}, 1st~part, + pp.~655--660) has maintained that Poisson intended to state his principle in a + less general way than that in which it has been generally taken, and that he was + misunderstood by Quetelet and others. If we attend only to Poisson's contributions + to \textit{Comptes Rendus} in 1835 and~1836 and to the examples he gives + there, it is possible to make out a good case for thinking that he intended his + law to extend only to cases where certain strict conditions were fulfilled. But + this is not the spirit of his more popular writings or of the passage quoted above. + At any rate, it is the fashion, in which Poisson influenced his contemporaries, + that is historically interesting; and this is certainly not represented by Von + Bortkiewicz's interpretation.} +``sont soumises à une loi universelle +qu'on peut appeler la loi des grands nombres\ldots. De ces +exemples de toutes natures, il résulte que la loi universelle des +grands nombres est déjà pour nous un fait général et incontestable, +résultant d'expériences qui ne se démentent jamais.'' This is +the language of exaggeration; it is also extremely vague. But +%% -----File: 345.png---Folio 334------- +it is exciting; it seems to open up a whole new field to scientific +investigation; and it has had a great influence on subsequent +thought. Poisson seems to claim that, in the whole field of chance +and variable occurrence, there really exists, amidst the apparent +disorder, a discoverable system. Constant causes are always +at work and assert themselves in the long run, so that each class +of event does eventually occur in a definite proportion of cases. +It is not clear how far Poisson's result is due to \textit{à~priori} reasoning, +and how far it is a natural law based on experience; but it is +represented as displaying a certain harmony between natural +law and the \textit{à~priori} reasoning of probabilities. + +Poisson's conception was mainly popularised through the +writings of Quetelet. In 1823 Quetelet visited Paris on an +\index{Quetelet}% +astronomical errand, where he was introduced to Laplace and +\index{Laplace!Quetelet@{and Quetelet}}% +came into touch with ``la grande école française.'' ``Ma jeunesse +et mon zèle,'' he wrote in later years, ``ne tardèrent pas à me +mettre en rapport avec les hommes les plus distingués de cette +époque; qu'on me permette de citer Fourier, Poisson, Lacroix, +spécialement connus, comme Laplace, par leurs excellents écrits +sux la théorie mathématique des probabilités\ldots. C'est donc +au milieu des savants, statisticiens, et économistes de ce temps +que j'ai commencé mes travaux.''\footnote + {For the details of the life of Quetelet and for a very full discussion of his + writings with special reference to Probability, see Lottin's \textit{Quetelet, statisticien et + sociologue}.} +Shortly afterwards began +his long series of papers, extending down to 1873, on the application +of Probability to social statistics. He wrote a text-book +on Probability in the form of letters for the instruction of the +Prince Consort. + +Before accepting in 1815 at the age of nineteen (with a view to +a livelihood) a professorship of mathematics, Quetelet had studied +as an art student and written poetry; a year later an opera, of +which he was part-author, was produced at Ghent. The character +of his scientific work is in keeping with these beginnings. There +is scarcely any permanent, accurate contribution to knowledge +which can be associated with his name. But suggestions, projects, +far-reaching ideas he could both conceive and express, and +he has a very fair claim, I think, to be regarded as the parent of +modern statistical method. + +Quetelet very much increased the number of instances of the +%% -----File: 346.png---Folio 335------- +\index{Mendelism and statistics}% +Law of Great Numbers, and also brought into prominence a +slightly variant type of it, of which a characteristic example is +the law of height, according to which the heights of any considerable +sample taken from any population tend to group themselves +according to a certain well-known curve. His instances were +chiefly drawn from social statistics, and many of them were of a +kind well calculated to strike the imagination---the regularity of +the number of suicides, ``l'effrayante exactitude avec laquelle +les crimes se reproduisent,'' and so forth. Quetelet writes +\index{Quetelet}% +with an almost religious awe of these mysterious laws, and +certainly makes the mistake of treating them as being as +adequate and complete in themselves as the laws of physics, +and as little needing any further analysis or explanation.\footnote + {Compare, for instance, the following passage from \textit{Recherches sur le penchant + au crime}: ``Il me semble que ce qui se rattache à l'espèce humaine, considérée + en masse, est de l'ordre des faits physiques; plus le nombre des individus est + grand, plus la volonté individuelle s'efface et laisse prédominer la série des faits + généraux qui dépendent des causes générales\ldots. Ce sont ces causes qu'il + s'agit de saisir, et dès qu'on les connaîtra, on en déterminera les effets pour la + société comme on détermine les effets par les causes dans les sciences physiques.''} +Quetelet's sensational language may have given a considerable +impetus to the collection of social statistics, but it also involved +statistics in a slight element of suspicion in the minds of some +who, like Comte, regarded the application of the mathematical +\index{Comte!statistics@{and statistics}}% +calculus of probability to social science as ``purement chimérique +\index{Calculus of Probability!Sociology@{and Sociology}}% +et, par conséquent, tout à fait vicieuse.'' The suspicion of +quackery has not yet disappeared. Quetelet belongs, it must be +admitted, to the long line of brilliant writers, not yet extinct, who +have prevented Probability from becoming, in the scientific salon, +perfectly respectable. There is still about it for scientists a +smack of astrology, of alchemy. + +The progress of the conception since the time of Quetelet has +been steady and uneventful; and long strides towards this perfect +respectability have been taken. Instances have been multiplied +and the conditions necessary for the existence of statistical +stability have been to some extent analysed. While the most +fruitful applications of these methods have still been perhaps, +as at first, in social statistics and in errors of observation, a +number of uses for them have been discovered in quite recent +times in the other sciences; and the principles of Mendelism +have opened out for them a great field of application throughout +biology. +%% -----File: 347.png---Folio 336------- +\index{Statistical frequency, theory of!stability of}% + +\Paragraph{2.} The existence of numerous instances of the Law of Great +Numbers, or of something of the kind, is absolutely essential for +the importance of Statistical Induction. Apart from this the more +precise parts of statistics, the collection of facts for the prediction +of future frequencies and associations, would be nearly useless. +But the `Law of Great Numbers' is not at all a good name for the +principle which underlies Statistical Induction. The `Stability +of Statistical Frequencies' would be a much better name for it. +The former suggests, as perhaps Poisson intended to suggest, but +\index{Poisson!great numbers@{and great numbers}}% +what is certainly false, that every class of event shows statistical +regularity of occurrence if only one takes a sufficient number of +instances of it. It also encourages the method of procedure, by +which it is thought legitimate to take any observed degree of +frequency or association, which is shown in a fairly numerous +set of statistics, and to assume with insufficient investigation +that, because the statistics are \emph{numerous}, the observed degree of +frequency is therefore \emph{stable}. Observation shows that some +statistical frequencies are, within narrower or wider limits, stable. +But stable frequencies are not very common, and cannot be +assumed lightly. + +The gradual discovery, that there are certain classes of +phenomena, in which, though it is impossible to predict what will +happen in each individual case, there is nevertheless a regularity +of occurrence if the phenomena be considered together in successive +sets, gives the clue to the abstract inquiry upon which we +are about to embark. +%% -----File: 348.png---Folio 337------- +\index{Bernoulli's Theorem|ifoll}% +\index{Probability, and relevant knowledge!truth frequency@{and truth frequency}|ifoll}% + + +\Chapter{XXIX}{The Use of \textit{à~priori} Probabilities for the Prediction of +Statistical Frequency---the Theorems of Bernoulli, +Poisson, and Tchebycheff} + +\begin{Quote} +Hoc igitur est illud Problema, quod evulgandum hoc loco proposui, postquam +jam per vicennium pressi, et cujus tum novitas, tum summa utilitas cum +pari conjuncta difficultate omnibus reliquis hujus doctrinae capitibus pondus +et pretium superaddere potest.---\textsc{Bernoulli}.\footnote + {\textit{Ars Conjectandi}, p.~227.} +\end{Quote} + +\Paragraph{1.} \First{Bernoulli's} Theorem is generally regarded as the central +theorem of statistical probability. It embodies the first attempt +to deduce the measures of statistical frequencies from the measures +of individual probabilities, and it is a sufficient fruit of the twenty +years which Bernoulli alleges that he spent in reaching his result, +if out of it the conception first arose of general laws amongst +masses of phenomena, in spite of the uncertainty of each particular +case. But, as we shall see, the theorem is only valid subject +to stricter qualifications, than have always been remembered, +and in conditions which are the exception, not the rule. + +The problem, to be discussed in this chapter, is as follows: +Given a series of occasions, the probability\footnote + {In the simplest cases, dealt with by Bernoulli, these probabilities are all + supposed equal.} +of the occurrence +of a certain event at each of which is known relative to certain +initial \textit{data}~$h$, on what proportion of these occasions may we +reasonably anticipate the occurrence of the event? Given, that +is to say, the individual probability of each of a series of events +\textit{à~priori}, what statistical frequency of occurrence of these events +is to be anticipated over the whole series? Beginning with +Bernoulli's Theorem, we will consider the various solutions of +this problem which have been propounded, and endeavour to +%% -----File: 349.png---Folio 338------- +determine the proper limits within which each method has +validity. + +\Paragraph{2.} Bernoulli's Theorem in its simplest form is as follows: If +the probability of an event's occurrence under certain conditions +is~$p$, then, if these conditions are present on $m$~occasions, the most +probable number of the event's occurrences is~$mp$ (or the nearest +integer to this), \ie\ the most probable \emph{proportion} of its occurrences +to the total number of occasions is~$p$: further, the probability +that the proportion of the event's occurrences will diverge from +the most probable proportion~$p$ by less than a given amount~$b$, +increases as $m$~increases, the value of this probability being +calculable by a process of approximation. + +The probability of the event's occurring $n$~times and failing +$m - n$~times out of the $m$~occasions is (subject to certain conditions +to be elucidated later) $p^n q^{m-n}$ multiplied by the coefficient of +this expression in the expansion of $(p + q)^m$, where $p + q = 1$. If +we write $n = mp - h$, this term is $\dfrac{m!}{(mp - h)!(mq + h)!} p^n q^{m-n}$. It +is easily shown that this is a maximum when $h = 0$, \ie\ when $n = mp$ +(or the nearest integer to this, where $mp$~is not integral). This +result constitutes the first part of Bernoulli's Theorem. + +For the second part of the theorem some method of approximation +is required. Provided that $m$~is large, we can simplify +the expression $\dfrac{m!}{(mp - h)!(mq + h)!} p^n q^{m - n}$ by means of Stirling's +Theorem, and obtain as its approximate value +\[ +\frac{1}{\sqrt{2\pi mpq}}e^{-\frac{h^2}{2mpq}} +\] +As before, this is a maximum when $h = 0$, \ie\ when $n = mp$. + +It is possible, of course, by more complicated formulae to +obtain closer approximations than this.\footnote + {See, \eg, Bowley, \textit{Elements of Statistics}, p.~298. The objection about to + be raised does not apply to these closer approximations.} +But there is an objection, +which can be raised to this approximation, quite distinct +from the fact that it does not furnish a result correct to as many +places of decimals as it might. This is, that the approximation +is independent of the sign of~$h$, whereas the original expression +is not thus independent. That is to say, the approximation +implies a symmetrical distribution for different values of~$h$ about +%% -----File: 350.png---Folio 339------- +\index{Czuber|inote}% +the value for $h = 0$; while the expression under approximation +is unsymmetrical. It is easily seen that this want of symmetry +is appreciable unless $mpq$ is large. We ought, therefore, to have +laid it down as a condition of our approximation, not only that +$m$~must be large, but also that $mpq$ must be large. Unlike most +of my criticisms, this is a mathematical, rather than a logical +point. I recur to it in §\;15. + +``Par une fiction qui rendra les calculs plus faciles'' (to quote +Bertrand), we now replace the integer~$h$ by a continuous variable~$z$ +\index{Bertrand!Bernoulli's Theorem@{and Bernoulli's Theorem}}% +and argue that the probability that the amount of the divergence +from the most probable value~$mp$ will lie between $z$~and~$z + dz$, +is +\[ +\frac{1}{\sqrt{2\pi mpq}} e^{-\frac{z^2}{2mpq}} dz . +\] +This `fiction' will do no harm so long as it is remembered that we +are now dealing with a particular kind of approximation. The +probability that the divergence~$h$ from the most probable value~$mp$ +will be less than some given quantity~$a$ is, therefore, +\[ +\frac{1}{\sqrt{2\pi mpq}} \int^{+a}_{-a} e^{-\frac{z^2}{2mpq}} dz . +\] +If we put $\dfrac{z}{\sqrt{2mpq}} = t$, this is equal to +\[ +\frac{2}{\sqrt{\pi}} \int^{\frac{a}{\sqrt{2mpq}}}_{0} e^{-t^2} dt . +\] +Thus, if we write $a = \sqrt{2mpq} \gamma$, the probability\footnote + {The replacement of the integer~$h$ by the continuous variable~$z$ may render + the formula rather deceptive. It is certain, for example, that the error does not + lie between $h$~and~$h + 1$.} +that the +number of occurrences will lie between +\[ +mp + \sqrt{2mpq}\gamma \text{ and } +mp - \sqrt{2mpq}\gamma +\] +is measured by\footnote + {The\Pagelabel{339} above proof follows the general lines of Bertrand's (\textit{Calcul des probabilités}, + chap.~iv.). Some writers, using rather more precision, give the result as + \[ + \frac{2}{\sqrt{\pi}} + \int^{\gamma}_{0} e^{-t^2}\, dt + \frac{e^{-\gamma^2}}{\sqrt{2\pi mpq}} + \] + (\eg\ Laplace, by the use of Euler's Theorem, and more recently Czuber, + \textit{Wahrscheinlichkeitsrechnung}, vol.~i.\ p.~121). As the whole formula is approximate, + the simpler expression given in the text is probably not less satisfactory in + practice. See also Czuber, \textit{Entwicklung}, pp.~76,~77, and Eggenberger, \textit{Beiträge + zur Darstellung des Bernoullischen Theorems}.} +$\displaystyle\frac{2}{\sqrt{\pi}} \int^{\gamma}_{0} e^{-t^2} dt$. This same expression measures +%% -----File: 351.png---Folio 340------- +\index{Eggenberger|inote}% +the probability that the \emph{proportion} of occurrences will lie +between +\[ +p + \sqrt\frac{2pq}{m}\gamma \text{ and } +p - \sqrt\frac{2pq}{m}\gamma. +\] +The different values of the integral $\dfrac{2}{\sqrt{\pi}}\displaystyle\int_0^te^{-t^2}dt=\Theta(t)$ are given +in tables.\footnote + {A list of the principal tables is given by Czuber, \textit{loc.\ cit.}\ vol.~i.\ p.~122.} +\index{Czuber!Bernoulli's Theorem@{and Bernoulli's Theorem}|inote}% + +The probability that the proportion of occurrences will lie +between given limits varies with the magnitude of~$\sqrt{\dfrac{2pq}{m}}$, and +this expression is sometimes used, therefore, to measure the +`precision' of the series. Given the \textit{à~priori} probabilities, the +precision varies inversely with the \emph{square root} of the number of +instances. Thus, while the probability that the \emph{absolute} divergence +will be less than a given amount~$a$ decreases, the probability +that the corresponding \emph{proportionate} divergence (\ie\ the absolute +divergence divided by the number of instances) will be less than +a given amount~$b$, increases, as the number of instances increases. +This completes the second part of Bernoulli's Theorem. + +\Paragraph{3.} Bernoulli himself was not acquainted with Stirling's +theorem, and his proof differs a good deal from the proof outlined +in §\;2. His final enunciation of the theorem is as follows: If in +each of a given series of experiments there are $r$~contingencies +favourable to a given event out of a total number of contingencies~$t$, +so that $\dfrac{r}{t}$ is the probability of the event at each experiment, +then, given any degree of probability~$c$, it is possible to make such +a number of experiments that the probability, that the proportionate +number of the event's occurrences will lie between +$\dfrac{r+1}{t}$ and $\dfrac{r-1}{t}$, is greater than~$c$.\footnote + {\textit{Ars Conjectandi}, p.~236 (I have translated freely). There is a brief account + of Bernoulli's proof in Todhunter's \textit{History}, pp.~71,~72. The problem is dealt +\index{Todhunter!Bernoulli's Theorem@{and Bernoulli's Theorem}|inote}% + with by Laplace, \textit{Théorie analytique}, livre~ii.\ chap.~iii. For an account of +\index{Laplace!Bernoulli's Theorem@{and Bernoulli's Theorem}}% + Laplace's proof see Todhunter's \textit{History}, pp.~548--553.} + +%% -----File: 352.png---Folio 341------- +\index{Ellis, Leslie!Bernoulli's Theorem@{and Bernoulli's Theorem}}% + +\Paragraph{4.} We seem, therefore, to have proved that, if the \textit{à~priori} +probability of an event under certain conditions is~$p$, the proportion +of times most probable \textit{à~priori} for the event's occurrence +on a series of occasions where the conditions are satisfied is also~$p$, +and that if the series is a long one the proportion is very unlikely +to differ widely from~$p$. This amounts to the principle +which Ellis\footnote + {\textit{On the Foundation of the Theory of Probabilities}: ``If the probability of a + given event be correctly determined, the event will on a long run of trials tend + to recur with frequency proportional to this probability. This is generally + proved mathematically. It seems to me to be true \textit{à~priori}\ldots. I have been + unable to sever the judgment that one event is more likely to happen than + another from the belief that in the long run it will occur more frequently.''} +and Venn have employed as the defining axiom of +\index{Venn!Bernoulli@{and Bernoulli}}% +probability, save that if the series is `long enough' the proportion, +\index{Laplace!Bernoulli's Theorem@{and Bernoulli's Theorem}}% +according to them, will \emph{certainly} be~$p$. Laplace\footnote + {\textit{Essai philosophique}, p.~53: ``On peut tirer du théorème précédent cette + conséquence qui doit être regardée comme une loi générale, savoir, que les + rapports des effets de la nature, sont à fort peu près constans, quand ces effets + sont considérés en grand nombre.''} +believed that the +theorem afforded a demonstration of a general law of nature, and +in his second edition published in 1814 he replaces\footnote + {Introduction, pp.~liii, liv.} +the eloquent +dedication, \textit{A Napoléon-le-Grand}, which prefaces the edition of +1812, by an explanation that Bernoulli's Theorem must always +bring about the eventual downfall of a great power which, drunk +with the love of conquest, aspires to a universal domination,---``c'est +encore un résultat du calcul des probabilités, confirmé +par de nombreuses et funestes expériences.'' + +\Paragraph{5.} Such is the famous Theorem of Bernoulli which some have +believed\footnote + {Even by Mr.~Bradley, \textit{Principles of Logic}, p.~214. After criticising Venn's +\index{Bradley!Bernoulli's Theorem@{and Bernoulli's Theorem}|inote}% + view he adds: ``It is false that the chances must be realised in a series. It is, + however, true that they most probably will be, and true again that this probability + is increased, the greater the length we give to our series.''} +to have a universal validity and to be applicable to \emph{all} +`properly calculated' probabilities. Yet the theorem exhibits +algebraical rather than logical insight. And, for reasons about +to be given, it will have to be conceded that it is only true of a +special class of cases and requires conditions, before it can be +legitimately applied, of which the fulfilment is rather the exception +than the rule. For consider the case of a coin of which +it is given that the two faces are either both heads or both tails: +at every toss, provided that the results of the other tosses are +unknown, the probability of heads is~$\frac{1}{2}$ and the probability of +tails is~$\frac{1}{2}$; yet the probability of $m$~heads and $m$~tails in $2m$~tosses +%% -----File: 353.png---Folio 342------- +is zero, and it is certain \textit{à~priori} that there will be either $2m$~heads +or none. Clearly Bernoulli's Theorem is inapplicable to +such a case. And this is but an extreme case of a normal +condition. + +For the first stage in the proof of the theorem assumes that, +if $p$~is the probability of one occurrence, $p^r$~is the probability of $r$~occurrences +running. Our discussion of the theorems of multiplication +\index{Multiplication!theorems of}% +will have shown how considerable an assumption this +involves. It assumes that a \emph{knowledge} of the fact that the event +has occurred on every one of the first $r-1$~occasions does not in +any degree affect the probability of its occurrence on the~$r$th. +Thus Bernoulli's Theorem is only valid if our initial \textit{data} are of +such a character that additional knowledge, as to the proportion +of failures and successes in one part of a series of cases is altogether +irrelevant to our expectation as to the proportion in another +part. If, for example, the initial probability of the occurrence +of an event under certain circumstances is one in a million, we +may only apply Bernoulli's Theorem to evaluate our expectation +over a million trials, if our original \textit{data} are of such a character +that, even after the occurrence of the event in every one of the +first million trials, the probability in the light of this additional +knowledge that the event will occur on the next occasion is still +no more than one in a million. + +Such a condition is very seldom fulfilled. If our initial probability +is partly founded upon experience, it is clear that it is +liable to modification in the light of further experience. It is, +in fact, difficult to give a concrete instance of a case in which the +conditions for the application of Bernoulli's Theorem are completely +fulfilled. At the best we are dealing in practice with a +good approximation, and can assert that no realised series of +moderate length can much affect our initial probability. If we +wish to employ the expression $\displaystyle\frac{2}{\sqrt \pi}\int_0^\gamma e^{-t^2}dt$ we are in a worse +position. For this is an approximate formula which requires for +its validity that the series should be \emph{long}; whilst it is precisely +in this event, as we have seen above, that the use of Bernoulli's +Theorem is more than usually likely to be illegitimate. + +\Paragraph{6.} The conditions, which have been described above, can be +expressed precisely as follows: +%% -----File: 354.png---Folio 343------- + +Let $_mx_n$ represent the statement that the event has occurred +on $m$ out of $n$ occasions and has not occurred on the others; and +let $_1x_1/h = p$, where $h$ represents our \textit{à~priori data}, so that $p$ is the +\textit{à~priori} probability of the event in question. Bernoulli's Theorem +then requires a series of conditions, of which the following is +typical: $_{m+1}x_{n+1}/_mx_n· h = _1x_1/h$, \ie\ the probability of the event +on the $n+1$th occasion must be unaffected by our knowledge of +its proportionate frequency on the first $n$~occasions, and must be +exactly equal to its \textit{à~priori} probability before the first occasion. + +Let us select one of these conditions for closer consideration. +If $y_r$ represents the statement that the event has occurred on each +of $r$ successive occasions, $y_r/h = y_r/y_{r-1}h· y_{r-1}/h$ and so on, so +that $y_r/h = \Prod\limits_{s=1}^{s=r}y_s/y_{s-1}h$. Hence if we are to have $y_r/h=p^r$, we +must have $y_s/y_{s-1}h=p$ for all values of~$s$ from $1$ to~$r$. But in +many particular examples $y_s/y_{s-1}h$ increases with~$s$, so that +$y_r/h>p^r$. Bernoulli's Theorem, that is to say, tends, if it is +carelessly applied, to exaggerate the rate at which the probability +of a given divergence from the most probable decreases as the +divergence increases. If we are given a penny of which we have +no reason to doubt the regularity, the probability of heads at +the first toss is~$\frac{1}{2}$; but if heads fall at every one of the first 999 +tosses, it becomes reasonable to estimate the probability of heads +at the thousandth toss at much more than~$\frac{1}{2}$. For the \textit{à~priori} +probability of its being a conjurer's penny, or otherwise biassed +so as to fall heads almost invariably, is not usually so infinitesimally +small as $(\frac{1}{2})^{1000}$. We can only apply Bernoulli's Theorem +with rigour for a prediction as to the penny's behaviour over a +series of a thousand tosses, if we have \textit{à~priori} such exhaustive +knowledge of the penny's constitution and of the other conditions +of the problem that $999$~heads running would not cause +us to modify in any respect our prediction \textit{à~priori}. + +\Paragraph{7.} It seldom happens, therefore, that we can apply Bernoulli's +Theorem with reference to a long series of natural events. For +in such cases we seldom possess the exhaustive knowledge which +is necessary. Even where the series is short, the perfectly +rigorous application of the Theorem is not likely to be legitimate, +and some degree of approximation will be involved in +utilising its results. + +Not so infrequently, however, artificial series can be devised +%% -----File: 355.png---Folio 344------- +in which the assumptions of Bernoulli's Theorem are relatively +legitimate.\footnote + {In the discussion in \Chapref{XVI}., \Pageref{170}, of the probability of a divergence + from an equality of heads and tails in coin-tossing, an example has been + given of the construction of an artificial series in which the application of + Bernoulli's Theorem is more legitimate than in the natural series.} +Given, that is to say, a proposition~$a_1$, \emph{some} series +$a_1a_2\ldots$ can be found, which satisfies the conditions: +\begin{align*} +\text{(i.) }&a_1/h = a_2/h\ldots = a_r/h.\\ +\text{(ii.) }&a_r/a_s\ldots\bar a_t\ldots h = a_r/h. +\end{align*} +Adherents of the Frequency Theory of Probability, who use the +\index{Frequency theory!Bernoulli's Theorem@{and Bernoulli's Theorem}}% +principal conclusion of Bernoulli's Theorem as the defining property +of \emph{all} probabilities, sometimes seem to mean no more than +that, relative to given evidence, every proposition belongs to +\emph{some} series, to the members of which Bernoulli's Theorem is +rigorously applicable. But the \emph{natural} series, the series, for +example, in which we are most often interested, where the $a$'s +are \emph{alike} in being accompanied by certain specified conditions~$c$, +is not, as a rule, rigorously subject to the Theorem. Thus `the +probability of~$a$ in certain conditions~$c$ is~$\frac{1}{2}$' is \emph{not} in general +equivalent, as has sometimes been supposed, to `It is $500$~to~$1$ +that in $40,000$~occurrences of~$c$, $a$~will not occur more than $20,200$~times, +and $500$~to~$1$ that it will not occur less than $19,800$~times.' + +\Paragraph{8.} Bernoulli's Theorem supplies the simplest formula by +which we can attempt to pass from the \textit{à~priori} probabilities of +each of a series of events to a prediction of the statistical frequency +of their occurrence over the whole series. We have seen that +Bernoulli's Theorem involves two assumptions, one (in the form +in which it is usually enunciated) tacit and the other explicit. +It is assumed, first, that a knowledge of what has occurred at +some of the trials would not affect the probability of what may +occur at any of the others; and it is assumed, secondly, that these +probabilities are all \emph{equal} \textit{à~priori}. It is assumed, that is to say, +that the probability of the event's occurrence at the $r$th~trial is +equal \textit{à~priori} to its probability at the $n$th~trial, and, further, that +it is unaffected by a knowledge of what may actually have +occurred at the $n$th~trial. + +A formula, which dispenses with the explicit assumption of +equal \textit{à~priori} probabilities at every trial, was proposed by +\index{Poisson!Theorem of}% +Poisson,\footnote + {\textit{Recherches}, pp.~246 \textit{et~seq.}} +and is usually known by his name. It does \emph{not} dispense, +%% -----File: 356.png---Folio 345------- +\index{Czuber|inote}% +however, with the other inexplicit assumption. The difference +between Poisson's Theorem and Bernoulli's is best shown by +reference to the ideal case of balls drawn from an urn. The +typical example for the valid application of Bernoulli's Theorem +is that of balls drawn from a single urn, containing black and +white balls in a known proportion, and replaced after each drawing, +or of balls drawn from a series of urns, each containing black +and white balls in the \emph{same} known proportion. The typical +example for Poisson's Theorem is that of balls drawn from a series +of urns, each containing black and white balls in \emph{different} known +proportions. + +Poisson's\Pagelabel{345} Theorem may be enunciated as follows:\footnote + {For the proof see Poisson, \textit{Recherches}, \textit{loc.\ cit.}, or Czuber, \textit{Wahrscheinlichkeitsrechnung}, + vol.~i.\ pp.~153--159.} +Let $s$~trials +be made, and at the $\lambda$th~trial ($\lambda=1,2\ldots s$) let the probabilities +for the occurrence and non-occurrence of the event be +$p_{\lambda}$,~$q_{\lambda}$ respectively. Then, if $\dfrac{\Sum p_{\lambda}}{s}=p$, the probability that the +number of occurrences~$m$ of the event in the $s$~trials will lie +between the limits~$sp ± l$ is given by +\[ +P = \frac{2}{k\sqrt{\pi s}}\int_0^l e^{-\frac{x^2}{k^2s}}\, dx + + \frac{e^{-\frac{l^2}{k^2s}}}{k\sqrt{\pi s}} +\] +where $k = \sqrt{\dfrac{2\Sum p_\lambda q_\lambda}{s}}$. + +By substituting $\dfrac{x}{k\sqrt{s}}=t$ and $\dfrac{l}{k\sqrt{s}}=\gamma$, this may be written +in a form corresponding to that of Bernoulli's Theorem,\footnote + {For the analogous form of Bernoulli's Theorem see \Pageref{339} (footnote).} +namely: + +The probability that the number of occurrences of the event +will lie between $sp ± \gamma k\sqrt{s}$ is given by +\[ +P = \frac{2}{\sqrt{\pi}} \int_0^\gamma e^{-t^2}\, dt + + \frac{e^{-\gamma^2}}{k\sqrt{\pi s}} +\] + +\Paragraph{9.} This is a highly ingenious theorem and extends the application +of Bernoulli's results to some important types of cases. It +embraces, for example, the case in which the successive terms of +a series are drawn from distinct populations known to be characterised +by differing statistical frequencies; no further complication +%% -----File: 357.png---Folio 346------- +being necessary beyond the calculation of two simple +functions of these frequencies and of the number of terms in the +series. But it is important not to exaggerate the degree to which +Poisson's method has extended the application of Bernoulli's +results. Poisson's Theorem leaves untouched all those cases in +which the probabilities of some of the terms in the series of events +can be influenced by a knowledge of how some of the other terms +in the series have turned out. + +Amongst these cases two types can be distinguished. In the +first type such knowledge would lead us to discriminate between +the conditions to which the different instances are subject. If, +for example, balls are drawn from a bag, containing black and +white balls in known proportions, and not replaced, the knowledge +whether or not the first ball drawn was black affects the +probability of the second ball's being black because it tells us +how the conditions in which the second ball is drawn differ +from those in which the first ball was drawn. In the second type +such knowledge does not lead us to discriminate between the +conditions to which the different instances are subject, but it leads +us to modify our opinion as to the nature of the conditions which +apply to all the terms alike. If, for instance, balls are drawn +from a bag, which is one, but it is not certainly known which, out +of a number of bags containing black and white balls in differing +proportions, the knowledge of the colour of the first ball drawn +affects the probabilities at the second drawing, because it throws +some light upon the question as to which bag is being drawn from. + +This last type is that to which most instances conform which +are drawn from the real world. A knowledge of the characteristics +of some members of a population may give us a clue to the +general character of the population in question. Yet it is this +type, where there is a change in knowledge but \emph{no change in the +material conditions} from one instance to the next, which is most +frequently overlooked.\footnote + {Numerous instances could be quoted. To take a recent English example, + reference may be made to Yule, \textit{Introduction to the Theory of Statistics}, +\index{Yule!coin@{and coin-tossing}|inote}% + p.~251. Mr.~Yule thinks that the condition of independence is satisfied if ``the + \emph{result} of any one throw or toss does not affect, and is unaffected by, the results + of the preceding and following tosses,'' and does not allow for the cases in which + \emph{knowledge} of the result is relevant apart from any change in the physical conditions.} +It will be worth while to say something +further about each of these two types.\footnote + {The types which I distinguish under four heads (the Bernoullian, the + Poissonian, and the two described above) Bachelier (\textit{Calcul des probabilités}, + p.~155) classifies as follows: + + (i.) When the conditions are identical throughout, the problem has \textit{uniformité}; + + (ii.) When they vary from stage to stage, but according to a law given from + the beginning and in a manner which does not depend upon what has happened + at the earlier stages, it has \textit{indépendance}; + + (iii.) When they vary in a manner which depends upon what has happened + at the earlier stages, it has \textit{connexité}. + + Bachelier gives solutions for each type on the assumption that the number of + trials is very great, and that the number of successes or failures can be regarded + as a continuous variable. This is the same kind of assumption as that made + in the proof of Bernoulli's Theorem given in §\;2, and is open to the same objections,---or + rather the value of the results is limited in the same way.} +%% -----File: 358.png---Folio 347------- +\index{Bachelier|inote}% +\index{Pearson, Karl!asymmetry@{and asymmetry}}% +\index{Pearson, Karl!generalised Probability@{and generalised Probability curves}}% + +\Paragraph{10.} For problems of the first type, where there is physical +or material dependence between the successive trials, it is not +possible, I think, to propose any general solution; since the +probabilities of the successive trials may be modified in all kinds +of different ways. But for particular problems, if the conditions +are precise enough, solutions can be devised. The problem, for +instance, of an urn, containing black and white balls in known +proportions, from which balls are drawn successively and \emph{not +replaced},\footnote + {It is of no consequence whether the balls are drawn successively and not + replaced, or are drawn simultaneously.} +\index{Czuber}% +is ingeniously solved by Czuber\footnote + {\textit{Loc.\ cit.}\ vol.~i.\ pp.~163,~164.} +with the aid of +Stirling's Theorem. If $\sigma$~is the number of balls and $s$~the number +of drawings, he reaches the interesting conclusion (assuming that +$\sigma$,~$s$ and~$\sigma-s$ are all large) that the probability of the number of +black balls lying within given limits is the same as it would be +if the balls were replaced after each drawing and the number +of drawings were $\dfrac{\sigma-s}{\sigma}s$ instead of~$s$. + +In addition to the assumptions already stated, Professor +Czuber's solution applies only to those cases where the limits, for +which we wish to determine the probability, are narrow compared +with the total number of black balls~$p\sigma$. Professor Pearson\footnote + {``Skew Variation in Homogeneous Material,'' \textit{Phil.\ Trans.}\ (1895), p.~360.} +has +worked out the same problem in a much more general manner, +so as to deal with the \emph{whole} range, \ie~the frequency or probability +of all possible ratios of black balls, even where $s>p\sigma$. The +various forms of curve, which result, according to the different +relations existing between $p$,~$s$, and~$\sigma$, supply examples of each +of the different types of frequency curve which arise out of a +%% -----File: 359.png---Folio 348------- +classification according to (i.)~skewness or symmetry, (ii.)~limitation +of range in one, both or neither direction; and he designates, +therefore, the curves which are thus obtained as \emph{generalised probability +curves}. His discussion of the properties of these curves is +interesting, however, to the student of descriptive statistics +rather than to the student of probability. The most generalised +and, mathematically, by far the most elegant treatment of this +problem, with which I am acquainted, is due to Professor +\index{Tschuprow!statistical frequency@{and statistical frequency}}% +Tschuprow.\footnote + {``Zur Theorie der Stabilität statistischer Reihen,'' p.~216, published in + the \textit{Skandinavisk Aktuarietidskrift} for~1919.} + +\index{Poisson!statistical frequency@{and statistical frequency}}% +Poisson, in attempting a somewhat similar problem,\footnote + {\textit{Loc.\ cit.}\ pp.~231,~232.} +arrives +at a result, which seems obviously contrary to good sense, by a +curious, but characteristic, misapprehension of the meaning of +`independence' in probability. His problem is as follows: +If $l$~balls be taken out from an urn, containing $c$~black and white +balls in known proportions, and not replaced, and if a further +number of balls~$\mu$ be then taken out, the probability that a given +proportion~$\dfrac{m}{m+n}$ of these $\mu$~balls will be black \emph{is independent of +the number and the colour of the $l$~balls originally drawn out}. For, +he argues, if $l+\mu$~balls are drawn out, the probability of a combination, +which is made up of $l$~black and white balls in given +proportions followed by $\mu$~balls, of which $m$~are white and $n$~black, +must be the same as that of a similar combination in which the +$\mu$~balls precede the $l$~balls. Hence the probability of $m$~white +balls in $\mu$~drawings, given that the $l$~balls have already been +drawn out, must be equal to the probability of the same result, +when no balls have been previously drawn out. The reader will +perceive that Poisson, thinking only of physical dependence, has +been led to his paradoxical conclusion by a failure to distinguish +between the cases where the proportion of black and white balls +amongst the $l$~balls originally drawn is \emph{known} and where it is not. +The \emph{fact} of their having been drawn in certain proportions, provided +that only the total number drawn is known and the proportions +are \emph{unknown}, does not influence the probability. Poisson +states in his conclusion that the probability is independent of the +number and colour of the $l$~balls originally drawn. If he had +added---as he ought---`provided the number of each colour is +%% -----File: 360.png---Folio 349------- +\index{Yule|inote}% +\emph{unknown},' the air of paradox disappears. This is an exceedingly +good example of the failure to perceive that a probability cannot +be influenced by the \emph{occurrence} of a material event but only by +such \emph{knowledge}, as we may have, respecting the occurrence of the +event.\footnote + {For an attempt to solve other problems of this type see Bachelier, \textit{Calcul +\index{Bachelier!statistical frequency@{and statistical frequency}|inote}% + des probabilités}, chap.~ix.\ \textit{(Probabilités connexes}). I think, however, that the + solutions of this chapter are vitiated by his assuming in the course of them + both that certain quantities are very large, and also, at a later stage, that the + same quantities are infinitesimal. On this account, for example, his solution + of the following difficult problem breaks down: Given an urn~$A$ with $m$~white + and $n$~black balls and an urn~$B$ with $m'$~white and $n'$~black balls, if at each move + a ball is taken from~$A$ and put into~$B$, and at the same time a ball is taken from~$B$ + and put into~$A$, what is the probability after $x$~moves that the urns $A$~and~$B$ + shall have a given composition?} + +\Paragraph{11.} For problems of the second type, where knowledge of the +result of one trial is capable of influencing the probability at the +next apart from any change in the material conditions, there is, +likewise, no general solution. The following artificial example, +however, will illustrate the sort of considerations which are involved. + +In the cases where Bernoulli's Theorem is applied to practical +questions, the \textit{à~priori} probability is generally obtained empirically +by reference to the statistical frequency of each alternative +in past experience under apparently similar conditions. Thus +the \textit{à~priori} probability of a male birth is estimated by reference +to the recorded proportion of male births in the past.\footnote + {Cf.\ Yule, \textit{Theory of Statistics}, p.~258: ``We are not able to assign an + \textit{à~priori} value to the chance~$p$ (\ie~of a male birth) as in the case of dice-throwing, + but it is quite sufficiently accurate for practical purposes to use the proportion + of male births actually observed if that proportion be based on a moderately + large number of observations.''} +The +validity of estimating probabilities in this manner will be discussed +later. But for the purposes of this example let us assume +that the \textit{à~priori} probability has been calculated on this basis. +Thus the \textit{à~priori} probability $p\left(=\dfrac{r}{s}\right)$ of an event is based on +the observation of its occurrence $r$~times out of $s$~occasions on +which the given conditions were present. Now, according to +Bernoulli's Theorem directly applied, the probability of the +event's occurring $n$~times running is~$p^n$ or~$\left(\dfrac{r}{s}\right)^n$. But, if the +event occurs at the first trial, the probability at the second +%% -----File: 361.png---Folio 350------- +becomes $\dfrac{r+1}{s+1}$, and so on. Hence the probability~$P$, properly +calculated, of $n$~successive occurrences is +\[ +\frac{r}{s}·\frac{r+1}{s+1}·\frac{r+2}{s+2}\ldots\frac{r+n-1}{s+n-1}. +\] +Hence +\begin{DPalign*}[m] +P &= \frac{(r+n-1)!\, (s-1)!}{(s+n-1)!\, (r-1)!} \\ + &= \frac{(r+n-1)^{r+n-\frac{1}{2}} e^{-(r+n-1)} s^{s-\frac{1}{2}} e^{-(s-1)}} + {(s+n-1)^{s+n-\frac{1}{2}} e^{-(s+n-1)} r^{r-\frac{1}{2}} e^{-(r-1)}} +\rintertext{by Stirling's} \\ + & +\rintertext{\llap{Theorem, provided that $r$~and~$s$ are large;}} \\ + &= \left(\frac{r}{s}\right)^n + \frac{\left(1 + \dfrac{n-1}{r}\right)^{r+n-\frac{1}{2}}} + {\left(1 + \dfrac{n-1}{s}\right)^{s+n-\frac{1}{2}}}\\ + &= p^{n}Q^{n}, \text{ where } + Q = \frac{\left(1 + \dfrac{n-1}{r}\right)^{\frac{r-\frac{1}{2}}{n} + 1.}} + {\left(1 + \dfrac{n-1}{s}\right)^{\frac{s-\frac{1}{2}}{n} + 1.}} +\end{DPalign*} +Thus, in this case, the assumption of Bernoulli's Theorem is +approximately correct, only if $Q$~is nearly unity. This condition +is not satisfied unless $n$~is small both compared with~$r$ and compared +with~$s$. It is very important to notice that \emph{two} conditions +are involved. Not only must the experience, upon which the +\textit{à~priori} probability is based, be extensive in comparison with the +number of instances to which we apply our prediction; but also +the number of previous instances multiplied by the probability +based upon them, \ie~$sp\, (=r)$, must be large in comparison with +the number of new instances. Thus, even where the prior experience, +upon which we found the initial probability~$P$, is very +extensive, we must not, if $P$~is very small, say that the probability +of $n$~successive occurrences is approximately~$p^n$, unless $n$~is also +small. Similarly if we wish to determine, by the methods of +Bernoulli, the probability of $n$~occurrences and $m$~failures on +$m+n$~occasions, it is necessary that we should have $m$~and~$n$ small +%% -----File: 362.png---Folio 351------- +\index{Pearson, Karl|inote}% +compared with~$s$, $n$~small compared with~$r$, and $m$~small compared +with~$s-r$.\footnote + {This paragraph is concerned with a different point from that dealt with + in Professor Pearson's article ``On the Influence of Past Experience on Future + Expectation,'' to which it bears a superficial resemblance. Professor Pearson's + article which deals, not with Bernoulli's Theorem, but with Laplace's ``Rule of +\index{Laplace!Rule of Succession@{and Rule of Succession}|inote}% + Succession,'' will be referred to in §\;16 of this chapter and in §\;12 of the next.