THE NEW LOGICS. i6i 



But this is not all. Mr. Russell's logic of propo- 

 sitions is the study of the laws in accordance with 

 which combinations are formed with the conjunctions 

 if, and, or, and the negative not. This is a consider- 

 able extension of the ancient logic. The properties of 

 the classical syllogism can be extended without any 

 difficulty to the hypothetical syllogism, and in the 

 forms of this latter we can easily recognize the 

 scholastic forms ; we recover what is essential in the 

 classical logic. But the theory of the syllogism is still 

 only the syntax of the conjunction if and, perhaps, 

 of the negative. 



By adding two other conjunctions, and and or, 

 Mr. Russell opens up a new domain to logic. The 

 signs and and or follow the same laws as the two 

 signs X and +, that is to say, the commutative, 

 associative, and distributive laws. Thus and repre- 

 sents logical multiplication, while or represents logical 

 addition. This, again, is most interesting. 



Mr. Russell arrives at the conclusion that a false 

 proposition of any kind involves all the other pro- 

 positions, whether true or false. M. Couturat says 

 that this conclusion will appear paradoxical at first 

 sight. However, one has only to correct a bad 

 mathematical paper to recognize how true Mr. 

 Russell's view is. The candidate often takes an 

 immense amount of trouble to find the first false 

 equation ; but as soon as he has obtained it, it is 

 no more than child's play for him to accumulate 

 the most surprising results, some of which may 

 actually be correct. 



(1.777) II 



