For, when P is true, then Q and R must be false; consequently, neither B nor C can be true, for then Q or R would be true. But either A, B, or C must be true, therefore A must be true; or, when P is true, A is true. In a similar way the remaining assertions may be proved.
| Case 1. If | When P is Q, | A is B |
| When P is not Q, | A is not B | |
| It follows that | When A is B, | P is Q |
| When A is not B, | P is not Q | |
| Case 2. If | When A is greater than B, | P is greater than Q |
| When A is equal to B, | P is equal to Q | |
| When A is less than B, | P is less than Q | |
| It follows that | When P is greater than Q, | A is greater than B |
| When P is equal to Q, | A is equal to B | |
| When P is less than Q, | A is less than B | |
We have hitherto supposed that the premises are actually true; and, in such a case, the logical conclusion is as certain as the premises. It remains to say a few words upon the case in which the premises are probably, but not certainly, true.
The probability of an event being about to happen, and that of an argument being true, may be so connected that the usual method of measuring the first may be made to give an easy method of expressing the second. Suppose an urn, or lottery, with a large number of balls, black or white; then, if there be twelve white balls to one black, we say it is twelve to one that a white ball will be drawn, or that a white ball is twelve times as probable as a black one. A certain assertion may be in the same condition as to the force of probability with which it strikes the mind: that is, the questions
Is the assertion true?
Will a white ball be drawn?
may be such that the answer, ‘most probably,’ expresses the same degree of likelihood in both cases.
We have before explained that logic has nothing to do with the truth or falsehood of assertions, but only professes, supposing them true, to collect and classify the legitimate methods of drawing inferences. Similarly, in this part of the subject, we do not trouble ourselves with the question, How are we to find the probability due to premises? but we ask: Supposing (happen how it may) that we have found the probability of the premises, required the probability of the conclusion. When the odds in favour of a conclusion are, say 6 to 1, there are, out of every 7 possible chances, 6 in favour of the conclusion, and 1 against it. Hence ⁶⁄₇ and ⅐ will represent the proportions, for and against, of all the possible cases which exist.
Thus we have the succession of such results as in the following table:—