} + +The case solved above is the simplest possible. The general +problem is as follows: If an event has occurred $x$~times in the +first $y$~trials, its probability at the~$y+1$th is~$\dfrac{r+x}{s+y}$; determine the +\textit{à~priori} probability of the event's occurring $p$~times in $q$~trials. +If the \textit{à~priori} probability in question is represented by~$\phi(p,q)$, we +have %[** TN: Displaying in-line formula] +\[ +\phi(p,q) + = \frac{r + p - 1}{s + q - 1}\, \phi(p-1, q-1) + + \frac{s + q - 1 - r - p}{s + q - 1}\, \phi(p, q-1). +\] +I know of no solution of this, even approximate. But we may +say that the conditions are those of supernormal dispersion as +compared with Bernoulli's conditions. That is to say, the probability +of a proportion differing widely from~$\dfrac{r}{s}$ is greater than +in Bernoullian conditions; for when the proportion begins to +diverge it becomes more probable that it will continue to diverge +in the same direction. If, on the other hand, the conditions of +the problem had been such, that when the proportion begins to +diverge it becomes more probable that it will recover itself and +tend back towards~$\dfrac{r}{s}$ (as when we draw balls without replacing +them from a bag of known composition), we should have subnormal +dispersion.\footnote + {Bachelier (\textit{Calcul des probabilités}, p.~201) classifies these two kinds of conditions +\index{Bachelier!statistical frequency@{and statistical frequency}}% + as \textit{conditions accélératrices} and \textit{conditions retardatrices}.} + +\Paragraph{12.}\Pagelabel{351} The condition elucidated in the preceding paragraph is +frequently overlooked by statisticians. The following example +\index{Czuber!statistical frequency@{and statistical frequency}}% +from Czuber\footnote + {\textit{Loc.\ cit.}\ vol.~ii.\ p.~15. I choose my example from Professor Czuber because + he is usually so careful an exponent of theoretical statistics.} +will be sufficient for the purpose of illustration. +Czuber's argument is as follows: + +In the period 1866--1877 there were registered in Austria +\[ +\begin{array}{@{}r@{ }l} +\llap{$m$} = 4,311,076 & \text{ male births} \\ + n = 4,052,193 & \text{ female births} \\ +\cline{1-1} +\rule{0pt}{12pt} + s = 8,363,269\rlap{\,;} & +\end{array} +\] +%% -----File: 363.png---Folio 352------- +for the succeeding period, 1877--1899, we are given only +\[ +m' = 6,533,961 \text{ male births}; +\] +what conclusion can we draw as to the number~$n'$ of female +births? We can conclude, according to Czuber, that the most +probable value +\[ +n_0' = \frac{nm'}{m} = 6,141,587, +\] +and that there is a probability $P = .9999779$ that $n'$~will lie +between the limits $6,118,361$ and~$6,164,813$. + +It seems in plain opposition to good sense that on +such evidence we should be able with practical certainty +$\left(P=.9999779 = 1-\dfrac{1}{45250}\right)$ to estimate the number of female +births within such narrow limits. And we see that the conditions +laid down in §\;11 have been flagrantly neglected. The +number of cases, over which the prediction based on Bernoulli's +Theorem is to extend, actually \emph{exceeds} the number of cases upon +which the \textit{à~priori} probability has been based. It may be added +that for the period, 1877--1894, the actual value of~$n'$ did lie +between the estimated limits, but that for the period, 1895--1905, +it lay \emph{outside} limits to which the same method had +attributed practical certainty. + +That Professor Czuber should have thought his own argument +plausible, is to be explained, I think, by his tacitly taking account +in his own mind of evidence not stated in the problem. He was +relying upon the fact that there is a great mass of evidence for +believing that the ratio of male to female births is peculiarly +stable. But he has not brought this into the argument, and he +has not used as his \textit{à~priori} probability and as his coefficient of +dispersion the values which the whole mass of this evidence would +have led him to adopt. Would not the argument have seemed +very preposterous if $m$~had been the number of males called +George, and $n$~the number of females called Mary? Would it not +have seemed rather preposterous if $m$~had been the number of +legitimate births and $n$~the number of illegitimate births? Clearly +we must take account of other considerations than the mere +numerical values of $m$~and~$n$ in estimating our \textit{à~priori} probability. +But this question belongs to the subject-matter of later chapters, +%% -----File: 364.png---Folio 353------- +\index{Tchebycheff, Theorem of}% +and, quite apart from the manner of calculation of the \textit{à~priori} +probability, the argument is invalidated by the fact than an +\textit{à~priori} probability founded on $8,363,269$ instances, without +corroborative evidence of a non-statistical character, cannot +be assumed stable through a calculation which extends over +$12,700,000$ instances. + +\Paragraph{13.} Before we leave the theorems of Bernoulli and Poisson, +it is necessary to call attention to a very remarkable theorem by +Tchebycheff, from which both of the above theorems can be +derived as special cases. This result is reached rigorously and +without approximation, by means of simple algebra and without +the aid of the differential calculus. Apart from the +beauty and simplicity of the proof, the theorem is so valuable +and so little known that it will be worth while to quote it in +full:\footnote + {From \textit{Journ.\ Liouville} (2),~xii., 1867, ``Des valeurs moyennes,'' an article + translated from the Russian of Tchebycheff. This proof is also quoted by + Czuber, \textit{loc.\ cit.}\ p.~212, through whom I first became acquainted with it. Most +\index{Czuber!Tchebycheff's Theorem@{and Tchebycheff's Theorem}|inote}% + of Tchebycheff's work was published previous to 1870 and appeared originally + in Russian. It was not easily accessible, therefore, until the publication + at Petrograd in 1907 of the collected edition of his works in French. + His theorems are, consequently, not nearly so well known as they deserve + to be, although his most important theorems were reproduced from time + to time in the Journals of Euler and Liouville. For full references see the + Bibliography.} + +Let $x, y, z\ldots$ represent certain magnitudes, of which $x$~can +take the values $x_1x_2\ldots x_k$ with probabilities $p_1p_2\ldots p_k$ +respectively, $y$~the values $y_1y_2\ldots y_l$ with probabilities $q_1q_2\ldots q_l$, +$z$~the values $z_1z_2\ldots z_m$ with probabilities $r_1r_2\ldots r_m$ and so on, +so that +\begin{DPalign*} +& \Sum_1^k p=1,\quad + \Sum_1^l q=1,\quad + \Sum_1^m r=1, \text{ etc.} \\ +\lintertext{Write} +& \Sum_1^k p_\kappa x_\kappa = a,\quad + \Sum_1^l q_\lambda y_\lambda = b,\quad + \Sum_1^m r_\mu z_\mu =c, \text{ etc.,} \\ +\lintertext{and} +& \Sum_1^k p_\kappa x_\kappa^2 = a_1,\quad + \Sum_1^l q_\lambda y_\lambda^2 = b_1,\quad + \Sum_1^m r_\mu z_\mu^2 = \DPtypo{c}{c_1}, \text{ etc.,} +\end{DPalign*} +so that we can describe~$a$ as the mathematical expectation or +average value of~$x$ and $a_1$~as the mathematical expectation or +average value of $x^2$,~etc. +%% -----File: 365.png---Folio 354------- + +Consider the expression: +\begin{DPalign*} +\Sum (x_\kappa + y_\lambda + z_\mu + \ldots - a - b - c - \ldots)^2 p_\kappa q_\lambda r_\mu \ldots \\ +\lintertext{Now} +\begin{aligned}[t] +\Sum_1^k (x_\kappa^2 - 2ax_\kappa + a^2)p_\kappa + &= \Sum p_\kappa x_\kappa^2 - 2a\Sum p_\kappa x_\kappa + a^2\Sum p_\kappa \\ + &= a_1 - 2a^2 + a^2 = a_1-a^2. +\end{aligned} +\end{DPalign*} +Also $\Sum q_\lambda r_\mu \ldots = 1$ summed for all values of $\lambda$,~$\mu~\ldots$, and +\begin{DPgather*}[m] +\Sum_1^k 2(x_\kappa - a)(y_\lambda - b)p_\kappa + = \Sum_1^k 2(x_\kappa y_\lambda - bx_\kappa - ay_\lambda + ab) p_\kappa \\ + = 2\left(y_\lambda\Sum p_\kappa x_\kappa - b\Sum p_\kappa x_\kappa + - a y_\lambda\Sum p_\kappa + ab\Sum p_\kappa\right) \\ + = 2(ay_\lambda - ab - ay_\lambda + ab) = 0. \\ +\lintertext{\rlap{Therefore}} +\Sum(x_\kappa + y_\lambda + z_\mu + \ldots - a - b - c\ldots)^2 p_\kappa q_\lambda r_\mu\ldots \\ + = a_1 + b_1 + c_1 + \ldots - a^2 - b^2 - c^2 - \ldots, \\ +\lintertext{whence} +\frac{\Sum(x_\kappa + y_\lambda + z_\mu + \ldots - a - b - c\ldots)^2 p_\kappa q_\lambda r_\mu\ldots} + {\alpha^2(a_1 + b_1 + c_1 + \ldots - a^2 - b^2 - c^2 - \ldots)} + = \frac{1}{\alpha^2}, +\end{DPgather*} +where the summation extends over all values of $\kappa$,~$\lambda$,~$\mu~\ldots$ and +$\alpha$~is some arbitrary number greater than unity. + +If we omit those terms of the sum on the left-hand side of +the above equation for which +\[ +\frac{(x_\kappa + y_\lambda + z_\mu + \ldots - a - b - c\ldots)^2} + {\alpha^2(a_1 + b_1 + c_1 + \ldots - a^2 - b^2 - c^2 - \ldots)} < 1, +\] +and write unity for this expression in the remaining terms, both +these processes diminish the magnitude of the left-hand side. +Hence $\Sum p_\kappa q_\lambda r_\mu \ldots < \dfrac{1}{\alpha^2}$, where the summation covers those sets +of values only for which +\[ +\frac{(x_\kappa + y_\lambda + z_\mu + \ldots - a - b - c\ldots)^2} + {\alpha^2(a_1 + b_1 + c_1 + \ldots - a^2 - b^2 - c^2 \ldots)} \geq 1. +\] + +If $P$~is the probability that +\[ +\frac{(x_\kappa + y_\lambda + z_\mu + \ldots - a - b - c\ldots)^2} + {\alpha^2(a_1 + b_1 + c_1 + \ldots - a^2 - b^2 - c^2 - \ldots)} +\] +is equal to or less than unity, it follows that +%% -----File: 366.png---Folio 355------- +\index{Tchebycheff, Theorem of}% +\begin{DPgather*} +1-P < \frac{1}{\alpha ^2} \\ +\lintertext{\ie} P > 1 - \frac{1}{\alpha ^2} +\end{DPgather*} +Hence the probability that the sum +\begin{DPgather*} +x_\kappa + y_\lambda + z_\mu \ldots +\rintertext{\llap{lies between the limits}} \\ +a + b + c + \ldots + - \alpha\sqrt{a_1 + b_1 + c_1 + \ldots - a^2 - b^2 - c^2 - \ldots} \\ +\lintertext{and} +a + b + c + \ldots + + \alpha\sqrt{a_1 + b_1 + c_1 + \ldots - a^2 - b^2 - c^2 - \ldots} +\end{DPgather*} +is greater than $1-\dfrac{1}{\alpha ^2}$, where $\alpha$~is some number greater than +unity. + +This result constitutes Tchebycheff's Theorem. It may also +be written in the following form: + +Let $n$ be the number of the magnitudes $x, y, z\ldots$, and +write $\alpha = \dfrac{\sqrt{n}}{t}$; then the probability that the arithmetic +mean $\dfrac{x_\kappa + y_\lambda + z_\mu+\ldots}{n}$ lies between the limits +\[ +\frac{a + b + c+ \ldots}{n} + ± \frac{1}{t}\sqrt{\frac{a_1 + b_1 + c_1 + \ldots}{n} + - \frac{a^2 + b^2 + c^2 + \ldots}{n}} +\] +is greater than $1-\dfrac{t^2}{n}$. + +It is also easy to show\footnote + {For a proof see Czuber, \textit{loc.\ cit.}\ vol.~i.\ p.~216.} +\index{Czuber!Tchebycheff's Theorem@{and Tchebycheff's Theorem}|inote}% +as a deduction from Tchebycheff's +Theorem that, if an amount~$A$ is won when an event of probability +$p\,[p=1-q]$ occurs and an amount~$B$ lost when it fails, then in +$s$~trials the probability that the total winnings (or losses) will lie +between the limits +\[ +s(pA - qB) ± \alpha(A + B)\sqrt{spq} +\] +is greater than $1-\dfrac{1}{\alpha^2}$. + +\Paragraph{14.} From this very general result for the probable limits of +a sum composed of a number of independently varying magnitudes, +Bernoulli's Theorem is easily derived. For let there be +%% -----File: 367.png---Folio 356------- +$s$~observations or trials, and $s$ magnitudes $x_1x_2\ldots x_s$ corresponding, +such that $x=1$ when the event under consideration +occurs, and $x=0$ when it fails. If the probability of the events +occurrence is~$p$, we have $a=p$, $b=p$, etc., and $a_1=p$, $b_1=p$, etc. +Hence the probability~$P$ that the number of the event's occurrences +will lie between the limits $sp ± \alpha \sqrt{sp - sp^2}$, \ie\ between +the limits $sp ± \alpha\sqrt{spq}$ where $q=1-p$, is $>1-\dfrac{1}{\alpha^2}$. If we +compare this formula with the formula for Bernoulli's +Theorem already given, we find that, where this formula +gives $P > 1 - \dfrac{1}{\alpha^2}$, Bernoulli's Theorem with greater precision +gives $P = \Theta\left(\dfrac{\alpha}{\sqrt{2}}\right)$. The degree of superiority in the matter +of precision supplied by the latter can be illustrated by the +following table: +\[ +\begin{array}{|>{\quad}r<{\quad}|>{\quad}l<{\quad}|>{\quad}l<{\quad}|} +\hline +\rule[-12pt]{0pt}{32pt} +\alpha^2. & +%[** TN: Added parentheses to heading] +\multicolumn{1}{c|}{\Theta\left(\frac{\alpha}{\sqrt{2}}\right).} & +\multicolumn{1}{c|}{1-\frac{1}{\alpha^2}.} \\ +\hline +\rule{0pt}{20pt} +1.5 &.7788 &.333 \\ +2\phantom{.5} &.8427 &.5 \\ +4.5 &.9661 &.7778 \\ +8\phantom{.5} &.9953 &.875 \\ +12.5 &.9996 &.92 \\ +\rule[-12pt]{0pt}{12pt} +18\phantom{.5} &.99998 &.9445 \\ +\hline +\end{array} +\] +Thus when the limits are narrow and $\alpha$~is small, Bernoulli's +formula gives a value of~$P$ very much in excess of $1-\dfrac{1}{\alpha^2}$. But +Bernoulli's formula involves a process of approximation which is +only valid when $s$~is large. Tchebycheff's formula involves no +such process and is equally valid for all values of~$s$. We have +seen in §\;11 that there are numerous cases in which for a +different reason Bernoulli's formula exaggerates the results, +and, therefore, Tchebycheff's more cautious limits may sometimes +prove useful. + +The deduction of a corresponding form of Poisson's Theorem +from Tchebycheff's general formula obviously follows on similar +lines. For we put\footnote + {I am using the same notation as that used for Poisson's Theorem in §\;8.} +$a=p_1$, $b=p_2$, etc., and $a_1=p_1$, $b_1=p_2$, etc., +%% -----File: 368.png---Folio 357------- +\index{Markoff, A. A.!Tchebycheff's Theorem@{and Tchebycheff's Theorem}}% +\index{Tchebycheff!Poisson's Theorem@{and Poisson's Theorem}}% +and find that the probability that the number of the event's +occurrences will lie between the limits +\[ +\Sum_1^\lambda p_\lambda ± \alpha \sqrt{\Sum_1^\lambda p_\lambda + - \Sum_1^\lambda p_\lambda^2}, +\] +\begin{DPgather*} +\lintertext{\rlap{\ie\ between the limits}} +sp ± \alpha\sqrt{\Sum^\lambda p_\lambda q_\lambda}, \\ +\lintertext{\rlap{\ie\ between the limits}} +sp ± \sqrt{2}\,\alpha k\sqrt{s}, +\end{DPgather*} +is greater than $t-\dfrac{1}{\alpha^2}$. + +In \textit{Crelle's Journal}\footnote + {Vol.~33 (1846), \textit{Démonstration élémentaire d'une proposition générale de la + théorie des probabilités}.} +Tchebycheff proves Poisson's Theorem +\index{Poisson!Tchebycheff@{and Tchebycheff}}% +directly by a method similar to his general method, and also +obtains several supplementary results such as the following: + +I\@. If the chances of an event~$E$ in $\mu$~consecutive trials are +$p_1p_2\ldots p_\mu$ respectively, and their sum is~$s$, the probability that +$E$~will occur at least $m$~times is less than +\[ +\frac{1}{2(m-s)} \sqrt\frac{m(\mu - m)}{\mu} + \left(\frac{s}{\mu}\right)^m + \left(\frac{\mu - s}{\mu - m}\right)^{\mu - m + 1} +\] +\begin{flushright} +provided that $m>s+1$; +\end{flushright} %[** TN: [sic] no \par] +II\@. and the probability that $E$~will not occur more than $n$~times +is less than +\[ +\frac{1}{2(s-n)} \sqrt\frac{\mu(\mu - n)}{\mu} + \left(\frac{\mu - s}{\mu - n}\right)^{\mu - n} + \left(\frac{s}{n}\right)^{n + 1} +\] +\begin{flushright} +provided that $n<s-1$. +\end{flushright} +III\@. Hence the probability that $E$~will occur less than $m$~times +and more than~$n$ is greater than +\begin{align*} +1 &- \frac{1}{2(m-s)}\sqrt\frac{m(\mu -m)}{\mu}\left(\frac{s}{m}\right)^m + \left(\frac{\mu -s}{\mu -m}\right)^{\mu -m+1}\\ + &- \frac{1}{2(s-n)}\sqrt\frac{n(\mu -n)}{\mu}\left(\frac{s}{n}\right)^{n+1} + \left(\frac{\mu -s}{\mu -n}\right)^{n-\mu} +\end{align*} +\begin{flushright} +provided $m>s+1$, $n<s-1$. +\end{flushright} + +\Paragraph{15.} Tchebycheff's methods have been set out and his results +admirably extended by A.~A. Markoff.\footnote + {The reader is referred to Markoff's \textit{Wahrscheinlichkeitsrechnung}, and particularly + to p.~67, for a striking development, along mathematical lines, of + Tchebycheff's leading idea. Further references to later memôirs, + which, being in the Russian language, are inaccessible to me, will + be found in the Bibliography.} +And some developments +%% -----File: 369.png---Folio 358------- +\index{Asymmetry, and Bernoulli's Theorem|ifoll}% +\index{Laplace!school of}% +along the same lines by Tschuprow (``Zur Theorie der +\index{Tschuprow}% +Stabilität statistischer Reihen,'' \textit{Skandinavisk Aktuarietidskrift}, +1919) have convinced me that Tchebycheff's discovery is far +more than a technical device for solving a special problem, and +points the way to the fundamental method for attacking these +questions on the mathematical side. The Laplacian mathematics, +although it still holds the field in most text-books, is +really obsolete, and ought to be replaced by the very beautiful +work which we owe to these three Russians. + +\Paragraph{16.} There is one other investigation relating to Bernoulli's +Theorem which deserves remark. I have already pointed out, +in §\;2, that the dispersion about the most probable value, even +when the conditions for the applicability of Bernoulli's Theorem +\index{Bernoulli's Theorem!asymmetry@{and asymmetry}|ifoll}% +in its non-approximate form are strictly fulfilled, is unsymmetrical. +The fact, that the usual approximation for the probability +of a divergence~$h$ from the most probable number of +occurrences (the notation is that of §\;2 above) takes the form +$\dfrac{1}{\sqrt{2\pi mpq}}\, e^{-\frac{h^2}{2mpq}}$, which is the same for~$+h$ as for~$-h$, has led +to this want of symmetry being very generally overlooked; +and it is not uncommon to assume that the probability of a +given divergence less than~$pm$ is equal to that of the same divergence +in excess of~$pm$, and, in general, that the probability of +the frequency's exceeding~$pm$ in a set of $m$~trials is \emph{equal} to that +of its falling short of~$pm$. + +That this is not strictly the case is obvious. If a die is cast +$60$~times, the most probable number of appearances of the ace +is~$10$; but the ace is more likely to appear $9$~times than $11$~times; +and much more likely (about $5$~times as likely) not to appear at +all than to appear exactly $20$~times. That this must be so will +be clear to the reader (without his requiring to trouble himself +with the algebra), when he reflects that the ace cannot appear +less often than not at all, whereas it may well appear more than +$20$~times, so that the smallness of the possible divergence in +defect from the most probable value~$10$, as compared with the +possible divergence in excess, must be made up for by the greater +%% -----File: 370.png---Folio 359------- +\index{Lexis, and asymmetry of statistical frequency|inote}% +\index{Pearson, Karl!asymmetry@{and asymmetry}|inote}% +\index{Rule of Succession|inote}% +\index{Simmons and asymmetry in Bernoulli's!Theorem}% +frequency of any given defection as compared with the corresponding +excess. Thus the actual frequency in a series of trials +of an event, of which the probability at each trial is less than~$\frac{1}{2}$, +is likely to fall short of its most probable value more often than +it exceeds it. What is in fact true is that the mathematical +expectation of deficiency is equal to the mathematical expectation +of excess, \ie\ that the sum of the possible deficiencies each +multiplied by its probability is equal to the sum of the possible +excesses each multiplied by its probability. + +The actual measurement of this want of symmetry and the +determination of the conditions, in which it can be safely +neglected, involves laborious mathematics, of which I am only +acquainted with one direct investigation, that published in the +\textit{Proceedings of the London Mathematical Society} by Mr.~T.~C. +Simmons.\footnote + { ``A New Theorem in Probability.'' Mr.~Simmons claimed novelty for + his investigation, and so far as I know this claim is justified; but recent + investigations obtaining closer approximations to Bernoulli's Theorem by means + of the Method of Moments are essentially directed towards the same problem. + + A somewhat analogous point has, however, been raised by Professor Pearson + in his article (\textit{Phil.\ Mag.}, 1907) on ``The Influence of Past Experience on Future + Expectation.'' He brings out an exactly similar want of symmetry in the + probabilities of the various possible frequencies about the most probable frequency, + when the calculation is based, not on Bernoulli's Theorem as in Mr.~Simmons's + investigation, but on Laplace's rule of succession (see next chapter). +\index{Laplace!Rule of Succession@{and Rule of Succession}|inote}% + The want of symmetry has also been pointed out by Professor Lexis (\textit{Abhandlungen}, + p. 120).} + +For the details of the proof I must refer the reader to Mr.~Simmons's +article. His principal theorem\footnote + {I am not giving his own enunciation of it.} +is as follows: If +$\dfrac{1}{a+1}$~is the probability of the event at each trial and $n(a+1)$ the +number of trials, $n$~and~$a$ being integers,\footnote + {Mr.~Simmons does not seem to have been able to remove this restriction + on the generality of his theorem, but there does not seem much reason to doubt + that it can be removed.} +the probability that the +frequency of occurrence will fall short of~$n$ is always greater than +the probability that it will exceed~$n$; the difference between the +two probabilities being a maximum when $n=1$, constantly +diminishing as $n$~increases, lying always between $\dfrac{1}{3}\,\dfrac{a-1}{a+1}$~times the +greatest term in $\left(\dfrac{a}{a+1}+\dfrac{1}{a+1}\right)^{n(a+1)}$ and $\dfrac{1}{3}\,\dfrac{a-1}{a+1}$ times the +%% -----File: 371.png---Folio 360------- +greatest term in $\left(\dfrac{a}{a+1} + \dfrac{1}{a+1}\right)^{(n+1)(a + 1)}$, and being approximately +equal, when $n$~is very large, to $\dfrac{1}{3}\,\dfrac{a-1}{\sqrt{2\pi na(a+1)}}$. + +The following table gives the value of the excess~$\Delta$ of the +probability of a frequency less than~$pm$ over the probability of +a frequency greater than~$pm$ for various values of~$p$ the probability +and $m$~the number of trials $\left[p = \dfrac{1}{a+1},\ m=n(a+1)\right]$, as +calculated by Mr.~Simmons: +\[ +\renewcommand{\arraystretch}{1.25}% +\begin{array}{|>{\quad}c<{\quad}|>{\quad}r<{\quad}|>{\quad}l<{\quad}|} +\hline +p. & \multicolumn{1}{c|}{m.} & \multicolumn{1}{c|}{\Delta.} \\ +\hline +\frac{1}{3} & 3 & .037037 \\ +\frac{1}{3} & 15 & .02243662 \\ +\frac{1}{3} & 24 & .0182706 \\ +\frac{1}{4} & 4 & .054687 \\ +\frac{1}{4} & 20 & .03201413 \\ +\frac{1}{10} & 10 & .084777 \\ +\frac{1}{10} & 20 & .068673713 \\ +\frac{1}{100} & 100 & .101813 \\ +\frac{1}{100} & 200 & .081324387 \\ +\frac{1}{1000} & 1000 & .103454\\ +\hline +\end{array} +\] +Thus unless not only $m$ but $mp$~also is large the want +of symmetry is likely to be appreciable. Thus it is easily found +that in $100$~sets of $4$~trials each, where $p=\dfrac{1}{4}$, the actual frequency +is likely to exceed the most probable $26$~times and to fall short of +it $31$~times; and in $100$~sets of $10$~trials each, where $p=\dfrac{1}{10}$, to +exceed $26$~times and to fall short $34$~times. + +Mr.~Simmons was first directed to this investigation through +%% -----File: 372.png---Folio 361------- +\index{Dice-tossing|ifoll}% +\index{Mode, and law of error!asymmetry about}% +\index{Yule|inote}% +noticing in the examination of sets of random digits that ``each +digit presented itself, with unexpected frequency, \emph{less} than $\dfrac{1}{10}$~of +the number of times. For instance, in $100$~sets of $150$~digits each, +I found that a digit presented itself in a set more frequently under +$15$~times than over $15$~times; similarly in the case of $80$~sets each +of $250$~digits, and also in other aggregations.'' Its possible +bearing on such experiments with dice and roulette, as are +\index{Roulette}% +described at the end of this chapter, is clear. But apart from +these artificial experiments, it is sometimes worth the statistician's +while to bear in mind this appreciable want of symmetry +in the distribution about the mode or most probable value in +many even of those cases in which Bernoullian conditions are +strictly fulfilled. + +\Paragraph{17.} I will conclude this chapter by an account of some of the +attempts which have been made to verify \textit{à~posteriori} the conclusions +of Bernoulli's Theorem. These attempts are nearly +\index{Bernoulli's Theorem!empirical verification of|ifoll}% +useless, first, because we can seldom be certain \textit{à~priori} that the +conditions assumed in Bernoulli's Theorem are fulfilled, and, +secondly, because the theorem predicts not what will happen +but only what is, on certain evidence, likely to happen. Thus +even where our results do not verify Bernoulli's Theorem, the +theorem is not thereby discredited. The results have bearing +on the conditions in which the experiments took place, rather +than upon the truth of the theorem. In spite, therefore, of the +not unimportant place which these attempts have in the history +of probability, their scientific value is very small. I record them, +because they have a good deal of historical and psychological +interest, and because they satisfy a certain idle curiosity from +which few students of probability are altogether free.\footnote + {Mr.~Yule (\textit{Introduction to Statistics}, p. 254) recommends its indulgence: +\index{Yule!coin@{and coin-tossing}|inote}% + ``The student is strongly recommended to carry out a few series of such experiments + personally, in order to acquire confidence in the use of the theory.'' + Mr.~Yule himself has indulged moderately.} + +\Paragraph{18.} The \textit{data} for these investigations have been principally +drawn from four sources---coin-tossing, the throw of dice, lotteries, +\index{Lotteries}% +and roulette; for in such cases as these the conditions for +Bernoulli's Theorem seem to be fulfilled most nearly. The earliest +recorded experiment was carried out by Buffon,\footnote + {\textit{Essai d'arithmètique morale} (see Bibliography), published + 1777, said to have been composed about 1760.} +who, assisted +%% -----File: 373.png---Folio 362------- +\index{De Morgan!pupil of}% +\index{Edgeworth|inote}% +\index{Petersburg Paradox!Buffon@{and Buffon}}% +\index{Poisson|inote}% +\index{Weldon and dice}% +\index{Wolf and dice}% +by a child tossing a coin into the air, played $2048$~\textit{partis} of the +Petersburg game, in which a coin is thrown successively until +the \textit{parti} is brought to an end by the appearance of heads. The +same experiment was repeated by a young pupil of De~Morgan's +`for his own satisfaction.'\footnote + {\textit{Formal Logic}, p.~185, published 1847. De~Morgan gives Buffon's results, +\index{Buffon!coin@{and coin-tossing}}% + as well as his pupil's, in full. Buffon's results are also investigated by Poisson, + \textit{Recherches}, pp.~132--135.} +In Buffon's trials there were $1992$ +tails to $2048$ heads; in Mr.~H.'s (De~Morgan's pupil) $2044$ tails to +$2048$ heads. A further experiment, due to Buffon's example, +\index{Quetelet!balls@{and balls}}% +was carried out by Quetelet\footnote + {\textit{Letters on the Theory of Probabilities} (Eng.\ trans.), p.~37.} +in 1837. He drew $4096$ balls from +an urn, replacing them each time, and recorded the result at +different stages, in order to show that the precision of the result +tended to increase with the number of the experiments. He +drew altogether $2066$ white balls and $2030$ black balls. Following +\index{Jevons!coin@{and coin-tossing}}% +in this same tradition is the experiment of Jevons,\footnote + {\textit{Principles of Science} (2nd~ed.), p.~208.} +who made +$2048$ throws of ten coins at a time, recording the proportion of +heads at each throw and the proportion of heads altogether. In +the whole number of $20,480$ single throws, he obtained heads +$10,353$ times; More recently Weldon\footnote + {Quoted by Edgeworth, ``Law of Error'' (\textit{Ency.\ Brit.}\ 10th~ed.), and by + Yule, \textit{Introduction to Statistics}, p.~254.} +threw twelve dice $4096$ +times, recording the proportion of dice at each throw which +showed a number greater than three. + +All these experiments, however, are thrown completely into +the shade by the enormously extensive investigations of the Swiss +astronomer Wolf, the earliest of which were published in 1850 +and the latest in~1893.\footnote + {See Bibliography. Of the earliest of these investigations I have no first-hand + knowledge and have relied upon the account given by Czuber, \textit{loc.\ cit.}\ +\index{Czuber!verification@{and verification of Bernoulli}|inote}% + vol.~i.\ p.~149. For a general account of empirical verifications of Bernoulli's + Theorem reference may be made to Czuber, \textit{Wahrscheinlichkeitsrechnung}, vol.~i.\ + pp\DPtypo{}{.}~139--152, and Czuber, \textit{Entwicklung der Wahrscheinlichkeitstheorie}, pp.~88--91.} +In his first set of experiments Wolf +completed $1000$ sets of tosses with two dice, each set continuing +until every one of the $21$~possible combinations had occurred at +least once. This involved altogether $97,899$ tosses, and he then +completed a total of $100,000$. These \textit{data} enabled him to work +out a great number of calculations, of which Czuber quotes the +following, namely a proportion of~$.83533$ of unlike pairs, as against +the theoretical value $.83333$, \ie~$\dfrac{5}{6}$. In his second set of experiments +%% -----File: 374.png---Folio 363------- +\index{Meissner, Otto, and dice-throwing}% +Wolf used two dice, one white and one red (in the first set +the dice were indistinguishable), and completed $20,000$ tosses, the +details of each result being recorded in the \textit{Vierteljahrsschrift der +Naturforschenden Gesellschaft in Zürich}. He studied particularly +the number of sequences with each die, and the relative frequency +of each of the $36$~possible combinations of the two dice. The +sequences were somewhat fewer than they ought to have been, +and the relative frequency of the different combinations very +different indeed from what theory would predict.\footnote + {Czuber quotes the principal results (\textit{loc.\ cit.}\ vol.~i.\ pp.~149--151). The + frequencies of only~$4$, instead of~$18$, out of the $36$~combinations lay within the + probable limits, and the standard deviation was~$76.8$ instead of~$23.2$.} +The explanation +of this is easily found; for the records of the relative +frequency of each face show that the dice must have been very +irregular, the six face of the white die, for example, falling $38$~per +cent more often than the four face of the same die. This, +then, is the sole conclusion of these immensely laborious experiments,---that +Wolf's dice were very ill made. Indeed the experiments +could have had no bearing except upon the accuracy +of his dice. But ten years later Wolf embarked upon one more +series of experiments, using \emph{four} distinguishable dice,---white, +yellow, red, and blue,---and tossing this set of four $10,000$ times. +Wolf recorded altogether, therefore, in the course of his life +$280,000$ results of tossing individual dice. It is not clear that +Wolf had any well-defined object in view in making these +records, which are published in curious conjunction with various +astronomical results, and they afford a wonderful example of the +pure love of experiment and observation.\footnote + {The latest experiment of the kind, of which I am aware, is that of Otto + Meissner (``Würfelversuche,'' \textit{Zeitschrift für Math.\ und Phys.}\ vol.~62 (1913), pp.~149--156), + who recorded $24$~series of $180$~throws each with four distinguishable + dice.} + +\Paragraph{19.} Another series of calculations have been based upon the +ready-made \textit{data} provided by the published results of lotteries +\index{Lotteries!published results of}% +\index{Roulette!published results of|inote}% +and roulette.\footnote + {For the publication of such returns there has always been a sufficient + demand on the part of gamblers. An \textit{Almanach romain sur la loterie royale de + France} was published at Paris in~1830, which contained all the drawings of the + French lottery (two or three a month) from 1758~to~1830. Players at Monte + Carlo are provided with cards and pins with which to record the results of + successive coups, and the results at the tables are regularly published in \textit{Le + Monaco}. Gamblers study these returns on account of the belief, which they + usually hold, that as the number of cases is increased the \emph{absolute} deviation from + the most probable proportion becomes less, whereas at the best Bernoulli's + Theorem shows that the \emph{proportionate} deviation decreases while the absolute + deviation \emph{increases}. Cf.~Houdin's \textit{Les Tricheries des Grecs dévoilées}: ``In a + game of chance, the oftener the same combination has occurred in succession, the + nearer we are to the certainty that it will not recur at the next cast or turn-up. + This is the most elementary of the theories on probabilities; it is termed the + \emph{maturity of the chances}.'' Laplace (\textit{Essai philosophique}, p.~142) quotes an + amusing instance of the same belief not drawn from the annals of gambling: + ``J'ai vu des hommes désirant ardemment d'avoir un fils, n'apprendre qu'avec + peine les naissances des garçons dans le mois où ils allaient devenir pères. + S'imaginant que le rapport de ces naissances à celles des filles devait être le + même à la fin de chaque mois, ils jugaient que les garçons déjà nés rendaient + plus probables les naissances prochaines des filles.'' + + The literature of gambling is very extensive, but, so far as I am acquainted + with it, excessively lacking in variety, the maturity of the chances and the + martingale continually recurring in one form or another. The curious reader + will find tolerable accounts of such topics in Proctor's \textit{Chance and Luck}, and + Sir Hiram Maxim's \textit{Monte Carlo Facts and Fallacies}.} +%% -----File: 375.png---Folio 364------- +\index{Houdin|inote}% +\index{Laplace!birth proportions@{and birth proportions}}% +\index{Lotteries|ifoll}% +\index{Maxim, Sir Hiram|inote}% +\index{Pearson, Karl!roulette@{and roulette}}% +\index{Proctor|inote}% + +\index{Czuber!lotteries@{and lotteries}}% +Czuber\footnote + {\textit{Zum Gesetz der grossen Zahlen}. The results are summarised in his \textit{Wahrscheinlichkeitsrechnung}, + vol.~i.\ p.~139.} +has made calculations based on the lotteries +of Prague ($2854$~drawings) and Brünn ($2703$~drawings) between +\index{Brunn and lotteries@{Brünn and lotteries}}% +the years 1754~and~1886, in which the actual results agree +\index{Fechner, and median!lotteries@{and lotteries}}% +very well with theoretical predictions. Fechner\footnote + {\textit{Kollektivmasslehre}, p.~229. These results also are summarised by Czuber, + \textit{loc.\ cit.}} +employed the +lists of the ten State lotteries of Saxony between the years 1843 +and~1852. Of a rather more interesting character are Professor +Karl Pearson's investigations\footnote + {\textit{The Chances of Death}, vol.~i.} +into the results of Monte Carlo +\index{Monte Carlo}% +Roulette as recorded in \textit{Le Monaco} in the course of eight weeks. +\index{Roulette}% +Applying Bernoulli's Theorem, on the hypothesis of the equiprobability +of all the compartments throughout the investigation, +he found that the actually recorded proportions of red and black +were not unexpected, but that alternations and long runs were +so much in excess that, on the assumption of the exact accuracy +of the tables, the \textit{à~priori} odds were at least a thousand millions +to one against some of the recorded deviations. Professor +Pearson concluded, therefore, that Monte Carlo Roulette is not +objectively a game of chance in the sense that the tables on which +it is played are absolutely devoid of bias. Here also, as in the +case of Wolf's dice, the conclusion is solely relevant, not to the +theory or philosophy of Chance, but to the material shapes of +the tools of the experiment. + +Professor Pearson's investigations into Roulette, which dealt +with $33,000$ Monte Carlo coups, have been overshadowed, just +%% -----File: 376.png---Folio 365------- +\index{Bortkiewicz, von, and great numbers!Marbe@{and Marbe}|inote}% +\index{Bromse and Marbel@{Brömse and Marbe}|inote}% +\index{Bruns and Marbe|inote}% +\index{D'Alembert|inote}% +\index{Grunbaum and Marbe@{Grünbaum and Marbe}|inote}% +\index{Lexis, and asymmetry of statistical frequency!Marbe@{and Marbe}|inote}% +\index{Marbe, Dr.\ Karl, and roulette}% +as all other tosses of coins and dice have been outdone by Wolf, +by Dr.~Karl Marbe,\footnote + {\textit{Naturphilosophische Untersuchungen zur Wahrscheinlichkeitstheorie}.} +who has examined $80,000$ coups from Monte +Carlo and elsewhere. Dr.~Marbe arrived at exactly opposite +conclusions; for he claims to have shown that long runs, so far +from being in excess, were greatly in defect. Dr.~Marbe introduces +this experimental result in support of his thesis that the +world is so constituted that long runs do not as a matter of fact +occur in it.\footnote + {Dr.~Marbe's monograph has given rise in Germany to a good deal of discussion, + not directed towards showing what a preposterous method this is for + demonstrating a natural law, but because the experimental result itself does not + really follow from the \textit{data} and is due to a somewhat subtle error in Marbe's + reasoning, by which he has been led into an incorrect calculation of the probable + proportions \text{à~priori} of the various sequences. The problem is discussed by + Von Bortkiewicz, Brömse, Bruns, Grimsehl, and Grünbaum (for exact references +\index{Grimsehl!Marbe@{and Marbe}|inote}% + to these see the Bibliography), and by Lexis (\textit{Abhandlungen}, pp.~222--226) and + Czuber (\textit{Wahrscheinlichkeitsrechnung}, vol.~i.\ pp.~144--149). Largely as a result +\index{Czuber!Marbe@{and Marbe}}% + of this controversy, Von Bortkiewicz has lately devoted a complete treatise + (\textit{Die Iterationen}) to the mathematics of `runs.' Dr.~Marbe has been given + far more attention by his colleagues in Germany than he conceivably deserves.} +Not merely are long runs very improbable. They +do not, according to him, occur at all. But we may doubt +whether roulette can tell us very much either of the laws of logic +or of the constitution of the universe. + +Dr.~Marbe's main thesis is identical, as he himself recognises, +\index{D'Alembert!Marbe@{and Marbe}}% +with one of the heterodox contentions of D'Alembert.\footnote + {D'Alembert's principal contributions to Probability are most accessible in + the volumes of his \textit{Opuscules mathématiques} (1761). Works on Probability + usually contain some reference to D'Alembert, but his sceptical opinions, rejected + rather than answered by the orthodox school of Laplace, have not always +\index{Laplace!school of}% + received full justice. D'Alembert has three main contentions to which in his + various papers he constantly recurs: + + (1) That a probability very small mathematically is really zero; + + (2) That the probabilities of two successive throws with a die are not + independent; + + (3) That `mathematical expectation' is not properly measured by the + product of the probability and the prize. + + The first and third of these were partly advanced in explanation of the + Petersburg paradox (see \Pageref{316}). The second is connected with the first, and + was also used to support his incorrect evaluation of the probability of heads + twice running; but D'Alembert, in spite of many of his results being wrong, + does not altogether deserve the ridicule which he has suffered at the hands of + writers, who accepted without sceptical doubts the hardly less incorrect conclusions + of the orthodox theory of that time.} +But this +principle of variety, precisely opposite to the usual principle of +Induction, can have no claim to be accepted \textit{à~priori} and, as a +\emph{general} principle, there is no adequate evidence to support it from +experience. Its origin is to be found, perhaps, in the fact that +%% -----File: 377.png---Folio 366------- +in a certain class of cases, especially where conscious human +agency comes in, it may contain some element of truth. The +fact of an act's having been done in a particular way once may +be a special reason for thinking that it will not be performed on +the next occasion in precisely the same manner. Thus in many +so-called random events some slight degree of causal and material +dependence between successive occurrences may, nevertheless, +exist. In these cases `runs' may be fewer and shorter than those +which we should predict, if a complete absence of such dependence +is assumed. If, for example, a pack of cards be dealt, collected, +and shuffled, to the extent that card-players do as a rule shuffle, +there may be a greater presumption against the second hand's +being identical with the first than against any other particular +distribution. In the case of croupiers long experience might +possibly suggest some psychological generalisation,---that they +are very mechanical, giving an excess of numbers belonging to a +particular section of the wheel, or, on the other hand, that when +a croupier sees a run beginning, he tends to vary his spin more than +usual, thus bringing runs to an end sooner than he ought.\footnote + {A good roulette table is, however, so delicate an instrument that no probable + degree of regularity of habit on the part of the spinner could be sufficient + to produce regularity in the result.} +At any +rate, it is worth emphasising once more that from such experiments +as these this is the only \emph{kind} of knowledge which we can +hope to obtain,---knowledge of the material construction of a +die or of the psychology of a croupier. +%% -----File: 378.png---Folio 367------- +\index{Probability, and relevant knowledge!from statistics|ifoll}% + + +\Chapter{XXX}{The Mathematical use of Statistical Frequencies for +the Determination of Probability \textit{à posteriori}---the +Methods of Laplace} + +\begin{Quote} +Utilissima est aestimatio probabilitatum, quanquam in exemplis juridicis +politicisque plerumque non tam subtili calculo opus est, quam accurata +omnium circumstantiarum enumeratione.---\textsc{Leibniz}. +\end{Quote} + +\Paragraph{1.} \First{In} the preceding chapter we have assumed that the probability +of an event at each of a series of trials is given, and have considered +how to infer from this the probabilities of the various possible +frequencies of the event over the whole series, without discussing +in detail by what method the initial probability had been determined. +In statistical inquiries it is generally the case that this +initial probability is based, not upon the Principle of Indifference, +\index{Principle of Indifference!statistics@{and statistics}}% +but upon the statistical frequencies of similar events +which have been observed previously. In this chapter, therefore, +we must commence the complementary part of our inquiry,---namely, +into the method of deriving a measure of probability +from an observed statistical frequency. + +I do not myself believe that there is any direct and simple +method by which we can make the transition from an observed +numerical frequency to a numerical measure of probability. +The problem, as I view it, is part of the general problem of founding +judgments of probability upon experience, and can only be +dealt with by the general methods of induction expounded in +\Partref{III}\@. The nature of the problem precludes any other method, +and direct mathematical devices can all be shown to depend +upon insupportable assumptions. In the next chapters we will +consider the applicability of general inductive methods to this +problem, and in this we will endeavour to discredit the mathematical +charlatanry by which, for a hundred years past, the basis +of theoretical statistics has been greatly undermined. +%% -----File: 379.png---Folio 368------- +\index{Bernoulli, Jac.}% + +\Paragraph{2.} Two direct methods have been commonly employed, +theoretically inconsistent with one another, though not in every +case noticeably discrepant in practice. The first and simplest of +these may be termed the Inversion of Bernoulli's Theorem, and +\index{Bernoulli's Theorem!Inverse of|ifoll}% +the other Laplace's Rule of Succession. +\index{Laplace!Rule of Succession@{and Rule of Succession}}% +\index{Rule of Succession}% + +The earliest discussion of this problem is to be found in the +\index{Leibniz}% +Correspondence of Leibniz and Jac.\ Bernoulli,\footnote + {For the exact references see \Bibref.} +and its true +nature cannot be better indicated than by some account of the +manner in which it presented itself to these very illustrious +philosophers. The problem is tentatively proposed by Bernoulli +in a letter addressed to Leibniz in the year 1703. We can determine +from \textit{à~priori} considerations, he points out, by how much it +is more probable that we shall throw~$7$ rather than~$8$ with two dice, +but we cannot determine by such means the probability that a +young man of twenty will outlive an old man of sixty. Yet is it +not possible that we might obtain this knowledge \textit{à~posteriori} +from the observation of a great number of similar couples, each +consisting of an old man and a young man? Suppose that the +young man was the survivor in $1000$~cases and the old man in $500$~cases, +might we not conclude that the young man is twice as likely +as the old man to be the survivor? For the most ignorant +persons seem to reason in this way by a sort of natural instinct, +and feel that the risk of error is diminished as the number of +observations is increased. Might not the solution tend asymptotically +to some determinate degree of probability with the +increase of observations? \textit{Nescio, Vir Amplissime, an speculationibus +istis soliditatis aliquid inesse Tibi videatur.} + +Leibniz's reply goes to the root of the difficulty. The calculation +of probabilities is of the utmost value, he says, but in statistical +inquiries there is need not so much of mathematical subtlety +as of a precise statement of all the circumstances. The possible +contingencies are too numerous to be covered by a finite number +of experiments, and exact calculation is, therefore, out of the +question. Although nature has her habits, due to the recurrence +of causes, they are general, not invariable. Yet empirical calculation, +although it is inexact, may be adequate in affairs of practice.\footnote + {Leibniz's actual expressions (in a letter to Bernoulli, December~3, 1703) are + as follows: Utilissima est aestimatio probabilitatum, quanquam in exemplis + juridicis politicisque plerumque non tam subtili calculo opus est, quam accurata + omnium circumstantiarum enumeratione. Cum empirice aestimamus probabilitates + per experimenta successuum, quaeris an ea via tandem aestimatio + perfecte obtineri possit. Idque a Te repertum scribis. Difficultas in eo mihi + inesse videtur, quod contingentia seu quae infinitis pendent circumstantiis, per + finita experimenta determinari non possunt; natura quidem suas habet consuetudines, + natas ex reditu causarum, sed non nisi \textgreek{<ws `ep`i t`o pol'u}. Novi morbi + inundant subinde humanum genus, quodsi ergo de mortibus quotcunque experimenta + feceris, non ideo naturae rerum limites posuisti, ut pro futuro variare + non possit. Etsi autem empirice non posset haberi perfecta aestimatio, non + ideo minus empirica aestimatio in praxi utilis et sufficiens foret.