374 BISHOP TERROT ON PROBABILITIES. 



Now ~^-j when rg)> sp, Or rq — rp /> sp — rp, 



or r(q-p)> p. (s-r), or j^>^ . 



When the ratios only are given, any conceivable case of the grounds upon 

 which the probabilities are given may be represented by mp, mq, nr, and ns. 



Hence the original probability is — - , the composite is — ^— r . and this is greater 



° l J mq L m q + n s 



than "HOlL^ when 



m q 



m?pq + mnqr\ m 2 p q + mnp s 



or r q ^> p s, as before. 



(10.) Valid objections may, I think, be made to the last paragraph "of the sec- 

 tion in the Encyclopsedia already referred to. As this is not long, I quote it en- 

 tire. " The following theorem will be readily admitted on its own evidence. If 

 any assertion appear neither likely nor unlikely in itself, then any logical argument 

 in its favour, however weak the premises, makes it in some degree more likely than 

 not. In the manner in which writers on Logic apply the calculus of probabilities, 

 this is never the consequence of their suppositions. For what we have called a 

 is their resulting probability of the argument. Suppose, for instance, a writer 



on logic presumed that the argument from analogy gave -r- to the probabilit} r 



that there is vegetation in the planets, which must be regarded as a thing neither 

 likely nor unlikely in regard of evidence from any other source, he would take 



— to be the probability of this result, that is, less after an argument in its favour 



1 1 3 13 " 

 than it was before. We substitute „ + « ttt=?^- 

 J 2 2 • 10 20 



This numerical equation is the value of the expression (a + e-ae), when 

 1 3 



«= g € = jo - I nave already shown that this expression does not truly represent 

 the composite force of the two probabilities a and e. But farther than this, the 

 argument from analogy, giving — as the probability of the affirmative, is an 



argument, not in favour of but against the proposition that there is vegetation 

 in the planets. It implies that for every three reasons for believing that there is, 

 there are seven for believing that there is not ; and, consequently, the effect of 

 the argument ought to be to diminish our disposition to believe the proposition, 

 or, in other words, to diminish its probability. 



(11.) But it may be worth while to examine whether the fraction - be, after 



At 



all, a true available expression for the probability of an event, which is neither 

 likely nor unlikely to happen, or to have happened, there being no evidence, no 

 reasons for belief, either for or against it. 



