148 



those in which the consequent Y is true ; and v denotes an 

 indeterminate symbol of election. The verbal enunciation, 

 therefore, of equation (1) is, that all the cases of the truth of 

 X are included amongst cases of the trtith of Y: a statement 

 which resembles the proposition, — If X be true, Y is true, — 

 more nearly than the verbal interpretation of equation (2), 

 which asserts that there are no cases of the truth of X in- 

 cluded amongst cases of the falsehood of Y. 



It is interesting to observe how readily the ordinary rules 

 for hypotheticals flow as mathematical consequences from 

 equation (I), regarded merely as an equation between commu- 

 tative and distributive symbols. 



1. Ify = 0, a; = 0. 



2. If a; = 0, it does not follow that y = 0. 



3. If a; be not = 0, r/ is not = 0. 



4. If y be not = 0, it does not follow that x is not = 0. 

 These mathematical results being interpreted, give the fol- 

 lowing well-known rules : 



1. If the consequent be false, the antecedent must be 



false. 



2. The falsehood of the antecedent does not prove the 



falsehood of the consequent. 



3. If the antecedent be true, the consequent must be 



true. 



4. The truth of the consequent does not prove the truth 



of the antecedent. 



The following example, treated by Mr. Boole in his book, 

 p. 57, illustrates the use of the form here proposed for the 

 equations of hypothetical propositions. 



If X be true, either Yis true or Z is true. 



But Y is not true. 



Therefore, if X be true, Z is true. 



The foregoing argument is succinctly expressed by means 

 of the following equations : 