} +%% -----File: 380.png---Folio 369------- +\index{Bernoulli, Jac.}% +\index{Inverse Probability!statistics@{and statistics}}% + +Bernoulli in his answer fell back upon the analogy of balls +drawn from an urn, and maintained that without estimating +each separate contingency we might determine within narrow +limits the \emph{proportion} favouring each alternative. If the true +proportion were~$2:1$, we might estimate it with moral certainty +\textit{à~posteriori} as lying between $201:100$ and $199:100$. ``Certus +sum,'' he concluded the controversy, ``Tibi placituram demonstrationem, +cum publicavero.'' But whether he was impressed by +the just caution of Leibniz, or whether death intercepted him, +he advances matters no further in the \textit{Ars Conjectandi}. After +dealing with some of Leibniz's objections\footnote + {The relevant passages are on pp.~224--227 of the \textit{Ars Conjectandi}.} +and seeming to +promise some mode of estimating probabilities \textit{à~posteriori} by an +inversion of his theorem, he proves the direct theorem only and +the book is suddenly at an end. + +\Paragraph{3.} In dealing with the correspondence of Leibniz and Bernoulli, +I have not been mainly influenced by the historical interest +of it. The view of Leibniz, dwelling mainly on considerations +of analogy, and demanding ``not so much mathematical subtlety +as a precise statement of all the circumstances,'' is, substantially, +the view which will be supported in the following chapters. +The desire of Bernoulli for an exact formula, which would derive +from the numerical frequency of the experimental results a +numerical measure of their probability, preludes the exact +formulas of later and less cautious mathematicians, which will be +examined immediately. + +\Paragraph{4.} During the greater part of the eighteenth century there is +no trace, I think, of the explicit use of the Inversion of Bernoulli's +Theorem. The investigations carried out by D'Alembert, Daniel +\index{D'Alembert}% +Bernoulli, and others relied upon the type of argument examined +in \Chapref{XXV}\@. They showed, that is to say, that certain +observed series of events would have been very improbable, if +we had supposed independence between some two factors or if +%% -----File: 381.png---Folio 370------- +\index{De Morgan!Inverse@{and Inverse of Bernoulli's Theorem}|inote}% +\index{Inverse Probability!statistics@{and statistics}|inote}% +\index{Munro|inote}% +\index{Todhunter|inote}% +some occurrence had been assumed to be as likely as not, and they +inferred from this that there was in fact a measure of dependence +or that the occurrence had probability in its favour. But they +did not endeavour to pass from the observed frequency of occurrence +to an exact measure of the probability. With the advent +of Laplace more ambitious methods took the field. +\index{Laplace!Bernoulli's Theorem@{and Bernoulli's Theorem}}% +\index{Laplace!unknown probabilities@{and unknown probabilities}}% + +Laplace began by assuming without proof a direct inversion +of Bernoulli's Theorem. Bernoulli's Theorem, in the form in +which Laplace proved it, states that, if $p$~is the probability \textit{à~priori}, +there is a probability~$P$ that the proportion of times $\dfrac{m}{m+n}$ +of the event's occurrence in $\mu\,(=m+n)$ trials will lie between +$p±\gamma\sqrt{\dfrac{2pq}{\mu}}$, where $P = \displaystyle\frac{2}{\sqrt{\pi}}\int_0^\gamma e^{-t^2}\, dt + \frac{1}{\sqrt{2\pi\mu pq}}e^{-\gamma^2}$. The inversion +of the theorem, which he assumes without proof, +states that, if the event is observed to happen $m$~times +in $\mu$~trials, there is a probability~$P$ that the probability +of the event~$p$ will lie between $\dfrac{m}{\mu} ± \gamma\sqrt{\dfrac{2mn}{\mu^3}}$, where +$P = \displaystyle\frac{2}{\sqrt{\pi}} \int_0^\gamma e^{-t^2}\, dt + \frac{1}{\sqrt{2\pi\mu\dfrac{m}{\mu^2}}}e^{-\gamma^2}$. The same result is also given +\index{Poisson!inverse@{and inverse of Bernoulli's Theorem}}% +by Poisson.\footnote + {For an account of the treatments of this topic both by Laplace and by + Poisson, see Todhunter's \textit{History}, pp.~554--557. Both of them also obtain a + formula slightly different from that given above by a method analogous to the + first part of the proof of Laplace's Rule of Succession; \ie~by an application of + the inverse principle of probability to the assumption that the probability of + the probability's lying within any interval is proportional to the length of the + interval. This discrepancy has given rise to some discussion. See Todhunter, + \textit{loc.\ cit.}; De~Morgan, \textit{On a Question in the Theory of Probabilities}; Monro, \textit{On the + Inversion of Bernoulli's Theorem in Probabilities}; and Czuber, \textit{Entwicklung}, +\index{Czuber!Inverse@{and Inverse of Bernoulli's Theorem}|inote}% + pp.~83,~84. But this is not the important distinction between the two mathematical + methods by which this question has been approached, and this minor + point, which is of historical interest mainly, I forbear to enter into.} +Thus, given the frequency of occurrence in $\mu$~trials, +these writers infer the probability of occurrence at +subsequent trials within certain limits, just as, given the +\textit{à~priori} probability, Bernoulli's Theorem would enable them +to predict the frequency of occurrence in $\mu$~trials within corresponding +limits. +%% -----File: 382.png---Folio 371------- + +If the number of trials is at all numerous, these limits are +narrow and the purport of the inversion of Bernoulli's Theorem +may therefore be put briefly as follows. By the direct theorem, +if $p$~measures the probability, $p$~also measures the most probable +value of the frequency; by the inversion of the theorem, if $\dfrac{m}{m+n}$ +measures the frequency, $\dfrac{m}{m+n}$ also measures the most probable +value of the probability. The simplicity of the process has recommended +it, since the time of Laplace, to a great number of +writers. Czuber's argument, criticised on \Pageref{351}, with reference +to the proportions of male and female births in Austria, is based +upon an unqualified use of it. But examples abound throughout +the literature of the subject, in which the theorem is employed in +circumstances of greater or less validity. + +The theorem was originally given without proof, and is indeed +incapable of it, unless some illegitimate assumption has been +introduced. But, apart from this, there are some obvious objections. +We have seen in the preceding chapter that Bernoulli's +Theorem itself cannot be applied to all kinds of \textit{data} indiscriminately, +but only when certain rather stringent conditions are fulfilled. +Corresponding conditions are required equally for the +inversion of the theorem, and it cannot possibly be inferred from +a statement of the number of trials and the frequency of occurrence +merely, that these have been satisfied. We must know, +for instance, that the examined instances are similar in the main +relevant particulars, both to one another and to the unexamined +instances to which we intend our conclusion to be applicable. +An unanalysed statement of frequency cannot tell us this. + +This method of passing from statistical frequencies to probabilities +is not, however, like the method to be discussed in a +moment, radically false. With due qualifications it has its place +in the solution of this problem. The conditions in which an +inversion of Bernoulli's Theorem is legitimate will be elucidated +in \Chapref{XXXI}\@. In the meantime we will pass on to Laplace's +second method, which is more powerful than the first and has +obtained a wider currency. The more extreme applications of +it are no longer ventured upon, but the theory which underlies +it is still widely adopted, especially by French writers upon +probability, and seldom repudiated. +%% -----File: 383.png---Folio 372------- +\index{Probability, and relevant knowledge!unknown@{`\textit{unknown},' and Laplace}}% +\index{Rule of Succession}% + +\index{Venn!Rule of Succession@{and Rule of Succession}}% +\Paragraph{5.} The formula in question, which Venn\footnote + {\textit{Logic of Chance}, p~190.} +has called the \textit{Rule +of Succession}, declares that, if we know no more than that an +event has occurred $m$~times and failed $n$~times under given conditions, +then the probability of its occurrence when those conditions +are next fulfilled is $\dfrac{m+1}{m+n+2}$. It is necessary, however, +before we examine the proof of this formula, to discuss in detail +the reasoning which leads up to it. + +This preliminary reasoning involves the Laplacian theory of +`unknown probabilities.' The postulate, upon which it depends, +\index{Unknown probabilities}% +is introduced to supplement the Principle of Indifference, and +\index{Principle of Indifference!Laplace@{and Laplace}}% +is in fact the extension of this principle from the probabilities +of arguments, when we know nothing about the arguments, to the +probabilities that the probabilities of arguments have certain +values, when we know nothing about the probabilities. Laplace's +enunciation is as follows: ``Quand la probabilité d'un événement +simple est inconnue, on peut lui supposer également toutes les +valeurs depuis zéro jusqu'à l'unité. La probabilité de chacune +de ces hypothèses tirée de l'événement observé est \ldots\ une +fraction dont le numérateur est la probabilité de l'événement dans +cette hypothèse, et dont le dénominateur est la somme des probabilités +semblables relatives à toutes les hypothèses\ldots.''\footnote + {\textit{Essai philosophique}, p.~16.} + +Thus when the probability of an event is unknown, we may +suppose all possible values of the probability between $0$~and~$1$ to +be equally likely \textit{à~priori}. The probability, \emph{after} the event has +occurred, that the probability \textit{à~priori} was~$\dfrac{1}{r}$ (say), is measured +by a fraction of which $\dfrac{1}{r}$~is the numerator and the sums of all the +possible \textit{à~priori} values the denominator. The origin of this rule +is evident. If we consider the problem in which a ball is drawn +from a bag containing an infinite number of black and white balls +in unknown proportions, we have hypotheses, corresponding to +each of the possible constitutions of the bag, the assumption of +which yields in turn every value between $0$~and~$1$ as the \textit{à~priori} +probability of drawing a white ball. If we could assume that +these constitutions are equally probable \textit{à~priori}, we should +obtain probabilities for each of them \textit{à~posteriori} according to +Laplace's rule. +%% -----File: 384.png---Folio 373------- +\index{Unknown probabilities}% + +On the analogy of this Laplace assumes in general that, where +everything is unknown, we may suppose an infinite number of +possibilities, each of which is equally likely, and each of which +leads to the event in question with a \emph{different} degree of probability, +so that for every value between $0$~and~$1$ there is one and only one +hypothetical constitution of things, the assumption of which +invests the event with a probability of that value. + +\Paragraph{6.} It might be an almost sufficient criticism of the above to +point out that these assumptions are entirely baseless. But the +theory has taken so important a place in the development of +probability that it deserves a detailed treatment. + +What, in the first place, does Laplace mean by an \emph{unknown} +probability? He does not mean a probability, whose value is in +fact unknown \emph{to us}, because we are unable to draw conclusions +which \emph{could be drawn} from the \textit{data}; and he seems to apply the +term to any probability whose value, according to the argument +of \Chapref{III}., is numerically indeterminate. Thus he assumes +that \emph{every} probability has a numerical value and that, in those +cases where there seems to be no numerical value, this value is +not non-existent but unknown; and he proceeds to argue that +where the numerical value is unknown, or as I should say where +there is \emph{no} such value, every value between $0$~and~$1$ is equally +probable. With the possible interpretations of the term `unknown +probability,' and with the theory that every probability +can be measured by one of the real numbers between $0$~and~$1$, +I have dealt, as carefully as I can, in \Chapref{III}\@. If the view +taken there is correct, Laplace's theory breaks down immediately. +But even if we were to answer these questions, not as they have +been answered in \Chapref{III}., but in a manner favourable to +Laplace's theory, it remains doubtful whether we could legitimately +attribute a value to the probability of an unknown probability's +having such and such a value. If a probability is +unknown, surely the probability, relative to the same evidence, +that this probability has a given value, is also unknown; and we +are involved in an infinite regress. + +\Paragraph{7.} This point leads on to the second objection; Laplace's +theory requires the employment of both of two inconsistent +methods. Let us consider a number of alternatives $a_1$,~$a_2$,~etc., +having probabilities $p_1$,~$p_2$,~etc.; if we do not know anything +about~$a_1$, we do not know the value of its probability~$p_1$, and we +%% -----File: 385.png---Folio 374------- +must consider the various possible values of~$p_1$, namely $b_1$,~$b_2$,~etc., +the probabilities of these possible values being $q_1$,~$q_2$,~etc.\ respectively. +There is no reason why this process should ever stop. +For as we do not know anything about~$b_1$, we do not know the +value of its probability~$q_1$, and we must consider the various +possible values of~$q_1$ namely $c_1$,~$c_2$,~etc., the probabilities of these +possible values being $r_1$,~$r_2$,~etc., respectively; and so on. This +method consists in supposing that, when we do not know anything +about an alternative, we must consider all the possible values of +the probability of the alternative; these possible values can form +in their turn a set of alternatives, and so on. But this method +\emph{by itself} can lead to no final conclusion. Laplace superimposes +on it, therefore, his other method of determining the probabilities +of alternatives about which we know nothing,---namely, the +Principle of Indifference. According to this method, when +\index{Principle of Indifference!Laplace@{and Laplace}}% +we know nothing about a set of alternatives, we suppose the +probabilities of each of them to be \emph{equal}. In some parts of +his writings---and this is true also of most of his followers---he +applies this method from the beginning. If, that is to say, we +know nothing about~$a_1$, since $a_1$~and its contradictory form a pair +of exhaustive alternatives two in number, the probability of these +alternatives is \emph{equal} and each is~$\dfrac{1}{2}$. But in the reasoning which +leads up to the Law of Succession he chooses to apply this method +at the second stage, having used the other method at the first +stage. If, that is to say, we know nothing about~$a_1$, its probability~$p_1$ +may have any of the values $b_1$,~$b_2$,~etc.\ where $b_1$~is any +fraction between $0$~and~$1$; and, as we know nothing about the +probabilities $q_1$,~$q_2$,~etc.\ of these alternatives $b_1$,~$b_2$,~etc., we may +by the Principle of Indifference suppose them to be \emph{equal}. This +account may seem rather confused; but it is not easy to give +a lucid account of so confused a doctrine. + +\Paragraph{8.} Turning aside from these considerations, let us examine +the theory, for a moment, from another side. When we reach the +Rule of Succession, it will be seen that the hypothetical \textit{à~priori} +\index{Rule of Succession}% +probabilities are treated as if they were possible \emph{causes} of the +event. It is assumed, that is to say, that the number of possible +sets of antecedent conditions is proportional to the number of +real numbers between $0$~and~$1$; and that these fall into equal +groups, each group corresponding to one of the real numbers +%% -----File: 386.png---Folio 375------- +\index{De Morgan!Rule of Succession@{and Rule of Succession}}% +between $0$~and~$1$, this number measuring the degree of probability +with which we could predict the event, if we knew that an antecedent +condition belonging to that group was fulfilled. It is +then assumed that all of these possible antecedent conditions are +\textit{à~priori} equally likely. The argument has arisen by false analogy +from the problem in which a ball is drawn from an urn containing +an infinite number of black and white balls. But for the assumption +that we have \emph{in general} the kind of knowledge which is +necessary about the possible antecedents, no reasonable foundation +has been suggested. + +De~Morgan endeavoured to deal with the difficulty in much +the same way in the following passage:\footnote + {\textit{Cabinet Encyclopaedia}, p.~87.} +``In determining the +chance which exists (under known circumstances) for the happening +of an event a number of times which lies between certain +limits, we are involved in a consideration of some difficulty, +namely, the \emph{probability of a probability}, or, as we have called it, +the presumption of a probability. To make this idea more clear, +remember that any state of probability may be immediately +made the expression of the result of a set of circumstances, which +being introduced into the question, the difficulty disappears. +The word presumption refers distinctly to an act of the mind, or a +state of the mind, while in the word probability we feel disposed +rather to think of the external arrangements on the knowledge +of which the strength of our presumption ought to depend, than +of the presumption itself.'' The point of this explanation lies +in the assumption that ``any state of probability may be immediately +made the expression of the result of a set of circumstances.'' +It cannot be allowed that this is generally true;\footnote + {For instance, it is not true even in the standard instance of balls drawn from + an urn containing black and white in unknown proportions, unless the number + of balls is infinite.} +and even in +those cases in which it is true we are thrown back on the \textit{à~priori} +probabilities of the various sets of circumstances which need not +be, as De~Morgan assumes, either equal or exhaustive alternatives. + +\Paragraph{9.} The proof of the Rule of Succession, which is based upon +\index{Rule of Succession!proof of}% +this theory of unknown probabilities, is, briefly, as follows: +\index{Unknown probabilities}% + +If $x$~stands for the \textit{à~priori} probability of an event in given +conditions, then the probability that the event will occur $m$~times +and fail $n$~times in these conditions is~$x^m(1-x)^n$. If, +however, $x$~is unknown, all values of it between $0$~and~$1$ +%% -----File: 387.png---Folio 376------- +\index{Whittaker, E. T., and Rule of Succession|inote}% +are \textit{à~priori} equally probable. It follows from these two +sets of considerations that, if the event has been observed +to occur $m$ times out of $m+n$, the probability \textit{à posteriori} that +$x$ lies between $x$ and $x+dx$ is proportional to $x^m(1-x)^ndx$, +and is equal, therefore, to $Ax^m(1-x)^ndx$ where $A$ is a constant. +Since the event has in fact occurred, and since $x$ must have +one of its possible values, $A$ is determined by the equation +\[ +\int_0Ax^m(1-x)^ndx=1\quad\therefore A=\dfrac{\Gamma(m+n+2)}{\Gamma(m+1)\Gamma(n+1)}. +\] +Hence the probability that the event will occur at the $(m+n+1)$th +trial, when we know that it has occurred $m$ times in $m+n$ +trials, is +\[ +A\int_0^1 x^{m+1}(1-x)^ndx. +\] +If we substitute the value of A found above, this is equal to +$\dfrac{m+1}{m+n+2}$.\footnote + {The theorem is sometimes enunciated by contemporary writers in a much + more guarded form, \eg\ by Czuber, \textit{Wahrscheinlichkeitsrechnung}, vol.~i.\ p.~197, +\index{Czuber!Rule of Succession@{and Rule of Succession}|inote}% + and by Bachelier, \textit{Calcul des probabilités}, p.~487. Bachelier, instead of assuming +\index{Bachelier!Rule of Succession@{and Rule of Succession}|inote}% + that the \textit{à~priori} probabilities of all possible values of the probability of the + event are equal, writes $\hat\omega(y)dy$ as the \textit{à~priori} probability that the probability is~$y$, + so that after $m$~occurrences \DPtypo{is}{in} $m+n$~trials the probability that the probability + lies between $y$~and~$y+dy$ is $\dfrac{y^m(1-y)^n\hat\omega(y)dy}{\int y^m(1-y)^n\hat\omega(y)dy}$. If one has no idea of $\hat\omega$ \textit{à~priori}, + he suggests that the simplest hypothesis is to put $\hat\omega=1$, which leads, as + above, to Laplace's Law of Succession. He also proposes the hypothesis + $\hat\omega(y)=a+a_1y+a_2y^2+\ldots$, in which case the denominator is a series of Eulerian + integrals. There is a discussion of the Law of Succession, and of the contradictions + and paradoxes to which it leads, by E.~T. Whittaker and others in + Part~VI. vol.~viii.\ (1920) of the \textit{Transactions of the Faculty of Actuaries in + Scotland}.} + +The class of problem to which the theorem is supposed to +apply is the following: There are certain conditions such that we +are ignorant \textit{à~priori} as to whether they do or do not lead to the +occurrence of a particular event; on $m$ out of $m+n$ occasions, +however, on which these conditions have been observed, the +event has occurred; what is the probability in the light of this +experience that the event will occur on the next occasion? The +answer to all such problems is $\dfrac{m+1}{m+n+2}$. In the cases where +$n=0$, \ie\ when the event has invariably occurred, the formula +%% -----File: 388.png---Folio 377------- +yields the result $\dfrac{m + 1}{m + 2}$. In the case where the conditions have +been observed once only and the event has occurred on that +occasion, the result is~$\dfrac{2}{3}$. If the conditions have \emph{never} been met +with at all, the probability of the event is~$\dfrac{1}{2}$. And even in the +case where on the only occasion on which the conditions were +observed, the event did \emph{not} occur, the probability is~$\dfrac{1}{3}$. + +Some of the flaws in this proof have been already explained. +One minor objection may be pointed out in addition. It is +assumed that, if $x$~is the \textit{à~priori} probability of the event's happening +once, then $x^n$~is the \textit{à~priori} probability of its happening $n$~times +in succession, whereas by the theorem's own showing the +knowledge that the event has happened once modifies the probability +of its happening a second time; its successive occurrences +are not, therefore, independent. If the \textit{à~priori} probability of the +event is~$\dfrac{1}{2}$, and if, after it has been observed once, the probability +that it will occur a second time is~$\dfrac{2}{3}$, then it follows that the \textit{à~priori} +probability of its occurring twice is not $\dfrac{1}{2} × \dfrac{1}{2}$, but $\dfrac{1}{2} × \dfrac{2}{3}$, +\ie~$\dfrac{1}{3}$; and in general the \textit{à~priori} probability of its happening +n times in succession is not $\left(\dfrac{1}{2}\right)^n$ but~$\dfrac{1}{n+1}$. + +\Paragraph{10.} But refinements of disproof are hardly needed. The +principle's conclusion is inconsistent with its premisses. We +begin with the assumption that the \textit{à~priori} probability of an event, +about which we have no information and no experience, is unknown, +and that all values between 0 and 1 are equally probable. +We end with the conclusion that the \textit{à~priori} probability of +such an event is~$\dfrac{1}{2}$. It has been pointed out in §\;7 that this +contradiction was latent, as soon as the Principle of Indifference +\index{Principle of Indifference!Rule of Succession@{and Rule of Succession}}% +was superimposed on the principle of unknown probabilities. + +The theorem's conclusions, moreover, are a \textit{reductio ad +absurdum} of the reasoning upon which it is based. Who could +suppose that the probability of a purely hypothetical event, of +%% -----File: 389.png---Folio 378------- +whatever complexity, in favour of which no positive argument +exists, the like of which has \emph{never} been observed, and which has +failed to occur on the one occasion on which the hypothetical +conditions were fulfilled, is no less than~$\dfrac{1}{3}$? Or if we do suppose it, +we are involved in contradictions,---for it is easy to imagine more +than three \emph{incompatible} events which satisfy these conditions. + +\Paragraph{11.} The theorem was first suggested by the problem of the urn +which contains black and white balls in unknown proportions: +$m$ white and $n$ black balls have been successively drawn and +replaced; what is the probability that the next draw will yield +a white ball? It is supposed that all compositions of the urn are +equally probable, and the proof then proceeds precisely as in the +case of the more general rule of succession. The rule of succession +\index{Rule of Succession!frequency theory@{and frequency theory}}% +has been, sometimes, directly deduced from the case of the urn, +by assimilating the occurrence of the event to the drawing of a +white ball and its non-occurrence to the drawing of a black ball. + +On the hypothesis that all compositions of the urn are equally +probable, an hypothesis to which in general there is nothing corresponding, +and on the further hypothesis that the number of balls +is infinite, this solution is correct.\footnote + {This second condition is often omitted (\eg\ Bertrand, \textit{Calcul des probabilités}, + p.~172).} +But the rule of succession +does not apply, as it is easy to demonstrate, even to the case of +balls drawn from an urn, if the number of balls is finite.\footnote + {The correct solution for the case of a finite number of balls, on the hypothesis + that each possible ratio is equally likely, is as follows: The probability + of a black ball at a further trial, after black balls have been successively withdrawn + and replaced $p$~times, is $\dfrac{1}{n}\,\dfrac{s_{p+1}}{s_p}$ where there are $n$~balls and $s_r$~represents + the sum of the $r$th~powers of the first $n$~natural numbers. This reduces to + $\dfrac{p+1}{p+2}$---the solution usually given,---when $n$~is infinite. More generally, if + $p$~black balls and $q$~white balls have been drawn and replaced, the chance + that the next ball will be black is + \[ %[** TN: Displaying in-line formula.] + \frac{1}{n}\, + \frac{ \Sum^{r=n}_{r=0} r^{p+1}(n-r)^q}{\Sum^{r=n}_{r=0} r^p (n-r)^q}. + \]} + +\Paragraph{12.} If the Rule of Succession is to be adopted by adherents of +\index{Frequency theory!Rule of Succession@{and Rule of Succession}}% +the Frequency Theory of Probability,\footnote + {See \Chapref{VIII}\@.} +it is necessary that they +should make some modification in the preliminary reasoning on +which it is based. By Dr.~Venn, however, the rule has been +\index{Venn!Rule of Succession@{and Rule of Succession}}% +%% -----File: 390.png---Folio 379------- +\index{Bayes, and Inverse Probability!Theorem of}% +\index{Pearson, Karl!Rule of Succession@{and Rule of Succession}}% +explicitly rejected on the ground that it does not accord with +experience.\footnote + {\textit{Logic of Chance}, p.~197.} +But Professor Karl Pearson, who accepts it, has +made the necessary restatement,\footnote + {``On the Influence of Past Experience on Future Expectation,'' \textit{Phil.\ + Mag.}\ 1907, pp.~365--378. The quotations given below are taken from this + article.} +and it will be worth while to +examine the reasoning when it is put in this form. Professor +Pearson's proof of the Rule of Succession is as follows: + +``I start, as most mathematical writers have done, with `the +equal distribution of ignorance,' or I assume the truth of Bayes' +Theorem. I hold this theorem not as rigidly demonstrated, but +\index{Edgeworth}% +I think with Edgeworth\footnote + {This reference is, no doubt, to Edgeworth's ``Philosophy of Chance'' + (\textit{Mind}, 1884, p.~230), when he wrote: ``The assumption that any probability-constant + about which we know nothing in particular is as likely to have one value + as another is grounded upon the rough but solid experience that such constants + do, as a matter of fact, as often have one value as another.'' See also \Chapref{VII}. + §\;6, above.} +that the hypothesis of the equal distribution +of ignorance is, within the limits of practical life, justified +by our experience of statistical ratios, which \textit{à~priori} are +unknown, \ie\ such ratios do not tend to cluster markedly round +any particular value. `Chances' lie between $0$~and~$1$, but our +experience does not indicate any tendency of actual chances to +cluster round any particular value in this range. The ultimate +basis of the theory of statistics is thus not mathematical but +observational. Those who do not accept the hypothesis of the +equal distribution of ignorance and its justification in observation +are compelled to produce definite evidence of the clustering of +chances, or to drop all application of past experience to the judgment +of probable future statistical ratios\ldots. + +``Let the chance of a given event occurring be supposed to lie +between $x$~and~$x + dx$, then if on $n = p + q$ trials an event has been +observed to occur $p$~times and fail $q$~times, the probability that +the true chance lies between $x$~and~$x + dx$ is, on the equal +distribution of our ignorance, +\[ +P_z = \frac{x^p (1-x)^q\, dx}{\int^1_0 x^p (1-x)^q\, dx}. +\] + +``This is Bayes' Theorem\ldots.\footnote + {Professor Pearson's use of this title for the above formula is not, I think, + historically correct. Bayes' Theorem is the Inverse Principle of Probability + itself, and not this extension of it.} +%% -----File: 391.png---Folio 380------- + +``Now suppose that a second trial of $m=r + s$ instances be +made, then the probability that the given event will occur $r$~times +and fail~$s$, is on the \textit{à~priori} chance being between $x$~and~$x+dx$ +\[ += P_x\frac{\Gamma m}{\Gamma r\Gamma s}n^r(1-x)^s, +\] +and accordingly the total chance~$C_r$, whatever $x$~may be of the +event occurring $r$~times in the second series, is +\[ +C_r = \frac{\Gamma m}{\Gamma r\Gamma s}\, + \dfrac{\int_0\DPtypo{}{^1} x^{p+r}(1-x)^{q+s}\, dx} + {\int_0^1 x^p(1-x)^{q}\, dx}. +\] + +This is, with a slight correction, Laplace's extension of Bayes' +\index{Laplace!Bayes' Theorem@{and Bayes' Theorem}}% +Theorem.''\footnote + {The rest of the article is concerned with the determination of the probable + error when Laplace's Rule of Succession is used not simply to yield the probability +\index{Rule of Succession!Pearson@{and Pearson}|inote}% + of a single additional occurrence, but to predict the probable limits within + which the frequency will lie in a considerable series of additional trials. Professor + Pearson's method applies more rigorous methods of approximation to + the fundamental formulae given above than have been sometimes used. As + my main purpose in this chapter is to dispute the general validity of the fundamental + formulae, it is not worth while to consider these further developments + here. If the validity of the fundamental formula were to be granted, Professor + Pearson's methods of approximation would, I think, be satisfactory.} + +\Paragraph{13.} This argument can be restated as follows. Of all the +objects which satisfy~$\phi(x)$, let us suppose that a proportion~$p$ +also satisfy~$f(x)$. In this case $p$~measures the probability that +any object, of which we know only that it is~$\phi$, is in fact also~$f$. +Now if we do not know the value of~$p$ and have no relevant information +which bears upon it, we can assume \textit{à~priori} that all +values of~$p$ between $0$~and~$1$ are equally likely. This assumption, +which is termed the `equal distribution of ignorance,' is justified +by our experience of statistical ratios. Our experience, that is +to say, leads us to suppose that of all the theories, which could be +propounded, there are just as many which are always true as +there are which are always false, just as many which are true once +in fifty times as there are which are true once in three times, and +so on. Professor Pearson challenges those who do not accept +this assumption to produce definite evidence to the contrary. + +The challenge is easily met. It would not be difficult to produce +$10,000$ positive theories which are always false corresponding +to every one which is always true, and $10,000$ correlations of positive +%% -----File: 392.png---Folio 381------- +qualities which hold less often than once in three times for +every one we can name which holds more often than once in three +times. And the converse is the case for negative theories and +correlations between negative qualities; for corresponding to +every positive theory which is true there is a negative theory +which is false, and so on. Thus experience, if it shows anything, +shows that there is a very marked clustering of statistical ratios +in the neighbourhoods of zero and unity,---of those for positive +theories and for correlations between positive qualities in the +neighbourhood of zero, and of those for negative theories and for +correlations between negative qualities in the neighbourhood of +unity. Moreover, we are seldom in so complete a state of ignorance +regarding the nature of the theory or correlation under +investigation as not to know whether or not it is a positive theory +or a correlation between positive qualities. In general, therefore, +whenever our investigation is a practical one, experience, if it +tells us anything, tells us not only that the statistical ratios cluster +in the neighbourhood of zero and unity, but in which of these two +neighbourhoods the ratio in this particular case is most likely +\textit{à~priori} to be found. If we seek to discover what proportion of +the population suffer from a certain disease, or have red hair, or +are called Jones, it is preposterous to suppose that the proportion +is as likely \textit{à~priori} to exceed as to fall short of (say) fifty per cent. +As Professor Pearson applies this method to investigations where +it is plain that the qualities involved are positive, he seems to +maintain that experience shows that there are as many positive +attributes which are shared by more than half of any population +as there are which are shared by less than half. + +It is also worthwhile to point out that it is formally impossible +that it should be true of all characters, simple and complex, that +they are as likely to have any one frequency as any other. For let +us take a character $c$ which is compound of two characters $a$~and~$b$, +between which there is no association, and let us suppose that +a has a frequency~$x$ in the population in question and that $b$~has +a frequency~$y$, so that, in the absence of association, the frequency~$z$ +of~$c$ is equal to~$xy$. Then it is easy to show that, if all values of +$x$~and~$y$ between $0$~and~$1$ are equally probable, all values of~$z$ +between $0$~and~$1$ are \emph{not} equally probable. For the value~$\dfrac{1}{2}$ +is more probable than any other, and the possible values of~$z$ +%% -----File: 393.png---Folio 382------- +\index{De Morgan!Rule of Succession@{and Rule of Succession}}% +\index{Pearson, Karl!Rule of Succession@{and Rule of Succession}}% +become increasingly improbable as they differ more widely +from~$\dfrac{1}{2}$. + +It may be added that the conclusions, which Professor +Pearson himself derives from this method, provide a \textit{reductio +ad absurdum} of the arguments upon which they rest. He considers, +for example, the following problem: A sample of~$100$ of a +population shows $10$~per cent affected with a certain disease. +What percentage may be reasonably expected in a second sample +of~$100$? By approximation he reaches the conclusion that the +percentage of the character in the second sample is as likely to +fall inside as outside the limits, $7.85$~and~$13.71$.\DPnote{** TN: Original uses centered dot.} Apart from the +preceding criticisms of the reasoning upon which this depends, +it does not seem reasonable upon general grounds that we should +be able on so little evidence to reach so certain a conclusion. The +argument does not require, for example, that we have any knowledge +of the manner in which the samples are chosen, of the +positive and negative analogies between the individuals, or indeed +anything at all beyond what is given in the above statement. +The method is, in fact, much too powerful. It invests any positive +conclusion, which it is employed to \emph{support}, with far too high +a degree of probability. Indeed this is so foolish a theorem +that to entertain it is discreditable. + +\Paragraph{14.} The Rule of Succession has played a very important part +in the development of the theory of probability. It is true that +\index{Boole!Rule of Succession@{and Rule of Succession}}% +it has been rejected by Boole\footnote + {\textit{Laws of Thought}, p.~369.} +on the ground that the hypotheses +on which it is based are arbitrary, by Venn\footnote + {\textit{Logic of Chance}, p.~197.} +on the ground that it +\index{Bertrand!Rule of Succession@{and Rule of Succession}}% +does not accord with experience, by Bertrand\footnote + {\textit{Calcul des probabilités}, p.~174.} +because it is +ridiculous, and doubtless by others also. But it has been very +widely accepted,---by De~Morgan,\footnote + {Article in \emph{Cabinet Encyclopaedia}, p\DPtypo{}{.}~64.} +\index{Jevons!Rule of Succession@{and Rule of Succession}}% +by Jevons,\footnote + {\textit{Principles of Science}, p.~297.} +\index{Lotze!Rule of Succession@{and Rule of Succession}}% +by Lotze,\footnote + {\textit{Logic}, pp.~373,~374; Lotze propounds a ``simple deduction'' ``as convincing'' + to him ``as the more obscure analysis, by which it is usually obtained.'' + The proof is among the worst ever conceived, and may be commended to those + who seek instances of the profound credulity of even considerable thinkers.} +by +\index{Czuber!Rule of Succession@{and Rule of Succession}}% +Czuber,\footnote + {\textit{Wahrscheinlichkeitsrechnung}, vol.~i.\ p.~199,---though much more guardedly + and with more qualifications than in the form discussed above.} +and by Professor Pearson,\footnote + {\textit{Loc.~cit.}}% +---to name some representative +writers of successive schools and periods. And, in any case, it +%% -----File: 394.png---Folio 383------- +\index{Bobek and Rule of Succession}% +is of interest as being one of the most characteristic results of a +way of thinking in probability introduced by Laplace, and never +thoroughly discarded to this day. Even amongst those writers +who have rejected or avoided it, this rejection has been due +more to a distrust of the particular applications of which the law +is susceptible than to fundamental objections against almost +every step and every presumption upon which its proof depends. + +Some of these particular applications have certainly been +surprising. The law, as is evident, provides a numerical measure +of the probability of any simple induction, provided only that our +ignorance of its conditions is sufficiently complete, and, although, +when the number of cases dealt with is small, its results are incredible, +there is, when the number dealt with is large, a certain +plausibility in the results it gives. But even in these cases +paradoxical conclusions are not far out of sight. When Laplace +proves that, account being taken of the experience of the human +race, the probability of the sun's rising to-morrow is $1,826,214$ to~$1$, +this large number may seem in a kind of way to represent our +state of mind about the matter. But an ingenious German, +Professor Bobek,\footnote + {\textit{Lehrbuch der Wahrscheinlichkeitsrechnung}, p.~208.} +has pushed the argument a degree further, and +proves by means of these same principles that the probability of +the sun's rising every day for the next $4000$ years, is not more, +approximately, than two-thirds,---a result less dear to our natural +prejudices. +%% -----File: 395.png---Folio 384------- +\index{Bortkiewicz, von, and great numbers!method of}% +\index{Lexis, and asymmetry of statistical frequency!method of}% + + +\Chapter{XXXI}{The Inversion of Bernoulli's Theorem} + +\Paragraph{1.} \First{I conclude}, then, that the application of the mathematical +methods, discussed in the preceding chapter, to the general +problem of statistical inference is invalid. Our state of knowledge +about our material must be positive, not negative, before +we can proceed to such definite conclusions as they purport to +justify. To apply these methods to material, unanalysed in +respect of the circumstances of its origin, and without reference +to our general body of knowledge, merely on the basis of arithmetic +and of those of the characteristics of our material with +which the methods of descriptive statistics are competent to +deal, can only lead to error and to delusion. + +But I go further than this in my opposition to them. Not +only are they the children of loose thinking, and the parents of +charlatanry. Even when they are employed by wise and competent +hands, I doubt whether they represent the most fruitful +form in which to apply technical and mathematical methods to +statistical problems, except in a limited class of special cases. +The methods associated with the names of Lexis, Von Bortkiewicz, +and Tschuprow (of whom the last named forms a link, to some +\index{Tschuprow!method of}% +extent, between the two schools), which will be briefly described +in the next chapter, seem to me to be much more clearly consonant +with the principles of sound induction. + +\Paragraph{2.} Nevertheless it is natural to suppose that the fundamental +ideas, from which these methods have sprung, are not wholly +\textit{égarés}. It is reasonable to presume that, subject to suitable conditions +and qualifications, an inversion of Bernoulli's Theorem +must have validity. If we \emph{knew} that our material could be +likened to a game of chance, we might expect to infer chances +from frequencies, with the same sort of confidence as that with +%% -----File: 396.png---Folio 385------- +which we infer frequencies from chances. This part of our +inquiry will not be complete, therefore, until we have endeavoured +to elucidate the conditions for the validity of an Inversion of +Bernoulli's Theorem. +\index{Bernoulli's Theorem!Inverse of|ifoll}% + +\Paragraph{3.} The problem is usually discussed in terms of the happening +of an event under certain conditions, that is to say, of the coexistence +of the conditions, as affecting a particular event, with +that event. The same problem can be dealt with more generally +and more conveniently as an investigation of the correlation +between two characters $A(x)$~and~$B(x)$, which, as in \Partref{III}., +are propositional functions which may be said to concur or coexist +when they are both true of the same argument~$x$. Given +that, within the field of our knowledge, $B(x)$~is true for a certain +proportion of the values of~$x$ for which $A(x)$~is true, what is the +probability for a further value~$a$ of~$x$ that, if $A(a)$~holds, $B(a)$~will +hold also? + +Let us suppose that the occurrence of an instance of~$A(x)$ is a +sign of one of the events $e_{1}(x), e_{2}(x)\ldots$ or $e_{m}(x)$, and that these +are exhaustive, exclusive, and ultimate alternatives. By \emph{exhaustive} +it is meant that, whenever there is an instance of~$A(x)$, +one of the~$e$'s is present; by \emph{exclusive}, that the presence of one +of the~$e$'s is not a sign of the presence of any other, but not that +the concurrence of two or more of the~$e$'s is in fact impossible; +by \emph{ultimate}, that no one of the~$e$'s is a disjunction of two or more +alternatives which might themselves be members of the~$e$'s. +Let us assume that these alternatives are initially and \emph{throughout +the argument} equally probable, which, subject to the above conditions, +is justified by the Principle of Indifference. We have no +reason, that is to say, and no part of our evidence ever gives us +one, for thinking that $A(a)$~is more likely to be a sign of one of the~$e$'s +than of any other, or even for thinking that some~$e$'s, although +we do not know which, are more likely to occur than others. +Let us also assume that, out of $e_{1}(x), e_{2}(x) \ldots e_{m}(x)$, the set +$e_{1}(x), e_{2}(x) \ldots e_{l}(x)$, and these only, are signs or occasions of~$B(x)$; +and further that we have no evidence bearing on the actual +magnitude of the integers $l$~and~$m$, so that the \emph{ratio}~$l/m$ is the +only factor of which the probability varies as the evidence +accumulates. Let us assume, lastly, that our knowledge of the +several instances of~$B(x)$ is adequate to establish a perfect analogy +between them; the instances~$a$, etc., of~$B(x)$, that is to say, must +%% -----File: 397.png---Folio 386------- +not have anything in common except~$B$, unless we have reason +to know that the additional resemblances are immaterial. Even +by these considerable simplifications not every difficulty has +been avoided. But a development along the usual lines with +the assistance of Bernoulli's Theorem is now possible. + +Let $l/m = q$. If the value of~$q$ were known, the problem would +be solved. For this numerical ratio would represent the probability +that $A$~is, in any random instance, a sign of~$B$; and no +further evidence, which satisfies the conditions of the preceding +hypothesis, can possibly modify it. But in the inverse problem +$q$~is not known; and our problem is to determine whether evidence +can be forthcoming of such a kind, that, as this evidence is increased +in quantity, the probability that $A$~will be in any instance +a sign of~$B$, tends to a limit which lies between two determinate +ratios, just as the probability of an inductive generalisation may +tend towards certainty, when the evidence is increased in a +manner satisfying given conditions. + +Let $f(q)$ represent the proposition that $q$~is the true value of~$l/m$. +Let $q'$~represent the ratio of the number of instances actually +before us in which $A$~has been accompanied by~$B$ to that of the +instances in which $A$~has not been accompanied by~$B$; and let +$f'(q')$ be the proposition which asserts this. Now if the ratio~$q$ +is known, then, subject to the assumptions already stated, the +number~$q$ must also represent the \textit{à~priori} probability in any +instance, both before and after the results of other instances are +known, that~$A$, if it occurs, will be accompanied by~$B$. We have, +in fact, the conditions as set forth in \Chapref{XXIX}., in which +Bernoulli's Theorem can be validly applied, so that this theorem +enables us to give a numerical value, for all numerical values of +$q$~and~$q'$, to the probability $f'(q')/h· f(q)$,---which expression represents +the likelihood \textit{à~priori} of the frequency~$q'$, given~$q$. + +An application of the inverse formula allows us to infer from +the above the \textit{à posteriori} probability of~$q$, given~$q'$, namely: +\[ +\frac{f(q)/h · f(q')/h · f(q)} +{\Sum f(q)/h · f(q')/h · f(q)} +\] +where the summation in the denominator covers all possible +values of~$q$. In rough applications of this inverse of Bernoulli's +Theorem it has been usual to suppose that $f(q)/h$~is constant for +all values of~$q$,---that, in other words, all possible values of the +%% -----File: 398.png---Folio 387------- +ratio~$q$ are \textit{à~priori} equally likely. If this supposition were +legitimate, the formula could be reduced to the algebraical expression +\[ +\frac{f(q')/h· f(q)}{\Sum f(q')/h· f(q)}, +\] +all the terms of which can be determined numerically by Bernoulli's +Theorem. It is easy to show that it is a maximum when +$q=q'$, \ie\ that $q'$~is the most probable value of~$l/m$, and that, +when the instances are very numerous, it is very improbable that +$l/m$~differs from $q'$ widely. If, therefore, the number of instances +is increased in such a manner that the ratio continues in the +neighbourhood of~$q'$, the probability that the true value of~$l/m$ +is nearly~$q'$ tends to certainty; and, consequently, the probability, +that $A$~is in any instance a sign of~$B$, also tends to a +magnitude which is measured by~$q'$. + +I see, however, no justification for the assumption that all +possible values of the ratio~$q$ are \textit{à~priori} equally likely. It is +not even equivalent to the assumptions that all integral values +of $l$~and~$m$ respectively are equally probable. I am not satisfied +\emph{either} that different values of~$q$, or that different values of~$m$, +satisfy the conditions which have been laid down in \Partref{I}. for +alternatives which are equal before the Principle of Indifference. +There seem, for instance, to be relevant differences between the +statement that $A$~can arise in exactly two ways and the statement +that it can arise in exactly a thousand ways. We must, +therefore, be content with some lesser assumption and with a +less precise form for our final conclusion. + +\Paragraph{4.