Part 9

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:—

Odds in favour of an event Probability for Probability against

1 to 1 ½ ½ 2 to 1 ⅔ ⅓ 3 to 1 ¾ ¼ 3 to 2 ⅗ ⅖ 4 to 1 ⅘ ⅕ 4 to 3 ⁴⁄₇ ³⁄₇ 5 to 1 ⅚ ⅙ &c. &c. &c.

Let the probability of a conclusion, as derived from the premises (that is on the supposition that it was never imagined to be possible till the argument was heard), be called the intrinsic probability of the argument. This is found by multiplying together the probabilities of all the assertions which are necessary to the argument. Thus, suppose that a conclusion was held to be impossible until an argument of a single syllogism was produced, the premises of which have severally five to one and eight to one in their favour. Then ⅚ × ⁸⁄₉, or ⁴⁰⁄₅₄, is the intrinsic probability of the argument, and the odds in its favour are 40 to 14, or 20 to 7.

But this intrinsic probability is not always that of the conclusion; the latter, of course, depending in some degree on the likelihood which the conclusion was supposed to have before the argument was produced. A syllogism of 20 to 7 in its favour, advanced in favour of a conclusion which was beforehand as likely as not, produces a much more probable result than if the conclusion had been thought absolutely false until the argument produced a certain belief in the possibility of its being true. The change made in the probability of a conclusion by the introduction of an argument (or of a new argument, if some have already preceded) is found by the following rule.

From the sum of the existing probability of the conclusion and the intrinsic probability of the new argument, take their product; the remainder is the probability of the conclusion, as reinforced by the argument. Thus, a + b − ab is the probability of the truth of a conclusion after the introduction of an argument of the intrinsic probability b, the previous probability of the said conclusion having been a.

Thus, a conclusion which has at present the chance ⅔ in its favour, when reinforced by an argument whose intrinsic probability is ¾, acquires the probability ⅔ + ¾ − ⅔ × ¾ or, ⅔ + ¾ − ½, or ¹¹⁄₁₂; or, having 2 to 1 in its favour before, it has 11 to 1 in its favour after, the argument.

When the conclusion was neither likely nor unlikely beforehand (or had the probability ½), the shortest way of applying the preceding rule (in which a + b − ab becomes ½ + ½b) is to divide the sum of the numerator and denominator of the intrinsic probability of the argument by twice the denominator. Thus, an argument of which the intrinsic probability is ¾, gives to a conclusion on which no bias previously existed, the probability ⅞ or (3 + 4)/(2 × 4).

THE END.

LONDON:—PRINTED BY JAMES MOYES, Castle Street, Leicester Square.

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TRANSCRIBER’S NOTES

1. Silently corrected obvious typographical errors and variations in spelling. 2. Retained archaic, non-standard, and uncertain spellings as printed. 3. Enclosed italics font in underscores.