262 bepoet— 1879. 



4. On tlie Fundamental Principles of the Algebra of Logic. 

 By Alexander Macfarlane, M.A., D.Sc, F.R.S.E. 



In a work recently published, entitled ' The Algebra of Logic,' I have investigated 

 anew the foundations of that branch of mathematical analysis which was originated 

 by Boole in hi3 celebrated treatise on ' The Laws of Thought.' In making this 

 inquiry I have studied the contributions to the subject made by Harley, Venn, 

 Jevons, and other philosophers. 



The difficulty and apparent irrationality of Boole's calculus is due to the fact 

 that it is founded on the old and inadequate theory of the operation of the mind in 

 reasoning about quality. That theory supposes that the mind, in forming a com- 

 pound conception out of two simple conceptions, necessarily considers the second of 

 these as limited by, and in a measure dependent upon, the first ; in the theory 

 which I advance it is maintained that the mind may, on the one hand, form com- 

 pound conceptions in which the second element is entirely dependent on the first ; 

 and, on the other hand, compound conceptions, in which the two elements are 

 mutually independent. 



I consider that the fundamental notion in this branch of analysis is that of a 

 collection of homogeneous objects having differentiating characters. The collection 

 of objects, so far forth as they are homogeneous, may be denoted by u (as they form 

 the universe considered in the particular investigation) ; a differentiating character 

 may be denoted by a small letter, as x. The symbol x applies to, and is entirely 

 dependent upon, u. The arithmetical value of u is the number of the objects 

 considered, and may be singular, plural, or infinitely great. The arithmetical 

 value of x is the ratio of the number of the objects which have the character x to 

 the whole number of objects considered. 



By x = y it is asserted that those of the objects which have the character x are 

 identical with those which have the character y. Hence the members of a logical 

 equation are also equal arithmetically, and have the same sign. When the cha- 

 racters equated are identical, the equation is an identity; when they are merely 

 equivalent, the equation is one of condition. 



The symbol + 1 denotes that mental operation which, when applied to ux, 

 takes them once and arranges them in the positive direction along the line in which 

 the mind moves in counting ; and —1 arranges them along the negative direction. 



These operations are connected by the relations +1-1=0. The symbols (—1)2 



and ( — 1)5 that is, ( — )r, and ( — )2, when applied to ux, arrange them along 

 another and independent line of counting in the positive and negative directions 

 respectively. 



In x + y the two parts are perfectly independent, and therefore are not neces- 

 sarily mutually exclusive. In the expression x — y, the two parts destroy one 

 another as far as possible in virtue of the relation + 1 — 1 = ; the result in general 

 consists of a positive part and a negative part. 



_ Thus a qualitative expression x is in general both positive and negative. When 

 it is positive and not negative, it satisfies the condition x % = x ; when negative and 

 not positive, it satisfies the condition x 2 =—x; and when neither positive nor 

 negative, it satisfies the two conditions x* = x and x*= —x. 



uxy properly denotes those of the objects which have the character x and 

 which have the character y. The expression xy is a function of x and y, in which 

 these symbols are co-ordinately dependent on u. According to Boole, x applies to 

 u, and y applies to ux. But y applied to ux has in general a different meaning and 

 a different arithmetical value from y applied to u ; hence it is necessary to denote 

 the change of subject by a mark, as x.y. This distinction appears in the 

 theory of probability, in the contrast between two events which are independent of 

 one another, and two events which are dependent one on the other. 



x 

 The function xy has a single meaning and arithmetical value. The function - » 



on the contrary, has a manifold meaning and arithmetical value. It means any 



