402 PROCEEDINGS OF THE AMERICAN ACADEMY 



Five hundred and eighty-fifth Meeting. 



September 10, 1867. — Adjourned Statute Meeting. 



The President in the chair. 



The President called the attention of the Academy to the 

 recent decease of Dr. J. Mason Warren and Dr. James Jack- 

 son, of the Resident Fellows ; of Jeremiah Day, former Presi- 

 dent of Yale College, of the Associate Fellows ; and of Sir 

 William Lawrence, Augustus Boeckh, and Michael Faraday, of 

 the Foreign Honorary Members. 



The following paper was presented : — 



Upon the Logic of Mathematics. By C. S. Peirce. 



Part I. 



The object of the present paper is to show that there are certain 

 general propositions from which the truths of mathematics follow syl- 

 logistically, and that these propositions may be taken as definitions 

 of the objects under the consideration of the mathematician without 

 involving any assumption in reference to experience or intuition. 

 That there actually are such objects in experience or pure intuition is 

 not in itself a part of pure mathematics. 



Let us first turn our attention to the logical calculus of Boole. I 

 have shown in a previous communication to the Academy, that this cal- 

 culus involves eight operations, viz. Logical Addition, Arithmetical 

 Addition, Logical Multiplication, Arithmetical Multiplication, and the 

 processes inverse to these. 



Definitions. 



1. Identity, a == b expresses the two facts that any a is b and any 

 b is a. 



2. Logical Addition, a -\r b denotes a member of the class which 

 contains under it all the a's and all the b's, and nothing else. 



3. Logical Multiplication, a , b denotes only whatever is -both a and b. 



4. Zero denotes nothing, or the class without extent, by which we mean 

 that if a is any member of any class, a -^ is a. 



5. Unity, denotes being, or the class without content, by which we 

 mean that, if a is a member of any class, a is a , 1 . 



6. Arithmetical Addition, a -)- b, if a , b = is the same as a -fr b, 

 but, if a and b are classes which have any extent in common, it 

 is not a class. 



