THE THEORY OF PROBABILITY. 241 



probabilities that A is B, ^ and that B is C, ^, we have 

 no right to suppose that the probability of A being C 

 is reduced by the argument in its favour. If the conclu- 

 sion is true on its own grounds, the failure of the argument 

 does not affect it ; thus its total probability is its ante- 

 cedent probability, added to the probability that this 

 failing, the new argument in question establishes it. 

 There is a probability ^ that we shall not require the 

 special argument ; a probability ^ that we shall, and 

 a probability ^ that the argument does in that case 

 establish it. Thus the complete result is ^ + ^ x ^, or -f . 

 In general language, if a be the probability formed on 

 a particular argument, and c the antecedent probability, 

 then the general result is 



i (i - a)(i c), or a + c ac. 



We may put it still more generaUy in this way : Let 

 a, 6, c, d, &c., be the probabilities of a conclusion grounded 

 on various arguments or considerations of any kind. It is 

 only when all the arguments fail that our conclusion 

 proves finally untrue ; the probabilities of each failing 

 are respectively i a, i 6, i c, &c. ; the probability 

 that they will all fail (i a)(i b)(i c)... ; therefore 

 the probability that the conclusion will not fail is 

 i (i a)(i 6)(i c)...&c. On this principle it follows 

 that every argument in favour of a fact, however flimsy 

 and slight, adds probability to it. When it is unknown 

 whether an overdue vessel has foundered or not, every 

 slight indication of a lost vessel will add some proba- 

 bility to the belief of its loss, and the disproof of any 

 particular evidence will not disprove the event. 



We must apply these principles of evidence with great 

 care, and observe that in a great proportion of cases the 

 adducing of a weak argument does tend to the disproof of 

 its conclusion. The assertion may have in itself great 

 inherent improbability as being opposed to other evidence 



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