80 INDUCTION. 



when he comes to its applications he no longer 

 restricts it to the ascertainment of causes alone, but, 

 without any previous notice substitutes for the idea 

 of causes that of hypotheses, or suppositions of any 

 kind. In this extended sense, I do not conceive 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, multiplied 

 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 previously 

 happened, is not the antecedent probability of M 

 multiplied by that of A, but the antecedent proba- 

 bility of A only. The antecedent probability of M 

 cannot be an element of a question into which the 

 occurrence of M enters not as a contingency but as a 

 certainty. What the product of the antecedent pro- 

 babilities of A and M does give, is, not the probability 

 of the one when the other is a known past event, but 

 the antecedent probability of the two together, consi- 

 dered as future events. 



This error of Laplace has not been harmless. We 

 shall see hereafter, in treating of the Grounds of 