} Since, in accordance with our hypothesis, $m$~cannot exceed +some finite number, and since $l$~must necessarily be less than~$m$, +the possible values of~$m$, and therefore of~$q$, are finite in number. +Perhaps we can assume, therefore, as one of our fundamental +assumptions, that there is \textit{à~priori} a finite probability in favour +of each of these possible values. Let $\mu$~be the finite number +which $m$~cannot exceed. Then there is a \emph{finite} probability for +each of the intervals\footnote + {The intervals are supposed to include their lower but not their upper + limit.} +\[ +\frac{1}{\mu} \text{ to } \frac{2}{\mu}, \quad +\frac{2}{\mu} \text{ to } \frac{3}{\mu}, \ \ldots \ +\frac{\mu - 1}{\mu} \text{ to } 1 +\] +%% -----File: 399.png---Folio 388------- +\index{Measurement of Probability!induction@{and induction}}% +that $q$~lies in this interval; but we cannot assume that there is +an \emph{equal} probability for each interval. + +We must now return to the formula +\[ +\frac{f{q}/h · f(q')/hf(q)}{\Sum f(q)/h · f(q')/hf(q)}, +\] +which represents the \textit{à posteriori} probability of~$q$, given~$q'$. Since +by sufficiently increasing the number of instances, the sum of +terms $f(q')/hf(q)$ for possible values of~$q$ within a certain finite +interval in the neighbourhood of~$q'$ can be made to exceed the +other terms by any required amount, and since the sum of the +values of $f(q)/h$ for possible values of~$q$ within this interval is +finite, it clearly follows that a finite number of instances can +make the probability, that $q$~lies in an interval of magnitude +$1/\mu$ in the neighbourhood of~$q'$, to differ from certainty by less +than any finite amount however small. + +\Paragraph{5.} We have, therefore, reached the main part of the conclusion +after which we set out---namely, that as the number of instances +is increased the probability, that $q$~is in the neighbourhood of~$q'$, +tends towards certainty; and hence that, subject to certain +specified conditions, if the frequency with which $B$~accompanies~$A$ +is found to be~$q'$ in a great number of instances, then the +probability that $A$~will be accompanied by~$B$ in any further +instance is also approximately~$q'$. But we are left with the same +vagueness, as in the case of generalisation, respecting the value +of~$\mu$ and the number of instances that we require. We know +that we can get as near certainty as we choose by a finite number +of instances, but what this number is we do not know. This is +not very satisfactory, but it accords very well, I think, with +what common sense tells us. It would be very surprising, in +fact, if logic could tell us exactly how many instances we want, +to yield us a given degree of certainty in empirical arguments. + +Nobody supposes that we can measure exactly the probability +of an induction. Yet many persons seem to believe that in the +weaker and much more difficult type of argument, where the +association under examination has been in our experience, not +invariable, but merely in a certain proportion, we can attribute +a definite measure to our future expectations and can claim +practical certainty for the results of predictions which lie within +relatively narrow limits. Coolly considered, this is a preposterous +%% -----File: 400.png---Folio 389------- +\index{Universal Induction and statistical!methods}% +claim, which would have been universally rejected long ago, +if those who made it had not so successfully concealed themselves +from the eyes of common sense in a maze of mathematics. + +\Paragraph{6.} Meantime we are in danger of forgetting that, in order to +reach even our modified conclusion, material assumptions have +been introduced. In the first place, we are faced with exactly +the same difficulties as in the case of universal induction dealt +with in \Partref{III}., and our original starting-point must be the +same. We have the same difficulty as to how our \emph{initial} probability +is to be obtained; and I have no better suggestion to offer +in this than in the former case---namely, the supposed principle +of a limitation of independent variety in experience. We have +to suppose that if $A$~and~$B$ occur together (\ie\ are true of the +same object), this is some just appreciable reason for supposing +that in \emph{this} instance they have a common cause; and that, if +$A$~occurs again, this is a just appreciable reason for supposing +that it is due to the \emph{same} cause as on the former occasion. But +in addition to the usual inductive hypothesis, the argument has +rested on two particularly important assumptions, first, that we +have no reason for supposing that some of the events of which +$A$~may be a sign are more likely to be exemplified in some of the +particular instances than in others, and secondly, that the analogy +amongst the examined~$B$'s is perfect. The first assumption +amounts, in the language of statisticians, to an assumption of +\emph{random sampling} from amongst the~$A$'s. The second assumption +corresponds precisely to the similar condition which we discussed +fully in connection with inductive generalisation. The instances +\index{Generalisation}% +of $A(x)$ may be the result of \emph{random sampling}, and yet it may +still be the case that there are material circumstances, common +to all the examined instances of~$B(x)$, yet not covered by the +statement $A(x)B(x)$. In so far as these two assumptions are not +justified, an element of doubt and vagueness, which is not easily +measured, assails the argument. It is an element of doubt +precisely similar to that which exists in the case of generalisation. +But we are more likely to forget it. For having overcome +the difficulties peculiar to correlation,\footnote + {I am here using this term in distinction to \emph{generalisation}; that is to say, + I call the statement that $A(x)$~is always accompanied by~$B(x)$ a \emph{generalisation}, + and the statement that $A(x)$~is accompanied by~$B(x)$ in a certain proportion + of cases a \emph{correlation}. This is not quite identical with its use by modern + statisticians.} +it is, possibly, not unnatural +%% -----File: 401.png---Folio 390------- +for a statistician to feel as if he had overcome \emph{all} the +difficulties. + +In practice, however, our knowledge, in cases of correlation +\index{Correlation}% +just as in cases of generalisation, will seldom justify the assumption +of perfect analogy between the~$B$'s; and we shall be faced +by precisely the same problems of analysing and improving our +knowledge of the instances, as in the general case of induction +already examined. If $B$~has invariably accompanied~$A$ in $100$~cases, +we have all kinds of difficulties about the exact character +of our evidence before we can found on this experience a valid +generalisation. If $B$~has accompanied~$A$, not invariably, but +only $50$~times in the $100$~cases, clearly we have just the same +kind of difficulties to face, and more too, before we can announce a +valid correlation. Out of the mere unanalysed statement that $B$~has +accompanied~$A$ as often as not in $100$~cases, without precise +particulars of the cases, or even if there were $1,000,000$ cases +instead of~$100$, we can conclude very little indeed. +%% -----File: 402.png---Folio 391------- +\index{Analogy, principle of!statistics@{and statistics}}% + + +\Chapter{XXXII}{The Inductive use of Statistical Frequencies for the +Determination of Probability \textit{à~posteriori}---the +Methods of Lexis} + +\Paragraph{1.} \First{No} one supposes that a good induction can be arrived at +merely by counting cases. The business of strengthening the +argument chiefly consists in determining whether the alleged +association is \emph{stable}, when the accompanying conditions are +varied. This process of improving the Analogy, as I have called +it in \Partref{III}., is, both logically and practically, of the essence of +the argument. + +Now in statistical reasoning (or inductive correlation) that +\index{Correlation!Quantitative}% +part of the argument, which corresponds to counting the cases +in inductive generalisation, may present considerable technical +difficulty. This is especially so in the particularly complex cases +of what in the next chapter (§\;9) I shall term \emph{Quantitative Correlation}, +which have greatly occupied the attention of +English statisticians in recent years. But clearly it would be an error to +suppose that, when we have successfully overcome the mathematical +or other technical difficulties, we have made any greater +progress towards establishing our conclusion than when, in the +case of inductive generalisation, we have counted the cases but +have not yet analysed or compared the descriptive and non-numerical +differences and resemblances. In order to get a good +scientific argument we still have to pursue precisely the same +scientific methods of experiment, analysis, comparison, and +differentiation as are recognised to be necessary to establish any +scientific generalisation. These methods are not reducible to a +precise mathematical form for the reasons examined in \Partref{III}. +of this treatise. But that is no reason for ignoring them, or for +pretending that the calculation of a probability, which takes into +%% -----File: 403.png---Folio 392------- +\index{Bernoulli, Jac.!statistical series@{and statistical series}}% +\index{Graunt|inote}% +\index{Statistical frequency, theory of!stability of|(}% +\index{Statistical frequency, theory of!fluctuation of}% +account nothing whatever except the numbers of the instances, +is a rational proceeding. The passage already quoted from +Leibniz (\textit{In exemplis juridicis politicisque plerumque non tamen +\index{Leibniz}% +subtili calculo opus est, quam accurata omnium circumstantiarum +enumeratione}) is as applicable to scientific as to political inquiries. + +Generally speaking, therefore, I think that the business of +statistical technique ought to be regarded as strictly limited to +preparing the numerical aspects of our material in an intelligible +form, so as to be ready for the application of the usual inductive +methods. Statistical technique tells us how to `count the cases' +when we are presented with complex material. It must not +proceed also, except in the exceptional case where our evidence +furnishes us from the outset with data of a particular kind, to +turn its results into probabilities; not, at any rate, if we mean +by probability a measure of rational belief. + +\Paragraph{2.} There is, however, one type of technical, statistical investigation +not yet discussed, which seems to me to be a valuable +aid to inductive correlation. This method consists in breaking +\index{Inductive correlation}% +up a statistical series, according to appropriate principles, into +a number of sub-series, with a view to analysing and measuring, +not merely the frequency of a given character over the aggregate +series, but the \emph{stability} of this frequency amongst the sub-series; +that is to say, the series as a whole is divided up by some +principle of classification into a set of sub-series, and the \emph{fluctuation} +of the statistical frequency under examination between the +various sub-series is then examined. It is, in fact, a technical +method of increasing the Analogy between the instances, in the +sense given to this process in \Partref{III}\@. + +\Paragraph{3.} The method of analysing statistical series, as opposed to +the Laplacian or \emph{mathematical} method, one might designate the +\emph{inductive} method. Independently of the investigations of +Bernoulli or Laplace, practical statisticians began at least as early +\index{Laplace!statistical series@{and statistical series}}% +as the end of the seventeenth century\footnote + {Graunt in his \textit{Natural and Political Observations upon the Bills of Mortality} + has been quoted as one of the earliest statisticians to pay attention to these + considerations.} +to pay attention to the +\emph{stability} of statistical series when analysed in this manner. +Throughout the eighteenth century, students of mortality +statistics, and of the ratio of male to female births (including +Laplace himself), paid attention to the degree of constancy of the +%% -----File: 404.png---Folio 393------- +\index{Bortkiewicz, von, and great numbers!Lexis@{and Lexis}|ifoll}% +\index{Lexis, and asymmetry of statistical frequency!method of|ifoll}% +ratios over different parts of their series of instances as well as +to their average value over the whole series. And in the early +part of the nineteenth century, Quetelet, as we have already +\index{Quetelet!statistical stability@{and statistical stability}}% +noticed, widely popularised the notion of the stability of various +social statistics from year to year. Quetelet, however, sometimes +asserted the existence of stability on insufficient evidence, and +involved himself in theoretical errors through imitating the +methods of Laplace too closely; and it was not until the last +quarter of the nineteenth century that a school of statistical +theory was founded, which gave to this way of approaching the +problem the system and technique which it had hitherto lacked, +and at the same time made explicit the contrast between this +analytical or inductive method and the prevailing mathematical +theory. The sole founder of this school was the German economist, +Wilhelm Lexis, whose theories were expounded in a series +of articles and monographs published between the years 1875 +and~1879. For some years Lexis's fundamental ideas did not +attract much notice, and he himself seems to have turned his +attention in other directions. But more recently a considerable +literature has grown up round them in Germany, and their full +purport has been expressed with more clearness than by Lexis +himself---although no one, with the exception of Ladislaus von +Bortkiewicz, has been able to make additions to them of any +great significance.\footnote + {A list of Lexis's principal writings on these topics will be found in the + Bibliography. There is little of first-rate importance which is not contained + either in the volume, \textit{Zur Theorie der Massenerscheinungen in der menschlichen + Gesellschaft}, or in the \textit{Abhandlungen zur Theorie der Bevölkerungs- und Moral-Statistik}. + In this latter volume the two important articles on ``Die Theorie der + Stabilität statistischer Reihen'' and on ``Das Geschlechtsverhältnis der + Geborenen und die Wahrscheinlichkeitsrechnung,'' originally published in Conrad's + \textit{Jahrbüche},\DPnote{** TN: Jahrbücher?} are reprinted.} +Lexis devised his theory with an immediate +view to its practical application to the problems of sex ratio and +mortality. The fact that his general theory is so closely intermingled +with these particular applications of it is, probably, a +part explanation of the long interval which elapsed before the +general theoretical importance of his ideas was widely realised. +I cannot help doubting how fully Lexis himself realised it in the +first instance. It would certainly be easy to read his earlier +contributions to the question without appreciating their generalised +significance. After 1879 Lexis added nothing substantial to +his earlier work, and later developments are mainly due to Von +%% -----File: 405.png---Folio 394------- +\index{Kries, von!School of Lexis@{and School of Lexis}}% +Bortkiewicz. Those of the latter's writings, which have an +important bearing on the relation between probability and +statistics, are given in the Bibliography.\footnote + {The reader may be specially referred to the \textit{Kritische Betrachtungen zur + theoretischen Statistik} (first instalment---the later instalments being of less interest +to the student of Probability), the \textit{Anwendungen der Wahrscheinlichkeitsrechnung + auf Statistik}, and \textit{Homogeneität und Stabilität in der Statistik}. Of other German + and Russian writers it will be sufficient to mention here Tschuprow, who in +\index{Tschuprow!statistical frequency@{and statistical frequency}|inote}% + ``Die Aufgaben der Theorie der Statistik'' (Schmoller's \textit{Jahrbuch}, 1905) and ``Zur + Theorie der Stabilität statistischer Reihen'' (\textit{Skandinavisk Aktuarietidskrift}) gives + by far the best and most lucid general accounts that are available of the doctrines + of the school, he alone amongst these authors writing in a style from which + the foreign reader can derive pleasure, and Czuber, who in his \textit{Wahrscheinlichtkeitsrechnung} +\index{Czuber!statistical frequency@{and statistical frequency}}% + (vol.~ii.\ part iv.\ section~1) supplies a useful mathematical + commentary.} + +On the logic and philosophy of Probability writers of the +school of Lexis are in general agreement with Von Kries; but this +seems to be due rather to the reaction which is common both to +him and to them against the Laplacian tradition, than to any +very intimate theoretical connection between Von Kries's main +contributions to Probability and those of Lexis, though it is true +that both show a tendency to find the ultimate basis of Probability +in physical rather than in logical considerations. I am not +acquainted with much work, which has been appreciably influenced +by Lexis, written in other languages than German (including +with Germans, that is to say, those Russians, Austrians, and Dutch +who usually write in German, and are in habitual connection with +\index{Dormoy}% +the German scientific world). In France Dormoy\footnote + {\textit{Journal des actuaires français}, 1874, and \textit{Théorie mathématique des assurances + sur la vie}, 1878; on the question of priority see Lexis, \textit{Abhandlungen}, p.~130.} +published +independently and at about the same time as Lexis some not +dissimilar theories, but subsequent French writers have paid +little attention to the work of either. Such typical French +treatises as that of Bertrand, or, more recently, that of Borel, +contain no reference to them.\footnote + {Though both these writers touch on closely cognate matters, where Lexis's + investigations would be highly relevant---Bertrand, \textit{Calcul}, pp.~312--314; Borel, + \textit{Éléments}, p.~160.} +In Italy there has been some +discussion recently on the work of Von Bortkiewicz. Among +Englishmen Professor Edgeworth has shown a close acquaintance +\index{Edgeworth!German statisticians@{and German statisticians}}% +with the work of the German school,\footnote + {See especially his ``Methods of Statistics'' in the \textit{Jubilee Volume of the + Stat.\ Journ.}, 1885, and ``Application of the Calculus of Probabilities to + Statistics,'' \textit{International Statistical Institute Bulletin}, 1910.} +he providing for nearly forty +years past, on this as on other matters where the realms of +%% -----File: 406.png---Folio 395------- +Statistics and Probability overlap, almost the only connecting +link between English and continental thought. + +Nevertheless, an account in English of the main doctrines of +this school is still lacking. It would be outside the plan of the +present treatise to attempt such an account here. But it may +be useful to give a short summary of Lexis's fundamental ideas. +After giving this account I shall find it convenient, in proceeding +to my own incomplete observations on the matter, to approach +it from a rather different standpoint from that of Lexis or of +Von Bortkewicz, though not for that reason the less influenced +or illuminated by their eminent contributions to this problem. + +\Paragraph{4.} It will be clearer to begin with some analysis due to Von~Bortkiewicz,\footnote + {What follows is a free rendering of some passages in his \textit{Kritiscke + Betrachungen}.} +and then to proceed to the method of Lexis himself, +although the latter came first in point of time. + +A group of observations may be made up of a number of subgroups, +to which different frequencies for the character under +investigation are properly applicable. That is to say, a proportion~$\dfrac{z_1}{z}$ +of the observations may belong to a group, for which, given +the frequency, the \textit{à~priori} probability of the character under +observation in a particular instance would be~$p_1$, a proportion~$\dfrac{z_2}{z}$ +may belong to a second group for which~$p_2$ is the probability, and +so on. In this case, given the frequencies for the \DPchg{sub-groups}{subgroups}, +the probability $p$ for the group as a whole would be made up as +follows: +\[ +p = \frac{z_1}{z}\, p_1 + \frac{z_2}{z}\, p_2 + \ldots. +\] + +We may call $p$ a \emph{general probability}, and~$p_1$, etc., \emph{special probabilities}. +But the special probabilities may in their turn be +general probabilities, so that there may be more than one way +of resolving a general probability into special probabilities. + +If $p_{1} = p_{2} = \ldots = p$, then~$p$, for that particular way of resolving +the total group into partial groups, is, in Bortkiewicz's terminology, +\emph{indifferent}. If $p$~is indifferent for all conceivable resolutions +into partial groups,\footnote + {This is clearly a very loose statement of what Bortkiewicz really means.} +then, borrowing a phrase from Von~Kries, +Bortkiewicz says of it that it has a \emph{definitive interpretation}. In +%% -----File: 407.png---Folio 396------- +dealing with \textit{à~priori} probabilities, we can resolve a total probability +until we reach the special probabilities of each individual +case; and if we find that all these special probabilities are equal, +then, clearly, the general probability satisfies the condition for +definitive interpretation. + +So far we have been dealing with \textit{à~priori} probabilities. But +the object of the analysis has been to throw light on the inverse +problem. We want to discover in what conditions we can regard +an observed frequency as being an adequate approximation to a +definitive general probability. + +If $p'$ is the empirical value of~$p$ (or, as I should prefer to call it, +the frequency) given by a series of $n$~observations, we may +have +\[ +p' = \frac{n_1}{n}\, p_1' + \frac{n_2}{n}\, p_2' + \ldots. +\] +Even if this particular way of resolving the series of observations +is indifferent, the \emph{actually observed} frequencies $p_1', p_2'$,~etc., may +nevertheless be unequal, since they may fluctuate round the +norm~$p'$ through the operation of `chance' influences. If, +however, $n_1, n_2$,~etc., are large, we can apply the usual Bernoullian +formula to discover whether, \emph{if} there was a norm~$p'$, the divergences +of $p_1', p_2'$,~etc., from it are within the limits reasonably attributable +on Bernoullian hypotheses to `chance' influences. We +can, however, only base a sound argument in favour of the +existence of a `definitive' probability~$p'$ by resolving our +aggregate of instances into sub-series in a great variety of ways, +and applying the above calculations each time. Even so, some +measure of doubt must remain, just as in the case of other +inductive arguments. + +Bortkiewicz goes on to say that probabilities having definitive +interpretation (\emph{definitive} \textit{Bedeutung}) may be designated elementary +probabilities (\textit{Elementarwahrscheinlichkeiten}). But the +probabilities which usually arise in statistical inquiries are not +of this type, and may be termed \emph{average probabilities} (\textit{Durchschnittswahrscheinlichkeiten}). +That is to say, a series of observed +frequencies (or, as he calls them, empirical probabilities) does not, +as a rule, group itself as it would if the series was in fact subject +to an elementary probability. + +\Paragraph{5.} This exposition is based on a philosophy of Probability +different from mine; but the underlying ideas are capable of +%% -----File: 408.png---Folio 397------- +\index{Lexis, and asymmetry of statistical frequency!method of|ifoll}% +translation. Suppose that one is endeavouring to establish an +inductive correlation, \eg\ that the chance of a male birth is~$m$. +\index{Inductive correlation}% +The conclusion, which we are seeking to establish, takes no +account of the place or date of birth or the race of the parents, +and assumes that these influences are irrelevant. Now, if we had +statistics of birth ratios for all parts of the world throughout the +nineteenth century, and added them all up and found that the +average frequency of male births was~$m$, we should not be justified +in arguing from this that the frequency of male births in England +next year is very unlikely to diverge widely from~$m$. For this +would involve the unwarranted assumption, in Bortkiewicz's +terminology, that the empirical probability~$m$ is elementary for +any resolution dependent on time or place, and is not an average +probability compounded out of a series of groups, relating to +different times or places, to each of which a distinct special +probability is applicable. And, in my terminology, it would +assume that variations of time and place were irrelevant to the +correlation, without any attempt having been made to employ +the methods of positive and negative Analogy to establish this. + +We must, therefore, break up our statistical material into +groups by date, place, and any other characteristic which our +generalisation proposes to treat as irrelevant. By this means +we shall obtain a number of frequencies $m_1', m_2', m_3', \ldots$ $m_1'', +m_2'', m_3'', \ldots$ etc., which are distributed round the average +frequency~$m$. For simplicity let us consider the series of frequencies +$m_1', m_2', m_3', \ldots$ obtained by breaking up our +material according to the \emph{date} of the birth. If the observed +divergences of these frequencies from their mean are not significant, +we have the beginnings of an inductive argument for +regarding \emph{date} as being in this connection irrelevant. + +\Paragraph{6.} At this point Lexis's fundamental contribution to the +problem must be introduced. He concentrated his attention on +the nature of the dispersion of the frequencies $m_1', m_2', m_3'\ldots$ +round their mean value~$m$; and he sought to devise a technical +method for measuring the degree of stability displayed by the series of sub-frequencies, which are yielded by the various possible +criteria for resolving the aggregate statistical material into a +number of constituent groups. + +For this purpose he classified the various types of dispersion +which could occur. It may be the case that some of the sub-frequencies +%% -----File: 409.png---Folio 398------- +show such wide and discordant variations from the +mean as to suggest that some significant Analogy has been overlooked. +In this event the lack of symmetry, which characterises +the oscillations, may be taken to indicate that some of the subgroups +are subject to a relevant influence, of which we must take +account in our generalisation, to which some of the other subgroups +are not subject. + +But amongst the various types of dispersion Lexis found one +class clearly distinguishable from all the others, the peculiarity +of which is that the individual values fluctuate in a `purely +chance' manner about a constant fundamental value. This +type he called typical (\textit{typische}) dispersion. He meant by this +that the dispersion conformed approximately to the distribution +which would be given by some normal law of error. +\index{Law of error!Lexis and}% + +The next stage of Lexis's argument\footnote + {I am here following fairly closely his paper, ``\textit{Über die Theorie der Stabilität + statisticher Reihen},'' reprinted in his \textit{Abhandlungen zur Theorie der Bevölkerungsund + Moral-Statistik}, pp.~170--212.} +was to point out that +series of frequencies which are typical in character may have as +their foundation either a constant probability,\footnote + {This mode of expression, which is not in accurate conformity with my + philosophy of Probability, is Lexis's, not mine. His meaning is intelligible.} +or one which is +itself subject to chance variations about a mean. The first case +is typified by the example of a series of sets of drawings of balls, +each set being drawn from a similar urn; the second case by the +example of a series of sets of drawings, the urns from which each +set is drawn being not similar, but with constitutions which vary +in a chance manner about a mean. + +As his \emph{measure} of dispersion Lexis introduces a formula, which +is evidently in part conventional (as is the case with so many +other statistical formulae, the particular shape of which is often +determined by mathematical convenience rather than by any +more fundamental criterion). He expresses himself as follows. +Where the underlying probability is constant, the probable error +in a particular frequency \textit{à~priori} is $r = \rho\sqrt{\dfrac{2v(1-v)}{g}}$, where +$\rho = .4769$, $v$~is the underlying probability, and $g$~is the number of +instances to which the frequency refers. This follows from the +usual Bernoullian assumptions. Now let $R$ be the corresponding +expression derived \textit{à posteriori} by reference to the actual deviations +of a series of observed frequencies from their mean, so that +%% -----File: 410.png---Folio 399------- +\index{Series of probabilities!organic}% +\index{Tschuprow|inote}% +$R = \rho\sqrt{\dfrac{2[\delta^2]}{n-1}}$, where $[\delta^2]$~is the sum of the squares of the deviations +of the individual frequencies from their mean and $n$~is their +number. Now, if the observed facts are due to merely chance +variations about a constant~$v$, we must have approximately +$R = r$, though, if $g$~is small, comparatively wide deviations between +$R$~and~$r$ will not be significant. If, on the other hand, $v$~itself +is not constant but is subject to chance variations, the case +stands differently. For the fluctuations of the observed frequencies +are now due to two components. The one which would +be present, even if the underlying probability were constant, +Lexis terms the ordinary or unessential component; the other +he terms the physical component. If $p$~is the probable deviation +of the various values of~$v$ from their mean, then, on the same +assumptions and as a deduction from the same theory as before, +$R$~will tend to equal not~$r$ but $\sqrt{r^2 + p^2}$. In this event $R$~cannot +be less than~$r$. If, therefore, $R < r$, one must suppose that the +individual instances of each several series on which each frequency +is based are not independent of one another. Such a series +Lexis terms an organic or dependent (\textit{gebundene}) series, and +explains that it cannot be handled by purely statistical methods. + +Since,\Pagelabel{399} therefore, we have three types of series, differing +fundamentally from one another according as $R = r$, $>r$, or~$<r$, +Lexis puts $\dfrac{R}{r} = Q$, and takes $Q$~as his measure of dispersion.\footnote + {In Tschuprow's notation (\textit{Die Aufgaben der Theorie der Statistik}, p.~45), + $Q = P/C$, where $P$~(the Physical modulus) $= \sqrt{\dfrac{2\Sum^{k=n}_{k=1} (p_k-p)^2}{n}}$ and $C$~(the Combinatorial + modulus) $=\sqrt{\dfrac{2p(1-p)}{M}}$, $M$~being the number of instances in each + set, $n$~the number of sets, $p_k$~the frequency for set~$k$, and $p$~the mean of the + $n$~frequencies.} +If +$Q = 1$, we have normal dispersion; if $Q > 1$, we have supernormal +dispersion; and if $Q < 1$, we have subnormal dispersion, which is +an indication that the series is `organic.' + +If the number of instances on which the frequencies are based +is very great, $r$~becomes negligible in comparison with~$p$ (the +physical component), and, therefore, $R = \sqrt{r^2 + p^2}$ becomes +approximately $R = p$. On the other hand, if $p$~is not very large +and the base number of instances is small, $p$~becomes negligible +%% -----File: 411.png---Folio 400------- +in comparison with~$r$, and we have a delusive appearance of +normal dispersion.\footnote + {This is part of the explanation of Bortkiewicz's \emph{Law of Small Numbers.} + See also \Pageref{401}.} +Lexis well illustrates the former point by +the example that the statistics of the ratio of male to female +births for the forty-five registration districts of England over the +years 1859--1871 approximately satisfy the relation $R = r$. But +if we take the figures for all England over those thirteen years, +although the extreme limits of the fluctuation of the ratio about +its mean~$1.042$ are $1.035$ and~$1.047$, nevertheless $R = 2.6$ and $r = 1.6$, +so that $Q = 1.625$; the explanation being that the base number +of instances, namely $730,000$, is so large that $r$~is very small, with +the result that it is swamped by the physical component~$p$. And +he illustrates the latter point by the assertion that, if in $20$~or $30$~series +each of $100$~draws from an urn containing black and white +balls equally, the number of black balls drawn each time were +only to vary between $49$~and~$51$, he would have confidence that +the game was in some way falsified and that the draws were not +independent. That is to say, undue regularity is as fatal to the +assumption of Bernoullian conditions as is undue dispersion. + +\Paragraph{7.} In a characteristic passage\footnote + {``On Methods of Statistics,'' \textit{Jubilee Volume of the Royal Statistical Society}, + p.~211.} +Professor Edgeworth has applied +\index{Edgeworth}% +these theories to the frequency of dactyls in successive extracts +from the \textit{Aeneid}. The mean for the line is~$1.6$, exclusive of the +fifth foot, thus sharply distinguishing the Virgilian line from the +Ovidian, for which the corresponding figure is~$2.2$. But there is +also a marked stability. ``That the Mean of any five lines +should differ from the general Mean by a whole dactyl is proved +to be an exceptional phenomenon, about as rare as an Englishman +measuring $5$~feet, or $6$~feet $3$~inches. An excess of two dactyls +in the Mean of five lines would be as exceptional as an Englishman +measuring $6$~feet $10$~inches.'' But not only so---the stability is +\emph{excessive}, and the fluctuation is less ``than that which is obtained +upon the hypothesis of pure sortition. If we could imagine +dactyls and spondees to be mixed up in the poet's brain in the +proportion of $16$~to~$24$ and shaken out at random, the modulus +in the number of dactyls would be~$1.38$, whereas we have constantly +obtained a smaller number, on an average (the square +root of the average fluctuation)~$1.2$.'' On Lexian principles +these statistical results would support the hypothesis that the +%% -----File: 412.png---Folio 401------- +\index{Bortkiewicz, von, and great numbers!Law of Small Numbers@{and Law of Small Numbers}|ifoll}% +\index{Lexis, and asymmetry of statistical frequency!Edgeworth@{and Edgeworth}}% +\index{Poetry and statistics}% +\index{Small Numbers, Law of|ifoll}% +series under investigation is `organic' and not subject to +Bernoullian conditions, an hypothesis in accordance with our +ideas of poetry. That Edgeworth should have put forward +this example in criticism of Lexis's conclusions, and that Lexis\footnote + {``Über die Wahrscheinlichkeitsrechnung,'' p.~444 (see \Bibref).} +should have retorted that the explanation was to be found in +Edgeworth's series' not consisting of an adequate number of +separate observations, indicates, if I do not misapprehend them, +that these authorities are at fault in the principles, if not of +Probability, of Poetry. + +The dactyls of the Virgilian hexameter are, in fact, a very +good example of what has been termed \textit{connexité}, leading to \DPchg{sub-normal}{subnormal} +dispersion. The quantities of the successive feet are not +independent, and the appearance of a dactyl in one foot \emph{diminishes} +the probability of another dactyl in that line. It is like the case +of drawing black and white balls out of an urn, where the balls +are not replaced. But Lexis is wrong if he supposes that a \emph{supernormal} +dispersion cannot also arise out of \textit{connexité}, or organic +connection between the successive terms. It might have been +the case that the appearance of a dactyl in one foot \emph{increased} +the probability of another dactyl in that line. He should, I +think, have contemplated the result $R > r$ as possibly indicating +a non-typical, organic series, and should not have assumed that, +where $R$~is greater than~$r$, it is of the form $\sqrt{r^{2} + p^{2}}$.\Pagelabel{411} + +In short, Lexis has not pushed his analysis far enough, and he +has not fully comprehended the character of the underlying +conditions. But this does not affect the fact that it was he who +made the vital advance of taking as the unit, not the single +observation, but the frequency in given conditions, and of conceiving +the nature of statistical induction as consisting in the +examination, and if possible the measurement, of the stability +of the frequency when the conditions are varied. + +\Paragraph{8.}\Pagelabel{401} There is one special piece of work illustrative of the above +methods, due to Von Bortkiewicz, which must not be overlooked, +and which it is convenient to introduce in this place---the so-called +\emph{Law of Small Numbers}.\footnote + {There are numerous references to this phenomenon in periodical literature; + but it is sufficient to refer the reader to Von Bortkiewicz's \textit{Das Gesetz der kleinen + Zahlen}.} + +Quetelet, as we have seen in \Chapref{XXVIII}., called attention +\index{Quetelet}% +%% -----File: 413.png---Folio 402------- +\index{Bortkiewicz, von, and great numbers!Quetelet@{and Quetelet}}% +to the remarkable regularity of comparatively \emph{rare} events. Von +Bortkiewicz has enlarged Quetelet's catalogue with modern +instances out of the statistical records of bureaucratic Germany. +The classic instance, perhaps, is the number of Prussian cavalrymen +killed each year by the kick of a horse. The \hyperref[table:1]{table} is worth +giving as a statistical curiosity. (The period is from 1875 to +1894; $G$~stands for the Corps of Guards, and I.--XV.~for the +$15$~Army Corps.) +\begin{sidewaystable}[hp] +\phantomsection\label{table:1} +\footnotesize\renewcommand{\arraystretch}{1.5}% +\begin{tabular}{|r|*{20}{c|}} +\hline + & 75 & 76 & 77 & 78 & 79 & 80 & 81 & 82 & 83 & 84 & 85 & 86 & 87 & 88 & 89 & 90 & 91& 92 & 93 & 94 \\ +\hline +G. & .. & 2 & 2 & 1 & .. & .. & 1 & 1 & .. & 3 & .. & 2 & 1 & .. & .. & 1 & .. & 1 & .. & 1\rule{0em}{1.5em}\\ +I. & .. & .. & .. & 2 & .. & 3 & .. & 2 & .. & .. & .. & 1 & 1 & 1 & .. & 2 & .. & 3 & 1 & ..\\ +II. & .. & .. & .. & 2 & .. & 2 & .. & .. & 1 & 1 & .. & .. & 2 & 1 & 1 & .. & .. & 2 & .. & ..\\ +III. & .. & .. & .. & 1 & 1 & 1 & 2 & .. & 2 & .. & .. & .. & 1 & .. & 1 & 2 & 1 & .. & .. & ..\\ +IV. & .. & 1 & .. & 1 & 1 & 1 & 1 & .. & .. & .. & .. & 1 & .. & .. & .. & .. & 1 & 1 & .. & ..\\ +V. & .. & .. & .. & .. & 2 & 1 & .. & .. & 1 & .. & .. & 1 & .. & 1 & 1 & 1 & 1 & 1 & 1 & ..\\ +VI. & .. & .. & 1 & .. & 2 & .. & .. & 1 & 2 & .. & 1 & 1 & 3 & 1 & 1 & 1 & .. & 3 & .. & ..\\ +VII. & 1 & .. & 1 & .. & .. & .. & 1 & .. & 1 & 1 & .. & .. & 2 & .. & .. & 2 & 1 & .. & 2 & ..\\ +VIII.& 1 & .. & .. & .. & 1 & .. & .. & 1 & .. & .. & .. & .. & 1 & .. & .. & .. & 1 & 1 & .. & 1\\ +IX. & .. & .. & .. & .. & .. & 2 & 1 & 1 & 1 & .. & 2 & 1 & 1 & .. & 1 & 2 & .. & 1 & .. & ..\\ +X. & .. & .. & 1 & 1 & .. & 1 & .. & 2 & .. & 2 & .. & .. & .. & .. & 2 & 1 & 3 & .. & 1 & 1\\ +XI. & .. & .. & .. & .. & 2 & 4 & .. & 1 & 3 & .. & 1 & 1 & 1 & 1 & 2 & 1 & 3 & 1 & 3 & 1\\ +%[** TN: [sic] No XII. and XIII.] +XIV. & 1 & 1 & 2 & 1 & 1 & 3 & .. & 4 & .. & 1 & .. & 3 & 2 & 1 & .. & 2 & 1 & 1 & .. & ..\\ +XV. & .. & 1 & .. & .. & .. & .. & .. & 1 & .. & 1 & 1 & .. & .. & .. & 2 & 2 & .. & .. & .. & ..\\ +\hline +\end{tabular} +\end{sidewaystable} + +The agreement of this table with the theoretical results of a +random distribution of the total number of casualties is remarkably +close:\footnote + {Bortkiewicz, \textit{op.\ cit}.\ p.~24.} +\begin{center} +\begin{tabular}{|*{3}{c|}} +\hline +\settowidth{\TmpLen}{Casualties in a}% +\parbox[c]{\TmpLen}{\centering Casualties in a\\ Year.} & +\multicolumn{2}{c|}{% + \settowidth{\TmpLen}{Number of Occasions on which the Annual}% + \parbox[c]{\TmpLen}{\medskip\centering% + Number of Occasions on which the Annual \\ + Casualties in a Corps reach the Figure \\ + in Column~1.\medskip}} \\ +\hline +\rule{0pt}{12pt} & Actual. & Theoretical. \\ +0 & 144 & 143.1 \\ +1 & \Z91 & \Z92.1 \\ +2 & \Z32 & \Z33.3 \\ +3 & \Z11 & \Z\Z8.9 \\ +4 & \Z\Z2 & \Z\Z2.0 \\ +5 and more & .. & \Z\Z0.6 \\ +\hline +\end{tabular} +\end{center} + +Other instances are furnished by the numbers of child suicides +in Prussia, and the like. + +It is Von Bortkiewicz's thesis that these observed regularities +%% -----File: 414.png---Folio 403------- +have a good theoretical explanation behind them, which he +dignifies with the name of the \emph{Law of Small Numbers}. + +The reader will recall that, according to the theory of Lexis, +his measure of stability~$Q$ is, in the more general case, made up +of two components $r$~and~$p$, combined in the expression $\sqrt{r^{2} + p^{2}}$, +of which one is due to fluctuations from the average of the conditions +governing all the members of a series, which furnishes us +with one of our observed frequencies, and of which the other is +due to fluctuations in the individual members of the series about +the true norm of the series. Bortkiewicz carries the same +analysis a little further, and shows that Lexis's~$Q$ is of the form +$\sqrt{1 + (n - 1)c^{2}}$, where $n$~is the number of times that the event +occurs in each series.\footnote + {I refer the reader to the original, \textit{op.\ cit.}\ pp.~29--31, for the interpretation + of~$c$ (which is a function of the mean square errors arising in the course of the + investigation) and for the mathematical argument by which the above result + is justified.} +That is to say, $Q$~increases with~$n$, and, +when $n$~is small, $Q$~is likely to exceed unity to a less extent than +when $n$~is large. To postulate that $n$~is small, is, when we are +dealing with observations drawn from a wide field, the same +thing as to say that the event we are looking for is a comparatively +rare one. This, in brief, is the mathematical basis of the Law +of Small Numbers. + +In his latest published work on these topics,\footnote + {``Homogeneität und Stabilität in der Statistik,'' published in the \textit{Skandinavisk + Aktuarielidskrift}, 1918. Those readers, who look up my references, + will, I think, agree with me that Von Bortkiewicz does not get any less + obscure as he goes on. The mathematical argument is right enough, and + often brilliant. But what it is all really about, what it all really amounts to, + and what the premisses are, it becomes increasingly perplexing to decide.} +Von Bortkiewicz +builds his mathematical structure considerably higher, without, +however, any further underpinning of the logical foundations +of it. He has there worked out further statistical constants, +arising out of the conceptions on which Lexis's~$Q$ is based (the +precise bearing of which is not made any clearer by his calling +them \emph{coefficients of syndromy}), which are explicitly dependent +on the value of $n$; and he elaborately compares the theoretical +value of the coefficients with the observed value in certain actual +statistical material. He concludes with the thesis, that Homogeneity +and Stability (defined as he defines them) are opposed +conceptions, and that it is not correct to premise, that the larger +statistical mass is as a rule more stable than the smaller, unless +%% -----File: 415.png---Folio 404------- +we also assume that the larger mass is less homogeneous. At this +point, it would have helped, if Von Bortkiewicz, excluding from +his vocabulary homogeneity, paradromy,~$\gamma_M'$, and the like, had +stopped to tell in plain language where his mathematics had led +him, and also whence they had started. But like many other +students of Probability he is eccentric, preferring algebra to earth. + +\Paragraph{9.