324 INDUCTION. 



beino- the only function of the two which obeys that particular law of 

 Variation.^ Therefore, the probability that M was produced by either 

 cause, is as the antecedent probability of the cause, multiplied by the 

 probability that if it existed it would produce M. Which was to bo 

 demonstrated. 



Or we may prove the third case as we proved the first and second. 

 Let A be twice as frequent as B ; and let them also be unequally likely, 

 when they exist, to produce M : let A produce it twice in four times, 

 B thrice in four times. The antecedent probability of A is to that of. 

 B as 2 to 1 ; the probabilities of their producing M are as 2 to 3 ; the 

 product of these ratios is the ratio of 4 to 3, which, therefore, if the 

 theorem be true, will be the ratio of the probabilities that A or B was. 

 the producing cause in the given instance. And such will that ratio 

 really be. For since A is twice as frequent as B, out of twelve cases in 

 which one or other exists, A exists in 8 and B in 4, But of its eight 

 cases. A, by the supposition, produces M in only 4, while B of its four 

 cases produces M in 3. M, therefore, is only produced at all m 

 seven of the twelve cases; but in four of these it is produced by A, in 

 three by B ; hence, the probabilities of its being produced by A and 

 by B are as 4 to 3, and are expressed by the fractions | and ^. Which 

 was to be demonstrated. 



It is here necessary to point out another serious oversight in La- 

 place's theory. When he first introduces the foregoing theorem, he 

 characterizes it correctly, as the principle for determining to which of 

 several causes we are to attribute a known fact. But after having con- 

 ceived the principle thus accurately, when he comes to its applications 

 he no longer restricts it to the ascertainment of causes alone, but, with- 

 out any previous notice substitutes for the idea of causes that o^ hypo- 

 theses, or suppositions of any kind. In this extended sense, I do not 

 cpnceive the proposition to be tenable. The hypotheses must be either 

 causes, or at least signs showing the existence of causes. If we could 

 be permitted to substitute mere suppositions affording no ground for 

 concluding that the effect would be produced, in the room of causes 

 capable of producing it, the theorem thus extended would stand as 

 follows. A fact, M, having happened, the probability of the truth of 

 any arbitrary supposition altogether unconnected with M, is as the 

 antecedent probability of the supposition, multiphed by the probability 

 that if the supposition was true M would happen ; that is, multiplied 

 by the antecedent probability of M, since M is neither more nor less 

 probable on account of a supposition which has nothing to do with the 

 causes of it. Now the proposition as thus stated, is an absurdity. The 

 probability that when M happened A had pre\aously happened, is not 

 the antecedent probability of M multiplied by that of A, but the ante- 

 cedent probability of A only. The antecedent probabiUty of M cannot 

 be an element of a question into which the occun'ence of M enters not 

 as a contingency but as a certainty. What the product of the antece- 

 dent probabilities of A and M does give, is, not the probability of the 

 the one when the other is a known past event, but the antecedent prob- 

 ability of the two together, considered as fiiture events. 



This error of Laplace has not been harmless. We shall see here- 

 after, in ti-eating of the Grounds of DisbeHef, that he has been led by 

 it into serious practical mistakes when attempting to pronounce upon 

 the circumstances which render any statement incredible. 



