x.] THE THEORY OF PROBABILITY. 211 



bability | that we shall not require the special argument ; 

 a probability ^ that we shall, and a probability J that the 

 argument does in that case establish it. Thus the com- 

 plete result is | + i x , or f . In general language, if a 

 be the probability founded on a particular argument, and 

 c the antecedent probability of the event, the general result 

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



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

 a, b, c, &c. be the probabilities of a conclusion grounded 

 on various arguments. It is only when all the arguments 

 fail that our conclusion proves finally untrue ; the proba- 

 bilities of each failing are respectively, i a, I &, ic, 

 &c. ; the probability that they will all fail is (i - a)(i - 1} 

 (i c)... ; therefore the probability that the conclusion 

 will not fail is I (i a)(i Z>)(i c)... &c. It follows 

 that every argument in favour of a conclusion, 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 

 or to the supposed law of nature, and every reasoner may 

 be assumed to be dealing plainly, and putting forward the 

 whole force of evidence which he possesses in its favour. 

 If he brings but one argument, and its probability a is 

 small, then in the formula I (i a)(i c) both a and c 

 are small, and the whole expression has but little value. 

 The whole effect of an argument thus turns upon the 

 question whether other arguments remain, so that we can 

 introduce other factors (i &), (i d), &c., into the above 

 expression. In a court of justice, in a publication having 

 an express purpose, and in many other cases, it is doubtless 

 right to assume that the whole evidence considered to 

 have any value as regards the conclusion asserted, is put 

 forward. 



To assign the antecedent probability of any proposition, 

 may be a matter of difficulty or impossibility, and one 



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