we have to believe that it has taken place, which is conceptualistic; but shortly after they state that it is the ratio of the number of cases favorable to the event to the total number of cases favorable or contrary, and all equally possible. Except that this introduces the thoroughly unclear idea of cases equally possible in place of cases equally frequent, this is a tolerable statement of the materialistic view. The pure conceptualistic theory has been best expounded by Mr. De Morgan in his "Formal Logic: or, the Calculus of Inference, Necessary and Probable."
The great difference between the two analyses is, that the conceptualists refer probability to an event, while the materialists make it the ratio of frequency of events of a species to those of a genus over that species, thus giving it two terms instead of one. The opposition may be made to appear as follows:
Suppose that we have two rules of inference, such that, of all the questions to the solution of which both can be applied, the first yields correct answers to 81100, and incorrect answers to the remaining 19100; while the second yields correct answers to 93100, and incorrect answers to the remaining 7100. Suppose, further, that the two rules are entirely independent as to their truth, so that the second answers correctly 93100 of the questions which the first answers correctly, and also 93100 of the questions which the first answers incorrectly, and answers incorrectly the remaining 7100 of the questions which the first answers correctly, and also the remaining 7100 of the questions which the first answers incorrectly. Then, of all the questions to the solution of which both rules can be applied—
| both answer correctly | 93 | of | 81 | or | 93 × 81 | ; |
| 100 | 100 | 100 × 100 | ||||
| the second answers correctly and the first incorrectly | 93 | of | 19 | or | 93 × 19 | ; |
| 100 | 100 | 100 × 100 | ||||
| the second answers incorrectly and the first correctly | 7 | of | 81 | or | 7 × 81 | ; |
| 100 | 100 | 100 × 100 | ||||
| and both answer correctly | 7 | of | 19 | or | 7 × 19 | ; |
| 100 | 100 | 100 × 100 |
Suppose, now, that, in reference to any question, both give the same answer. Then (the questions being always such as are to be answered by yes or no), those in reference to which their answers agree are the same as those which both answer correctly together with those which both answer falsely, or
| 93 × 81 | + | 7 × 19 |
| 100 × 100 | 100 × 100 |
of all. The proportion of those which both answer correctly out of those their answers to which agree is, therefore—
| 93 | × | 81 | or | |||||||||||
| 100 | × | 100 | 93 | × | 81 | |||||||||
| 93 | × | 81 | + | 7 | × | 19 | (93 | × | 81) | + | (7 | × | 19) | |
| 100 | × | 100 | 100 | × | 100 |
