286 ELEMENTARY ILLUSTRATIONS. [CHAP. XVIII. 



but to every form of deductive ratiocination. Given the proba- 

 bilities separately attaching to the premises of any train of ar- 

 gument ; it is always possible by the above method to determine 

 the consequent probability of the truth of a conclusion legitimately 

 drawn from such premises. It is not needful to remind the 

 reader, that the truth and the correctness of a conclusion are dif- 

 ferent things. 



8. One remarkable circumstance which presents itself in such 

 applications deserves to be specially noticed. It is, that propo- 

 sitions which, when true, are equivalent, are not necessarily 

 equivalent when regarded only as probable. This principle will 

 be illustrated in the following example. 



Ex. 6. Given the probability p of the disjunctive proposition 

 " Either the proposition Yis true, or both the propositions X and 

 Fare false," required the probability of the conditional propo- 

 sition, " If the proposition X is true, Yis true." 



Let x and y be appropriated to the propositions X and Y 

 respectively. Then we have 



Prob.y + (l-x) (1 -#)=;>, 



from which it is required to find the value of -^ ^ . 



Prob. x 



Assume y + (1 - x) (1 - y) = t. (1) 



Eliminating y we get 



(1 - a) (1 - = 0. 



Whence "' 



and proceeding in the usual way, 



Prob. x = 1 - p + cp. (2) 



Where c is the probability that if either Y is true, or X and Y 

 false, X is true. 



Next to find Prob. ocy. Assume 



xy = iv. (3) 



Eliminating y from (1) and (3) we get* 



* (1 - = ; 



