CHAP. II.] SIGNS AND THEIR LAWS. 31 



tioned by the intimate laws of thought, but for reasons of conve- 

 nience not exercised in the ordinary use of language. 



Thirdly, The law expressed by (1) may be characterized by 

 saying that the literal symbols a?, y, z, are commutative, like the 

 symbols of Algebra. In saying this, it is not affirmed that the 

 process of multiplication in Algebra, of which the fundamental 

 law is expressed by the equation 



xy = yx, 



possesses in itself any analogy with that process of logical com- 

 bination which xy has been made to represent above ; but only 

 that if the arithmetical and the logical process are expressed in 

 the same manner, their symbolical expressions will be subject to 

 the same formal law. The evidence of that subjection is in the 

 two cases quite distinct. 



9. As the combination of two literal symbols in the form xy 

 expresses the whole of that class of objects to which the names 

 or qualities represented by x and y are together applicable, it 

 follows that if the two symbols have exactly the same significa- 

 tion, their combination expresses no more than either of the 

 symbols taken alone would do. In such case we should there- 

 fore have 



xy = x. 



As y is, however, supposed to have the same meaning as #, we 

 may replace it in the above equation by x, and we thus get 



xx = x. 



Now in common Algebra the combination xx is more briefly re- 

 presented by x 2 . Let us adopt the same principle of notation 

 here ; for the mode of expressing a particular succession of mental 

 operations is a thing in itself quite as arbitrary as the mode of 

 expressing a single idea or operation (II. 3). In accordance with 

 this notation, then, the above equation assumes the form 



(2) 



and is, in fact, the expression of a second general law of those 

 symbols by which names, qualities, or descriptions, are symboli- 

 cally represented. 