} Where, then, though an admirer, do I criticise all this? I +think that the argument has proceeded so far from the premisses, +that it has lost sight of them. If the limitations prescribed by +the premisses are kept in mind, I do not contest the mathematical +accuracy of the results. But many technical terms have been +introduced, the precise signification and true limitations of which +will be misunderstood if the conclusion of the argument is allowed +to detach itself from the premisses and to stand by itself. I will +illustrate what I mean by two examples from the work of Von +Bortkiewicz described above. + +Von Bortkiewicz enunciates the seeming paradox that the +larger statistical mass is only, as a rule, more stable if it is less +homogeneous. But an illustration which he himself gives shows +how misleading his aphorism is. The opposition between +stability and homogeneity is borne out, he says, by the judgment +of practical men. For actuaries have always maintained that +their results average out better, if their cases are drawn from a +wide field subject to \emph{variable} conditions of risk, whilst they are +chary of accepting too much insurance drawn from a single +\index{Insurance}% +homogeneous area which means a concentration of risk. But +this is really an instance of Von Bortkiewicz's own distinction +between a general probability~$p$ and special probabilities $p_{1}$~etc., +where +\[ +p = \frac{z_1}{z}\, p_1 + \frac{z_2}{z}\, p_2 + \ldots\DPtypo{}{.} +\] +If we are basing our calculations on~$p$ and do not know $p_1, p_2$, +etc., then these calculations are more likely to be borne out by +the result if the instances are selected by a method which spreads +them over all the groups $1, 2$, etc., than if they are selected by a +method which concentrates them on group~$1$. In other words, +the actuary does not like an undue proportion of his cases to be +drawn from a group which may be subject to a common relevant +influence \emph{for which he has not allowed}. If the \textit{à~priori} calculations +are based on the average over a field which is not homogeneous +%% -----File: 416.png---Folio 405------- +in all its parts, greater stability of result will be obtained if the +instances are drawn from all parts of the non-homogeneous +total field, than if they are drawn now from one homogeneous +\DPchg{sub-field}{subfield} and now from another. This is not at all paradoxical. +Yet I believe, though with hesitation, that this is all that Von +Bortkiewicz's elaborately supported mathematical conclusion +really amounts to. + +My second example is that of the Law of Small Numbers. +Here also we are presented with an apparent paradox in the +statement that the regularity of occurrence of rare events is more +stable than that of commoner events. Here, I suspect, the +paradoxical result is really latent in the particular measure of +stability which has been selected. If we look back at the figures, +which I have quoted above, of Prussian cavalrymen killed by +the kick of a horse, it is evident that a measure of stability could +be chosen according to which exceptional instability would be +displayed by this particular material; for the frequency varies +from $0$~to~$4$ round a mean somewhat less than unity, which is a +very great \emph{percentage} fluctuation. In fact, the particular measure +of stability which Von Bortkiewicz has adopted from Lexis has +about it, however useful and convenient it may be, especially for +mathematical manipulation, a great deal that is arbitrary and +conventional. It is only one out of a great many possible +formulae which might be employed for the numerical measurement +of the conception of stability, which, quantitatively at +least, is not a perfectly precise one. The so-called Law of Small +Numbers is, therefore, little more than a demonstration that, +where rare events are concerned, the Lexian measure of stability +does not lead to satisfactory results. Like some other formulae +which involve a use of Bernoullian methods in an approximative +form, it does not lead to reliable results in all circumstances, +I should add that there is one other element which may contribute +to the total psychological reaction of the reader's mind to the +Law of Small Numbers, namely, the surprising and \emph{piquant} +examples which are cited in support of it. It is startling and +even amusing to be told that horses kick cavalrymen with the +same sort of regularity as characterises the rainfall. But our +surprise at this particular example's fulfilling the Law of Great +Numbers has little or nothing to do with the exceptional stability +about which the Law of Small Numbers purports to concern itself. +%% -----File: 417.png---Folio 406------- +\index{Statistical inference!induction|ifoll}% +\index{Universal Induction and statistical!methods|(}% + + +\Chapter{XXXIII}{Outline of a Constructive Theory} + +\Paragraph{1.} \First{There} is a great difference between the proposition. ``It is +probable that \emph{every} instance of this generalisation is true'' and +the proposition ``It is probable of \emph{any} instance of this generalisation +taken at random that it is true.'' The latter proposition +may remain valid, even if it is certain that some instances of the +generalisation are false. It is more likely than not, for example, +that any number will be divisible either by two or by three, but +it is not more likely than not that all numbers are divisible either +by two or by three. + +The first type of proposition has been discussed in \Partref{III}.~under + the name of \emph{Universal Induction}. The latter belongs to +\index{Induction!universal}% +\index{Induction!statistical|ifoll}% +\emph{Inductive Correlation} or \emph{Statistical Induction}, an attempt at the +\index{Correlation!Inductive}% +\index{Inductive correlation}% +logical analysis of which must be my final task. + +\Paragraph{2.} What advocates of the Frequency Theory of Probability +wrongly believe to be characteristic of \emph{all} probabilities, namely, +that they are essentially concerned not with single instances but +with series of instances, is, I think, a true characteristic of +statistical induction. A statistical induction either asserts the +probability of an instance \emph{selected at random} from a series of +propositions, or else it assigns the probability of the assertion, +that the truth frequency of a series of propositions (\ie\ the +\index{Truth frequency}% +proportion of true propositions in the series) is in the neighbourhood +of a given value. In either case it is asserting a characteristic +of a \emph{series} of propositions, rather than of a particular +proposition. + +Whilst, therefore, our unit in the case of Universal Induction +is a single instance which satisfies both the condition and the +conclusion of our generalisation, our unit in the case of Statistical +%% -----File: 418.png---Folio 407------- +\index{Analogy, principle of!statistics@{and statistics}}% +Induction is not a single instance, but a set or series of instances, +all of which satisfy the condition of our generalisation but +which satisfy the conclusion only in a certain proportion of cases. +And whilst in Universal Induction we build up our argument by +examining the known positive and negative Analogy shown in a +series of single instances, the corresponding task in Statistical +Induction consists in examining the Analogy shown in a \emph{series of +series} of instances. + +\Paragraph{3.} We are presented, in problems of Statistical Induction, with +a set of instances all of which satisfy the conditions of our generalisation, +and a proportion~$f$ of which satisfy its conclusion; and +we seek to generalise as to the probable proportion in which +further instances will satisfy the conclusion. + +Now it is useless merely to pay attention to the proportion (or +frequency) $f$ discovered in the aggregate of the instances. For +any collection whatever, comprising a definite number of objects, +must, if the objects be classified with reference to the presence +or absence of any specified characteristic whatever, show some +definite proportion or statistical frequency of occurrence; so that +a mere knowledge of what this frequency is can have no appreciable +bearing on what the corresponding frequency will be for +some other collection of objects, or on the probability of finding +the characteristic in an object which does not belong to the +original collection. We should be arguing in the same sort of +way as if we were to base a universal induction as to the +concurrence of two characteristics on a single observation of this +concurrence, and without any analysis of the accompanying +circumstances. + +Let the reader be clear about this. To argue from the \emph{mere} +fact that a given event has occurred invariably in a thousand +instances under observation, without any analysis of the circumstances +accompanying the individual instances, that it is likely +to occur invariably in future instances, is a feeble inductive +argument, because it takes no account of the Analogy. Nevertheless +an argument of this kind is not entirely worthless, as we have +seen in \Partref{III}\@. But to argue, without analysis of the instances, +from the mere fact that a given event has a frequency of $10$~per +cent in the thousand instances under observation, or even in a +million instances, that its probability is~$1/10$ for the next instance, +or that it is likely to have a frequency near to~$1/10$ in a further +%% -----File: 419.png---Folio 408------- +set of observations, is a far feebler argument; indeed it is hardly +an argument at all. Yet a good deal of statistical argument is not +free from this reproach;---though persons of common sense often +conclude better than they argue, that is to say, they select for +credence, from amongst arguments similar in form, those in +favour of which there is in fact other evidence tacitly known to +them though not explicit in the premisses as stated. + +\Paragraph{4.} The analysis of statistical induction is not fundamentally +different from that of universal induction already attempted in +\Partref{III}\@. But it is much more intricate; and I have experienced +exceptional difficulty, as the reader may discover for himself in +the following pages, both in clearing up my own mind about it +and in expounding my conclusions precisely and intelligibly. I +propose to begin with a few examples of what commonly impresses +us as good arguments in this field, and also of the attendant +circumstances which, if they were known to exist, might be held +to justify such a mode of reasoning; and, having thus attempted +to bring before the reader's mind the character of the subject-matter, +to proceed to an abstract analysis. + +\textit{Example One}.---Let us investigate the generalisation that the +proportion of male to female births is~$m$. The fact that the +aggregate statistics for England during the nineteenth century +yield the proportion~$m$ would go no way at all towards justifying +the statement that the proportion of male births in Cambridge +next year is likely to approximate to~$m$. Our argument would +be no better if our statistics, instead of relating to England during +the nineteenth century, covered all the descendants of Adam. +But if we were able to break up our aggregate series of instances +into a series of sub-series, classified according to a great variety +of principles, as for example by date, by season, by locality, by +the class of the parents, by the sex of previous children, and so +forth, and if the proportion of male births throughout these sub-series +showed a significant stability in the neighbourhood of $m$, +then indeed we have an argument worth something. Otherwise +we must either abandon our generalisation, amplify its conditions, +or modify its conclusion. + +\textit{Example Two}.---Let us take a series of objects~$s$ all alike in +some specified respect, this resemblance constituting membership +of the class~$F$; let us determine of how many members of the +series a certain property~$\phi$ is true, the frequency of which is to be +%% -----File: 420.png---Folio 409------- +the subject of our generalisation; and if a proportion~$f$ of the +series~$s$ have the property~$\phi$ we may say that the series~$s$ has a +frequency~$f$ for the property~$\phi$. + +Now if the whole field~$F$ has a finite number of constituents, +it must have some determinate frequency~$p$, and if, therefore, +we increase the comprehensiveness of $s$ until eventually it +includes the whole field, $f$~must come in the end to be equal +to~$p$. This is obvious and without interest and not what we +mean by the law of great numbers and the stability of statistical +frequency. + +Let us now divide up the field~$F$, according to some determinate +principle of division~$D$, into subfields $F_1, F_2$,~etc.; and +let the series~$s_1$ be taken from~$F_1$, $s_2$~from~$F_2$, and so on. Where +$F_1, F_2$,~etc., have a finite number of constituents, $s_1, s_2$~etc., may +possibly coincide with them; if $s_1, s_2$~etc., do not coincide with +$F_1, F_2$,~etc., but are chosen from them, let us suppose that they are +chosen according to some principle of random or unbiassed +selection---$s_1$, that is to say, will be a random sample from~$F_1$. +Now it may happen that the frequencies $f_1, f_2$,~etc., of the series +$s_1, s_2$,~etc., thus selected cluster round some mean frequency~$f$. If +the frequencies show this characteristic (the measurement and precise +determination of which I am not now considering), then the +series of series $s_1, s_2$,~etc., has a stable frequency for the classification~$D$. +`Great numbers' only come in because it is difficult to +ascertain the existence of stable frequency unless the series $s_1, s_2$,~etc., +are themselves numerous and unless each of these comprises +numerous individual instances. + +Let us then apply a different principle of division~$D'$, leading +to series $s_1', s_2'$,~etc., and to frequencies $f_1', f_2'$,~etc.; and then again +a third principle of division~$D''$ leading to frequencies $f_1'', f_2''$,~etc.; +and so on, to the full extent that our knowledge of the differences +between the individual instances permits us. If the frequencies +$f_1, f_2$,~etc., $f_1', f_2'$,~etc., $f_1'', f_2''$,~etc., and so on are all stable about~$f$, +we have an inductive ground of some weight for asserting a +statistical generalisation. + +Let the field~$F$, for example, comprise all Englishmen in their +sixtieth year, and let the property~$\phi$, about the frequency of +which we are generalising, be their death in that year of their age. +Now the field~$F$ can be divided into subfields $F_1, F_2$,~etc., on innumerable +different principles. $F_1$~might represent Englishmen +%% -----File: 421.png---Folio 410------- +in their sixtieth year in 1901, $F_2$~in 1902, and so on; or we might +classify them according to the districts in which they live; or +according to the amount of income tax they pay; or according as +they are in workhouses, in hospitals, in asylums, in prisons, or at +large. Let us take the second of these classifications and let the +subfields $F_1, F_2$,~etc., be constituted by the districts in which they +live. If we take large random selections $s_1, s_2$,~etc., from $F_1, F_2$,~etc., +respectively, and find that the frequencies, $f_1, f_2$,~etc., fluctuate +closely round a mean value~$f$, this can be expressed by the +statement that there is a stable frequency~$f$ for death in the +sixtieth year in different English districts. We might also find +a similar stability for all the other classifications. On the other +hand, for the third and fourth classifications we might find no +stability at all, and for the first a greater or less degree of stability +than for the second. In the latter case the form of our statistical +generalisation must be modified or the argument in its favour +weakened. + +\textit{Example Three}.---Let us return to the example given in \Chapref{XXVII}. +of the dog which is fed sometimes by scraps at table +and so judges it reasonable to be there. From one year to another, +let us assume, the dog gets scraps on a proportion of days more +or less stable. What sorts of explanation might there be of +this? First, it might be the case that he was fed on the movable +feasts of the Church; there would be the same number of these +in each year, but it would not be easy for any one who had not +the clue to discover any regularity in the occasions of their +individual occurrence. Second, it might be the case that he +was given scraps whenever he looked thin, and that the scraps +were withheld whenever he looked fat, so that if he was given +scraps on one day, this diminished the likelihood of his getting +scraps on the next day, whilst if they were withheld this would +increase the likelihood; the dog's constitution remaining constant, +the number of days for scraps would tend to fluctuate from +year to year about a stable value. Third, it might be the case +that the company at table varied greatly from day to day, and +that some days people were there of the kind who give dogs +scraps and other days not; if the set of people from whom +the company was drawn remained more or less the same from +year to year, and it was a matter of chance (in the objective sense +defined in §\;8 of \Chapref{XXIV}. above) which of them were +%% -----File: 422.png---Folio 411------- +there from day to day, the proportion of days for scraps might +again show some degree of stability from year to year. Lastly, +a combination between the first and third type of circumstance +gives rise to a variant deserving separate mention. It might be +the case that the dog was only given scraps by his master, that +his master generally went away for Saturday and Sunday, and +was at home the rest of the week unless something happened +to the contrary, and that ``chance'' causes would sometimes +intervene to keep him at home for the week-end and away in +the week; in this case the frequency of days for scraps would +probably fluctuate in the neighbourhood of five-sevenths. In +circumstances of this third type, however, the degree of stability +would probably be less than in circumstances of the first two +types; and in order to get a really stable frequency it might +be necessary to take a longer period than a year as the basis +for each series of observations, or even to take the average for +a number of dogs placed in like circumstances instead of one +dog only. + +It has been assumed so far that we have an opportunity of +observing what happens on \emph{every} day of the year. If this is +not the case and we have knowledge only of a random sample +from the days of each year, then the stability, though it will be +less in degree, may be nevertheless observable, and will increase +as the number of days included in each sample is increased. +This applies equally to each of the three types. + +\Paragraph{5.} What is the correct logical analysis of this sort of reasoning? +If an inductive generalisation is a true one, the conclusion which +it asserts about the instance under inquiry is, so far as it goes, +definite and final, and cannot be modified by the acquisition of +more detailed knowledge about the particular instance. But a +statistical induction, when applied to a particular instance, is +not like this; for the acquisition of further knowledge might +render the statistical induction, though not in itself less probable +than before, \emph{inapplicable} to that particular instance. + +This is due to the fact that a statistical induction is not really +about the particular instance at all, but has its subject, about +which it generalises, a \emph{series}; and it is only applicable to the +particular instance, in so far as the instance is relative to our +knowledge, a \emph{random member} of the series. If the acquisition of +new knowledge affords us additional relevant information about +%% -----File: 423.png---Folio 412------- +\index{Randomness}% +\index{Variables in Probability|inote}% +the particular instance, so that it ceases to be a random member +of the series, then the statistical induction ceases to be applicable; +but the statistical induction does not for that reason become +any less probable than it was---it is simply no longer indicated +by our data as being the statistical generalisation appropriate +to the instance under inquiry. The point is illustrated by the +familiar example that the probability of an unknown individual +posting a letter unaddressed can be based on the statistics of +the Post Office, but \emph{my} expectation that \emph{I} shall act thus, cannot +be so determined. + +Thus a statistical generalisation is always of the form: `The +probability, that an instance taken at random from the series~$S$ +will have the characteristic~$\phi$, is~$p$;' or, more precisely, if $a$~is +a random member of~$S(x)$, the probability of~$\phi (a)$ is~$p$. + +It will be convenient to recapitulate from \Chapref{XXIV}. §\;11 +the definition of `an instance taken at random': Let $\phi (x)$ +stand for `$x$~has the characteristic~$\phi$,' and $S(x)$~for `$x$~is a member +of the class~$S$'; then, on evidence~$h$, $a$~is a random member +of the class~$S$ for characteristic~$\phi$, if `$x$~is~$a$' is irrelevant to +$\phi(x) / S(x)· h$,\footnote + {The use of variables in probability, as has been pointed out on \Pageref{58}, is + very dangerous. It might therefore be better to enunciate the above: $a$~is a + random member of~$S$ for characteristic~$\phi$, if $\phi(a)/S(a)· h = \phi(b)/S(b)· h$ where +$S(b)· h$ contains no information about~$b$, except that $b$~is a member of~$S$\@.} +\ie~if we have no information about $a$ relevant +to~$\phi(a)$ except~$S(a)$. + +Or alternatively we might express our definition as follows: +Consider a particular instance~$a$, where the object of our inquiry +is the probability of $\phi (a)$ relative to evidence~$h$. Let us discard +that part of our knowledge~$h(a)$ which is irrelevant to~$\phi (a)$, +leaving us with relevant knowledge~$h'(a)$. Let the class of +instances $a_1, a_2$,~etc., which satisfy~$h'(x)$ be designated by~$S$\@. Then, +relative to evidence~$h$, $a$~is a random member of the class or +series~$S$ for the characteristic~$\phi$. + +Let us denote the proposition `$x$~is, on evidence~$h$, a random +member of~$S$ for characteristic~$\phi$' by $R(x, S, \phi, h)$; then our +statistical generalisation is of the form $\phi(x)/R(x, S, \phi, h)· h = p$. + +If $R (a, S, \phi, h)$ holds, then, on evidence~$h$, $S$~is the appropriate +statistical series to which to refer $a$ for the purposes of the characteristic~$\phi$. + +It is not always the case that the evidence indicates any +series at all as `appropriate' in the above sense. In particular, +%% -----File: 424.png---Folio 413------- +if evidence~$h$ indicates $S$~as the appropriate series, and evidence~$h'$ +indicates~$S'$ as the appropriate series, then relative to evidence~$hh'$ +(assuming these to be not incompatible), it may be the case +that no determinate series is indicated as appropriate. In this +case the method of statistical induction fails us as a means of +determining the probability under inquiry. + +\Paragraph{6.} We can now remove our attention from the individual +instance~$a$ to the properties of the series~$S$\@. What sort of evidence +is capable of justifying the conclusion that $p$~is the probability +that a random member of the series~$S$ will have the characteristic~$\phi$? + +In the simplest case, $S$~is a finite series of which we know the +truth frequency for the characteristic~$\phi$, namely~$f$.\footnote + {\Ie\ if $f$~is the proportion of the members of the series for which $\phi(x)$~is true.} +Then by a +straightforward application of the Principle of Indifference we +have $p = f$, so that +$\phi(x)/R(x, S, \phi, h)· h = f$. + +In another important type~$S$ is a series, with an indefinite +number of members which, however, group themselves in such +a way that for every member of which $\phi(x)$~is true, there corresponds +a determinate number of members of which $\phi(x)$~is +false. The series, that is to say, contains an indefinite number +of atoms, but each atom is made up of a set of molecules of +which $\phi(x)$~is true and false respectively in fixed and determinate +proportions. If this determinate proportion is known to be~$f$, we +have, as before, $p = f$. The typical instance of this type is afforded +by games of chance. Every possible state of affairs which might +lead to a divergence in one direction is balanced by another +probability leading in the opposite direction; and these alternative +possibilities are of a kind to which the Principle of Indifference +is applicable. Thus for every poise of the dice box which leads +to the fall of the six-face, there is a corresponding poise which +leads to the fall of each of the other faces; so that if $S$ is the +series of possible poises, we may equate~$p$ to~$\frac{1}{6}$ where $\phi$~is the fall +of the six-face. It is not necessary, in order to obtain this +result, to assert that $S$~is a finite series with an actual determinate +frequency~$f$ for the fall of each face. + +So far no inductive element enters in. But in general we do +not know the constitution of~$S$ for certain, and can only infer it +inductively from its resemblance to other series of which we know +the constitution. This presents a normal inductive problem---the +%% -----File: 425.png---Folio 414------- +determination by an analysis of the positive and negative +analogies as to whether the respects in which $S$~differs or may +differ from the other series is or is not relevant in the particular +context~$\phi$; and it involves the same sort of considerations as +those discussed in \Partref{III}\@. + +There is, however, a further difficulty to be introduced before +we have reached the typical statistical problem. In the case +now to be considered our actual data do not consist of positive +knowledge of the constitutions either of $S$~itself or of other series +more or less resembling~$S$, but only of the frequency of the +characteristic in actually observed sets of \emph{selections}, great or +small, either from $S$~itself or from other series more or less +resembling~$S$\@. + +Thus in the most general case our inquiry falls into two parts. +We are given the observed frequency in statistical sets \emph{selected} +from $S_1, S_2$,~etc., respectively. The first part of our inquiry is +the problem of arguing from these observed frequencies to the +probable constitutions of $S_1, S_2$,~etc., \ie\ of determining the values +of $\phi(x)/R(x, S_1, \phi, h)· h$, etc.; we may call this part the statistical +problem. The second part of our inquiry is the problem of +arguing from the probable constitutions of $S_1, S_2$,~etc., to the +probable constitution of~$S$, where $S, S_1, S_2$ resemble one another +more or less, and we have to determine whether the differences +are or are not relevant to our inquiry; we may call this part the +inductive problem. + +Now if the observed statistical sets are made up of random +instances of $S_1, S_2$,~etc., we can argue in certain conditions from +the observed frequencies to the probable constitutions of the +series, out of which the random selections have been made, by +an inverse application of Bernoulli's Theorem on the lines explained +in \Chapref{XXXI}\@. Moreover, if the series $S_1, S_2$,~etc., +are finite series and the observed selections cover a great part +of their members, we can reach an at least approximate conclusion +without raising all the theoretical difficulties or satisfying +all the conditions of \Chapref{XXXI}\@. The commonly received +opinions as to the bearing of the observed frequencies in a +random sample on the constitution of the universe out of which +the sample is drawn, though generally stated too precisely and +without sufficient insistence on the assumptions they involve, +our actual evidence not warranting in general more than an +%% -----File: 426.png---Folio 415------- +\index{Analogy, principle of!negative}% +\index{Analogy, principle of!positive}% +\index{Analogy, principle of!statistics@{and statistics}|ifoll}% +\index{Lexis, and asymmetry of statistical frequency!statistical stability@{and statistical stability}}% +approximate result, are not, I think, fundamentally erroneous. +The most usual error in modern method consists in treating too +lightly what I have termed above the \emph{inductive} problem, \ie\ the +problem of passing from the series $S_1, S_2$,~etc., of which we +have observed samples, to the series~$S$ of which we have not +observed samples. + +Let us, then, assume that we have ascertained $p_1, p_2$,~etc., with +more or less exactness, by examining either all the instances of +the series $S_1, S_2$,~etc., or random selections from them, \ie\ $\phi(x)/R +(x, S_1, \phi, h)· h = p_1$, etc. This can be expressed for short by saying +that the series $S_1, S_2$,~etc., are subject to probable-frequencies +$p_1, p_2$,~etc., for the characteristic~$\phi$. Our problem is to infer from +this the probable-frequency~$p$ of the unexamined series~$S$\@. The +class characteristics of the series $S_1, S_2$,~etc., will be partly the same +and partly different. Using the terminology of \Partref{III}. we +may term the class characteristics which are common to all of +them the Positive Analogy, and the class characteristics which +are not common to all of them the Negative Analogy. + +Now, if the observed or inferred probable-frequencies of +the series $S_1, S_2$, are to form the basis of a statistical induction, +they must show a \emph{stable} value; that is to say, either we must +have $p_1 = p_2 =$~etc., or at least $p_1, p_2$,~etc., must be stably grouped +about their mean value. Our next task, therefore, must be +to discover whether the probable-frequencies $p_1, p_2$,~etc., display +a significant stability. It is the great merit of Lexis that he was +the first to investigate the problem of stability and to attempt its +measurement. For, until a \textit{primâ facie} case has been established +for the existence of a stable probable-frequency, we have but +a flimsy basis for any statistical induction at all; indeed we are +limited to the class of case where the instance under inquiry is +a member of identically the \emph{same} series as that from which our +samples were drawn, \ie\ where $S = S_1$, which in social and scientific +inquiries is seldom the case. + +What is the meaning of the assertion that $p_1, p_2$,~etc., are +\emph{stably} grouped about their mean value? The answer is not +simple and not perfectly precise. We could propound various +formulae for the measurement of stability and dispersion, respectively, +and the problem of translating the conception of stability, +which is not quantitatively precise, into a numerical formula +involves an arbitrary or approximative element. For practical +\index{Statistical frequency, theory of!stability of|)}% +%% -----File: 427.png---Folio 416------- +purposes, however, I doubt if it is possible to improve on Lexis's +measure of stability~$Q$, the mathematical definition of which +has been given above on \Pageref{399}. Lexis describes the stability +as subnormal, normal, or supernormal according as $Q$~is less than, +equal to, or greater than~$1$. This is too precise, and it is better +perhaps to say that the stability about the mean is normal if +the dispersion is such as would not be improbable \textit{à~priori}, if +we had assumed that the members of $S_1, S_2$,~etc., were obtained +by random selection out of a single universe~$U$, that it is subnormal +if the dispersion is less than one would have expected on +the same hypothesis, and that it is supernormal if the dispersion +is greater than one would have expected. + +Let us suppose that we find that on this definition $p_1, p_2$,~etc., +are stable about~$p$, and let us postpone consideration of the cases +of subnormal or supernormal dispersion. This is equivalent to +saying that the frequencies of $S_1, S_2$,~etc., are within limits which +we should expect \textit{à~priori}, if we had knowledge relative to which +their members were chosen at random from a universe~$U$ of which +the frequency was~$p$ for the characteristic under inquiry. We +next seek to extend this result to the unexamined series~$S$ and to +justify anticipations about it on the basis of the members of~$S$ +also being chosen at random from the universe~$U$. This leads us +to the strictly inductive part of our inquiry. + +The class characteristics of the several series $S_1, S_2$,~etc., will be +partly the same and partly different, those that are the same +constituting the positive analogy and those that are different +constituting the negative analogy, as stated above. The series~$S$ +will share part of the positive analogy. The argument for +assimilating the properties of~$S$, in relation to the characteristic +under inquiry, to the properties of $S_1, S_2$,~etc., in relation to this +characteristic depends on the differences between $S, S_1, S_2$,~etc., +being \emph{irrelevant} in this particular connection. The method of +strengthening this argument seems to me to be the same as the +general inductive method discussed in \Partref{III}. and to present +the same, but not greater, difficulties. + +In general this inductive part of our inquiry will be best +advanced by classifying the aggregate series of instances with +which we are presented in such a way as to analyse most clearly +the significant positive and negative analogies, to group them, +that is to say, into sub-series $S_1, S_2$,~etc., which show the most +%% -----File: 428.png---Folio 417------- +marked and definite class characteristics. Our knowledge of the +differences between the particular observed instances which +constitute our original data will suggest to us one or more +principles of classification, such that the members of each sub-series +all have in common some set of positive or negative characteristics, +not all of which are shared in common by all the +members of any of the other sub-series. That is to say, we +classify our whole set of instances into a series of series $S_1, S_2$,~etc., +which have frequencies $f_1, f_2$,~etc., for the characteristic under +inquiry; and then again we classify them by another principle or +criterion of classification into a second series of series $S_1', S_2'$,~etc., +with frequencies $f_1', f_2'$,~etc.; and so on, so far as our knowledge of +the possible relevant differences between the instances extends; +the whole result being then summed up in a statement of the +positive and negative analogies of the series of series. If we then +find that all the frequencies $f_1, f_2$,~etc., $f_1', f_2'$~etc., are stable about +a value~$p$, and if, on the basis of the above positive and negative +analogies, we have a normal inductive argument for assimilating +the unexamined series~$S$ to the examined series $S_1, S_2$,~etc., $S_1', S_2'$,~etc., +in respect of the characteristic under inquiry, in this case we +have, not conclusive grounds, but grounds of some weight for +asserting the probability~$p$, that an instance taken at random +from~$S$ will have the characteristic in question. + +Let me recapitulate the two essential stages of the argument. +We first find that the observed frequencies in a set of +series are such as would have been not improbable \textit{à~priori} if, +relative to our knowledge, these series had all been made up of +random members of the same universe~$U$; and we next argue +that the positive and negative analogies of this set of series +furnish an inductive argument of some weight for supposing that +a further unexamined series~$S$ resembles the former series in +having a frequency for the characteristic under inquiry such as +would have been not improbable \textit{à~priori} if, relative to our knowledge, +$S$~was also made up of random members of the hypothetical +universe~$U$. + +\Paragraph{7.} It is very perplexing to decide how far an argument of +this character involves any new and theoretically distinct +difficulties or assumptions, beyond those already admitted +as inherent in Universal Induction. I believe that the foregoing +\index{Induction!universal}% +analysis is along the right lines and that it carries the +\index{Universal Induction and statistical!methods|)}% +%% -----File: 429.png---Folio 418------- +\index{Chance, objective}% +inquiry a good deal further than it has been carried hitherto. +But it is not conclusive, and I must leave to others its more +exact elucidation. + +There is, however, a little more to be said about the half-felt +reasons which, in my judgment, recommend to common sense +some at least of the scientific (or semi-scientific) arguments +which run along the above lines. In expressing these reasons I +shall be content to use language which is not always as precise as +it ought to be. + +I gave in \Chapref{XXIV}. §§\;7--9 an interpretation of what is +meant by an `objectively chance' occurrence, in the sense in +which the results of a game, such as roulette, may be said to be +governed by `objective chance.' This interpretation was as +follows: ``An event is due to objective chance if in order to +predict it, or to prefer it to alternatives, at present \DPchg{equi-probable}{equiprobable}, +with any high degree of probability, it would be necessary to +know a great many more facts of existence about it than we +actually do know, and if the addition of a wide knowledge of +general principles would be little use.'' The ideal instance of +this is the game of chance; but there are other examples afforded +by science in which these conditions are fulfilled with more or +less perfection. Now the field of statistical induction is the class +of phenomena which are due to the combination of two sets of +influences, one of them constant and the other liable to vary in +accordance with the expectations of objective chance,---Quetelet's +\index{Quetelet}% +`permanent causes' modified by `accidental causes.' In social +and physical statistics the ultimate alternatives are not as a rule +so perfectly fixed, nor the selection from them so purely random, +as in the ideal game of chance. But where, for example, we find +stability in the statistics of crime, we could explain this by +supposing that the population itself is stably constituted, that +persons of different temperaments are alive in proportions more +or less the same from year to year, that the motives for crime are +similar, and that those who come to be influenced by these +motives are selected from the population at large in the same +kind of way. Thus we have stable causes at work leading to the +several alternatives in fixed proportions, and these are modified +by random influences. Generally speaking, for large classes of +social statistics we have a more or less stable population including +different kinds of persons in certain proportions and on the other +%% -----File: 430.png---Folio 419------- +\index{Lexis, and asymmetry of statistical frequency!statistical stability@{and statistical stability}|inote}% +\index{Mendelism and statistics}% +hand sets of environments; the proportions of the different +kinds of persons, the proportions of the different kinds of environments, +and the manner of allotting the environments to the +persons vary in a \emph{random} manner from year to year (or, it may be, +from district to district). In all such cases as these, however, +prediction beyond what has been observed is clearly open to +sources of error which can be neglected in considering, for +example, games of chance;---our so-called `permanent' causes +are always changing a little and are liable at any moment to +radical alteration. + +Thus the more closely that we find the conditions in scientific +examples assimilated to those in games of chance, the more +confidently does common sense recommend this method. The +rather surprising frequency with which we find apparent stability +in human statistics may possibly be explained, therefore, if the +biological theory of Mendelism can be established. According to +this theory the qualities apparent in any generation of a given +race appear in proportions which are determined by methods +very closely analogous to those of a game of chance. To take a +specific example (I am giving not the correct theory of sex but an +artificially simplified form of it), suppose there are two kinds of +spermatozoa and two kinds of ova and of the four possible kinds +of union two produce males and two females, then if the kinds of +spermatozoa and ova exist in equal numbers and their union is +determined by random considerations in precisely the same sense +in which a game of chance such as roulette depends upon random +considerations, we should expect the observed proportions to +vary from equality, as indeed they do, in the same manner as +variations from equality of red and black occur at roulette.\footnote + {The fluctuations in the proportion of the sexes which, as is well known, + is not in fact one of equality, correspond, as Lexis has shown, to what one + would expect in a game of chance with an astonishing exactitude. But + it is difficult to find any other example, amongst natural or social phenomena, + in which his criteria of stability are by any means as equally well satisfied.} +If +the sphere of influence of Mendelian considerations is wide, we +have both an explanation in part of what we observe and also a +large opportunity in future of using with profit the methods of +statistical analysis. + +This is all familiar. This is the way in which in fact we do +think and argue. The inquiry as to how far it is covered by the +abstract analysis of the preceding paragraphs, and by what +%% -----File: 431.png---Folio 420------- +\index{Series of probabilities!independent}% +\index{Series of probabilities!organic}% +logical principle the use of this analysis can be justified as rational, +I have pushed as far as I can. It deserves a profounder study +than logicians have given it in the past. + +\Paragraph{8.} Two subsidiary questions remain to be mentioned. The +first of these relates to the character of series which, in the +terminology of Lexis, show a subnormal or supernormal stability; +for I have pressed on to the conclusion of the argument on the +assumption that the stabilities are normal. Subnormal stability +conceals two types: the one in which there is really no stability +at all and the results are in fact chaotic; and the other in which +there is mutual dependence between the successive instances of +such a kind that they tend to resemble one another so that any +divergence from the normal tends to accentuate itself. Super-*normal +stability corresponds in the other direction to the second +of these two types; that is to say, there is mutual dependence of +a regulative kind between the successive instances which tends +to prevent the frequency from swinging away from its mean +value. The case, where the dog was fed with scraps when he +looked thin and not fed when he looked fat, illustrated this. +The typical example of this type is where balls are drawn from +urns, containing black and white balls in certain proportions and +\emph{not} replaced; so that every time a black ball is drawn the next +ball is more likely than before to be white, and there is a tendency +to redress any excess of either colour beyond the proper proportions. +Possibly the aggregate annual rainfall may afford a +further illustration. + +Where there is no stability at all and the frequencies are chaotic, +the resulting series can be described as `non-statistical.' Amongst +`statistical series,' we may term `independent series' those of +which the instances are independent and the stability normal, +and `organic series,' those of which the instances are mutually +dependent and the stability abnormal, whether in excess or in +defect. `Organic series' have been incidentally discussed elsewhere +in this volume. I shall not pursue them further now, +because I do not think that they introduce any new \emph{theoretical} +difficulty into the general problem of statistical inference; +although the problem of fitting them into the general theoretical +scheme is not easy.\footnote + {The following more precise definitions bring these ideas into line with what + has gone before: consider the terms $a_1, a_2, \ldots a_n$ of a series~$s(x)$; let `$a_r$~is~$g$' + $\equiv g_r$ and let $g_r/h=p_r$, where $h$~is our data. Then, if $g_r/g_s \ldots g_t \ldots h=p_r$ for all + values of $r, s, \ldots, t \ldots$, the terms of the series are \emph{independent} relative to~$h$. If + $p_1=p_2= \ldots =p$ the terms are \emph{uniform}. If the terms are both independent and + uniform, the series may be called an \emph{independent Bernoullian series}, subject to + a \emph{Bernoullian probability}~$p$. If the terms are independent but not uniform, the + series may be called an \emph{independent compound series}, subject to a \emph{compounded + probability} $1/n\Sum p_r$. If the terms are not independent, the series is an \emph{organic + series}. + + The same terminology can then be applied to the series $S_1, S_2, \ldots S_n$, regarded + as members of the series of series~$S(x)$. Let the frequencies of the series for the + characteristic under inquiry be $x_1, x_2, \ldots x_n$, and let $x_1/h=\theta_1(x_1)$, \ie~$\theta_1(x_1)$ is the + probability of a frequency~$x_1$ in the first series. Then if $x_r/x_s \ldots h =\theta_r(x_r)$ for all + values of $r$,~$s$,~etc., the frequencies are \emph{independent}; and if $\theta_1(x)= \theta_2(x_2)= \dots \theta (x)$, + the frequencies are \emph{stable}. If the frequencies are stable and independent, the + series of series may be called \emph{Gaussian}. If the frequencies are stable and + independent, and if in addition each individual series is subject to a Bernoullian + probability, the probable dispersion of the frequency is normal and symmetrical. + If the individual series are organic, the dispersion of the frequencies may be + normal, subnormal, or supernormal. If the series of series is Gaussian, and the + individual series Bernoullian, we have the type of the perfect statistical series.} +%% -----File: 432.png---Folio 421------- +\index{Coefficient of Credibility!of Correlation|ifoll}% +\index{Series of probabilities!Gaussian|inote}% + +\Paragraph{9.