250 PROCEEDINGS OF THE AMERICAN ACADEMY 



On an Improvement in Boole's Calculus of Logic. By C. S. 



Peirce. 



The principal use of Boole's Calculus of Logic lies in its applica- 

 tion to problems concerning probability. It consists, essentially, in a 

 system of signs to denote the logical relations of classes. The data of 

 any problem may be expressed by means of these signs, if the letters 

 of the alphabet are allowed to stand for the classes themselves. From 

 such expressions, by means of certain rules for transformation, ex- 

 pressions can be obtained for the classes (of events or things) whose 

 frequency is sought in terms of those whose frequency is known. 

 Lastly, if certain relations are known between the logical relations and 

 arithmetical operations, these expressions for events can be converted 

 into expressions for their probability. 



It is proposed, first, to exhibit Boole's system in a modified form, 

 and second, to examine the difference between this form and that 

 given by Boole himself. 



Let the letters of the alphabet denote classes whether of things or 

 of occurrences. It is obvious that an event may either be singular, 

 as " this sunrise," or general, as " all sunrises." Let the sign of 

 equality with a comma beneath it express numei'ical identity. Thus 

 a == i is to mean that a and h denote the same class, — the same 

 collection of individuals. 



Let a -\r h denote all the individuals contained under a and b tosreth- 

 er. The operation here performed will differ from arithmetical addition 

 in two respects: 1st, that it has reference to identity, not to equality; 

 and 2d, that what is common to a and h is not taken into account twice 

 over, as it would be in arithmetic. The first of these differences, how- 

 ever, amounts to nothing, inasmuch as the sign of identity would indi- 

 cate the distinction in which it is founded ; and therefore we may 

 say that 



(1.) If No a is 6 a^b = a-\-b 



It is plain that 

 (2.) a -jr a == a 



and also, that the process denoted by -jr, and which I shall call the 

 process of logical addition, is both commutative and associative. 

 That is to say 



(3.) a^b=:b^a 



