CHAP. XVIII.] ELEMENTARY ILLUSTRATIONS. 287 



whence, proceeding as above, 



Prob. z = cp, 

 c having the same interpretation as before. Hence 



Prob. xy cp 



Prob. x 1 - p + cp ' 



for the probability of the truth of the conditional proposition 

 given. 



Now in the science of pure Logic, which, as such, is conver- 

 sant only with truth and with falsehood, the above disjunctive 

 and conditional propositions are equivalent. They are true and 

 they are false together. It is seen, however, from the above in- 

 vestigation, that when the disjunctive proposition has a proba- 

 bility p, the conditional proposition has a different and partly in- 



cv 

 definite probability - . Nevertheless these expressions 



are such, that when either of them becomes 1 or 0, the other as- 

 sumes the same value. The results are, therefore, perfectly con- 

 sistent, and the logical transformation serves to verify the formula 

 deduced from the theory of probabilities. 



The reader will easily prove by a similar analysis, that if the 

 probability of the conditional proposition were given as /?, that 

 of the disjunctive proposition would be 1 - c + cp, where c is the 

 arbitrary probability of the truth of the proposition X. 



9. Ex. 7. Required to determine the probability of an event 

 #, having given either the first, or the first and second, or the 

 first, second, and third of the following data, viz. : 



1st. The probability that the event x occurs, or that it alone 

 of the three events x, y, z, fails, is p. 



2nd. The probability that the event y occurs, or that it alone 

 of the three events x, y 9 z, fails, is q. 



3rd. The probability that the event z occurs, or that it alone 

 of the three events x, y, z, fails, is r. 



SOLUTION OF THE FIRST CASE. 



Here we suppose that only the first of the above data is 

 given. 