} The second question is concerned with the relation between +the Inductive Correlation, which has been the subject-matter of +\index{Correlation!coefficient|ifoll}% +this chapter, and the Correlation Coefficient or, as I should prefer +to call it, the \emph{Quantitative Correlation}, with which recent English +statistical theory has chiefly occupied itself. I do not propose +to discuss this theory in detail, because I suspect that it is much +more concerned, at any rate in its present form, with statistical +description than with statistical induction. The transition from +defining the `correlation coefficient' as an algebraical expression +to its employment for purposes of inference is very far from +clear even in the work of the best and most systematic writers +on the subject, such as Mr.~Yule and Professor Bowley. +\index{Bowley}% +\index{Yule!correlation@{and correlation}}% + +In the notation employed in the earlier part of this chapter I +have classified each examined instance~$a$ according as it did or +did not possess the characteristic~$\phi$, \ie~satisfy the propositional +function~$\phi (x)$, or, in other words, according as $\phi (a)$~was true or +false. Thus only two possible alternatives were contemplated, +and $\phi$~was not considered as a quantitative characteristic which +the instance could satisfy in greater or less degree. Equally the +common element in all the instances, required to constitute them +as instances for the purpose of our statistical generalisation (or, +as I have sometimes put it, required to satisfy the \emph{condition} of the +generalisation), was regarded as definite and unique and not +capable of quantitative variation. That is to say, all the instances +satisfied a function~$\psi (x)$, and the question was, what proportion +%% -----File: 433.png---Folio 422------- +of them also satisfied the function~$\phi(x)$. A typical example was +that of sex-ratio,---$\psi (x)$ being the birth of a child and $\phi (x)$~its +sex, where there is no question of \emph{degree} in either $\psi (x)$~or~$\phi(x)$. + +It might be the case, however; that the characteristics under +examination were capable of degree or quantitative variation; +for example $\psi (x)$~might be the age of the mother and $\phi(x)$~the +weight of the child at birth, in this case we should have a series +$\psi_1(x), \psi_2(x)$,~etc., corresponding to the various age-periods of the +mothers, and a series $\phi_1(x),~\phi_2(x)$,~etc., corresponding to the various +weights of the children. Now if we concentrated our attention +on $\psi_1(x)$~and~$\phi_1(x)$ alone, \ie~on mothers of a particular age and +the proportions of their children which had a particular weight +at birth, we have a one-dimensional problem of the same kind as +before; out of all the instances which satisfy~$\psi_1(x)$ a certain +proportion satisfy $\psi_1(x)$ also. But clearly we can push our +observations further and we can take note what proportion of the +instances which satisfy $\psi_1(x)$ satisfy $\phi_2(x), \phi_3(x)$, and so on, respectively; +and then we can do the same as regards the instances +which satisfy $\psi_2(x), \psi_3(x)$,~etc. The total results of this two-dimensional +set of observations can then be tabulated in what is +called a twofold correlation table. Thus if $f_{rs}$~is the proportion +of instances satisfying~$\psi_s(x)$ which also satisfy~$\phi_{r}(x)$ we have a +table as follows: +\[ +\renewcommand{\arraystretch}{1.5} +\begin{array}{|*{5}{>{\ }c<{\ }|}} +\hline + & \psi_1(x)& \psi_2(x) & \psi_3(x) & \ldots\\ +\hline +\phi_1(x) & f_{11} & f_{12} & f_{13} & \\ +\hline +\phi_2(x) & f_{21} & f_{22} & f_{23} & \\ +\hline +\phi_3(x) & f_{31} & f_{32} & f_{33} & \\ +\hline +\vdots & \vdots & \vdots & \vdots & \\ +\hline +\end{array} +\] + +We could, further, increase the complexity and completeness +of our observations to any required degree. For example we +might take account also of~$\theta(x)$, the age of the father, and construct +a threefold table where $f_{rst}$~is the proportion of instances +satisfying $\phi_{r}(x), \psi_{s}(x), \theta_{t}(x)$; and so on up to an $n$-fold table. + +Clearly it is not necessary for the construction of tables of +%% -----File: 434.png---Folio 423------- +this kind that $\phi(x)$~and~$\psi(x)$ should stand for degrees of the same +quantitative characteristic; they might be any set of exclusive +alternatives; for example, $\psi(x)$~might be the colour of the baby's +eyes, and $\phi(x)$~its Christian name. + +But in order that the correlation table may be of any +practical interest for the purposes of inference, it is necessary---and +this, I think, is one of the critical assumptions of correlation---that +$\psi_{1}(x), \psi_{2}(x), \ldots$ and also $\phi_{1}(x), \phi_{2}(x), \ldots$ should +be arranged in an order that is \emph{significant}, \ie~such that we have +some \textit{à~priori} reason for expecting some connection to exist +between the \emph{order} of the~$\psi$'s and the \emph{order} of the~$\phi$'s. The point +of this will be illustrated by concentrating our attention on the +simplest type of case where $\psi(x)$~and~$\phi(x)$ are quantitative +characteristics arranged in order of magnitude. Now suppose +it were the case that the younger mothers tended to bear heavier +babies, then, if $\psi_{1}(x)~\psi_{2}(x)$ are the ages increasing upwards and +$\phi_1(x)~\phi_{2}(x)$ the weights diminishing downwards, $f_{11}$~would probably +be the greatest of the~$f_{r1}$'s and, generally speaking, $f_{r1}$~would be +greater than~$f_{r+1,1}$; also $f_{22}$~might be the greatest of the~$f_{r2}$'s, and +so on; so that the frequencies lying on the diagonal of the table +would be the greatest and the frequencies would tend to be less +the farther they lay from the diagonal. If we had some reason +\textit{à~priori} (\ie~based on our pre-existing knowledge), if only a +slight one, for supposing that there might be some connection +between the age of the mother and the weight of the baby, then, +if in a particular set of instances the frequencies were grouped +about the diagonal as suggested above, this might be taken as +affording some inductive support for the hypothesis. + +Now the theory of correlation, as it is expounded in the +text-books, is almost entirely concerned with measuring how +nearly the observed frequencies are grouped about the diagonal +of the table (though the complete theory is not, of course, so +restricted as this). The `coefficient of correlation' is an algebraical +formula which may be regarded as measuring this phenomenon +in a way that is sufficiently satisfactory for all ordinary purposes. +If it is defined thus, it is simply a statistical description of a +particular set of observations arranged in a particular order. +How can we make use of this coefficient for the purposes of +inference? + +Dr.~Bowley faces this problem a little more definitely than do +\index{Bowley|ifoll}% +%% -----File: 435.png---Folio 424------- +most statistical writers. Mr.~Yule warns the student that the +\index{Yule!correlation@{and correlation}}% +problem exists,\footnote + {\textit{Introduction to the Theory of Statistics}, p.~191: ``The coefficient of correlation, + like an average or a measure of dispersion, only exhibits in a summary + and comprehensible form one particular aspect of the facts on which it is based, + and the real difficulties arise in the interpretation of the coefficient when + obtained.''} +but he does not himself attack it systematically +or do more than apply common sense to particular problems. +So much greater emphasis, however, has been laid hitherto on +the mathematical complications, that many statistical students +hazily float from defining the correlation coefficient as a statistical +description to employing it as a measure of the probability of a +statistical generalisation as to the association between quantitative +variations of $\phi(x)$~and~$\psi(x)$ respectively. If, for example, +it is found in a particular set of observations of +mothers' ages and babies' weights that the frequencies are +closely ranged about the diagonal, this is considered a sufficiently +good reason for attributing probability to a generalisation as to +the `correlation' (\ie~tendency to quantitative correspondence) +between the age of the mother and the weight of the baby. + +Dr.~Bowley's line of thought is as follows. He begins by +defining the correlation coefficient~$r$ merely as a statistical description +(\textit{Elements of Statistics}, p.~354). He then shows (p.~355), +as an illustration of the nature of~$r$, that if $x$~and~$y$ are two +variable quantities which depend (more strictly, \emph{are known} to +depend) on other variables $U$,~$V$,~$W$ in such a way that +\begin{align*} +X_t &= {}_1U_t + {}_2U_t + \ldots + {}_pU_t + {}_1V_t + {}_2V_t + \ldots + {}_qV_t \\ +Y_t &= {}_1U_t + {}_2U_t + \ldots + {}_pU_t + {}_1W_t + {}_2W_t + \ldots \DPtypo{}{+} {}_qW_t +\end{align*} +where ${}_1U_t,~{}_2U_t \ldots$ ${}_1V_t,~{}_2V_t \ldots$ ${}_1W_t,~{}_2W_t \ldots$ are selected +at random each from an independent group of quantities (more +strictly, are \emph{relative to our data}, random members of independent +groups); then, if we know \textit{à~priori} certain statistical coefficients +descriptive of the constitution of these groups, the value of~$r$ +will probably tend towards a certain value. So far we are on +fairly safe, but not very fruitful, ground. We have no basis +for arguing backwards from the observed value of~$r$; but, +provided we have rather extensive and peculiar knowledge +\textit{à~priori} as to how $X_t$~and~$Y_t$ are constituted, then we have +calculable expectations as to the limits within which the value +%% -----File: 436.png---Folio 425------- +of~$r$, namely the correlation coefficient between $X$~and~$Y$, will +probably turn out to lie, when we have observed it. + +Dr.~Bowley's next move is more dubious. If the constitutions +of the independent groups are similar in a certain, statistical +respect (\ie~if they have the same standard deviations), then, +Dr.~Bowley concludes, $r = \dfrac{p}{(p + q)}$, which ``expressed in words +shows that the correlation coefficient tends to be the ratio of +the number of causes common in the genesis of two variables +to the whole number of independent causes on which each +depends.'' By this time the student's mind, unless anchored +by a more than ordinary scepticism, will have been well launched +into a vague, fallacious sea. + +Neglecting, however, the \textit{dictum} just quoted, we find that the +second stage of the argument consists in showing that, \emph{if} we +have a certain sort of knowledge \textit{à~priori} as to how our variables +are constituted, then the various possible values for the coefficients +of correlation, which would be yielded by actual sets of observations +made in prescribed conditions, will have, \textit{à~priori}, and +before the observations have been made, calculable probabilities, +certain ranges of values being probable and others improbable. + +As a rule, however, we are not arguing from knowledge about +the variables to anticipations about their correlation coefficient; +but the other way round, that is from observations of their +correlation coefficients to theories about the nature of the variables. +Dr.~Bowley perceives that this involves a third stage +of the argument, and appeals accordingly (p.~409) to ``the +difficult and elusive theory of inverse probability.'' He apprehends +\index{Inverse Probability!Bowley@{and Bowley}}% +the difficulty but he does not pursue it; and, like Mr.~Yule, +he really falls back for practical purposes on the criteria +of common sense, an expedient well enough in his case, but not +a universal safeguard. + +The general argument from inverse probability to which Dr.~Bowley +makes his vague appeal is doubtless on the following +lines: If there is no causal connection between the two sets of +quantities, then a close grouping of the frequencies about the +diagonal would be \textit{à~priori} improbable (and the greater the +number of the individual observations, the greater the improbability +since, if the quantities are independent, there is, then, all +the more opportunity for `averaging out'); therefore, inversely, +%% -----File: 437.png---Folio 426------- +if the frequencies do group themselves about the diagonal, we +have a presumption in favour of a causal connection between +the two sets of quantities. + +But if the reader recalls our discussion of the principle of +inverse probability, he will remember that this conclusion cannot +be reached unless \textit{à~priori}, and quite apart from the observations +in question, we have some reason for thinking that there may be +such a causal connection between the quantities. The argument +can only strengthen a pre-existing presumption; it cannot +create one. And in the absence of reasons peculiar to the +particular inquiry, we have no choice but to fall back on the +general methods and the general presumptions of induction. + +It is apparent that, where the correlation argument seems +\index{Correlation!Quantitative}% +plausible, some tacit assumption must have slipped in, if we return +to the case where our correlation table relates to the weights of +the babies and their Christian names. Either by accident or +because we had arranged the order of the Christian names to +suit, it might happen with a particular set of observations, even +a fairly numerous set, that the correlation coefficient was large. +Yet on that evidence alone we should hardly assert a generalisation +connecting the weights of babies with their Christian names. + +The truth is that sensible investigators only employ the +correlation coefficient to test or confirm conclusions at which +they have arrived on other grounds. But that does not validate +the crude way in which the argument is sometimes presented, +or prevent it from misleading the unwary,---since not all investigators +are sensible. + +If we abandon the method of inverse probability in favour of +the less precise but better founded processes of induction, +`quantitative correlation,' as I should like to term this particular +branch of statistical induction, is more complicated than, but not +theoretically distinct from, the kind of arguments which have +occupied the earlier paragraphs of this chapter. The character +of the additional complication can be described by saying that +we are presented with a two-dimensional problem instead of a +one-dimensional problem. The mere existence of a particular +correlation coefficient as descriptive of a group of observations, +even of a large group, is not in itself a more conclusive or significant +argument than the mere existence of a particular frequency +coefficient would be. Of course if we have a considerable body +%% -----File: 438.png---Folio 427------- +\index{Lucretius}% +of pre-existing knowledge relevant to the particular inquiry, the +calculation of a small number of correlation coefficients may be +crucial. But otherwise we must proceed as in the case of frequency +coefficients; that is to say we must have before us, in +order to found a satisfactory argument, many sets of observations, +of which the correlation coefficients display a significant +stability in the midst of variation in the non-essential class +characteristics (\ie~those class characteristics which our generalisation +proposes to neglect) of the different sets of observations. + +\Paragraph{10.} I am now at the conclusion of an inquiry in which, +beginning with fundamental questions of logic, I have endeavoured +to push forward to the analysis of some of the actual arguments +which impress us as rational in the progress of knowledge and the +practice of empirical science. In writing a book of this kind the +author must, if he is to put his point of view clearly, pretend sometimes +to a little more conviction than he feels. He must give +his own argument a chance, so to speak, nor be too ready to +depress its vitality with a wet cloud of doubt. It is a heavy task +to write on these problems; and the reader will perhaps excuse +me if I have sometimes pressed on a little faster than the difficulties +were overcome, and with decidedly more confidence than +I have always felt. + +In laying the foundations of the subject of Probability, I have +departed a good deal from the conception of it which governed +the minds of Laplace and Quetelet and has dominated through +\index{Laplace}% +\index{Quetelet}% +their influence the thought of the past century,---though I believe +that Leibniz and Hume might have read what I have written with +\index{Hume}% +\index{Leibniz}% +sympathy. But in taking leave of Probability, I should like to +say that, in my judgment, the practical usefulness of those modes +of inference, here termed Universal and Statistical Induction, +on the validity of which the boasted knowledge of modern science +depends, can only exist---and I do not now pause to inquire +again whether such an argument \emph{must} be circular---if the universe +of phenomena does in fact present those peculiar characteristics +of atomism and limited variety which appear more and more +\index{Variety!limitation of}% +clearly as the ultimate result to which material science is tending: +\begin{center} +\settowidth{\TmpLen}{materiem quoque \textit{finitis} differre figuris.}% +\begin{minipage}{\TmpLen}% + \begin{flushright} + fateare necessest\\ + materiem quoque \textit{finitis} differre figuris. + \end{flushright} +\end{minipage} +\end{center} +The physicists of the nineteenth century have reduced matter to +%% -----File: 439.png---Folio 428------- +\index{Calculus of Probability}% +\index{Mendelism and statistics}% +the collisions and arrangements of particles, between which the +ultimate qualitative differences are very few; and the Mendelian +biologists are deriving the various qualities of men from the +collisions and arrangements of chromosomes. In both cases the +analogy with the perfect game of chance is really present; and +the validity of some current modes of inference may depend on the +assumption that it is to material of this kind that we are applying +them. Here, though I have complained sometimes at their want +of logic, I am in fundamental sympathy with the deep underlying +conceptions of the statistical theory of the day. If the contemporary +doctrines of Biology and Physics remain tenable, we may +have a remarkable, if undeserved, justification of some of the +methods of the traditional Calculus of Probabilities. Professors +of probability have been often and justly derided for arguing as +if nature were an urn containing black and white balls in fixed +proportions. Quetelet once declared in so many words---``l'urne +\index{Quetelet}% +que nous interrogeons, c'est la nature.'' But again in the +history of science the methods of astrology may prove useful to +the astronomer; and it may turn out to be true---reversing +Quetelet's expression---that ``La nature que nous interrogeons, +c'est une urne.'' +%% -----File: 440.png---Folio 429------- + +\cleardoublepage +\pagestyle{empty} +\thispagestyle{plain} +\phantomsection +\pdfbookmark[-1]{Back Matter}{Back Matter} +\null\vfil\vfil +{\LARGE BIBLIOGRAPHY} +\vfil\vfil\vfil +\cleardoublepage + +%% -----File: 441.png---Folio 430------- +%[Blank Page] +%% -----File: 442.png---Folio 431------- + + +\Bibliography + +\Section{INTRODUCTION} + +\begin{Quote} +There is no opinion, however absurd or incredible, which has not been +maintained by some one of our philosophers.---\textsc{Descartes}. +\end{Quote} + +\First{The} following Bibliography does not pretend to be complete, +but it contains a much longer list of what has been written +about Probability than can be found elsewhere. I have +hesitated a little before burdening this volume with the titles +of many works, so few of which are still valuable. But I was +myself much hampered, when first I embarked on the study of +this subject, by the absence of guide-posts to the scattered but +extensive literature of the subject; and a list which I drew up +for my own convenience, without much attention to bibliographical +nicety or to exact uniformity in the style of entry, +may be useful to others. + +It is rather an arbitrary matter to decide what to include +and what to exclude. Probability overlaps many other topics, +and some of the most important references to it are to be +found in books, the main topic of which is something else. On +the other hand it would be absurd to include every casual +reference; and no useful purpose would have been served by +cataloguing the very numerous volumes dealing with Insurance, +Games of Chance, Statistics, Errors of Observation, and Least +Squares, which treat in detail these various applications of the +Theory of Probability. It has been a matter of some difficulty, +therefore, to know precisely where to draw the line. Where +the main subject of a book or paper is Probability proper, I +have included it, nearly regardless of my own view as to its +importance, and have not attempted to act as censor; but +where Probability is not the main subject or where an application +of Probability is concerned, the chief interest of which is +%% -----File: 443.png---Folio 432------- +solely in the application itself, I have only included the entry +where I think it important, intrinsically or historically or +from the celebrity of the author. In particular, the existence +of Professor Mansfield Merriman's very extensive bibliography, +published in the \textit{Transactions of the Connecticut Academy} for +1877, has made it possible to deal very lightly (and to the +extent of but few entries) with the inordinately large literature +of Least Squares. This list comprises 408~titles of writings +relating to the Method of Least Squares and the theory of +accidental errors of observation, and is sufficiently exhaustive +so far as relates to memoirs on this topic published before +1877. + +Of bibliographical sources for Probability proper, Todhunter's +\textit{History of the Mathematical Theory of Probability} +and Laurent's \textit{Calcul des probabilités} are alone important. Of +\emph{mathematical} works published before the time of Laplace, +Todhunter's list, and also his commentary and analysis, are +complete and exact,---a work of true learning, beyond criticism. +The bibliographical catalogue at the conclusion of Laurent's +\textit{Calcul} (published in 1873) is the longest list published hitherto +of general works on Probability. But it is unduly swollen by +the inclusion of numerous items on Insurance and Errors of +Observation, the bearing of which on Probability is very +slight;\footnote + {Laurent's list contains 310~titles, of which I have excluded~174 from my + list as being insufficiently relevant.} +it is chiefly mathematical in bias; and it is now +nearly fifty years old. + +I have not read all these books myself, but I have read +more of them than it would be good for any one to read again. +There are here enumerated many dead treatises and ghostly +memoirs. The list is too long, and I have not always successfully +resisted the impulse to add to it in the spirit of a +collector. There are not above a hundred of these which it +would be worth while to preserve,---if only it were securely +ascertained which these hundred are. At present a bibliographer +takes pride in numerous entries; but he would be a +more useful fellow, and the labours of research would be +lightened, if he could practise deletion and bring into existence +an accredited \textit{Index Expurgatorius}. But this can only be +accomplished by the slow mills of the collective judgment of +%% -----File: 444.png---Folio 433------- +the learned; and I have already indicated my own favourite +authors in copious footnotes to the main body of the text. + +The list is long; yet there is, perhaps, no subject of equal +importance and of equal fascination to men's minds on which +so little has been written. It is now fifty-five years since +Dr.~Venn, still an accustomed figure in the streets and courts +of Cambridge, first published his \textit{Logic of Chance}; yet amongst +systematic works in the English language on the logical foundations +of Probability my Treatise is next to his in chronological +order. + +The student will find many famous names here recorded. +The subject has preserved its mystery, and has thus attracted +the notice, profound or, more often, casual, of most speculative +minds. Leibniz, Pascal, Arnauld, Huygens, Spinoza, Jacques +and Daniel Bernoulli, Hume, D'Alembert, Condorcet, Euler, +Laplace, Poisson, Cournot, Quetelet, Gauss, Mill, Boole, +Tchebychef, Lexis, and Poincaré, to name those only who are +dead, are catalogued below. + +\begin{center}\rule{10em}{.5pt}\end{center} + +\begin{Biblio} + +\Bibsect[noskip]{A} + +\BibItem[Abbott, T.~K.] ``On the Probability of Testimony and Arguments.'' Phil.\ +Mag.\ (4), vol.~27, 1864. + +\BibItem[Adrain, R.] ``Research concerning the Probabilities of the Errors which happen in making Observations.'' The Analyst or Math.\ Museum, vol.~1, +pp.~93--109, 1808. + +\Bibnote[This paper, which contains the first deduction of the normal law of error, was partly reprinted by Abbé with historical notes in Amer.\ Journ.\ +Sci.\ vol.~i.\ pp.~411--415, 1871.] + +\BibItem[Ammon, O.] ``Some Social Applications of the Doctrine of Probability.'' +Journ.\ Pol.\ Econ.\ vol.~7, 1899. + +\BibItem[Ampère.] Considérations sur la théorie mathématique du jeu. Pp.~63. 4to. Lyon, 1802. + +\BibItem[Ancillon.] ``Doutes sur les bases du calcul des probabilités.'' Mém.\ Ac.\ Berlin, pp.~3--32, 1794--5. + +\BibItem[Arbuthnot, J.] Of the Laws of Chance, or a Method of Calculation of the +Hazards of Game plainly Demonstrated. 16mo. London, 1692. + +\Bibnote[Contains a translation of Huygens, De ratiociniis in ludo aleae.] + +\BibItem[\bysame] 4th~edition revised by John Hans. By whom is added a demonstration of +the gain of the banker in any circumstance of the game call'd Pharaon, etc. +Sm.~8vo. London, 1738. + +\Bibnote[For a full account of this book and discussion of the authorship, see +Todhunter's History, pp.~48--53.] + +\BibItem[\bysame] ``An Argument for Divine Providence, taken from the constant Regularity +observ'd in the Births of both Sexes.'' Phil.\ Trans, vol.~27, pp.~186--190 +(1710--12). + +\Bibnote[Argues that the excess of male births is so invariable, that we may conclude +that it is not an even chance whether a male or female be born.] +%% -----File: 445.png---Folio 434------- + +\BibItem[Aristotle.] Anal.\ Prior.\ ii.~27, 70\textsuperscript{a}~3. +Rhet.\ i.~2, 1357~a 34. %[** TN: Semantic re-breaking] + +\Bibnote[See Zeller's \textit{Aristotle} for further references.] + +\BibItem[Arnauld.] (The Port Royal Logic.) La Logique ou l'Art de penser. 12mo. +Paris, 1662. Another ed.\ C.~Jourdain, Hachette, 1846. Transl.\ into +Eng.\ with introduction by T.~S. Baynes. London, 1851. xlvii~+~430. +See especially pp.~351--370. + +\Bibsect{B} + +\BibItem[Babbage, C.] An Examination of some Questions connected with Games of +Chance. 4to. 25~pp. Trans.\ R.~Soc.\ Edin., 1820. + +\BibItem[Bachelier, Louis.] Calcul des probabilités. Tome~i. 4to. Pp.~vii~+~517. +Paris, 1912. + +\BibItem[\bysame] Le Jeu, la chance, et le hasard. Pp.~320. Paris, 1914. + +\BibItem[[Bailey, Samuel.]] Essays on the pursuit of truth, on the progress of knowledge +and on the fundamental principle of all evidence and expectation. +Pp.~xii~+~302. London, 1829. + +\BibItem[Baldwin.] Dictionary of Philosophy. Bibliographical volumes; \textit{s.v.}~``Probability.'' + +\BibItem[Baniol, A.] ``Le Hasard.'' Revue Internationale de Sociologie. Pp.~16. 1912. + +\BibItem[Barbeyrac.] Traité du jeu. 1st~ed.\ 1709. 2nd~ed.\ 1744. + +\Bibnote[Todhunter states (p.~196) that Barbeyrac is said to have published a +discourse ``Sur la nature du sort.''] + +\BibItem[Bayes, Thomas.] An Essay towards solving a Problem in the Doctrine of +Chances. Phil.\ Trans.\ vol.~liii.\ pp.~370--418, 1763. A demonstration, +etc. Phil.\ Trans.\ vol.~liv.\ pp.~296--325, 1764. + +\Bibnote[Both the above were communicated by the Rev.\ Richard Price, and +the second is partly due to him.] + +\BibItem[\bysame] German transl. Versuch zur Lösung eines Problems der Wahrscheinlichkeitsrechnung. +Herausgegeben von H.~E. Timerding. Sm. 8vo. +Leipzig, 1908. Pp.~57. + +\BibItem[Béguelin.] ``Sur les suites ou séquences dans le loterie de Gênes.'' Hist.\ de +l'Acad. Pp.~231--280. Berlin, 1765. + +\BibItem[\bysame] `` Sur l'usage du principe de la raison suffisante dans le calcul des probabilités.'' +Hist.\ de l'Acad. Pp.~382--412. Berlin, 1767. (Publ.~1769.) + +\BibItem[Bellavitis.] ``Osservazioni sulla theoria delle probabilità.'' Atti del Instituto +Veneto di Scienze, Lettere, ed Arti, Venice, 1857. + +\BibItem[Benard.] ``Note sur une question de probabilités.'' Journal de l'École +royale politechnique. Vol.~15, Paris, 1855. + +\BibItem[Bentham, J.] Rationale of Judicial Evidence. \\ +See Introductory View, chap.~xii., and Bk.~i, chaps.\ v.,~vi.,~vii. + +\BibItem[Bernoulli, Daniel.] ``Specimen theoriae novae de mensura sortis.'' Comm.\ +Acad.\ Sci.\ Imp.\ Pet.\ vol.\ v.~pp.~175--192, 1738. + +\BibItem[\bysame] Germ.~transl.~1896, by A.~Pringsheim: Die Grundlage der modernen +Wertlehre. Versuch einer neuen Theorie der Wertbestimmung von Glücksfällen +(Einleitung von Ludvig Fick). Pp.~60. Leipzig, 1896. + +\BibItem[\bysame] ``Recueil des pièces qui ont remporté le prix de l'Académie Royale des +Sciences.'' 1734. iii.\ pp.~95--144. + +\BibItem[\bysame] [On ``La cause physique de l'inclinaison des plans des orbites des planètes +par rapport au plan de l'équateur de la révolution du soleil autour de son +axe.''] + +\BibItem[\bysame] ``Essai d'une nouvelle analyse de la mortalité causée par la petite +vérole.'' Hist.\ de l'Acad.\ pp.~1--45. Paris, 1760. + +\BibItem[\bysame] De usu algorithmi infinitesimalis in arte conjectandi specimen. Novi +Comm.\ Petrop., 1766. xii.\ pp.~87--98. A 2nd~memoir. Petrop., 1766. +xii.\ pp.~99--126. See a oriticism by Trembley, Mem.\ de l'Acad., Berlin, +1799. +%% -----File: 446.png---Folio 435------- + +%\BibItem[Bernoulli, Daniel.]---\emph{continued}. + +\BibItem[\bysame] Disquisitiones analytiquae de novo problemate conjecturali. Novi +Comm.\ Petrop.\ xiv.\ pp.~1--25, 1769. A 2nd~memoir, Petrop.\ xiv.\ pp.~26--45, 1769. + +\BibItem[\bysame] ``Dijudicatio maxime probabilis plurium observationum discrepantium +atque verisimillima inductio inde formanda.'' Acta Acad., pp.~3--23. +Petrop., 1777. Crit.\ by Euler, pp.~24--33\DPtypo{}{.} + +\BibItem[Bernoulli, Jac.] Ars conjectandi, opus posthumum. Pp.~ii~+~306~+~35. +Sm. 4to, Basileae, 1713. + +\Bibnote[Published by N.~Bernoulli eight years after Jac.\ Bernoulli's death.] + +\BibItem[\bysame] Part~I\@. Reprint with notes and additions of Huygens, De ratiociniis in +ludo aleae. + +\BibItem[\bysame] Part~II\@. Doctrina de permutationibus et combinationibus. + +\BibItem[\bysame] Part~III\@. Explicans usum praecedentis doctrinae in variis sortitionibus +et ludis aleae. [Twenty-four problems.] + +\BibItem[\bysame] Part~IV\@. Tradens usum et applicationem praecedentis doctrinae in +civilibus, moralibus et oeconomicis. + +\BibItem[\bysame] Tractatus de seriebus infinitis. [Not connected with the subject of +Probability.] + +\BibItem[\bysame] Lettre à un amy, sur les partis du jeu de paume. + +\Bibnote[The most important sections, including Bernoulli's Theorem, are in +Part~IV\@. For a very full account of the whole volume see Todhunter's +\textit{History}, chap.~vii.] + +\BibItem[\bysame]%[** TN: Retaining inconsistent capitalization of "transl."] +Engl.\ Transl.\ of Part~II. only, vide \textit{Maseres}. + +\BibItem[\bysame] Fr.\ transl.\ of Part~I. only, vide \textit{Vastel}. + +\BibItem[\bysame] Germ.\ transl.: Wahrscheinlichkeitsrechnung. 4~Teile mit dem Anhange: +Brief an einem Freund über das Ballspiel, übers.\ u.~hrsg.\ v.~R. Haussner. +2~vols. Sm.~8vo. 1899. + +\Bibnote[See also \textit{Leibniz}.] + +\BibItem[Bernoulli, John.] De alea, sive arte conjectandi, problemata quaedam. +Collected ed.\ vol.~iv.\ pp.~28--33. 1742. + +\BibItem[Bernoulli, John (grandson).] ``Sur les suites ou séquences dans la loterie de +Gènes.'' Hist.\ de l'Acad., pp.~234--253. Berlin, 1769. + +\BibItem[\bysame] ``Mémoire sur un problème de la doctrine du hasard.'' Hist.\ de l'Acad., +pp.~384--408. Berlin, 1768. + +\BibItem[Bernoulli, Nicholas.] Specimina artis conjectandi, ad quaestiones juris +applicatae. Basel, 1709. Repr.\ Act.\ Erud.\ Suppl., pp.~159--170, 1711. + +\BibItem[Bertrand, J.] Calcul des probabilités. Pp.~lvii~+~332. Paris, 1889. + +\BibItem[\bysame] ``Sur l'application du calcul des probabilités à la théorie des jugements.'' +Comptes rendus, 1887. + +\BibItem[\bysame] ``Les Lois du hasard.'' Rev.\ des Deux Mondes, p.~758. Avril 1884. + +\BibItem[Bessel.] ``Untersuchung über die Wahrscheinlichkeit der Beobachtungsfehler.'' +Astr.\ Nachrichten, vol.~xv.\ pp.~369--404, 1838. + +\BibItem[\bysame] Also Abhandl.\ von Bessel, vol.~ii.\ pp.~372--391. Leipzig, 1875. + +\BibItem[Bicquilley, C.~F.~de.] Du calcul des probabilités. 164~pp., 1783. 2nd~ed.\ 1805. + +\BibItem[\bysame] Germ.\ transl.\ by C.~F. Rüdiger. Leipzig, 1788. + +\BibItem[Bienaymé, J.] ``Sur un principe que Poisson avait cru découvrir et qu'il avait +appelé loi des grands nombres.'' Comptes rendus de l'Acad.\ des Sciences +morales, 1855. + +\Bibnote[Reprinted in Journal de la Soc.\ de Statistiques de Paris, pp.~199--204, +1876.] + +\BibItem[\bysame] ``Probabilité de la constance des causes conclue des effets observés.'' +Procès-verbaux de la Soc.\ Philomathique, 1840. + +\BibItem[\bysame] ``Sur la probabilité des résultats moyens des observations, etc.'' Sav.\ +Étrangers, v.,~1838. +%% -----File: 447.png---Folio 436------- + +% \BibItem[Bienaymé, J.]---\emph{continued}. + +\BibItem[\bysame] ``Théorème sur la probabilité des résultats moyens des observations.'' +Procès-verbaux de la Soc.\ Philomathique, 1839. + +\BibItem[\bysame] ``Considérations à l'appui de la découverte de Laplace sur la loi de probabilité +dans la méthode des moindres carrés.'' Comptes rendus des +séances de l'Académie des Sciences, vol.~xxxvii., 1853. + +\Bibnote[Reprinted in Journal de Liouville, 2nd~series, vol.~xii., 1867, pp.~158--176.] + +\BibItem[\bysame] Remarques sur les différences qui distinguent l'interpolation de Cauchy +de la méthode des moindres carrés.'' Comptes rendus, 1853. + +\BibItem[\bysame] ``Probabilité des erreurs dans la méthode des moindres carrés.'' Journ.\ Liouville, vol.~xvii., 1852. + +\BibItem[Binet.] ``Recherches sur une question de probabilité'' (Poisson's Theorem). +Comptes rendus, 1844. + +\BibItem[Blaschke, E.] Vorlesungen über mathematische Statistik. Pp.~viii~+~268. +Leipzig, 1906. + +\BibItem[Bobek, K.~J.] Lehrbuch der Wahrscheinlichkeitsrechnung. Nach System +Kleyer. Pp.~296. Stuttgart 1891. + +\BibItem[Bohlmann, G.] ``Die Grundbegriffe der Wahrscheinlichkeitsrechnung in ihrer +Anwendung auf die Lebensversicherung.'' Atti del IV~Congr.\ intern.\ dei +matematici, Rome, 1909. + +\BibItem[Boole, G.] Investigations of Laws of Thought on which are founded the +Mathematical Theories of Logic and Probabilities. Pp.~ix~+~424. London, +1854. + +\BibItem[\bysame] ``Proposed Questions in the Theory of Probabilities.'' Cambridge and +Dublin Math.\ Journal, 1852. + +\BibItem[\bysame] ``On the Theory of Probabilities, and in particular on Michell's Problem +of the Distribution of the Fixed Stars.'' Phil.\ Mag., 1851. + +\BibItem[\bysame] ``On a General Method in the Theory of Probabilities.'' Phil.\ Mag., +1852. + +\BibItem[\bysame] ``On the Solution of a Question in the Theory of Probabilities.'' Phil.\ Mag., 1854. + +\BibItem[\bysame] ``Reply to some Observations published by Mr.~Wilbraham in the Phil.\ +Mag.~vii.\ p.~465, on Boole's `Laws of Thought.'\,'' Phil.\ Mag., 1854. + +\BibItem[\bysame] ``Further Observations in reply to Mr.~Wilbraham.'' Phil.\ Mag., 1854. + +\BibItem[\bysame] ``On the Conditions by which the Solutions of Questions in the Theory +of Probabilities are limited.'' Phil.\ Mag., 1854. + +\BibItem[\bysame] ``On certain Propositions in Algebra connected with the Theory of +Probabilities.'' Phil.\ Mag., 1855. + +\BibItem[\bysame] ``On the Application of the Theory of Probabilities to the Question of +the Combination of Testimonies or Judgments.'' Edin.\ Phil.\ Trans, vol.~xxi.\ pp.~597--652, 1857. + +\BibItem[\bysame] ``On the Theory of Probabilities.'' Roy.\ Soc.\ Proc.\ vol.~xii.\ pp.~179--184, +1862--1863. + +\BibItem[Borchardt, B.] Einführung in die Wahrscheinlichkeitslehre. vi~+~86. +Berlin, 1889. + +\BibItem[Bordoni, A.] Sulle probabilità. 4to. Giorn.\ dell' I.~R. Instit.\ Lombardo di +Scienze. T.~iv. Nuova Serie. Milano, 1852. + +\BibItem[Borel, E.] Éléments de la théorie des probabilités. 8vo, pp.~vii~+~191. +Paris, 1909. 2nd~ed.\ 1910. + +\BibItem[\bysame] Le Hasard. Pp.~iv~+~312. Paris, 1914. + +\BibItem[\bysame] ``Le Calcul des probabilités et la méthode des majorités.'' L'Année +psychologique, vol.~14, pp.~125--151. Paris, 1908. + +\BibItem[\bysame] ``Les Probabilités dénombrables et leurs applications arithmétiques.'' +Rendiconti del Circolo matematico di Palermo, 1909. + +\BibItem[\bysame] ``Le Calcul des probabilités et la mentalité individualiste.'' Revue du +Mois, vol.~6, pp.~641--650, 1908. +%% -----File: 448.png---Folio 437------- + +%\BibItem[Borel, E.]---\emph{continued}. + +\BibItem[\bysame] +``La Valeur practique du calcul des probabilités.'' Revue du Mois, vol.~1, pp.~424--437, 1906. + +\BibItem[\bysame] ``Les Probabilités et M.~le Dantec.'' Revue du Mois, vol.~12, pp.~77--91, +1911. + +\BibItem[Bortkiewicz, L.~von.] Das Gesetz der kleinen Zahlen. 8vo, pp.~viii~+~52, +Leipzig, 1898. + +\BibItem[\bysame] ``Anwendungen der Wahrscheinlichkeitsrechnung auf Statistik.'' Encyklopädie +der mathematischen Wissenschaften, Band~1, Heft~6. + +\BibItem[\bysame] ``Wahrscheinlichkeitstheorie und Erfahrung.'' Zeitschrift für Philosophie +und philosophische Kritik, vol.~121, pp.~71--81. Leipzig, 1903. + +\Bibnote [With reference to Marbe, Brömse, and Grimsehl, \textit{q.v.}] + +\BibItem[\bysame] ``Kritische Betrachtungen zur theoretischen Statistik.'' Jahrb.\ f.~Nationalök.\ +u.~Stat.~(3), vol.~8, pp.~641--680, 1894; vol.~10, pp.~321--360, 1895; vol.~11, pp.~671--705, 1896. + +\BibItem[\bysame] ``Die erkenntnistheoretischen Grundlagen der Wahrscheinlichkeitsrechnung.'' +Jahrb.\ f.~Nationalök.\ u.~Stat.~(3), vol.~17, pp.~230--244, +1899. + +\Bibnote [Criticised by Stumpf, \textit{q.v.}, who is answered by Bortkiewicz, \textit{loc.\ cit.}, +vol.~18, pp.~239--242, 1899.] + +\BibItem[\bysame] ``Zur Verteidigung des Gesetzes der kleinen Zahlen.'' Jahrb.\ f.~Nationalök.\ +u.~Stat.~(3), vol.~39, pp.~218--236, 1910. + +\Bibnote [The literature of this topic is not fully dealt with in this Bibliography, +but very full references to it will be found in the above article.] + +\BibItem[\bysame] ``Über den Präzisionsgrad des Divergenzkoeffizientes.'' Mitteil.\ des Verbandes +der österr.\ und ungar.\ Versicherungstechniker, vol.~5. + +\BibItem[\bysame] ``Realismus und Formalismus in der mathematischen Statistik.'' Allg.\ +Stat.\ Archiv, vol.~ix.\ pp.~225--256. Munich, 1915. + +\BibItem[\bysame] Die Iterationen: ein Beitrag zur Wahrscheinlichkeitstheorie. Pp.~xii~+~205. +Berlin, 1917. + +\BibItem[\bysame] Die radioaktive Strahlung als Gegenstand wahrscheinlichkeitstheoretischer +Untersuchungen. Pp.~84. Berlin, 1913. + +\BibItem[\bysame] ``Wahrscheinlichkeitstheoretische Untersuchungen über die Knabenquote +bei Zwillings Gebieten.'' Sitzungsber.\ der Berliner Math.\ Ges., vol.~xvii.\ pp.~8--14, 1918. + +\BibItem[\bysame] Homogeneität und Stabilität in der Statistik. Pp.~81. (Extracted from +the Skandinavisk Aktuarietidskrift.) Uppsala, 1918. + +\BibItem[Bostwick, A.~E.] ``The Theory of Probabilities.'' Science,~iii., 1896, +p.~66. + +\BibItem[Boutroux, Pierre.] ``Les Origines du calcul des probabilités.'' Revue du +Mois, vol.~5, pp.~641--654, 1908. + +\BibItem[Bowley, A.~L.] Elements of Statistics. Pp.~xi~+~459. 4th~ed. London, +1920. + +\BibItem[Bradley, T.~H.] The Principles of Logic. Bk.~i.\ chap.~8, §§\;32--63, pp.~201--20. +London, 1883. + +\BibItem[Bravais.] ``Analyse mathématique sur les probabilités des erreurs de situation +d'un point.'' Mém.\ Sav.\ vol.~9, pp.~255--332, Paris, 1846. + +\BibItem[Brendel.] Wahrscheinliclikeitsrechnung mit Einschluss der Anwendungen. +Göttingen, 1907. + +\BibItem[Broad, C.~D.] ``The Relation between Induction and Probability.'' Mind, +vol.~xxvii.\ (1918). Pp.~389--404, and vol.~xxix.\ (1920) pp.~11--45. + +\BibItem[\DPtypo{Bromse}{Brömse}, H.] Untersuchungen zur Wahrscheinlichkeitslehre. (Mit besonderer +Beziehung auf Marbes Schrift (\textit{q.v.}).) + +\BibItem[\bysame] Zeitschrift für Philosophie und philosophische Kritik. Band~118.\ +Leipzig, 1901. Pp.~145--153. + +\Bibnote (See also Marbe, Grimsehl, and v.~Bortkiewicz.) +%% -----File: 449.png---Folio 438------- + +\BibItem[\DPtypo{Brunn}{Brünn}, Dr.\ Hermann.] ``Über ein Paradoxon der Wahrscheinlichkeitsrechnung.'' +Sitzungsberichte der philos.-philol. Klasse der K.~bayrische +Akademie, pp.~692--712, 1892. + +\BibItem[Bruns, H.] Wahrscheinlichkeitsrechnung und Kollektivmasslehre. 8vo. +Pp.~viii~+ 310~+~18. Leipzig, 1906. + +\BibItem[\bysame] ``Das Gruppenschema für zufällige Ereignisse.'' Abhandl.\ d.~Leipz.\ +Ges.\ d.~Wissensch.\ vol.~xxix.\ pp.~579--628, 1906. + +\BibItem[Bryant, Sophie.] ``On the Failure of the Attempt to deduce inductive Principles +from the Mathematical Theory of Probabilities.'' Phil.\ Mag.\ S.~5, +No.~109, Suppl.\ vol.~17. + +\BibItem[Buffon.] ``Essai d'arithmétique morale.'' Supplément à l'Histoire Naturelle, +vol.~4, 103~pp. 4to. 1777. Hist.\ Ac.\ Par.\ pp.~43--45, 1733. + +\BibItem[Bunyakovski.] Osnovaniya, etc. (Principles of the Mathematical Theory of +Probabilities.) Petersburg, 1846. + +\BibItem[Burbury, S.~H.] ``On the Law of Probability for a System of correlated +variables.'' Phil.\ Mag.~(6), vol.~17, pp.~1--28, 1909. + +\Bibsect{C} + +\BibItem[Campbell, R.] ``On a Test for ascertaining whether an observed Degree of +Uniformity, or the reverse, in tables of Statistics is to be looked upon as +remarkable.'' Phil.\ Mag., 1859. + +\BibItem[\bysame] ``On the Stability of Results based upon average Calculations.'' Journ.\ +Inst.\ Act.\ vol.~9, p.~216. + +\BibItem[\bysame] A popular Introduction to the Theory of Probabilities. Pp.~16, Edinburgh, +1865. + +\BibItem[Cantelli, F.~P.] ``Sulla applicazione delle probabilità parziali alla statistica.'' +Giornale di Matematica finanziaria, vol.~i.\ (1919), pp.~30--44. + +\BibItem[Cantor, G.] Historische Notizen über die Wahrscheinlichkeitsrechnung. 4to. +8~pp. Halle, 1874. + +\BibItem[Cantor, M.] Politische Arithmetik oder die Arithmetik des täglichen Lebens. +Pp.~x~+~155. Leipzig, 1898, 2nd~ed.\ 1903. + +\BibItem[Canz, E.~C.] Tractatio synoptica de probabilitate juridica sive de praesumtione. +4to. Tübingen, 1751. + +\BibItem[Caramuel, John.] Kybeia, quae combinatoriae genus est, de alea, et ludis +fortunae serio disputans. 1670. [Includes a reprint of Huygens, which +is attributed to Longomontanus.] + +\BibItem[Cardan.] De ludo aleae.\ fo.,~15~pp. 1663. [Cardan ob.\ 1576.] + +\BibItem[Carvello, E.] Le Calcul des probabilités et ses applications. 8vo. Pp.~ix~+~169. +Paris, 1912. + +\BibItem[Castelnuovo, Guido.] Calcolo delle probabilità. Large 8vo. Pp.~xxiii~+~373. +Rome, 1919. + +\BibItem[Catalan, E.] ``Solution d'un problème de probabilité, relatif au jeu de +rencontre.'' Journ. Liouville, vol.~ii., 1837. + +\BibItem[\bysame] ``Deux problèmes de probabilités.'' Journ.\ Liouville, vol.~vi. + +\BibItem[\bysame] Problèmes et théorèmes de probabilités. 4to. 1884. + +\BibItem[Cauchy.] Sur le système de valeurs qu'il faut attribuer à divers éléments +déterminés par un grand nombre d'observations. 4to. Paris, 1814. + +\BibItem[Cayley, A.] ``On a Question in the Theory of Probabilities.'' Phil.\ Mag., 1853. + +\BibItem[Cesàro, E.] ``Considerazioni sul concetto di probabilità.'' Periodico di +Matematica,~vi., 1891. + +\BibItem[Charlier, C.~V.~L.] Researches into the Theory of Probability. Publ.\ in +Engl.\ in Meddelanden from Lund's Astronom. Observatorium, Series~ii., +No.~24. 4to. 51~pp. Lund, 1906. + +\BibItem[\bysame] ``Contributions to the Mathematical Theory of Statistics,'' Arkiv för +matematik, astronomi och fysik, vols.\ 7,~8,~9,~\textit{passim}. + +\BibItem[\bysame] Vorlesungen über die Grandzüge der mathematischen Statistik. Sm.\ +4to. Pp.~125. Lund, 1920. +\end{Biblio} +%% -----File: 450.png---Folio 439------- +\begin{Biblio} + +\BibItem[Charpentier, T.~V.] ``Sur la nécessité d'instituer la logique du probable.'' +Comptes rendus de l'Acad.\ des Sciences morales, vol.~i.\ p.~103, 1875. + +\BibItem[\bysame] ``La Logique du probable.'' Rev.\ phil.\ vol.~vi.\ pp.~23--38, 146--163, 1878. + +\BibItem[Chrystal, G.] On some Fundamental Principles in the Theory of Probability. +London, 1891. + +\BibItem[Clark, Samuel.] The Laws of Chance: or a Mathematical Investigation of +the Probability arising from any proposed Circumstance of Play, etc. +Pp.~ii~+~204, 1758. + +\BibItem[Cohen, J.] Chance: A Comparison of 4~Facts with the Theory of Probabilities. +Pp.~47. London, 1905. + +\BibItem[Condorcet, Marquis de.] Essai sur l'application de l'analyse à la probabilité +des décisions rendues à la pluralité des voix. 4to. Pp.~cxci~+~304. +Paris, 1785. Another edition, 1804. + +\BibItem[\bysame] ``Sur les événements futurs.'' Acad.\ des Sc., 1803. + +\BibItem[\bysame] Memoir on Probabilities in six parts: + +\BibItem[\bysame] 1. ``Réflexions sur la règle générale qui prescrit de prendre pour valeur +d'un événement incertain la probabilité de cet événement, multipliée par +la valeur de l'événement en lui-même.'' Hist.\ de l'Acad.\ pp.~707--728. +Paris, 1781. + +\BibItem[\bysame] 2. ``Application de l'analyse à cette question: Déterminer la probabilité +qu'un arrangement régulier est l'effet d'une intention de le produire.'' +Hist.\ de l'Acad., Paris, 1781. With Part~i. + +\BibItem[\bysame] 3. Sur l'évaluation des droits éventuels. 1782, pp.~674--691. + +\BibItem[\bysame] 4. Réflexions sur la méthode de déterminer la probabilité des événements +futurs, d'après l'observation des événements passés. 1783, pp.~539--559. + +\BibItem[\bysame] 5. Sur la probabilité des faits extraordinaires. 1783, with Part~4. + +\BibItem[\bysame] 6. Application des principes de l'article précédent à quelques questions +de critique. 1784, pp.~454--468. + +\BibItem[Coover, J.] Experiments in Psychical Research at Leland Stanford Junior +University. Pp.