376 MEMOIRS OF THE AMERICAN ACADEMY. 



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The sum of them, therefore, which is kam is "betrayer of some man to nothing but 

 an enemy of him." In the same way it is obvious that 6 «m is " betrayer of nothing 



but a man to nothing but an enemy of him." We have k a m = Il(\ — a)^ - m ), or 

 " betrayer of all non-men to a non-enemy of all non-men." This is the same as " that 

 which stands to something which is an enemy of nothing but a man in the relation of 

 betrayer of nothing but men to what is not it." The interpretation of £«m is obvi- 

 ously "betrayer of nothing but a man to an enemy of him." It is equally plain 

 that &a m is " betrayer of no man to anything but an enemy of him," and that k a m is 

 " betrayer of nothing but a man to every enemy of him. By putting li am in the form 

 ^,(1— a)(i-m) we find that it denotes "betrayer of something besides a man to all 

 things which are enemies of nothing but men." When an absolute term is put in 



place of a, the interpretations are obtained in the same way, with greater facility. 



The sign of an operation is plainly a conjugative term. Thus, our commutative 

 multiplication might be denoted by the conjugative 



For we have, 



/,. 



/ 5 $w = tV,sw 



As conjugatives can all be reduced to conjugatives of two correlates, they might be 

 expressed by an operative sign (for which a Hebrew letter might be used) put 

 between the symbols for the two correlates. There would often be an advantage in 

 doing this, owing to the intricacy of the usual notation for conjugatives. If these 

 operational signs happened to agree in their properties with any of the signs of 



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algebra, modifications of the algebraic signs might be used in place of Hebrew letters. 



For instance, if & were such that 



f-xf-yz = fizt'yz, ■ 



then, if we were to substitute for fi the operational sign 1 we have 



which is the expression of the associative principle. So, if 



we may write, 



f-xy = f-yx 



xiy = y~\x 



which is the commutative principle. If both these equations held for any conjugative, 

 we might conveniently express it by a modified sign +• For example, let us consider 

 the conjugative " what is denoted by a term which either denotes — or else — ." 



