Testimony and Arguments, 23 



the argument a is invalid, b invalid =a(l— b) ; 



a invalid, b valid =6(1— a); 



neither is valid =(1— «)(1— b) ) 



Sum . . =1— ab. 

 Hence the probability of the conclusion to which a leads 

 fl(l-ft) 



Now the case of arguments suggests an important question. 

 It often happens that when a proposition has been established 

 on probable evidence, it is found to lead to a further inference 

 which admits of being tested directly to some extent. Thus the 

 proposition A may assert that a certain narrative is the work of 

 a credible witness, and B that a particular circumstance contained 

 in it is true. Our hypothesis is, If A is true, B is true; and 

 from this we infer conversely, If B is false, A is false. Logically 

 these two propositions are convertible ; yet there lurks a great 

 fallacy in the inference when we have to do, not with certain, but 

 with probable propositions. For example, if the truth of B does 

 not follow with certainty from that of A, wc have the following 

 cases possible :- — 



A true and B true ; let the probability of this be X, 

 A true and B false ; „ „ == ft, 



A false and B true ; „ „ = v, 



A false and B false ; „ „ = p 3 



where 



\ + fJb+V + p = l. 



Then if A is true, the probability that B is true is 



\ 

 X + p 

 If B is false, the probability that A is false is 



= _P 



p + fj, 



The former expression represents the probability of the state- 

 ment, If A is true, B is true : the latter that of the converse, If 

 B is false, A is false ; and it appears at once that these are really 

 independent. Suppose it to be known that the third case is 

 impossible, or nearly so, i. e. v=0; then, in order to determine 

 the probability of the second inference, we must know besides 

 that of the first, the absolute probability of A (or B). Let the 

 probability of A be v(=\ + fi) ; then the probability of the in- 