~641. Stanford University, California, 1917. + +\Bibnote [See Psychical Research and Statistical Method by F.~Y. Edgeworth, +Stat.\ JL., vol.~lxxxii. (1919), p.~222.] + +\BibItem[Corbaux, F.] Essais métaphysiques et mathématiques sur le hasard. 8vo. +Paris, 1812. + +\BibItem[Costa.] Probabilité du tir. 8vo. Paris, 1825. + +\BibItem[\bysame] ``Question de probabilité applicable aux décisions rendues par les +jurés.'' Liouv.\ J.~(1), vii., 1842. + +\BibItem[Courcy, Alph.\ de.] Essai sur les lois du hasard suivi d'étendus sur les assurances. +8vo. Paris, 1862. + +\BibItem[Cournot, A.] Revue de Métaphysique et de Morale, May 1905. Numéro +spécialement consacré à Cournot. See especially: + +\BibItem[\bysame] F. Faure: ``Les Idées de Cournot sur la statistique,'' pp.~395--411. + +\BibItem[\bysame] D. Parodi: ``Le Criticisme de Cournot,'' pp.~451--484. + +\BibItem[\bysame] +F. Mentré: ``Les Racines historiques du probabilisme rationnel de +Cournot,'' pp.~485--508. + +\BibItem[\bysame] +Art.\ ``Probabilités.'' Dictionnaire de Franck. + +\BibItem[\bysame] +``Sur la probabilité des jugements et la statistique.'' Journal de Liouville, +t.~iii p.~257. + +\BibItem[\bysame] ``Mémoire sur les applications du calcul des chances à la statistique +judiciaire.'' Liouv.\ J.~(1) iii., 1838. + +\BibItem[\bysame] +Exposition de la théorie des chances et des probabilités. Pp.~viii~+~448. +Paris, 1843. + +\BibItem[\bysame] +German translation by C.~H. Schnuse. 8vo. Braunschweig, 1849. + +\BibItem[Couturat, L.] La Logique de Leibniz d'après des documents inédits. +Pp~xiv.~+~608. Paris, 1901. +%% -----File: 451.png---Folio 440------- + +%\BibItem[Couturat, L.]---\emph{continued}. + +\Bibnote [See especially chap.~vi.\ for references to Leibniz's views on Probability.] + +\BibItem[\bysame] Opuscules et fragments inédits de Leibniz. Paris, 1903. + +\BibItem[Craig.] Theologiae Christianae principia mathematica. 4to. London, 1699. +Reprinted Leipzig, 1755. + +\BibItem[[Craig] (?).] ``A Calculation of the Credibility of Human Testimony.'' Phil.\ +Trans.\ vol.~xxi.\ pp.~359--365, 1699. + +\Bibnote [Also attributed to \textsc{Halley}.] + +\BibItem[Crakanthorpe, R.] Logica. 1st~ed.\ London, 1622. 2nd~ed.\ London, 1641 +(auctior et emendatior). 3rd~ed.\ Oxon., 1677. + +\Bibnote [Book~v.\ ``De syllogismo probabili.''] + +\BibItem[Crofton, M.~W.] ``On the Theory of Local Probability, applied to Straight +Lines drawn at random in a Plane.'' Phil.\ Trans.\ vol.~158, pp.~181--199, +1869. + +\Bibnote [Summarised in Proc.\ Lond.\ Math.\ Soc.\ vol.~2, pp.~55--57, 1868.] + +\BibItem[\bysame] ``Probability.'' Encycl.\ Brit.\ 9th~\ ed., 1885. + +\BibItem[\bysame] ``Geometrical Theorems relating to Mean Values.'' Proc.\ Lond.\ Math.\ +Soc.\ vol.~8, pp.~304--309, 1877. + +\BibItem[Czuber, E.] Zum Gesetz der grossen Zahlen. Prag, 1889. + +\BibItem[\bysame] Geometrische Wahrscheinlichkeiten und Mittelwerte. Pp.~vii~+~244. +Leipzig, 1884. + +\BibItem[\bysame] Theorie der Beobachtungsfehler. Pp.~xiv~+~418. Leipzig, 1891. + +\BibItem[\bysame] Die Entwicklung der Wahrscheinlichkeitstheorie und ihrer Anwendungen. +Pp.~viii~+~279. Leipzig, 1899. + +\BibItem[\bysame] Wahrscheinlichkeitsrechnung und ihre Anwendung auf Fehlerausgleichung, +Statistik und Lebensversicherung. Leipzig, 1903. + +\BibItem[\bysame] Ditto. 2~vols, 8vo. \DPtypo{}{pp.}~x~+~410~+~x~+~470. Leipzig, 1908--10. Second +edition, revised and enlarged. Vol.~i. Warscheinlichkeitstheorie, Fehlerausgleichung, +Kollektivmasslehre, 1908. Vol.~ii. Mathematische +Statistik, mathematische Grundlagen der Lebensversicherung, 1910. + +\Bibsect{D} + +%[** TN: Not exactly matching original indentation of these items.] +\BibItem[D'Alembert.] Opuscules mathématiques: Paris, 1761--1780. + +\BibItem[\bysame] [Réflexions sur le calcul des probabilités, ii.\ pp.~1--25, 1761. \\ +Sur l'application du c.~des~p.\ à l'inoculation, ii.\ pp.~26--95. \\ +Sur le calcul des probabilités, etc., iv.\ pp.~73--105; iv.\ pp.~283--341; +v.\ pp.~228--231; v.\ pp.~508--510; vii.\ pp.~39--60.] + +\BibItem[\bysame] Mélanges de littérature, d'histoire et de philosophie. Amsterdam, 1770. + +\BibItem[\bysame] [Doutes et questions sur le calcul des probabilités, vol.~v.\ pp.~223--246. \\ +Réflexions sur l'inoculation. Vol.~v. (These two papers were reprinted +in the first volume of D'Alembert's collected works published at Paris in +1821 (pp.~451--514).)] + +\BibItem[\bysame] Articles in Encyclopédie ou Dictionnaire raisonné: + +\BibItem[\bysame] ``Croix ou Pile,'' 1754. + +\BibItem[\bysame] ``Gageure,'' 1757. + +\BibItem[\bysame] Article in Encyclopédie méthodique: ``Cartes.'' + +\BibItem[D'Anières.] ``Réflexions sur les jeux de hasard.'' Mém.\ de l'Acad.\ pp.~391--398. +Berlin, 1784. + +\BibItem[Dantec, Félix le.] ``Le Hasard et la question d'échelle.'' Revue du Mois, +vol.~4, pp.~257--288, 1907. + +\BibItem[\bysame] Le Chaos et l'harmonie universelle. Paris, 1911. + +\BibItem[Darbyshire, A.~D.] Some Talks illustrating Statistical Correlation. (Reprinted +from Memoirs of the Manchester literary and Philosophical Society.) +21~pp.\ and plates. 8vo. 1907. + +\BibItem[Darbon, A.] Le Concept du hasard dans la philosophie de Cournot. Étude +critique. Pp.~60. Paris, 1911. + +\BibItem[Davenport, C.~B.] Statistical Methods. 1904. +%% -----File: 452.png---Folio 441------- + +\BibItem[De Moivre, A.] ``De mensura sortis, seu, de probabilitate eventuum in ludis +a casu fortuito pendentibus.'' Phil.\ Trans.\ vol.~xxvii.\ pp.~213--264, 1711. + +\BibItem[\bysame] Doctrine of Chances, or A Method of Calculating the Probabilities of +Events in Play. 1st~ed. 4to. Pp.~xiv~+~175. 1718. 2nd~ed. Large +4to. Pp.~xiv~+~258. 1738. 3rd~ed. Large 4to. Pp.~xii~+~348. 1756. + +\BibItem[\bysame] La dottrina d.~azzardi applic.\ ai problemi d.~probabilità di vita, di pensi, +ecc., trad.\ da R.~Gaeta e G.~Fontana. Milan, 1776. + +\BibItem[\bysame] Miscellanea analytica de seriebus et quadraturis. 4to. Pp.~250~+~22. +London, 1730. + +\BibItem[De Morgan, A.] Essay on Probabilities and their Application to life Contingencies +and Insurance Offices. 1838. + +\BibItem[\bysame] Formal Logic: or the Calculus of Inference Necessary and Probable. +1847. + +\BibItem[\bysame] Theory of Probabilities. 4to. 1849. + +\Bibnote [From the Encyclopaedia Metropolitana.] + +\BibItem[\bysame] On the Structure of the Syllogism and on the Application of the Theory +of Probabilities to Questions of Argument and Authority. 4to. Camb.\ +Phil.\ Soc.\ pp.~393--405, 1847 (read Nov.~9, 1846). + +\BibItem[\bysame] On the Symbols of Logic, the Theory of the Syllogism, and in particular +of the Copula, and the Application of the Theory of Probabilities to some +Questions of Evidence. 4to. Camb.\ Phil.\ Soc.\ vol.~ix.\ pp.~116--125, 1851. + +\BibItem[De Witt, John.] De vardye van de lif-renten na proportie van de los-renten. +La\ Haye, 1671. + +\BibItem[\bysame] English transl.: Contributions to the History of Insurance, by Frederick +Hendriks in the Assurance Magazine, vol.~2, p.~231 (1852). + +\BibItem[\bysame] [For an abstract see N.~Struyck, Inleiding tot het algemeine geography, +etc. 4to. Amsterdam, 1740. P.~345.] + +\BibItem[Dedekind, R.] Bemerkungen zu einer Aufgabe der Wahrscheinlichkeitsrechnung. +Pp.~268--271. Crelle~J.\ vol.~1., 1855. + +\BibItem[Degen, C.~F.] Tabularum ad faciliorem probabilitatis computationem utilem +Enneas. Kiobenhavn, 1824. + +\BibItem[Diderot.] Art.\ ``Probabilité'' in the Encyclopédie. + +\BibItem[Didion, J.] Calcul des probabilités appliqué au tir des projectiles. 8vo. 1858. + +\BibItem[Dodson, James.] Mathematical Repository. 3~vols. 1753. Vol.~ii.\ pp.~82--136. + +\BibItem[Donkin, W.~F.] ``Sur la théorie de la combinaison des observations.'' Liouv.\ J.~(1), +vol.~xv. 1850. + +\BibItem[\bysame] ``On Certain Questions relating to the Theory of Probabilities.'' Phil.\ +Mag., May 1851. + +\BibItem[Dormoy, E.] Théorie mathématique des assurances sur la vie. 2~vols. Paris, +1878. + +\BibItem[Drobisch, A.] ``Über die nach der Wahrscheinlichkeitsrechnung zu erwartende +Dauer der Ehen.'' Berichte über die Verhandlungen der Königl.\ Sächsischen +Gesellschaft der Wissenschaften mathem.-physik. 1880. + +\BibItem[Drobisch, M.~W.] Neue Darstellung der Logik. 2nd~ed.\ Leipzig, 1851. 3rd~ed.\ +1863. 4th~ed.\ 1875. 5th~ed.\ 1887. + +\Bibnote [Probability, pp.~181--209, §§\;145--157 (references to 4th~ed).] + +\Bibsect{E} + +\BibItem[Edgeworth, F.~Y.] ``Calculus of Probability applied to Psychical Research.'' +Proceedings of Soc.\ for Psych.\ Res. Parts viii.~and~x. + +\BibItem[\bysame] `` On the Method of ascertaining a Change in the Value of Gold.'' Roy.\ +Stat.\ Soc.\ J.~xlvi.\ pp.~714--718. 1883. + +\BibItem[\bysame] ``Law of Error.'' Phil.\ Mag.~(5) vol.~xvi.\ pp.~300--309, 1883. + +\BibItem[\bysame] ``Method of least Squares.'' Phil.\ Mag.~(5) vol.~xvi.\ pp.~360--375, 1883. + +\BibItem[\bysame] ``Physical Basis of Probability.'' Phil.\ Mag.\ vol.~xvi.\ pp.~433--435, +1883. +%% -----File: 453.png---Folio 442------- + +%\BibItem[Edgeworth, F.\ Y.]---\emph{continued}. + +\BibItem[\bysame] `` Chance and Law.'' Hermathene (Dublin), 1884. + +\BibItem[\bysame] ``On the Reduction of Observations.'' Phil.\ Mag.~(5) vol.~xvii.\ pp.~135--141, +1884. + +\BibItem[\bysame] ``Philosophy of Chance.'' Mind, April 1884. + +\BibItem[\bysame] ``\textit{A~priori} Probabilities.'' Phil.\ Mag.~(5) vol.~xviii.\ pp.~209--210, 1884. + +\BibItem[\bysame] ``On Methods of Statistics.'' Stat.\ Journ.\ Jub.~vol.\ pp.~181--217, 1885. + +\Bibnote [Criticised by Bortkiewicz and defended by Edgeworth, Jahrb.\ f.~nat.\ +Ök.\ u.~Stat.~(3), vol.~10, pp.~343--347; vol.~11, pp.~274--277, 701--705, +1896.] + +\BibItem[\bysame] `` Observations and Statistics.'' Phil.\ Soc. 1885. + +\BibItem[\bysame] ``Law of Error and Elimination of Chance.'' Phil.\ Mag., 1886, vol.~xxi.\ pp.~308--324. + +\BibItem[\bysame] ``Problems in Probabilities.'' Phil.\ Mag., 1886, vol.~xxii.\ pp.~371--384, +and 1890, vol.~xxx.\ pp.~171--188. + +\BibItem[\bysame] Metretike: or the Method of Measuring Probability and Utility. 8vo. +1887. + +\BibItem[\bysame] ``On Discordant Observations.'' Phil.\ Mag.~(5) vol.~xxiii.\ pp.~\DPtypo{}{364--375,} 1887. %[** TN: Page numbers found in google excert of http://www.jstor.org/pss/2335027 on April 24, 2010] + +\BibItem[\bysame] ``The Empirical Proof of the Law of Error.'' Phil.\ Mag.~(5) vol.~xxiv.\ +pp.~330--342, 1887. + +\BibItem[\bysame] ``The Element of Chance in Competitive Examinations.'' Roy.\ Stat.\ +Soc.\ Journ.~liii.\ pp.~460--475 and 644--663, 1890. + +\BibItem[\bysame] ``The Law of Error and Correlated Averages.'' Phil.\ Mag.~(5) vol.~xxxv.\ +pp.~63--64, 1893. + +\BibItem[\bysame] ``Statistical Correlation between Social Phenomena.'' Roy.\ Stat.\ Soc.\ +Journ.~lvi.\ pp.~670--675, 1893. + +\BibItem[\bysame] ``The Asymmetrical Probability-Curve.'' 1896. Phil.\ Mag.\ vol~xli.\ +pp.~90--99. + +\BibItem[\bysame] ``Miscellaneous Applications of the Calculus of Probabilities.'' Roy.\ +Stat.\ Soc. Journ.~lx.\ pp.~681--698, 1897; lxi.\ pp.~119--131 and 534--544, +1898. + +\BibItem[\bysame] ``Law of Error.'' Phil.\ Trans.\ vol.~xx. + +\BibItem[\bysame] ``The Generalised Law of Error.'' Stat.\ Journ.\ vol.~lxix., 1906. + +\BibItem[\bysame] ``On the Probable Errors of Frequency-Constants.'' Stat.\ Journ.\ vol.~lxxi.\ +pp.~381--397, 499--512, 651--678, 1908; and vol.~lxxii.\ pp.~81--90, 1909. + +\BibItem[\bysame] ``On the Application of the Calculus of Probabilities to Statistics.'' +Bulletin xviii.\ of the International Statistical Institute, Paris, 1910, 32~pp. + +\BibItem[\bysame] ``Applications of Probabilities to Economics.'' Economic Journal, vol.~xx.\ +pp.~284--304, 441--465, 1910. + +\BibItem[\bysame] ``Probability.'' Encyclopaedia Britannica, 11th~ed.\ vol.~22, pp.~376--403, +1911. + +\BibItem[\bysame] ``On the Application of Probabilities to the Movement of Gas-Molecules.'' +Phil.\ Mag., vol.~xl., pp.~249--272, 1920. + +\BibItem[\bysame] Molecular Statistics.'' Roy.\ Stat.\ Soc.\ Journ., vol.~lxxxiv.\ pp.~71--89, +1921. + +\BibItem[Eggenberger, J.] ``Beiträge zur Darstellung des bernoullischen Theorems.'' +Berner Mitth.\ vol.~50 (1894); and Zeitschr.\ f.~Math.\ u.~Ph.~45 (1900), p.~43. + +\BibItem[Elderton, W.~P.] Frequency-Curves and Correlation. 8vo. London, 1907. +xii~+~172. + +\Bibnote [Contains a useful list of papers on Correlation, p.~163.] + +\BibItem[Ellis, R.~L.] ``On the Foundations of the Theory of Probability.'' 4to. +Camb.\ Phil.\ Soc.\ vol.~viii., 1843. + +\Bibnote [Reprinted in ``Mathematical and other Writings,'' 1863.] + +\BibItem[\bysame] +``On a Question in the Theory of Probabilities.'' Camb.\ Math.\ Journ.\ +No.~xxi.\ vol~iv., 1844. + +\Bibnote [Reprinted in ``Mathematical and other Writings,'' 1863.] +%% -----File: 454.png---Folio 443------- + +%\BibItem[ Ellis, R.~L.]---\emph{continued}. + +\BibItem[\bysame] ``On the Method of Least Squares.'' Trans.\ Camb.\ Phil.\ Soc.\ vol.~viii., +1844. + +\Bibnote [Reprinted in ``Mathematical and other Writings,'' 1863.] + +\BibItem[\bysame] ``Remarks on an alleged Proof of the `Method of Least Squares.'\,'' +Phil.\ Mag.~(3) vol.~xxxvii., 1850. + +\Bibnote [Reprinted in ``Mathematical and other Writings,'' 1863.] + +\BibItem[\bysame] ``Remarks on the Fundamental Principle of the Theory of Probabilities.'' +Trans.\ Camb.\ Phil.\ Soc.\ vol.~ix., 1854. + +\Bibnote [Reprinted in ``Mathematical and other Writings,'' 1863.] + +\BibItem[Elsas, A.] ``Kritische Betrachtungen über die Wahrscheinlichkeitsrechnung.'' +Philos.\ Monatssch.\ vol.~xxv.\ pp.~557--584, 1889. + +\BibItem[Emerson, William.] Miscellanies, 1776. [See espec.\ pp.~1--48.] + +\BibItem[Encke, J.~F.] Methode der kleinsten Quadrate. Fehler theoret.\ Untersuchungen. +Berlin, 1888. + +\BibItem[Engel, G.] ``Über Möglichkeit und Wirklichkeit.'' Philos.\ Monatssch.\ vol.~v.\ +pp.~241--271, 1875. + +\BibItem[Ermakoff, W.~P.] Wahrscheinlichkeitslehre (in Russian). + +\BibItem[Euler.] ``Calcul de la probabilité dans le jeu de rencontre.'' Hist.\ Ac.\ Berl.\ +(1751), pp.~255--270, 1753. + +\BibItem[\bysame] ``Sur l'avantage du banquier au jeu de pharaon.'' Hist.\ Ac.\ Berl.\ +(1764), pp.~144--164, 1766. + +\BibItem[\bysame] ``Sur la probabilité des séquences dans la loterie génoise.'' Hist.\ Ac.\ +Berl.\ (1765), pp.~191--230, 1767. + +\BibItem[\bysame] +``Solution d'une question très difficile dans le calcul des probabilités.'' +Hist.\ Ac.\ Berl.\ (1769), pp.~285--302, 1771. + +\BibItem[\bysame] ``Solutio quarundam quaestionum difficiliorum in calculo probabilium.'' +Opuscula analytica, vol.~ii.\ pp.~331--346, 1785. + +\BibItem[\bysame] ``Solutio quaestionis ad calculum probabilitatis pertinentis: Quantum +duo conjuges persolvere debeant, ut suis haeredibus post utriusque mortem +certa argenti summa persolvatur.'' Opuscula analytica, vol.~ii, pp.~315--330, +1785. + +\BibItem[\bysame] ``Wahrscheinlichkeitsrechnung.'' Opera omnia, ser.~1,~A, vol.~iv.\ +Leipzig. + +\Bibsect{F} + +\BibItem[Fahlbeck.] ``La Régularité dans les choses humaines, ou les types statistiques +et leurs variations.'' Journ.\ Soc.\ Stat.\ de Paris, pp.~188--200, 1900. + +\BibItem[Fechner, G.~Th.] Kollektivmasslehre. (Edited by G.~F. Lipps.) 1897. + +\BibItem[Fick, A.] Philosophischer Versuch über die Wahrscheinlichkeiten. Pp.~46. +Würzburg, 1883. + +\BibItem[Fisher, A.] The Mathematical Theory of Probabilities. Translated from the +Danish. Pp.~xx~+~171 New York, 1915. + +\BibItem[Forbes, J.~D.] ``On the alleged Evidence for a Physical Connexion between +Stars forming Binary or Multiple Groups, deduced from the Doctrine of +Chances.'' Phil.\ Mag., Dec.\ 1850. (See also Phil.\ Mag., Aug.\ 1849.) + +\BibItem[Forncey.] The Logic of Probabilities. Transl.\ from the French. 8vo. +London, n.d.~(?~1760.) + +\BibItem[Förster, W.] Wahrheit und Wahrscheinlichkeit. Pp.~40. Berlin, 1875. + +\BibItem[Fries, J.~J.] Versuch einer Kritik der Principien der Wahrscheinlichkeitsrechnung. +Braunschweig, 1842. + +\BibItem[Frömmichen.] Über Lehre der Wahrscheinlichkeit. 4to. Braunschweig, +1773. + +\BibItem[Fuss, N.] ``Recherches sur un problème du calcul des probabilités.'' Act.\ Ac.\ +Petr.\ (1779), pars posterior, pp.~81--92, 1783. + +\BibItem[\bysame] ``Supplément au mémoire sur un problème du calcul des probabilités.'' +Act.\ Ac.\ Petr.\ (1780), pars posterior, pp.~91--96, 1784. +%% -----File: 455.png---Folio 444------- + +\Bibsect{G} + +\BibItem[Galileo, G.] ``Considerazioni sopra il giuoci dei dadi.'' Opere, vol.~iii.\ pp.~119--121, +1718. Also, Opere, vol.~xiv.\ pp.~293--296. Firenze, 1855. + +\BibItem[\bysame] ``Lettere intorno le stima di un cavallo.'' Opere, vol.~xiv.\ pp.~231--284. +Firenze, 1855. + +\BibItem[Galloway, T.] A Treatise on Probability. 8vo. Edinburgh, 1839. (From +the 7th~edition of the Encyclopaedia Britannica.) + +\BibItem[Galton, F.] ``Correlations and their Measurement.'' Proc.\ Roy.\ Soc., vol.~xlv.\ +pp.~136--145. + +\BibItem[\bysame] Probability, the Foundation of Eugenics. Herbert Spencer Lecture, +1907. (Reprinted---Essays in Eugenics. 8vo. ii~+~109~pp. London, 1909.) + +\BibItem[Gardon, C.] Antipathies des 90~nombres, probabilités, et observations comparatives, +sur les loteries de France et de Bruxelles. 8vo. Paris, 1801. + +\BibItem[\bysame] Traité élémentaire des probabilités, etc. Paris, 1805. + +\BibItem[\bysame] L'investigateur des chances \ldots\ pour obtenir souvent des succès aux +loteries impériales de France. Paris. + +\BibItem[Garve, C.] De nonnullis quae pertinent ad logicam probabilium. 4to. Halae, +1766. + +\BibItem[Gataker, T.] On the Nature and Use of Lots. 4to. 1619. + +\BibItem[Gauss, C. F.] Theoria motus corporum coelestium. 4to. Hamburg, 1809. + +\BibItem[\bysame] ``Theoria combinationis observationum erroribus minimis obnoxiae.'' +Comm.\ Soc.\ Göttingen, vol.~v.\ pp.~33--90. 1823. + +\BibItem[\bysame] Méthode des moindres carrés. Traduit en français par J.~Bertrand. +8vo. 1855. + +\Bibnote [A translation of part of the above.] + +\BibItem[\bysame] Wahrscheinlichkeitsrechnung. Werke, vol.~iv.\ pp.~1--53. 4to. Göttingen, +1873. + +\BibItem[Geisenheimer, L.] Über Wahrscheinlichkeitsrechnung. 8vo. Berlin, 1880. + +\BibItem[Gilman, B.~I.] ``Operations in Relative Number with Applications to Theory of +Probability,'' Johns Hopkins Studies in Logic, 1883. + +\BibItem[Gladstone, W.~E.] ``Probability as a Guide to Conduct.'' Nineteenth Cent.\ +vol.~v.\ pp.~908--934, 1879; and in ``Gleanings,'' vol.~ii.\ pp.~153--200. + +\BibItem[Glaisher, J.~W.~L.] ``On the Rejection of Discordant Observations.'' Monthly +Notices R.~Astr.~S. vol.~xxiii., 1873. + +\BibItem[\bysame] ``On the Law of Facility of Errors of Observation, and on the Method +of Least Squares.'' Mem.\ R.~Astr.~S. vol.~xxxix., 1872. + +\BibItem[Goldschmidt, L.] ``Wahrscheinlichkeit und Versicherung.'' Bull.\ du Comité +permanent des Congrès Internationaux d'Actuaires, 1897. + +\BibItem[\bysame] Die Wahrscheinlichkeitsrechnung: Versuch einer Kritik. Pp.~279. +Hamb., 1897. + +\Bibnote [Cf.\ Zeitschr.\ f.~Philos.\ u.~phil.~Kr., cxiv., pp.~116--119.] + +\BibItem[Gonzalez, T.] Fundamentum theologiae moralis, id est tractatus theologicus +de recto usu opinionum probabilium. 4to. Dillingen, 1689. Naples, 1694. + +\Bibnote [An abridgement entitled: Synopsis tract.\ theol.\ de recto usu opin.\ +prob., concinnata a theologo quodam Soc.\ Jesu: cui accessit logistica +probabilitatum. 3rd~ed. 8vo. Venice, 1696. See Migne, Theol.\ Cur.\ +Compl., vol.~xi., p.~1397.] + +\BibItem[Gourand, Ch.] Histoire du calcul des probabilités depuis ses origines jusqu'à +nos jours. 8vo. Paris, 1848, 148~pp. + +\Bibnote [His history seems to be a portion of a very extensive essay in 3~folio +volumes containing 1929~pp., written when he was very young, in competition +for a prize proposed by the Fr.\ Acad.\ on a subject entitled ``Théorie +de la certitude''; see Séances et Travaux de l'Académie des Sciences +morales et politiques, vol.~x.\ pp.~372,~382, vol.~xi.\ p.~137. See \textsc{Todhunter}.] + +\BibItem[Gravesande, W.~J.~'S.] Introductio ad philosophiam, metaphysicam et logicam +continens. 8vo. Venetiis, 1737. +%% -----File: 456.png---Folio 445------- + +% \BibItem[Gravesande, W.~J.~'S]---\emph{continued}. + +\BibItem[\bysame] {\OE}uvres philosophiques et mathématiques. 4to. Amsterdam, 1774, +2~vols. 4to. ii.\ pp.~82--93, 221--248. + +\BibItem[Grellings, K.] ``Die philosophischen Grundlagen der Wahrscheinlichkeitsrechnung.'' +Abhandlungen der Friesschen Schule, N.~F., vol.~iii., 1910. + +\BibItem[Grimsehl, E.] ``Untersuchungen zur Wahrscheinlichkeitslehre. (Mit besonderer +Beziehung auf Marbes Schrift (\textit{q.v.}).)'' ``Zeitschrift für Philosophie und +philosophische Kritik. Band~118, pp.~154--167. Leipzig, 1901. + +\Bibnote [See also \textsc{Brömse, Marbe}, and \textsc{v.~Bortkiewicz}.] + +\BibItem[Grolous] ``Sur une question de probabilité appliquée à la théorie des +nombres.'' Journal de l'Institut, 1872. + +\BibItem[Groschius, J.~A.] Logica probabilium in artium practicarum subsidium +adornata. Sm.\ 8vo. Halae, 1764. Pp.~xvi~+~352. + +\BibItem[Grünbaum, H.] Isolierte und reine Gruppen und die Marbesche Zahl~``$p$.'' +Würzburg, 1904. + +\BibItem[Guibert, A.] ``Solution d'une question relative à la probabilité des jugements +rendus à une majorité quelconque.'' Liouv.\ J.~(1) vol.~iii., 1838. + +\Bibsect{H} + +\BibItem[Hack.] Wahrscheinlichkeitsrechnung. Leipzig, 1911. + +\BibItem[Hagen, G.~F.] Meditationes philosophicae de methodo mathematico. Norimbergae, +1734. + +\BibItem[\bysame] Fortsetzung einiger aus der Mathematic abgenommenen Regeln, nach +welchen sich der menschliche Verstand bei Erfindung der Wahrheiten +richtet. Halle, 1737. + +\BibItem[Hagen, G.] Grundzüge der Wahrscheinlichkeitsrechnung. Berlin, 1837. +(2nd~ed.\ 1867, 3rd~ed.\ 1882.) + +\BibItem[\bysame] Der constante wahrscheinliche Fehler: Nachtrag zur 3ten Auflage der +Grundzüge der Wahrscheinlichkeitsrechnung. 38~pp. Berlin, 1884. + +\BibItem[Halley.] See \textsc{Craig}. + +\BibItem[Hans, John.] See \textsc{J.~Arbuthnot}. + +\BibItem[Hansdorff, F.] ``Beiträge zur Wahrscheinlichkeitsrechnung.'' Leipz.\ Ber., +vol.~53, pp.~152--178, 1901. + +\BibItem[\bysame] ``Das Risiko bei Zufallsspielen.'' Leipz.\ Ber., vol~49, pp.~497--548, 1897. + +\BibItem[Hansen, P.~A.] ``Über die Anwendung der Wahrscheinlichtkeitsrechnung auf +geodätische Vermessungen.'' Astr.~N. vol.~ix.\ 1831. + +\BibItem[Hartmann, E.~von.]\DPnote{** TN: "von" not small-capped in orig} ``Die Grundlage der Wahrscheinlichkeitsurteils.'' +Vierteljahrsschr.\ f.~wiss.\ Phil.\ u.~Soz., vol.~xxviii., 1904. + +\BibItem[Hauteserve, Gauthier~d'.] Traité élémentaire sur les probabilités. Paris, +1834. + +\BibItem[\bysame] Application de l'algèbre élémentaire au calcul des probabilités. Paris, +1840. + +\BibItem[Hélie] Mémoire sur la probabilité du tir. 8vo. 1854. + +\BibItem[Helm.] ``Eine Anwendung der Theorie des Tauschwerthes auf die Wahrscheinlichtkeitsrechnung.'' +Zeitschr.\ f.~Math.\ u.~Phys., vol.~38, pp.~374--376. +Leipzig, 1893. + +\BibItem[\bysame] ``Die Wahrscheinlichkeitslehre als Theorie der Kollektiv-begriffe.'' +Annalen der Naturphilosophie, vol.~1. + +\BibItem[Henry, Charles.] La Loi des petits nombres. Recherches sur le sens de +l'écart probable dans les chances simples à la roulette, au trente-et-quarante +etc., en général dans les phénomènes dépendant de causes purement +accidentales. 72~pp. 8vo. Paris, 1908. + +\BibItem[Herschel, W.] ``On the Theory of Probabilities.'' Journal of Actuaries, +1869. + +\BibItem[\bysame] ``Quetelet on Probabilities.'' Edin.\ Rev., 1850. + +\Bibnote [Reprinted in Quetelet's Physique Sociale, vol.~i.\ pp.~1--89, 1869.] + +\BibItem[\bysame] ``On an Application of the Rule of Succession.'' Edin.\ Rev., 1850. +%% -----File: 457.png---Folio 446------- + +\BibItem[Herz, N.] Wahrscheinlichkeits- und Ausgleichungsrechnung. Pp.~iv~+~381. +Leipzig, 1900. + +\BibItem[Hibben, J.~G.] Inductive Logic. London, 1896. + +\Bibnote [See chaps.\ xv.,~xvi.] + +\BibItem[Hobhouse, L.~T.] Theory of Knowledge. + +\Bibnote [See Part~II., chaps.\ x.,~xi.] + +\BibItem[Hoyle.] An Essay towards making the Doctrine of Chances easy to those who +understand vulgar Arithmetic only. Pp.~viii~+~73, 1754, 1758, 1764. + +\BibItem[Huberdt, A.] Die Principien der Wahrscheinlichkeitsrechnung. 4to. Berlin, +1845. + +\BibItem[Hume, David.] Treatise on Human Nature. 1st~ed.\ 1739. + +\Bibnote [See especially Part~III.] + +\BibItem[\bysame] An Enquiry concerning Human Understanding. + +\Bibnote [See specially Section~vi.] + +\BibItem[\bysame] Essays, Part~I.,~XIV\@. On the Rise and Progress of the Arts and Sciences, + pp.~115,~116, 1742. + +\BibItem[Huygens, Ch.] ``De ratiociniis in ludo aleae.'' Schooten's Exercitat.\ math.\ +pp.~519--534. 4to. Lugd.\ Bat., 1657. + +\Bibnote [Written by Huygens in Dutch and translated into Latin by Schooten.] + +\BibItem[\bysame] Engl.\ transl.\ by W.~Browne. Sm.\ 8vo, pp.~24. London, 1714. + +\Bibnote [See also \textsc{Jac.\ Bernoulli, Arbuthnot} (Engl.\ Transl.), and \textsc{Vastel} (Fr.\ Transl.).] + +\Bibsect{J} + +\BibItem[Jahn, G.~A.] Die Wahrscheinlichkeitsrechnung und ihre Anwendung auf das +wissenschaftliche und praktische Leben. Leipzig, 1839. + +\BibItem[Janet.] La Morale. Paris, 1874. [See Bk.~iii.\ chap.~3 for Probabilism.] +Engl.\ transl. The Theory of Morals. New York, 1883, pp.~292--308. + +\BibItem[Jevons, W.~S.] Principles of Science, 2~vols. 1874. + +\BibItem[Jordan, C.] ``De quelques formules de probabilité (sur les causes).'' Comptes +rendus, 1867. + +\BibItem[Jourdain, P.~E.~B.] ``Causality, Induction, and Probability.'' Mind, vol.~xxviii.\ +pp.~162--179, 1919. + +\Bibsect{K} + +\BibItem[Kahle, L.~M.] Elementa logicae probabilium methodo mathematica, in usu +scientiarum et vitae adornata. Pp.~10~+~xxii~+~245. Sm.\ 8vo. Halae, +1735. + +\BibItem[Kanner, M.] ``Allgemeine Probleme der Wahrscheinlichkeitsrechnung und ihre +Anwendung auf Fragen der Statistik.'' Journ.\ des Collegiums für Lebens-Versicherungs-Wissenschaft. +Berlin, 1870. + +\BibItem[Kaufmann, Al.] Theorie und Methoden der Statistik. [Translated from the +Russian.] Pp.~xii~+~540. Tübingen, 1913. + +\BibItem[Kepler, J.] ``De stella nova in pede serpentarii.'' 1606. See J.~Kepler's +Astr.\ Op.\ Omn.\ edidit Frisch, ii.\ pp.~714--716. + +\BibItem[Kirchmann, J.~H.~von.] Über die Wahrscheinlichkeit. Pp.~60. Leipzig, +1878. + +\BibItem[Knapp.] ``Quetelet als Theoretiker.'' Jahrb.\ f.~nat. Ök.\ und~Stat. (New Series), +vol.~xviii. + +\BibItem[Kozak, Josef.] Grundlehren der Wahrscheinlichkeitsrechnung als Vorstufe für +das Studium der Fehlerausgleichung, Schiesstheorie, und Statistik. Vienna, 1912. + +\BibItem[\bysame] Theorie des Schiesswesens auf Grundlage der Wahrscheinlichkeitsrechnung +und Fehlertheorie. Vienna, 1908. + +\BibItem[Kries, J.~von.] Die Principien der Wahrscheinlichkeitsrechnung. Eine +logische Untersuchung. Pp.~298. 8vo. Freiburg, 1886. + +\Bibnote [See also \textsc{Lexis, Meinong} and \textsc{Sigwart}.] +%% -----File: 458.png---Folio 447------- + +\Bibsect{L} + +\BibItem[Lacroix, S.~F.] Traite élémentaire du calcul des probabilités. Pp.~viii~+~299. +8vo. Paris, 1816. + +\Bibnote [2nde~éd., revue et augmentée, 1822; 4th~éd.\ 1864.] + +\Bibnote [Translated into German: E.~S. Unger, Erfurt, 1818.] + +\BibItem[Lagrange.] ``Mémoire sur l'utilité de la méthode de prendre le milieu entre +les résultats de plusieurs observations, dans lequel on examine les avantages +de cette méthode par le calcul des probabilités, et où l'on résout +différents problèmes relatifs à cette matière.'' Misc.\ Taurinensia, vol.~5, +pp.~167--232, 1770--1773. {\OE}uvres complètes, vol.~2, Paris, 1867--1877. + +\BibItem[\bysame] ``Recherches sur les suites recurrentes \ldots\ et sur l'usage de ces équations +dans la théorie des hasards.'' Nouv.\ Mém.\ Ac.\ Berl.\ (1775), pp.~183--272, +1777. {\OE}uvres complètes, vol.~4. Paris, 1867--1877. + +\BibItem[Laisant, C.~A.] Algèbre. Théorie des nombres, probabilités, géométrie de +situation. Paris, 1895. + +\BibItem[Lambert, J.~H.] ``Examen d'une espèce de superstition ramenée au calcul +des probabilités.'' Nouv.\ Mém.\ Ac.\ Berl., 1771, pp.~411--420. + +\BibItem[Lämmel, R.] Untersuchungen über die Ermittelung von Wahrscheinlichkeiten. +(Inaug.-Dissert.) Pp.~80. Zürich, 1904. + +\BibItem[Lampe, E.] ``Über eine Aufgabe aus der Wahrscheinlichkeitsrechnung.'' Grun.\ +Arch., vol.~70, 1884. + +\BibItem[Lange, F.~A.] Logische Studien. + +\BibItem[Laplace.] Essai philosophique sur les probabilités. (Printed as introduction +to Théorie analytique des probabilités, from 2nd~ed.\ of the latter onwards.) +4to. Paris, 1814. + +\BibItem[\bysame] German translation by Tönnies. Heidelberg, 1819. German translation +by N.~Schwaiger. Leipzig, 1886. + +\BibItem[\bysame] A Philosophical Essay on Probabilities: transl.\ from the 6th~French +ed.\ by E.~W. Truscott and F.~L. Emory. 8vo. New York, 1902, 196~pp. + +%[** TN: Next entry contains full-stop spaces after "ed.", not consistent with other entries; fixed.] +\BibItem[\bysame] Théorie analytique des probabilités. +1st~ed. 4to. Paris, 1812, 1st~and 2nd~Suppl., 1812--1820. 2nd~ed. +4to. cxi~+~506~+~2, Paris, 1814. 3rd~Suppl.\ 1820. 3rd~ed. Paris, 1820. +4th~Suppl.\ after 1820. {\OE}uvres complètes, vol.~7, pp.~cxcv~+~691, Paris, +1847. {\OE}uvres complètes, vol.~7, pp.~832, Paris, 1886. + +\BibItem[\bysame] ``Recherches sur l'intégration des équations différentielles aux différences +finies, et sur leur usage dans la théorie des hasards.'' Mém.\ prés.\ +à~l'Acad.\ des~Sc., pp.~113--163, 1773. + +\BibItem[\bysame] ``Mémoire sur les suites récurro-récurrentes et sur leurs usages dans la +théorie des hasards,'' Mém.\ prés.\ à~l'Acad.\ des~Sc., vol.~6, pp.~353--371, +1774. + +\BibItem[\bysame] ``Mémoire sur la probabilité des causes par les événements.'' Mém.\ +prés.\ à~l'Acad.\ des~Sc., vol.~6, pp.~621--656, 1774. + +\BibItem[\bysame] ``Mémoire sur les probabilités.'' Mém.\ prés.\ à~l'Acad.\ des~Sc., pp.~227--332, +1780. + +\BibItem[\bysame] ``Mémoire sur les approximations des formules qui sont fonctions de +très grands nombres, et sur leurs applications aux probabilités.'' Mém.\ +de l'Inst., pp.~353--415, 539--565, 1810. + +\BibItem[\bysame] ``Mémoire sur les intégrales définies, et leur application aux probabilités.'' +Mém.\ de l'Inst., pp.~279--347, 1810. + +\Bibnote [The above memoirs are reprinted in {\OE}uvres complètes, vols.\ 8,~9, and~12, +Paris, 1891--1898.] + +\BibItem[\bysame] +Sur l'application du calcul des probabilités appliqué à la philosophie +naturelle. Conn.\ des temps. {\OE}uvres complètes, vol.~13. Paris, 1904. + +\BibItem[\bysame] ``Applications du calcul des probabilités aux observations et spécialement +aux opérations du nivellement.'' Annales de Chimie. {\OE}uvres +complètes, vol.~14, Paris, 1913. + +\BibItem[La Placette, J.] Traité des jeux de hasard. 18mo. 1714. +%% -----File: 459.png---Folio 448------- + +\BibItem[Laurent, H.] Traité du calcul des probabilités. Paris, 1873. + +\BibItem[\bysame] [A la fin une liste des principaux ouvrages (320) ou mémoires publiés +sur le calcul des probabilités.] + +\BibItem[\bysame] ``Application du calcul des probabilités à la vérification des répartitions.'' +Journ.\ des Actuaires français, vol.~i. + +\BibItem[\bysame] ``Sur le théorème de J.~Bernoulli.'' Journ.\ des Actuaires français, vol.~i. + +\BibItem[Lechalas, G.] ``Le Hasard.'' Rev. Néo-scolastique, 1903. + +\BibItem[\bysame] ``A propos de Cournot: hasard et déterminisme.'' Rev.\ de~Mét.\ et +de~Mor., 1906. + +\BibItem[Legendre.] ``Méthode des moindres carrés.'' Mém.\ de l'Inst., 1810, 1811. + +\BibItem[\bysame] Nouvelles méthodes pour la détermination des orbites des comètes. +Paris. 1805--6. + +\BibItem[Lehr.] ``Zur Frage der Wahrscheinlichkeit von weiblichen Geburten und von +Totgeburten.'' Zeitschrift f.~des ges.\ Staatsw., vol.~45, p.~172, and p.~524, 1889. + +\BibItem[Leibniz.] Nouveaux Essais. Liv.~ii.\ chap.~xxi.; liv.~iv.\ chaps.~ii.\ §\;14, xv.,~xvi., +xviii.,~xx. + +\BibItem[\bysame] Opera omnia, ed.\ Dutens, v.~17, 22,~28,~29, 203,~206; vi.\ pt.~i., 271, +304, 36,~217; iv.\ pt.~iii.~264. + +\BibItem[\bysame] Correspondence between Leibnitz and Jac. Bernoulli. L.'s~Gesammelte +Werke (ed.\ Pertz and Gerhardt), vol.~3, pp.~71--97, \textit{passim}. Halle, 1855. + +\Bibnote [These letters were written between 1703 and~1705.] + +\BibItem[\bysame] See also s.v.~\textsc{Couturat}. + +\BibItem[Lemoine, E.] ``Solution d'un problème sur les probabilités.'' Bulletin de la +Soc.\ math.\ de~Paris, 1873. + +\BibItem[\bysame] Questions de probabilités et valeurs relatives des pièces du jeu des +échecs. 8vo. 1880. + +\BibItem[\bysame] ``Quelques questions de probabilités résolues géométriquement.'' Bull.\ +de~la Soc.\ math.\ de~France, 1883. + +\BibItem[\bysame] ``Divers problèmes de probabilité.'' Ass.\ française pour l'Avancement +des Sciences, 1885. + +\BibItem[Lexis, W.] Abhandlungen zur Theorie der Bevölkerungs- und Moral-statistik. +Pp.~253. Jena, 1903. + +\BibItem[\bysame] Zur Theorie der Massenerscheinungen in der menschlichen Gesellschaft. +Pp.~95. Freiburg, 1877. + +\BibItem[\bysame] +``Über die Wahrscheinlichkeitsrechnung und deren Anwendung auf die +Statistik.\DPtypo{}{''} Jahrb.\ f.~nat.\ Ök.\ u.~Stat.~(2), vol.~13, pp.~433--450, 1886. + +\Bibnote [Contains a review of v.~Kries's ``Principien.''] + +\BibItem[\bysame] ``Über die Theorie der Stabilität statistischer Reihen.'' Jahrb.\ f.~nat.\ Ök.\ u.~Stat.~(1), +vol.~32, p.~604, 1879. + +\Bibnote [Reprinted in Abhandlungen.] + +\BibItem[\bysame] ``Das Geschlechtsverhältnis der Geborenen und die Wahrscheinlichkeitsrechnung.'' +Jahrb.\ f.~nat.\ Ök.\ u.~Stat.~(1), vol.~27, p.~209, 1876. + +\Bibnote [Reprinted in Abhandlungen.] + +\BibItem[\bysame] Einleitung in die Theorie der Bevölkerungsstatistik. Strassburg, 1875. + +\BibItem[Liagre, J.~B.~J.] Calcul des probabilités et théorie des erreurs avec des applications +aux sciences d'observation en général et à la géodésie en particulier. +416~pp. Brussels, 1852. 2nd~ed. 8vo. 1879. + +\BibItem[\bysame] ``Sur la probabilité d'une cause d'erreur régulière, etc.'' Bull.\ de l'Acad.\ +de Belgique, 1855. + +\BibItem[Liapounoff, A.] ``Sur une proposition de la théorie des probabilités.'' Bull.\ +de l'Acad.\ des~Sc.\ de~Saint-Pét., v.~série, vol.~xiii. + +\BibItem[\bysame] ``Nouvelle Forme du théorème sur la limite de probabilité.'' Mém.\ de +l'Acad.\ des~Sc.\ de~Saint-Pét., viii.~série, vol.~xiii. (1901). + +\BibItem[Liebermeister, C.] ``Über Wahrscheinlichkeitsrechnung in Anwendung auf +therapeutische Statistik.'' Sammlung klinische Vorträge, Nr.~110. 1877. +%% -----File: 460.png---Folio 449------- + +\BibItem[Lilienfeld, J.] ``Versuch einer strengen Fassung des Begriffs der mathematischen +Wahrscheinlichkeit.'' Zeitschr.\ f.~Philos.\ u.~phil.\ Kr., vol.~cxx, +pp.~58--65, 1902. + +\BibItem[Lipps, G.~F.] Kollectivmasslehre. 1897. + +\BibItem[Littrow, J.~J.] Die Wahrscheinlichkeitsrechnung in ihrer Anwendung auf das +wissenschaftliche und praktische Leben. 8vo. Wien, 1833. + +\BibItem[Lobatchewsky, N.~J.] ``Probabilité des résultats moyens tirés d'observations +répétées.'' Crelle~J.\ 1824. + +\BibItem[\bysame] Reprinted. Liouv.~J. vol.~24. 1842. + +\BibItem[Lottin, J.] Le Calcul des probabilités et les régularités statistiques. 32~pp. +8vo. Louvain, 1910. (Originally published in the Revue Néo-scolastique, +Feb.~1910.) + +\BibItem[\bysame] Quetelet, statisticien et sociologue. Louvain, 1912. Pp.~xxx~+~564. + +\Bibnote [Contains a very full discussion of Quetelet's Work on Probability.] + +\BibItem[Lotze, H.] Logik. 1st~ed.\ 1874, 2nd~ed.\ 1880. + +\BibItem[\bysame] Engl.\ transl.\ by B.~Bosanquet. Oxford, 1884. + +\Bibnote [See Bk.~ii.\ chap.~ix.: Determination of Single Facts and Calculus of +Chances.] + +\BibItem[Lourié, S.] Die Prinzipien der Wahrscheinlichkeitsrechnung. Tübingen, 1910. + +\BibItem[Lubbock, J.~W. \textrm{and} Drinkwater.] Treatise on Probability. [Library of +Useful Knowledge.] + +\Bibnote [Often wrongly ascribed to \textsc{De Morgan}.] + +\Bibsect{M} + +\BibItem[Macalister, Donald.] The Law of the Geometric Mean. Phil.\ Trans., 1879. + +\BibItem[McColl, Hugh.] Symbolic Logic. 1906. [Especially chaps.\ xvii.,~xviii.] + +\BibItem[\bysame] The Calculus of Equivalent Statements. Proc.\ Lond.\ Math.\ Soc. Six +papers. + +\Bibnote [See particularly 1877, vol.~ix.\ pp.~9--20; 1880, xi.~113--121, 4th~paper; +1897, xxviii.~p.~556, 6th~paper.] + +\BibItem[\bysame] ``Growth and Use of a Symbolical Language.'' Memoirs Manchester +Lit.\ Phil.\ Soc.\ series~iii.\ vol.~7, 1881. + +\BibItem[\bysame] ``Symbolical or Abbreviated Language with an Application to Mathematical +Probability.'' Math.\ Questions, vol.~28, pp.~20--23. + +\BibItem[\bysame] Various Papers in Mathematical Questions from the Journal of Education, +vols.\ 29,~33,~etc. + +\BibItem[\bysame] ``A Note on Prof.\ C.~S. Peirce's Probability Notation of 1867.'' Proc.\ +Lond.\ Math.\ Soc.\ vol~xii.\ p.~102. + +\BibItem[Macfarlane, Alexander.] Principles of the Algebra of Logic. + +\Bibnote [See especially chaps.\ ii.,~iii.,~v., xx.,~xxi, xxii.,~xxiii., and the examples.] + +\BibItem[\bysame] Various Papers in Mathematical Questions from the Journal of Education, +vols.\ 32,~36,~etc. + +\BibItem[MacMahon, P.~A.] ``On the Probability that the Successful Candidate at an +Election by Ballot may never at any time have fewer Votes than the one +who is unsuccessful, etc.'' Phil.\ Trans.~(A), vol.~209, pp.~153--175, 1909. + +\BibItem[Maldidier, Jules.] ``Le Hasard.'' Rev.\ Philos.~xliii., 1897, pp.~561--588. + +\BibItem[Malfatti, G.~F.] ``Esame critico di un problema di probabilità del Sig.\ Daniele +Bernoulli, e~soluzione d'un altro problema analogo al bernuiliano.'' Memorie +di Matematica e~Fisica della Società Italiana, vol.~I, pp.~768--824, 1782. + +\BibItem[Mallet.] ``Sur le calcul des probabilités.'' Act.\ Helv.\ Basileae, 1772, +vii.\ pp.~133--163. + +\BibItem[Mansions, P.] ``Sur la portée objective du calcul des probabilités.'' Bulletin +de l'Académie de Belgique (Classe des sciences), pp.~1235--1294, 1903. + +\BibItem[Marbe, Dr.\ Karl.] Naturphilosophische Untersuchungen zur Wahrscheinlichkeitslehre. +50~pp. Leipzig, 1899. + +\BibItem[\bysame] Die Gleichförmigkeit in der Welt\DPtypo{}{.} Munich, 1916. +%% -----File: 461.png---Folio 450------- + +\BibItem[Markoff, A.~A.] ``Über die Wahrscheinlichkeit \textit{à~posteriori}'' (in Russian) +Mitteilungen der Charkowv Math.\ Gesell. 2~Serie, vol.~iii. 1900. + +\BibItem[\bysame] ``Untersuchung eines wichtigen Falles abhängiger Proben'' (in Russian). +Abh.\ der K.\ Russ.\ Ak.\ d.~W., 1907. + +\BibItem[\bysame] ``Über einige Fälle der Theoreme vom Grenzwert der mathematischen +Hoffnungen und vom Grenzwert der Wahrscheinlichkeiten'' (in Russian). +Abh.\ der K.\ Russ.\ Ak.\ d.~W., 1907. + +\BibItem[\bysame] ``Erweiterung des Gesetzes der grossen Zahlen auf von einander +abhängige Grössen'' (in Russian). Mitt.\ d.~phys.-math.\ Ges.\ Kazan, 1907. + +\BibItem[\bysame] ``Über einige Fälle des Theorems vom Grenzwert der Wahrscheinlichkeiten'' +(in Russian). Abh.\ der K.\ Russ.\ Ak.\ d.~W., 1908. + +\BibItem[\bysame] ``Erweiterung gewisser Sätze der Wahrscheinlichkeitsrechnung auf eine +Summe verketteter Grössen'' (in Russian). Abh.\ der K.\ Russ.\ Ak.\ d.~W., +1908. + +\BibItem[\bysame] ``Untersuchung des allgemeinen Falles verketteter Ereignisse'' (in +Russian). Abh.\ der K.\ Russ.\ Ak.\ d.~W., 1910. + +\BibItem[\bysame] ``Über einen Fall von Versuchen, die eine komplizierte \DPtypo{zusammenhängendes}{zusammenhängende} +Kette bilden,'' and ``Über zusammenhängende Grössen, die +keine echte Kette bilden'' (both in Russian). Bull.\ de l'Acad.\ des Sciences. +Petersburg, 1911\DPtypo{}{.} + +\BibItem[\bysame] Wahrscheinlichkeitsrechnung. Transl.\ from 2nd~Russian edition by H.~Liebmann. +Leipzig, 1912. Pp.~vii~+~318. + +\BibItem[\bysame] Démonstration du second théorème---limite du calcul des probabilités par +la méthode des moments. Saint-Pétersbourg, 1913. Pp.~66. + +\Bibnote [Supplement to the 3rd~Russian edition of Wahrscheinlichkeitsrechnung, +in honour of the bicentenary of the Law of Great Numbers, with a Portrait +of Jacques Bernoulli.] + +\BibItem[Masaryk, T.~G.] David Hume's Skepsis und die Wahrscheinlichkeitsrechnung. +Wien, 1884. + +\BibItem[Maseres, F.] The Doctrine of Permutations and Combinations, being an +Essential and Fundamental Part of the Doctrine of Chances: As it is +delivered by Mr.~James Bernoulli, in his excellent Treatise on the Doctrine +of Chances, intitled, Ars conjectandi \ldots\ 8vo. London, 1795. + +\BibItem[Meinong, A.] Review of Von Kries's ``Die Principien der Wahrscheinlichkeitsrechnung.'' +Göttingische Gelehrte Anzeigen, vol.~2,\ pp.~56--75, 1890. + +\BibItem[\bysame] Über Möglichkeit und Wahrscheinlichkeit: Beiträge zur Gegenstandstheorie +und Erkenntnistheorie. Pp.~xvi~+~760. Leipzig, 1915. + +\BibItem[Meissner (Otto).] Wahrscheinlichkeitsrechnung: I.~Grundlehren; II.~Anwendungen. +Leipzig, 1912; 2nd~ed., 1919. Pp.~56~+~52. + +\Bibnote [An elementary primer.] + +\BibItem[Mendelssohn, Moses.] Philos.\ Schriften, 2~Tle. 12mo. Pp.~xxii~+~278~+~283. +Berlin, 1771. (\textit{Vide} especially vol.~ii.\ pp.~243--283, entitled ``Ueber die +Wahrscheinlichkeit.'') + +\BibItem[Mentrè, F.] ``Rôle du hasard dans les inventions et découvertes.'' Rev.\ +de Phil., 1904. + +\BibItem[\bysame] ``Les Racines historiques du probabilisme rationnel de Cournot.'' Rev.\ +de Métaphysique et de Morale, pp.~485--508, May 1905. + +\BibItem[\bysame] Cournot et la renaissance du probabilisme au xixe\DPnote{** TN: Ordinal, no special formatting in orig}~siècle. Paris, 1908. + +\BibItem[Merriman, M.] A Text-book of the Method of Least Squares. New York, +1884. Pp.~vii~+~198. 6th~ed., 1894. + +\BibItem[\bysame] ``List of Writings relating to the Method of Least Squares, with Historical +and Critical Notes.'' Trans.\ Connecticut Acad.\ vol.~4, pp.~151--232, 1877. + +\BibItem[Mertz.] Die Wahrscheinlichkeitsrechnung und ihre Anwendung, etc. Frankfort, +1854. + +\BibItem[Messina, I.] ``Intorno a un nuovo teorema di calcolo delle probabilità.'' 20~pp.\ 4to. +Giornale di Matematiche di Battaglini, vol.~lvi.\ (1918). Naples. +%% -----File: 462.png---Folio 451------- + +%\BibItem[Messina, I.]---\emph{continued}. + +\Bibnote [Described Stat.\ Jl.\ vol.~lxxxii. (1919), p.~612.] + +\BibItem[\bysame] ``Su di un nuovo teorema di calcolo delle probabilità, sul teorema di +Bernoulli e sui postulati empirici per la loro applicazione.'' Boll.\ del Lavoro +et della Presidenza, vol.~xxxiii. (1920). + +\BibItem[Meyer, A.] Essai sur une exposition nouvelle de la théorie analytique des +probabilités \textit{à~posteriori}. 4to. Pp.~122. Liége, 1857. + +\BibItem[\bysame] Cours de calcul des probabilités fait à l'université de Liége de 1849 à~1857. +Publié sur les mss.\ de l'auteur par F.~Folie. Bruxelles, 1874. + +\BibItem[\bysame] Vorlesungen über Wahrscheinlichkeitsrechnung. (Translation of the +above by E.~Czuber.) Pp.~xii~+~554. Leipzig, 1879. + +\BibItem[Michell.] ``An Inquiry into the Probable Parallax and Magnitude of the Fixed +Stars, from the Quantity of light which they afford us, and the particular +Circumstances of their Situation.'' Phil.\ Trans.\ vol.~57, pp.~234--264, +1767. + +\BibItem[Milhaud, G.] ``Le Hasard chez Aristote et chez Cournot.'' Revue de Méta.\ +et de Mor.\ vol.~x.\ pp.~667--681, 1902. + +\BibItem[Mill, J.~S.] System of Logic. Bk.~iii.\ chaps.\ 18,~23. + +\BibItem[Mondésir.] ``Solution d'une question qui se présente dans le calcul des probabilités.'' +Liouville Journ.\ vol.~ii. + +\BibItem[Monro, C.~J.] ``Note on the Inversion of Bernoulli's Theorem in Probabilities.'' +Proc.\ Lond.\ Math.\ Soc.\ vol.~5, pp.~74--78 and~145, 1874. + +\BibItem[Montessus, R.~de.] Leçons élémentaires sur le calcul des probabilités. Pp.~191. +Paris, 1908. (Reviewed Stat.\ Journ., 1909, p.~113.) + +\BibItem[\bysame] ``Le Hasard.'' Rev.\ du Mois, March 1907. + +\BibItem[Montessus, R.~de, \textrm{and} Lechalas, G.] ``Un Paradoxe du calcul des probabilités.'' +Nouv.\ Ann.\ iv.~(3), 1903. + +\BibItem[Montmort, P.~de.] Essai d'analyse sur les jeux de hasard. 4to. Pp.~xxiv~+~189. +Paris, 1708. + +\BibItem[\bysame] Ditto. 4to. \DPtypo{pp}{Pp}.~414. Paris, 1714. (The 2nd~ed.\ is increased by a +treatise on Combinations, and the correspondence between M.~and Nicholas +Bernoulli.) + +\BibItem[Montuola, J.~T.] Histoire des mathématiques. 4~vols. 4to. Paris, 1799--1802. + +\BibItem[\bysame] Vol.~iii.\ pp.~380--426. + +\Bibsect{N} + +\BibItem[Newcomb, Simon.] A Statistical Inquiry into the Probability of Causes of the +Production of Sex in Human Offspring. (Published by the Carnegie +Institution of Washington.) Pp.~34. 8vo. Washington, 1904. + +\BibItem[Nicole, F.] ``Examen et résolution de quelques questions sur les jeux.'' Hist.\ +Ac.\ Par.\ pp.~45--56, 331--344, 1730. + +\BibItem[Nieuport, C.~F.~de.] Un peu detort ou amusemens d'un sexagenaire. 8vo. +Bruxelles, 1818. Containing ``Conversations sur la théorie des probabilités.'' + +\BibItem[Nitsche, A.] ``Die Dimensionen der Wahrscheinlichkeit und die Evidenz der +Ungewissheit.'' Vierteljahrsschr.\ f.~wissensch.\ Philos.\ vol.~16, pp.~20--35, +1892. + +\BibItem[Nixon, J.~W.] ``An Experimental Test of the Normal Law of Error.'' Stat.\ +Journ.\ vol.~76, pp.~702--706, 1913. + +\Bibsect{O} + +\BibItem[Oettinger, L.] Die Wahrscheinlichkeitslehre. 4to. Berlin, 1852. + +\Bibnote [Reprinted from Crelle\DPtypo{,}{}~J., vols.\ 26,~30, 34,~36, under the title, Untersuchungen +über Wahrscheinlichkeitsrechnung.] + +\BibItem[Ostrogradsky.] ``Probabilité des jugements.'' Acad.\ de St-Pétersbourg, 1834. + +\BibItem[\bysame] ``Sur la probabilité des hypothèses.'' Mélanges math.\ et astr., 1859. +%% -----File: 463.png---Folio 452------- + +\Bibsect{P} + +\BibItem[Pagano, F.] Logica dei probabili. Napoli, 1806. + +\BibItem[Parisot, S.~A.] Traité du calcul conjectural ou l'art de raisonner sur les choses +futures et inconnues. 4to. Paris, 1810. + +\BibItem[Pascal, B.] ``Letters to Fermat.'' Varia opera mathematica D.~Petri de +Fermat.\ pp.~179--188, Toulouse, 1678. + +\BibItem[\bysame] {\OE}uvres, vol.~4, pp.~360--388, Paris, 1819. + +\BibItem[Patavio.] Probabilismus methodo mathematico demonstratus. 1840. + +\BibItem[Paulhan, Fr.] ``L'erreur et la sélection.'' Rev.\ Philos.\ vol.~viii.\ pp.~72--86, +179--190, 290--306, 1879. + +\BibItem[Peabody, A.~P.] ``Religious Aspect of the Logic of Chance and Probability.'' +Princeton Rev.\ vol.~v.\ pp.~303--320, 1880. + +\BibItem[Pearson, K.] ``On a Form of Spurious Correlation which may arise when +Indices are used, etc.'' Proc.\ Roy.\ Soc.\ vol.~lx.\ pp.~489--498. + +\BibItem[\bysame] ``On the Criterion that a given System of Deviations from the Probable +in the case of a Correlated System of Variables is such that it can be +reasonably supposed to have arisen from Random Sampling.'' Phil.\ Mag.~(5), +vol.~50, pp.~157--160, 1900. + +\BibItem[\bysame] ``On some Applications of the Theory of Chance to Racial Differentiation.'' +Phil.\ Mag.~(6), vol.~1, pp.~110--124, 1901. + +\BibItem[\bysame] Contributions to the Mathematical Theory of Evolution. + +\Bibnote [The main interest of the twelve elaborate memoirs published in the +Phil.\ Trans.\ under the above title is in every case statistical. References +are given below to those of them which have most reference to the theory +of Probability and in which Professor Pearson's general theory is mainly +developed.] + +\BibItem[\bysame] II. ``Skew Variation in Homogeneous Material.'' Phil.\ Trans.~(A), +vol.~186, Part~i.\ pp.~343--414, 1895. + +\BibItem[\bysame] III. ``Regression, Heredity, and Panmixia.'' Phil.\ Trans.~(A), vol.~187, +pp.~253--318, 1897. + +\BibItem[\bysame] V. ``On the Probable Errors of Frequency Constants and on the +Influence of Random Selection on Variation and Correlation.'' Phil.\ +Trans.~(A), vol.~191, pp.~229--311, 1898. (With L.~N.~G. Filon.) + +\BibItem[\bysame] VII. ``On the Correlation of Characters not quantitatively measurable.'' +Phil.\ Trans.~(A), vol.~195, pp.~1--47, 1901. + +\BibItem[\bysame] ``Mathematical Contributions to the Theory of Evolution.'' Roy.\ Stat.\ +Soc.\ Journ.\ lvi.,~1893, pp.~675--679; lix.,~1896, pp.~398--402; lx.,~1897, pp.~440--449. + +\BibItem[\bysame] ``On the Mathematical Theory of Errors of Judgment, with special +reference to the Personal Equation.'' Phil.\ Trans.~(A), vol.~198, pp.~235--299, +1902. + +\BibItem[\bysame] On the Theory of Contingency and its relation to Association and +Normal Correlation. Pp.~35. 4to. London, 1904. + +\BibItem[\bysame] On the General Theory of Skew Correlation and Non-linear Regression. +Pp.~54. 4to. London, 1905. + +\BibItem[\bysame] On further Methods of determining Correlation. London, 1907. (Reviewed +by G.~U. Yale Journ.\ Roy.\ Stat.\ Soc., Dec.\ 1907.) + +\BibItem[\bysame] ``On the Influence of Past Experience on Future Expectation.'' Phil.\ +Mag.~(6), vol.~13, pp.~365--378, 1907. + +\BibItem[\bysame] ``The Fundamental Problem of Practical Statistics.'' Biometrika, vol.~xiii.\ +pp.~1--16, 1920. + +\Bibnote [On Inverse Probability.] + +\BibItem[\bysame] ``Notes on the History of Correlation.'' Biometrika, vol.~xiii.\ pp.~25--45, +1920. + +\BibItem[\bysame] ``The Chances of Death'' and other essays. 2~vols. 8vo, London, +1897. + +\BibItem[\bysame] The Grammar of Science. London, 1892. +%% -----File: 464.png---Folio 453------- + +\BibItem[Peirce, C.~S.] ``A Theory of Probable Inference.'' Johns Hopkins ``Studies in +Logic,'' 1883. + +\BibItem[\bysame] ``On an Improvement in Boole's Calculus of Logic.'' Proc.\ Amer.\ Acad.\ +Arts and Sci.\ vol.~vii.\ pp.~250--261, 1867. Pp.~62. Camb., 1870. + +\BibItem[Perozzo.] ``Nuove applicazioni del calcolo delle probabilità allo studio dei +fenomeni statistici.'' Proceedings of Academia dei Lincei, 1881--82. + +\BibItem[\bysame] Germ.\ transl.\ by O.~Elb. Neue Anwendungen der Wahrscheinlichkeitsrechnung +in der Statistik. Pp.~33. 4to. Dresden, 1883. + +\BibItem[Pièron, H.] ``Essai sur le hasard. La Psychologie d'un concept.'' Rev.\ de +Méta.\ et~de Mor.\ vol.~x.\ pp.~682--693, 1902. + +\BibItem[Pinard, H.] ``Sur la Convergence des Probabilités.'' Rev.\ Néo-Schol.\ de +Phil.\ No.~84 (1919) and No.~85 (1920). + +\BibItem[Pincherle, S.] ``Il calcolo delle probabilità e~l'intuizione.'' Scientia, vol.~xix.\ +pp.~417--426, 1916. + +\BibItem[Pizzetti, P.] I fondamenti matematici per la critica dei risultati sperimentali. +Atti della R.~Univ., Genova, 1892. + +\BibItem[Plaats, J.~D. van~der.] Over de toepassing der waarschijnlijkheidsrekening +op medische statistick. 1895. + +\BibItem[Plana, G.] ``Mémoire sur divers problèmes de probabilité.'' Mémoires de +l'Académie de Turin for 1811--12, vol.~xx.\ pp.~355--408, 1813. + +\BibItem[\DPtypo{Poincarè}{Poincaré}, H.] Calcul des probabilités. Pp.~274. Paris, 1896. + +\BibItem[\bysame] 2nd~edition (with additions). Pp.~333. Paris, 1912. + +\BibItem[\bysame] Science et hypothèse. Paris. + +\BibItem[\bysame] Engl.\ transl., London, 1905. + +\BibItem[\bysame] Science et méthode. Paris. (Includes a chapter on ``Le Hasard.'') + +\BibItem[\bysame] Eng.\ transl.\ (by F.~Maitland). Pp.~288. London, 1914. + +\BibItem[\bysame] ``Le Hasard.'' Rev.\ du~Mois, March 1907. + +\BibItem[Poisson, S.~D.] Recherches sur la probabilité des jugements en matière +criminelle et en matière civile, précédés des règles générales du calcul des +probabilités. 4to. Pp.~ix~+~415. Paris, 1837. + +\BibItem[\bysame] Lehrbuch der Wahrscheinlichkeitsrechnung. German translation of the +above by H.~Schnuse. Braunschweig. 8vo. 1841. + +\BibItem[\bysame] ``Sur la probabilité des résultats moyens des observations.'' Conn.\ +des Temps. Pp.~273--302, 1827. Pp.~3--22, 1832. + +\BibItem[\bysame] ``Formules relatives aux probabilités qui dépendent de très grand +nombres.'' Compt.\ Rend., Acad.\ Paris, vol.~2, pp.~603--613, 1836. + +\BibItem[\bysame] ``Sur le jeu de trente et quarante,'' Annal.\ de Gergonne,~xv. + +\BibItem[\bysame] ``Solution d'un problème de probabilité.'' Liouv.~J.~(1), vol.~2, 1837. + +\BibItem[\bysame] ``Mémoire sur la proportion des naissances des filles et des garçons.'' +Mém.\ Acad.\ Paris, vol.~9, pp.~239--308, 1830. + +\BibItem[Pondra \textrm{et} Hossard.] Question de probabilité résolue par la géométrie. 8vo. +Paris, 1819. + +\BibItem[Poretzki, Platon.~S.] ``Solution of the general Problem of the Theory of +Probability by means of Mathematical Logic.'' (In Russian.) Bull.\ of +the physico-mathematical Academy of Kasan, 1887. + +\BibItem[Prevost, P.] ``Sur les principes de la théorie des gains fortuits.'' Nouv.\ Mém.\ +Pp.~430--472. Berlin, 1780. + +\BibItem[Prevost, P., \textrm{and} Lhuilier, S.~A.] ``Sur les probabilités.'' Mém.\ Ac.\ Berl.\ +(1796), pp.~117--142, 1799. + +\BibItem[\bysame] ``Sur l'art d'estimer la probabilité des causes par les effets.'' Mém.\ Ac.\ +Berl.\ (1796), pp.~3--24, 1799. + +\BibItem[\bysame] ``Remarques sur l'utilité et l'étendue du principe par lequel on estime +la probabilité des causes.'' Mém.\ Ac.\ Berl.\ (1796), pp.~25--41, 1799. + +\BibItem[\bysame] Note on last. Mém.\ Ac.\ Berl.\ (1797), p.~152, 1800. + +\BibItem[\bysame] ``Mémoire sur l'application du calcul des probabilités à la valeur du +témoignage.'' Mém.\ Ac.\ Berl.\ (1797), pp.~120--151, 1800. +%% -----File: 465.png---Folio 454------- + +\BibItem[Price, R.] See \textsc{Bayes}. + +\BibItem[Pringsheim, A.] See \textsc{Daniel Bernoulli}. + +\BibItem[\bysame] ``Weiteres zur Geschichte des Petersburger Problems.'' Grunert, Archiv, +77, 1881. + +\BibItem[Proctor, R.~A.] Chance and Luck. A Discussion of the Laws of Luck, Coincidences, +Wagers, Lotteries, and the Fallacies of Gambling, with Notes on +Poker and Martingales. Pp.~vii~+~263. London, 1887. + +\BibItem[Protimalethes.] Miracle \textit{versus} Nature: being an Application of Certain Propositions +in the Theory of Chances to the Christian Miracles. 8vo. Cambridge, +1847. + +\Bibsect{Q} + +\BibItem[Quetelet, A.] Instructions populaires sur le calcul des probabilités. 12mo. +Bruxelles, 1828. + +\BibItem[\bysame] Engl.\ transl.: Popular Instructions on the Calculation of Probabilities, +transl.\ with notes by R.~Beamish. 1839. + +\BibItem[\bysame] Dutch transl.\ by H.~Strootman. Breda, 1834. + +\BibItem[\bysame] Lettres sur la théorie des probabilités appliquée aux sciences, morales +et politiques. Bruxelles, 1846. + +\BibItem[\bysame] Engl.\ transl.: Letters on the Theory of Probabilities as applied to the +Moral and Political Sciences, transl.\ by O.~G. Downes. 8vo. 1849. + +\BibItem[\bysame] ``Sur la possibilité de mesurer l'influence des causes qui modifient les +\DPtypo{élémens}{éléments} sociaux.'' Corresp.\ mathém.\ et~phys.\ vol.~vii.\ pp.~321--346. +Bruxelles, 1832. + +\BibItem[\bysame] ``Sur la constance qu'on observe dans le nombre des crimes qui se committent.'' +Corresp.\ mathém.\ et phys.\ vol.~vi.\ pp.~214--217. Brussels, 1830. + +\BibItem[\bysame] ``Théorie des probabilités.'' (In the Encycl.\ populaire.) Brussels, 1853. + +\BibItem[\bysame] ``Sur le calcul des probabilités appliqué à la science de l'homme.'' Bull.\ +de l'Acad.\ roy.\ vol.~xxvi.\ pp.~19--32. Brussels, 1873. + +\Bibnote [For a full bibliography and discussion of Quetelet's writings on these +topics see Lottin's Quetelet.] + +\Bibsect{R} + +\BibItem[Rayleigh, Lord.] ``On James Bernouilli's Theorem in Probabilities.'' Phil.\ +Mag.~(5), vol.~47, pp.~246--251, 1890. + +\BibItem[Regnault.] Calcul des chances et philosophie de la bourse. 8vo. Paris, +1863. + +\BibItem[Renouvier, Ch.] L'Homme: la raison, la passion, la liberté, la certitude, la +probabilité morale. 8vo. 1859. + +\BibItem[Revel, P.~Camille.] Esquisse d'un système de la nature fondé sur la loi du +hasard. 1890. 2nd~ed.\ (corrigée), 1892. + +\BibItem[\bysame] Le Hasard, sa loi et ses conséquences dans les sciences et en philosophie. +Paris, 1905. 2nd~ed.\ (corrigée et augmentee). Pp.~249. Paris, 1909. + +\BibItem[Rizzetti, J.] ``Ludorum scientia, sive artis conjectandi elementa ad alias +applicata.'' Act.\ Erud.\ Suppl.\ vol.~9, pp.~215--229, 295--307. Leipzig, +1729. + +\BibItem[Roberts, Hon.~Francis.] ``An Arithmetical Paradox concerning the Chances +of Lotteries.'' Phil.\ Trans.\ vol.~xvii.\ pp.~677--681, 1693. + +\BibItem[Roger.] ``Solution d'un problème de probabilité.'' Liouv.~J.~(1), vol.~17, +1852. + +\BibItem[Rouse, W.] Doctrine of Chances, or the Theory of Gaming made easy to every +Person---Lotteries, Cards, Horse-Racing, Dice, etc. 1814. + +\BibItem[Rudiger, Andreas.] De sensu falsi et veri libri~iv. [Lib.~i.\ cap.~xii.\ et~lib.~iii.] +Editio Altera. 4to. Lipsiae, 1722. + +\BibItem[Ruffini.] Critical Reflexions on the Essai philosophique of Laplace (in +Italian). Modena, 1821. +%% -----File: 466.png---Folio 455------- + +\Bibsect{S} + +\BibItem[Sabudski-Eberhard.] Die Wahrscheinlichkeitsrechnung, ihre Anwendung auf +das Schiessen und auf die Theorie des Einschiessens. Stuttgart, 1906. + +\BibItem[Sawitsch, A.] Die Anwendung der Wahrscheinlichkeitstheorie auf die Berechnung +der Beobachtungen und geodätischen Messungen oder die Methode +der kleinsten Quadrate. (Translated into German from the Russian by +Lais.) Leipzig, 1863. + +\BibItem[Schell, W.] Über Wahrscheinlichkeit. 8vo. + +\BibItem[Schnuse, H.] Vid.\ \textsc{Poisson}. + +\BibItem[Schweigger, F.] Berechnung der Wahrscheinlichkeit beim Würfeln. + +\BibItem[Scott, John.] The Doctrine of Chance: the Arithmetic of Gambling. 56~pp.\ +8vo. 1908. + +\BibItem[Segueri, Paolo.] Lettere sulla materia del probabile. 12mo. Colonia, 1732. + +\BibItem[Sextus Empiricus.] Works. + +\BibItem[Sheldon, W.~H.] ``Chance.'' Journal of Phil., Psych., and Sci.\ Meth., vol.\ ix.\ +pp.~281--290. 1912. + +\BibItem[Sheppard, W.~F.] ``On the Application of the Theory of Error to Cases of +Normal Distribution and Normal Correlation.'' Phil.\ Trans.~A.\ vol.~192, +pp.~101--167, 1899. + +\BibItem[\bysame] ``On the Calculation of the most Probable Values of the Frequency +Constants for Data arranged according to Equidistant Divisions of a Scale.'' +Proc.\ Lond.\ Math.\ Soc.\ vol.~xxix.\ pp.~353--380. + +\BibItem[\bysame] ``Normal Correlation.'' Camb.\ Phil.\ Soc.\ vol.~xix. + +\BibItem[\bysame] ``Normal Distribution and Correlation.\DPtypo{}{''} Roy.\ Soc.\ Trans., 1898. + +\BibItem[Sigwart, C.] Review of von Kries in Vierteljahrsschr.\ für Wiss.\ Phil.~xiv.\ +p.~90. + +\BibItem[\bysame] Logik. Tübingen, 1878. + +\BibItem[\bysame] 2nd~ed. Freiburg, i.~B., 1893. English~ed., 1895. + +\BibItem[\bysame] Vol,~ii.\ Part~3, chap.~3, §\;85, Die Wahrscheinlichkeitsrechnung; 5,~§\;102. +Die Wahrscheinlichkeit auf statischem Boden. + +\Bibnote References in English ed.: + +\BibItem[\bysame] Probability, vol.~ii.\ pp.~216--230, 261--271 (errors of observation), 303--309 +(induction), 504--507 (statistics). + +\BibItem[Simmons, T.~C.] ``A New Theorem in Probability.'' Proc.\ Lond.\ Math.\ Soc.\ +vol.~26, pp.~290--323, 1895. + +\BibItem[\bysame] ``Sur la probabilité des événements composés.'' Ass.\ Franc.\ pour +l'Avancement des Sciences. 1896. + +\BibItem[Simon.] ``Exposition des principes du calcul des probabilités.'' Journ.\ des +Actuaires français,~i. + +\BibItem[Simpson, T.] ``A Letter to the Right Honourable George, Earl of Macclesfield, +President of the Royal Society, as to the Advantage of taking the Mean +of a Number of Observations in Practical Astronomy.'' Phil.\ Trans.\ vol.~xlix.\ +pp.~82--93, 1755. + +\BibItem[\bysame] ``An Attempt to show the Advantage arising by taking the Mean of a +Number of Observations in Practical Astronomy.'' (Miscellaneous tracts +on some curious subjects, pp.~64--75). London, 4to, 1757. + +\Bibnote [A reprint of the above with some new matter. The probability, assuming +positive and negative errors to be equally likely, that the mean is +nearer to the truth than a single observation taken at random, is here +investigated for the first time.] + +\BibItem[\bysame] Treatise on the Nature and Laws of Chance. 4to. London, 1740. + +\BibItem[\bysame] Another edition. 8vo. 1792. + +\BibItem[Sorel, G.] ``Le Calcul des probabilités et l'expérience.'' Rev.\ Philos.\ vol.~xxiii.\ +pp.~50--66, 1887. + +\BibItem[Spehr, F.~W.] Vollständiger Lehrbegriff der reinen Combinationslehre mit +Anwendungen derselben auf Analysis und Wahrscheinlichkeitsrechnung. 2.~wohlfeile +Ausg. 4to. Braunschweig, 1840. +%% -----File: 467.png---Folio 456------- + +\BibItem[Spinoza.] ``Letter to Jan van der Meer.'' Opera ed.\ Van Vleten and Land, +vol.~ii.\ pp.~145--149, Ep.~38 (in Latin and Dutch). + +\BibItem[\bysame] See also Spinoza's Briefwechsel in J.~H. v.~Kirchmann's Philos.\ Bibliothek, +vol.~xlvi.\ pp.~145--147. + +\BibItem[Sprague, T.~B.] On Probability and Chance and their Connexion with the +Business of Insurance. 8vo. 1892. + +\BibItem[Stamkart, F.~J.] Over de waarschijnlijkheidsrekening. 8vo. + +\BibItem[Sterzinger, O.] Zur Logik und Naturphilosophie der Wahrscheinlichkeitslehre. +Leipzig, 1911. + +\BibItem[Stewart, Dugald.] ``On the Calculus of Probabilities, in reference to the +Preceding Argument for the Existence of God, from Final Causes.'' Philosophy +of the Moral Powers, vol.~ii.\ pp.~108--119. (Sir W.~Hamilton's ed., +Edin., 1860.) 1st~ed., 1828. + +\BibItem[Stieda, L.] ``Über die Anwendung der Wahrscheinlichkeitsrechnung in der +anthropologischen Statistik.\ Arch.\ f.~Anthrop., 1882. +2nd~ed.\ 8vo. Braunschweig, 1892. + +\BibItem[Streeter, T.~E.] The Elements of the Theory of Probabilities. 31~pp.\ 8vo. +1908. + +\BibItem[Struve.] Catalogus novus stellarum duplicium et multiplicium. Dorpati, +1827, pp.~xxxvii--xlviii. + +\BibItem[Stumpf, C.] ``Bemerkung zur Wahrscheinlichkeitslehre.'' Jahrb.\ f.~national. +Ök.\ u.~Stat.~(3), vol.~17, pp.~671,~672, 1899; vol.~18, p.~243, 1899. + +\Bibnote [In criticism of Bortkiewicz, \textit{q.v.}] + +\BibItem[Stumpf, K.] ``Über den Begriff der mathematischen Wahrscheinlichkeit.'' +Ber.\ bayr.\ Ak.\ (Phil.\ Cl.), pp.~37--120, 1892. + +\BibItem[\bysame] ``Über die Anwendung des mathematischen Wahrscheinlichkeitsbegriffes +auf Teile eines Continuums.'' Ber.\ bayr.\ Ak.\ (Phil.\ Cl.), pp.~681--691, +1892. + +\BibItem[Suppantschitsch.] Einführung in die Wahrscheinlichkeitsrechnung. Leipzig. + +\Bibsect{T} + +\BibItem[Tait, P.~G.] ``Law of Frequency of Error.'' Edin.\ Phil.\ Trans.\ vol.~4, 1865. + +\BibItem[\bysame] On a Question of Arrangement and Probabilities. 1873. + +\BibItem[Tchebychef, P.~L.] Essai d'analyse élémentaire de la théorie des probabilités. +4to. Moscow, 1845 (in Russian, degree thesis). Pp.~ii~+~61~+~iii. + +\BibItem[\bysame] ``Démonstration élémentaire d'une proposition générale de la théorie +des probabilités.'' Crelle~J.\ vol.~33, pp.~259--267, 1846. + +\BibItem[\bysame] ``Des valeurs moyennes.'' Liouv.~J.~(2), vol.~12, pp.~177--184, 1867. +(Extrait du Recueil des Sciences mathématiques, vol.~ii.) + +\BibItem[\bysame] ``Sur deux théorèmes relatifs aux probabilités.'' Petersb.\ Abh.\ vol.~55, +1887. (In Russian.) French translation by J. Lyon: Act.\ Math.\ +Petr.\ vol.~14, pp.~305--315, 1891. + +\BibItem[\bysame] {\OE}uvres. 2~vols. 4to. St-Pétersbourg, 1907. + +\Bibnote (The three memoirs preceding are here reprinted in French.) + +\BibItem[Terrot, Bishop.] ``Summation of a Compound Series and its Application to a +Problem in Probabilities.'' Edin.\ Phil.\ Trans., 1853, vol.~xx.\ pp.~541--545. + +\BibItem[\bysame] ``On the Possibility of combining two or more Probabilities of the same +Event, so as to form one Definite Probability.'' Edin.\ Phil.\ Trans., 1856, +vol.~xxi.\ pp.~369--376. + +\BibItem[Thiele, T.~N.] Theory of Observations. Pp.~6~+~143. 4to. London, 1903. + +\BibItem[Thomson, Archbishop.] Laws of Thought. §\;124, Syllogisms of Chance (13~pp.). + +\BibItem[Thubeuf.] Élémens\DPnote{** TN: [sic] Appears not to be a typo for Éléments} et principes de la royale arithmétique aux jettons, etc.\ +12mo. Paris, 1661. + +\BibItem[Timerding.] Die Analyse des Zufalls. Pp.~ix~+~168. Braunschweig, 1915. + +\BibItem[Todhunter, I.] ``On the Method of Least Squares.'' Camb.\ Phil.\ Trans.\ vol.~ii.\ + +\BibItem[\bysame] A History of the Mathematical Theory of Probability from the Time of +Pascal to that of Laplace. Lge.\ 8vo. pp.~xvi~+~624, Camb.\ and~Lond., 1865. +%% -----File: 468.png---Folio 457------- + +\BibItem[Tozer, J.] On the Measure of the Force of Testimony in Cases of Legal Evidence. +4to. Camb.\ Phil.\ Soc.\ vol.~viii. Part~II. 16~pp.\ (read Nov.~27, +1843). 1844. + +\BibItem[Trembley.] ``Observations sur le calcul d'un jeu de hasard.'' Mém.\ Ac.\ Berl.\ +(1802), pp.~86--102. + +\BibItem[\bysame] ``Recherches sur une question relative au calcul des probabilités.'' +Mém.\ Ac.\ Berl.\ (1794--5), pp.~69--108, 1799. + +\BibItem[\bysame] (On Euler's memoir, ``Solutio quarundam quaestionum difficiliorum in +calculo probabilitatum.'') + +\BibItem[\bysame] ``De probabilitate causarum ab effectibus oriunda.'' Comm.\ Soc.\ Reg.\ +Gott.\ (1795--8), vol.~13, pp.~64--119, 1799. + +\BibItem[\bysame] ``Observations sur la méthode de prendre les milieux entre les observations.'' +Mem.\ Ac.\ Berl.\ (1801), pp.~29--58, 1804. + +\BibItem[\bysame] ``Disquisitio elementaris circa calculum probabilium.'' Comm.\ Soc.\ +Reg.\ Gott.\ (1793--4), vol.~12, pp.~99--136, 1796. + +\BibItem[Tschuprow, A.~A.] ``Die Aufgaben der Theorie der Statistik.'' Jahrb.\ f.~gesetzg.\ +Verwalt.\ u.~Volkswirtsch.\ vol.~29, pp.~421--480, 1905. + +\BibItem[\bysame] ``Zur Theorie der Stabilität statistischer Reihen.'' Skandinavisk +Aktuarietidskrift, pp.~199--256, 1918; pp.~80--133, 1919. + +\BibItem[Twardowski, K.] ``Über sogenannte relative Wahrheiten.'' Arch.\ f.~syst.\ +Philos.\ vol.~viii.\ pp.~439--447, 1902. + +\Bibsect{U} + +\BibItem[Urban, F.~M.] ``Über den Begriff der mathematischen Wahrscheinlichkeit.'' +Vierteljahrsschr.\ f.~wiss.\ Phil.\ und~Soz., vol.~x.\ (N.S.), 1911. + +\Bibsect{V} + +\BibItem[Vastel, L.~G.~F.] L'Art de conjecturer. Traduit du latin de J.~Bernoulli, avec +observations, éclaircissemens et additions. Caen, 1801. + +\Bibnote [Translation of Part~I. only of Bernoulli's Ara Conjectandi (\textit{q.v.})\ containing +a commentary on and reprint of Huygens, De ratiociniis in ludo +aleae.] + +\BibItem[Venn, J.] The Logic of Chance. 1866. 2nd~ed., 1876. 3rd~ed., 1888. + +\BibItem[\bysame] ``The Foundations of Chance.'' Princeton Rev.\ vol.~2, pp.~471--510, +1872. + +\BibItem[\bysame] ``On the Nature and Uses of Averages.'' Stat.\ Journ.\ vol.~54, pp.~429--448, +1891. + +\Bibsect{W} + +\BibItem[Wagner, A.] Die Gesetzmässigkeit in den scheinbar willkürlichen Handlungen +des Menschen. Hamburg, 1864. + +\BibItem[\bysame] ``Wahrscheinlichkeitsrechnung und Lebensversicherung.'' Zeitschr.\ f.~d.\ +ges.\ Versicherungswissenschaft. Berlin, 1906. + +\BibItem[Waring, E.] (M.D.~Lucasian Prof.) On the Principles of translating Algebraic +Quantities into Probable Relations and Annuities, etc. Pp.~59. Cambridge, +1792. + +\BibItem[\bysame] An Essay on the Principles of Human Knowledge. Pp.~244. Cambridge, +1794. + +\BibItem[Welton, J.] Manual of Logic. (Probability, vol.~ii pp.~165--185.) London, +1896. + +\BibItem[Westergaard.] Grundzüge der Theorie der Statistik. + +\BibItem[Whitaker, Lucy.] ``On the Poisson Law of Small Numbers.'' Biometrika, +vol.~x. 1914. + +\BibItem[Whittaker (E.~T.).] ``On Some Disputed Questions of Probability.'' Transactions +of the Faculty of Actuaries in Scotland, vol.~viii (1920), pp.~163--206. + +\Bibnote [Problems of Inverse Probability including the Law of Succession. This +paper is followed by others on the same subject by various writers.] +%% -----File: 469.png---Folio 458------- + +\BibItem[Whitworth, W.~A.] Choice and Chance, An Elementary Treatise on Permutations, +Combinations, and Probability, with 300~Exercises. 1867. 2nd~ed., +1870. 3rd~ed.\ pp.~viii~+~244. Cambridge, 1878. + +\BibItem[\bysame] Expectations of Parts into which a Magnitude is divided at Random. +1898. + +\BibItem[Wicksell, S.~D.] ``Some Theorems in the Theory of Probabilities.'' Skandinavisk +Aktuarietidskrift, p.~196 (1910). + +\BibItem[Wijnne, H.~A.] De leer der waarschijnlijkheid in hare toepassing op het +dagelijksche leven. 1862. + +\BibItem[Wilbraham, H.] ``On the Theory of Chances developed in Prof.\ Boole's +`Laws of Thought.''' Phil.\ Mag., 1854. + +\BibItem[Wild, A.] Die Grundsätze der Wahrscheinlichkeitsrechnung und ihre Anwendungen. +München, 1862. + +\BibItem[Windelband, W.] Die Lehren vom Zufall. Berlin, 1870. + +\BibItem[Wolf, A.] ``The Philosophy of Probability.'' Proc.\ Arist.\ Soc.\ vol.~xiii.\ +pp.~29. London, 1913. + +\BibItem[Wolf, R.] ``Über eine neue Serie von Würfelversuchen.'' Vierteljs.\ Naturforsch.\ +Gesellsch.\ in Zürich, vol.~26, pp.~126--136 and 201--224, 1881; vol.~27, +pp.~241--262, 1882; vol.~28, pp.~118--124, 1883. + +\BibItem[\bysame] ``Neue Serie von Würfelsversuchern.'' \textit{Ibid.}\ vol.~38, pp.~10--32, 1893. + +\BibItem[\bysame] ``Versuche zur Vergleichung der Erfahrungswahrscheinlichkeit mit der +mathematischen Wahrscheinlichkeit.'' Mitth.\ d.~Naturforsch.\ Gesellsch., +Bern, 1849--1851, 1853. + +\BibItem[Wolff, Christian.] Philosophia rationalis sive logica. Leipzig, 1732. + +\BibItem[Woodward, R.~S.] Higher Mathematics, chap.~x.\ pp.~467,~507. ``Probability +and Theory of Error.'' New York, 1900. + +\BibItem[\bysame] Probability and Theory of Errors. New York, 1906. + +\BibItem[Wyrouboff, G.] ``Le Certain et le probable.'' La Philos.\ posit.\ p.~165, 1867. + +\Bibsect{Y} + +\BibItem[Young, J.~R.] Elementary Treatise on Algebra, Theoretical and Practical, with +an Appendix on Probabilities and Life Annuities. 4th~ed.\ enlarged, post +8vo. 1844. + +\BibItem[Young, Rev.~M.] ``On the Force of Testimony in establishing Facts contrary +to Analogy.'' Trans.\ Roy.\ Ir.\ Acad.\ vol.~vii.\ pp.~79--118, 1800. + +\BibItem[Young, T.] ``Remarks on the Probabilities of Error in Physical Observations, +etc.'' Phil.\ Trans., 1819. + +\BibItem[Yule, G.~U.] ``On the Theory of Correlation.'' Journ.\ Stat.\ Soc.\ vol.~lx.\ +p.~812, 1897. + +\BibItem[\bysame] ``On the Association of Attributes in Statistics.'' Phil.\ Trans.~(A), vol.~194, +pp.~257--319, 1900. + +\BibItem[\bysame] ``On the Theory of Consistence of Logical Class-frequencies.'' Phil.\ +Trans.~(A), vol.~197, pp.~91--132, 1901. + +\BibItem[\bysame] An Introduction to the Theory of Statistics. Pp.~xiii.~+~376. London, +1911. + +\BibItem[Yule \textrm{and} Galton.] ``The Median.'' Stat.\ Journ.\ pp.~392--398, 1896. +\end{Biblio} +%% -----File: 470.png---Folio 459------- + +\cleardoublepage +\phantomsection +\pdfbookmark[0]{Index}{Index} +\printindex + +\iffalse +Acquaintance, direct 12 + +Addition, of probabilities 37, 135 + definition of 120 + Theorem of 104, 121, 144 + and measurement 158 + +Analogy, principle of 68 + and induction 218, 222 + negative 219, 223, 233, 415 + positive 220, 223, 415 + and generalisation 223 + logical foundation of 258 + and Bacon 268 + and Leibniz 272 + and Jevons 273 + and statistics 391, 407 + and statistics|ifoll 415 + +Ancillon|inote#Ancillon 5 + +Ancillon 82 + +Apprehension, direct, and ethical judgment 316 + +Argument 13 + +Aristotle 80, 92 + and induction 274 + +Arithmetic mean (or average) 205 + and laws of error 197 + Laplace on 206 + Gauss on 206 + +Astronomers and Least Squares 210 + +Asymmetry, and Bernoulli's Theorem|ifoll 358 + +Atomic Uniformity 249 + +Averages|ifoll 205 + +Averages + weighting of 211 + and discordant observations 214 + +Axioms|ifoll 135 + +Axioms + non-self-evident 299 + +Bachelier|inote 347 + +Bachelier + and statistical frequency|inote 349 + and statistical frequency 351 + and Rule of Succession|inote 376 + +Bacon|ifoll 265 + +Bacon + tables of 269 + and limited variety 270 + +Bayes, and Inverse Probability 174 + Theorem of 379 + +Belief, rational|ifoll 4 + +Belief, rational 10, 16, 307 + degrees of 11 + +Bentham, measurement of Probability 20 + +Bernoulli, Daniel, and Inverse Probability 174 + and planets|inote 293 + and planets 294 + and Petersburg Paradox 316, 317 + +Bernoulli, Jac.|inote 15 + +Bernoulli, Jac. 41, 76, 81, 83, 86, 368, 369 + weight of evidence 313 + second axiom of 322 + and regular frequency 333 + and statistical series 392 + +Bernoulli's Theorem|ifoll 337 + +Bernoulli's Theorem|inote 319 + +Bernoulli's Theorem 109, 314, 333 + and asymmetry|ifoll 358 + empirical verification of|ifoll 361 + Inverse of|ifoll 368, 385 + +Bertrand|inote 48 + +Bertrand 49 + on multiplication 136 + and Maxwell|inote 172 + and independence 173 + and Law of Error|inote 208 + and chance 284 + and Petersburg Paradox 317 + and Bernoulli's Theorem 339 + and Rule of Succession 382 + +Bicquilley and testimony|inote 184 + +Bobek and Rule of Succession 383 + +Bode's Law 304 + +Boole|inote 43, 50, 294 + +Boole 84 + and German logicians 87 + and relation of Probability 90 + and symbolic probability 155 + and approximation 161 + and independence 167 + and Whately 179 + and combination of premisses 179 +%% -----File: 471.png---Folio 460------- +%\item Boole (\textit{contd.})--- + and testimony 180 + and Challenge Problem 187 + and Cournot|inote 284 + and Rule of Succession 382 + +Borel|inote 47 + +Borel 48 + +Bortkiewicz, von, and great numbers|inote 333 + +Bortkiewicz, von, and great numbers + and Marbe|inote 365 + method of 384 + and Lexis|ifoll 393 + and Law of Small Numbers|ifoll 401 + and Quetelet 402 + +Boscovitch and Least Squares 210 + +Bowley|ifoll 424 + +Bowley 421, 423 + +Bradley|inote 319 + +Bradley + and relativity of Probability 91 + and Bernoulli's Theorem|inote 341 + +Broad, C.~D.|inote 257 + +Brömse and Marbe|inote 365 + +Brünn and lotteries 364 + +Bruns and Marbe|inote 365 + +Buffon 317, 322 + and coin-tossing 362 + +Butler, Bishop 79, 80, 309, 310 + and risk 321 + +Calculus of Probability|inote 83 + +Calculus of Probability 149, 164, 303, 428 + and Psychical Research 302 + and Sociology 335 + +Casual@{`\textit{Casual}'} 288 + +Causality 263, 276 + and independence 164 + +Cause@{`\textit{Cause}'} 275 + +Cause, final 297 + +Cayley, and tradition 185 + and Challenge Problem 187 + +Certainty 10, 127, 128 + and truth 15 + Kahle and|inote 90 + definition of 120 + relation of 134 + and Bacon 267 + and Leibniz 272 + +Chance, objective|ifoll 286 + +Chance, objective 281, 295, 418 + Couturat on 283 + Poincaré on 284, 289 + Condorcet on 284 + definition of 287 + and planets 293 + and binary stars 295 + +Coefficient of Credibility 183 + of Correlation|ifoll 421 + +Combination of premisses 149, 178 + +Comte + and `\textit{seven}'#seven 246 + and statistics 335 + +Condorcet|inote 83 + +Condorcet 317 + and testimony 180 + and chance 282, 284 + and ethics 313, 316 + and gambling 319 + +Conduct and Probability 307 + +Consistence and group theory 124 + +Contradiction 143 + +Coover, J.|inote 298 + +Correlation 329, 390 + and statistical frequency 330 + Quantitative 391, 426 + Inductive 406 + coefficient|ifoll 421 + +Cournot, and frequency theory 92 + and independence 166 + on testimony 180 + and causality 275 + and chance 282, 283 + +Couturat|inote 272, 311 + +Craig and tradition 184 + +Cramer and Petersburg Paradox 318 + +Crofton|inote 47 + +Cumulative Formula 150 + Johnson and 121 + +Czuber|inote 47, 339, 345 + +Czuber 78, 82, 86, 347 + and symbolic probability 156 + and cause@{and `\textit{cause}'}|inote 275 + and risk|inote 315 + and Bernoulli's Theorem|inote 340 + and statistical frequency 351, 394 + and Tchebycheff's Theorem|inote 353, 355 + and verification of Bernoulli|inote 362 + and lotteries 364 + and Marbe 365 + and Inverse of Bernoulli's Theorem|inote 370 + and Rule of Succession|inote 376 + and Rule of Succession 382 + +D'Alembert|inote 170, 365 + +D'Alembert 82, 321, 369 + and chance 282 + and planets 293 + and mathematical expectation 314 + and ethics 316 + and Petersburg Paradox 317 + and Marbe 365 + +Darbon, A., and Cournot 284 + +Darwin 108 + and Lyell 161 + and Mill 265 + +Dedekind and `\textit{Challenge Problem}'|inote#Dedekind 187 +%% -----File: 472.png---Folio 461------- + +Definitions|ifoll 134 + +Definitions + summary of 120 + de la Placette, Jean, and chance 283 + +De Morgan 21, 74, 83 + and inference 139 + and independence 168 + and Inverse Probability 178 + and combination of premisses 179 + and tradition|inote 184 + and planets 293 + pupil of 362 + and Inverse of Bernoulli's Theorem|inote 370 + and Rule of Succession 375, 382 + +De Witt and arithmetic averages 206 + +Dice-tossing|ifoll 361 + +Diderot on testimony 183 + +Discordant observations, rejection of 213 + +Donkin, W. F. 20 + and Inverse Probability 176 + +Dormoy 394 + +Edgeworth|inote 29, 362 + +Edgeworth 84, 85, 379, 400 + use of `\textit{Probability}'|inote 96 + and randomness 290 + and Psychical Research|inote 298 + and ethics 316 + and German statisticians 394 + +Eggenberger|inote 340 + +Ellis, Leslie 84, 85 + and frequency theory 92 + and Least Squares|inote 207 + and Least Squares 209 + and Bacon|inote 265, 266, 269, 271, 274 + and Bernoulli's Theorem 341 + +Empirical School 85, 86 + +Epistemology 302 + and inductive hypothesis 261 + +Equiprobability 41, 63, 65 + +Equivalence, definition of 120, 134 + axiom of 135 + principle of 141 + +Error, probable 329 + +Ethics|ifoll 307 + +Euler and Least Squares 210 + +Event, probability of 5 + +Evidence, and measurement of Probability 7, 35 + relevant and irrelevant 53, 54 + independent and complementary 55 + external 57 + addition of 66, 68 + weight of 71 + and Induction 221 + +Excluded Middle, Law of 143 + +Experience and the Principle of + Indifference 100 + +Fechner, and median 201 + and law of sensation 208 + and lotteries 364 + +Fermat, formula of 242 + +Forbes, J. D.|inote 20, 294 + +Forbes, J. D. 21 + +Frazer, Sir J. 245 + +Frequency curves 199 + and statistics 328 + +Frequency, statistical 330 + +Frequency theory|ifoll 92 + +Frequency theory + and randomness 290 + and Bernoulli's Theorem 344 + and Rule of Succession 378 + +Fresnel and simplicity 206 + +Fries|inote 15 + +Galton 321 + and Fechner's law 208 + +Gambling 319 + +Gauss, and laws of error|inote 196 + +Gauss, and laws of error 198 + and arithmetic mean 206 + and Least Squares 210 + +Generalisation 389 + +definition of 222 + from statistics 328 + +Generator properties 253 + plurality of 254, 256, 257 + +Geometrical probability 47, 62 + +German logicians 87 + +Gibbon 29, 322, 333 + +Gilman, B. I., and symbolic probability 156 + +Goldschmidt|inote 29 + +Goodness, organic nature of 310 + +Graunt|inote 392 + +Great Numbers, Law of|ifoll 333 + +Great Numbers, Law of 82, 330 +% \item Greville, Fulke, 466 + +Grimsehl|inote 248 + +Grimsehl + and Marbe|inote 365 + +Groups, of propositions 117, 124 + definition of 120, 125 + real and hypothetical 129 + +Grünbaum and Marbe|inote 365 + +Hagen, and error 207 + and discordant observations|inote 214 + +Halley and mortality statistics 332 + +Herodotus 307 + +Herschell and binary stars 294 + +Houdin|inote 364 + +Hudson, W. H., and animism|inote 247 + +Hume 52, 70, 80, 81, 82, 83, 239, 427 + and testimony 182 +%% -----File: 473.png---Folio 462------- +%\item Hume (\textit{contd.})--- + and Induction 218, 233, 265, 272 + and analogy 222, 224 + and chance 282 + +Huyghens 82 + and six@{and `\textit{six}'}#six 247 + +Hypothesis 7 + +Hypothetical entities 299 + +Implication 124 + +Impossibility 15 + definition of 120 + relation of 134 + +Inconsistency, definition of 120 + +Independence, for knowledge 107, 165 + definition of 120, 138 + Theorem of 121, 146 + of events 164 + and law of error 195 + and measurement 204 + and averages 212 + and discordant observations 214 + and chance 283 + +Index numbers 211 + +Induction@{`\textit{Induction}'}#induction 274 + +Induction 97 + Principle of 68 + and frequency theory 98, 99, 107 + and Logic 217 + pure 218 + universal 220, 406, 417 + validity of 221 + and statistics|ifoll 327 + statistical|ifoll 406 + +Inductive correlation 220, 257, 258, 392, 397, 406 + +Inductive hypothesis 260, 264 + +Inductive method 260 + +Inference 129 + necessary 120, 139 + hypothetical and assertoric 130 + statistical|ifoll 327 + +Insurance 22, 285, 404 + +Intuition \textit{versus} experience 86 + and ethical judgment 312 + +Inverse Probability 149, 174 + and Venn 100 + and frequency theory 106 + Theorem of 121 + and statistics|inote 370 + and statistics 369 + and Bowley 425 + +Irrelevance 255 + judgments of 54 + definition of 55, 120, 138 + Theorem of 121, 146 + +James, W., and spirits 301 + +Jesuits 308 + +Jevons|inote 244 + +Jevons + and equiprobability|inote 42 + and Inverse Probability 178 + and index numbers 212 + and Induction 222, 238, 243, 265, 273, 274 + and analogy 246 + and coin-tossing 362 + and Rule of Succession 382 + +Johnson, W. E. 116 + and propositions|inote 11 + and added evidence 68 + and cumulative formula 121, 150, 153, 155 + and groups 124 + and testimony 183 + +Judgments 54 + of preference and relevance 65 + direct 70 + disjunctive 77 + +Kahle and the Probability relation 90 + +Kant 333 + and Hume 272 + +Kapteyn, Prof.\ J. C.#Kapteyn + and law of error 199 + +Knowledge 10 + kinds of 3, 4 + direct and indirect 12, 262 + incomplete and proper 13 + of logical relations 14 + probable and vague 17 + relativity of 17 + vague and distinct 53 + homologic and ontologic 276, 288 + and ignorance 281 + and chance 289 + +Kries, von|inote 44, 45, 46, 67 + +Kries, von 42, 50, 84 + and equiprobability 87 + and Principle of Indifference 172 + and independence 173 + and Inverse Probability 176 + and knowledge 276 + and Cournot|inote 284 + and School of Lexis 394 + +Lacroix|inote 184 + +Lambert and Least Squares 210 + +Lämmel|inote 47 + +Lämmel + and symbolic probability 156 + +Laplace|inote 15, 28 + +Laplace 31, 82, 83, 84, 318, 427 + school of 44, 51, 86, 358, 365 +%% -----File: 474.png---Folio 463------- +%\item Laplace (\textit{contd.})--- + and relation of Probability 91 + and independence 170 + and Inverse Probability 175, 178 + and testimony 180, 182 + and doctrine of averages 202 + and arithmetic mean 206 + and Least Squares 210 + and Induction 220, 239, 265, 273 + and chance 282 + and planets|inote 293 + and Quetelet 334 + and Bernoulli's Theorem 340, 341, 370 + and Rule of Succession|inote 351, 359 + and Rule of Succession 368 + and birth proportions 364 + and unknown probabilities 370 + and Bayes' Theorem 380 + and statistical series 392 + +Laurent and gambling 319 + +Law|inote 311 + +Law of error|ifoll 194 + +Law of error + and arithmetic mean 197 + and geometric mean 198 + and median 200 + and mode 203 + normal law 199, 202, 205 + Lexis and 398 + +Least Squares and Venn 206 + method of 202, 205, 206, 209 + +Lee and tradition|inote 184 + +Legendre and Least Squares 210 + +Leibniz|inote 24 + +Leibniz 308, 368, 392, 427 + and arithmetic average 206 + and Induction 272 + +Lexis, and asymmetry of statistical frequency|inote 359 + +Lexis, and asymmetry of statistical frequency + and Marbe|inote 365 + method of|ifoll 393, 397 + method of 384 + and Edgeworth 401 + and statistical stability|inote 419 + and statistical stability 415 + +Locke 76, 80, 82, 83, 308, 323 + on tradition 184 + and weight of evidence 313 + +Logic, academic 3 + of probability 8 + of implication 58 + and Induction 217, 245 + and initial probability 299 + +Logical priority 129 + +Lotteries|ifoll 364 + +Lotteries|inote 333 + +Lotteries 361 + published results of 363 + +Lotze 89 + and Rule of Succession 382 + +Lucretius 427 + +M'Alister, Sir Donald, and laws of error 198 + +Macaulay and Bacon 266 + +McColl, and symbolic probability 155 + and Boole|inote 167 + and Inverse Probability 176 + and Challenge@{and `\textit{Challenge Problem}'}|inote#Challenge 188 + +Macfarlane, and independence|inote 169 + +Macfarlane, and independence + and tradition 185 + and Challenge@{and `\textit{Challenge Problem}'}|inote#Challenge 187 + +Maclaurin, Theorem of 207 + +Marbe, Dr.\ Karl, and roulette 365 + +Marginal utility 318 + +Markoff, A. A.|inote 177 + +Markoff, A. A. + and Inverse Probability 176 + and Tchebycheff's Theorem 357 + +Mathematical Expectation 311, 315, 316 + +Mathematicians, and probability 84 + and cumulative formula 152 + and laws of error 207 + and ethics 316 + +Maxim, Sir Hiram|inote 364 + +Maxwell|inote 172 + +Maxwell + and theory of gases 172 + +Mayer and Least Squares 210 + +Means and laws of error|ifoll 194 + +Measurement of Probability 34, 158, 311 + and frequency theory 94 + and induction 259, 388 + and psychical research 302 + and ethics 311 + +Median and laws of error 200 + +Meinong 78 + +Meissner, Otto, and dice-throwing 363 + +Memory 14 + +Mendelism and statistics 335, 419, 428 + +Merriman, Mansfield, and Least Squares 209 + +Metaphysics and certainty 239 + +Method of Difference 246 + +Michell 302 + and Inverse Probability 174 + and binary stars 294 + +Middle Term, Fallacy of 68, 155 + +Mill, and inductive correlations 220 + and induction|ifoll 265 + and plurality of causes|inote 267 + and probability|inote 268 + and pure induction 269 + methods of 270 + and limited variety 271 + +Modality and probability|inote 16 +%% -----File: 475.png---Folio 464------- +%\item Modality (\textit{contd.})--- + +Modality and probability + Venn and 98 + +Mode, and law of error 203 + asymmetry about 361 + +Monte Carlo 364 + +Moore, G. E.|inote 240 + +Moore, G. E. 19, 309 + +Morgan, \textit{vide} De Morgan 0 + +Multiplication 135 + definition of 120 + theorems of 121, 148, 342 + of instances|ifoll 233 + +Munro|inote 370 + +Necessary connection, law of 251 + +Newton, and induction 244 + ans seven@{and `\textit{seven}'}#seven 247 + and Bacon 265 + +Nitsche, A.|inote 45, 50, 172 + +Nitsche, A. 78 + +Occurrences, remarkable 302 + +Pascal 82 + +Pearson, Karl|inote 351 + +Pearson, Karl 84 + and frequency theory 100 + and arithmetic mean 208 + and stars 297 + and asymmetry|inote 359 + and asymmetry 347 + and generalised Probability curves 347 + and roulette 364 + and Rule of Succession 379, 382 + +Peirce|inote 50 + +Peirce 304 + and randomness 290 + +Petersburg Paradox 316 + psychology of 318 + and Buffon 362 + +Peterson and tradition 184 + +Physics and initial probability 299 + +Planets, movements of 293 + +Playfair, Dr.\ Lyon 305 + +Plurality of causes and Mill 267 + +Poetry and statistics 401 + +Poincaré, Henri 48, 84 + and independence 173 + and chance 284, 289 + +Poisson|inote 51, 362 + +Poisson + on testimony 180 + and least errors 207 + and Petersburg Paradox 317 + and gambling 319 + and great numbers 333, 336 + Theorem of 344 + and statistical frequency 348 + and Tchebycheff 357 + and inverse of Bernoulli's Theorem 370 + +Poretzki, Platon S., and symbolic + probability 157 + +Port Royal logic 70, 80, 321 + and probabilism 308 + +Prediction, value of 305 + +Price and Bayes|inote 174 + +Primitive people and rational belief 245 + +Principle of compelling reason 86 + +Principle of Indifference|ifoll 83 + +Principle of Indifference 42, 81, 87, 104, 107, 171 + analysis of 53 + modification of 55, 58 + and induction 99 + and measurement 160 + and Psychical Research 302 + and ethics 310 + and statistics 367 + and Laplace 372, 374 + and Rule of Succession 377 + +Principle of Non-Sufficient Reason 41, 85 + +Principle of superposition of small + effects 249 + +Probabilism 308 + +Probability@{`\textit{Probability}'} 8 + Venn's use of 95 + Edgeworth's use of|inote 96 + +Probability, and relevant knowledge 4 + objective relation of 5, 8, 281 + mathematical 6 + dependent on evidence 7 + philosophical definition of 8 + three senses of 11 + measurement of|ifoll 20 + measurement of 37 + and law 24 + and similarity 28, 36 + comparison of 34, 66, 160 + series of 35, 38 + geometrical@{`\textit{geometrical}'} 47, 48, 62 + and rational belief 97 + and statistical frequency 98 + and truth frequency|ifoll 101, 337 + Inverse 106, 149 + and truth 116, 322 + negative 139 + finite 237 + and randomness 291 + and planetary orbits 293 + and binary stars 294 + and star drifts 296 + and final causes 297 + and spirits 300, 301 + and telepathy 300 + and ethics 307 +%% -----File: 476.png---Folio 465------- +%\item Probability (\emph{contd}.)--- + from statistics|ifoll 367 + unknown@{`\textit{unknown},' and Laplace}#Laplace 372 + +Probability relation 4, 8, 13, 134 + +intuition of 52 + +Probable error 74 + +Proctor|inote 364 + +Proposition, characterisation of 3, 4 + primary and secondary 11, 13 + knowledge of 12 + self-evident 17 + classes of|ifoll 101 + groups of 117, 124 + sub-groups of 126, 129 %[** TN: Regularized to subgroups] + disjunction and conjunction of 134 + synthetic 263 + existential 276 + +Propositional function 56 + and induction 222 + and randomness 291 + +Psychical Research|ifoll 278 + +Psychology and probability 52 + +Pythagoras and `\textit{seven}' 246 + +Quetelet|inote 333 + +Quetelet 334, 335, 401, 418, 427, 428 + and arithmetic mean 208 + and balls 362 + and statistical stability 393 + +Randomness 281, 290, 412 + Pearson's use of 297 + +Relation, of probability 6 + of `\textit{between}'#between 35, 39 + +Relativity, of knowledge 17 + of probabilities 102 + doctrine of, and the Law of Uniformity|inote 248 + +Relevance, judgments of 54 + and frequency theory 104 + theorems of 147 + +Remarkableness 302 + +Requirement 129 + +Risk 315 + and ethics 313 + and Petersburg Paradox 319 + `\textit{moral}'#moral 320, 322 + `\textit{physical}'#physical 322 + +Roulette 361, 364 + published results of|inote 363 + +Rule of Succession|inote 359 + +Rule of Succession 368, 372, 374 + proof of 375 + and frequency theory 378 + and Pearson|inote 380 + +Russell, Bertrand|inote 124 + +Russell, Bertrand 19, 115, 126 + and inference 117 + and implication 124 + +Schematisation 67 + +Schröder and symbolic probability 157 + +Selection, random 292 + +Series of probabilities 35, 38 + and frequency theory 93 + independent 283, 420 + organic 399, 420 + Gaussian|inote 421 + +Sigwart 88 + and inverse probability 178 + and induction 273 + +Simmons and asymmetry in Bernoulli's + Theorem 359 + +Simpson and Least Squares 210 + +Small Numbers, Law of|ifoll 401 + +Society for Psychical Research|inote 298 + +Space 255 + and uniformity 226 + irrelevance of 301 + +Spedding and Ellis and Bacon|inote 265, 266 + +Spielräume, doctrine of 88 + +Spinoza|inote 116, 282 + +Spirits, probability of 300 + +Star drifts 296 + +Stars, binary 294 + +Statistical frequency, theory of|ifoll 93 + +Statistical frequency, theory of + generalisation of 101 + criticism of 103 + stability of 336, 392-415 + fluctuation of 392 + +Statistical inference|ifoll 327 + +Statistical inference + induction|ifoll 406 + +Statistics, and prediction 306 + descriptive and inductive 327 + +Stumpf|inote 44, 50, 172 + +Sub-analogies 223, 229 + +Sub-groups of propositions 126, 129 %[** TN: Regularized to subgroups] + +Succession, Law of 82 + \textit{See} Rule of 0 + +Süssmilch and regular frequencies 333 + +Taylor, Jeremy|inote 308 + +Tchebycheff, Theorem of 353, 355 + and Poisson's Theorem 357 + +Telepathy, probability of 300 + +Terrot, Bishop|inote 43 + +Terrot, Bishop + and Whately|inote 179 + and combination of premisses 179 + +Testimony, theory of 180 +%% -----File: 477.png---Folio 466------- + +Time 255 + and uniformity 226 + irrelevance of 301 + +Todhunter|inote 294, 318, 370 + +Todhunter + and Bayes 175 + and Craig 184 + and Petersburg Paradox 316 + and Bernoulli's Theorem|inote 340 + +Truth and probability|inote 116 + +Truth and probability 322 + +Truth frequency 101, 406 + +Tschuprow|inote 399 + +Tschuprow 358 + and statistical frequency|inote 394 + and statistical frequency 348 + method of 384 + +Uniformity of Nature, Law of 226, 248, 255, 263, 276 + and Mill 270 + +Universal Causation, Law of 248 + +Universal Induction and statistical + methods 389, 406-417 + +Universe of reference 117, 129, 130 + +Unknown probabilities 372, 373, 375 + +Variables in Probability|inote 412 + +Variables in Probability 58, 123 + +Variety 234 + and induction 219 + limitation of 258, 260, 427 + +Venn|inote 106, 294 + +Venn 84 + and experience 85 + and Bernoulli 86, 341 + and frequency theory|ifoll 93 + and inverse probability 100 + and Least Squares|inote 206 + and induction 273 + and chance 288 + and `\textit{random}'#random 290 + and Rule of Succession 372, 378 + +Weight, of evidence 312 + and ethics 315 + +Weighting of averages 211 + +Weldon and dice 362 + +Whately and combination of misses 178 + +Whitehead, and frequency theory + and invalid inference|inote 329 + +Whittaker, E. T., and Rule of Succession|inote 376 %[** TN: "cession" in orig] + +Wilbraham, H., and Boole|inote 167 + +Wolf and dice 362 + +Yule|inote 349, 361 + +Yule + and approximation 161 + and independence 166 + and `\textit{statistics}'#statistics 327 + and coin-tossing|inote 346, 361 + and correlation 421, 424 +\fi + +%[** TN: index environment prints this at end] +\iffalse +\begin{center} +\begin{minipage}{0.55\linewidth}{\footnotesize O False and treacherous Probability,\\ +Enemy of truth, and friend to wickednesse;\\ +With whose bleare eyes Opinion learnes to see,\\ +Truth's feeble party here, and barrennesse.} +\end{minipage} \\ +\small THE END\rule{0em}{10em}\\ +\textsc{\footnotesize PRINTED BY R. \& R. CLARK, LTD., EDINBURGH\rule{0em}{12em}} +\end{center} +\fi +%% -----File: 478.png---Folio 467------- +%[Blank Page] + +\cleardoublepage + +%%%% LICENSE %%%% +\pagenumbering{Alph} +\pagestyle{fancy} +\phantomsection +\pdfbookmark[0]{Project Gutenberg License}{License} +\fancyhf{} +\fancyhead[C]{\CtrHeading{Project Gutenberg License}} +\SetPageNumbers + +\begin{PGtext} +End of Project Gutenberg's A Treatise on Probability, by John Maynard Keynes + +*** END OF THIS PROJECT GUTENBERG EBOOK A TREATISE ON PROBABILITY *** + +***** This file should be named 32625-pdf.pdf or 32625-pdf.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/2/6/2/32625/ + +Produced by Andrew D. 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