50 DERIVATION OF THE LAWS. [CHAP. III. 



The above interpretation has been introduced not on account 

 of its immediate value in the present system, but as an illustration 

 of a significant fact in the philosophy of the intellectual powers, 

 viz., that what has been commonly regarded as the fundamental 

 axiom of metaphysics is but the consequence of a law of thought, 

 mathematical in its form. I desire to direct attention also to the 

 circumstance that the equation (1) in which that fundamental 

 law of thought is expressed is an equation of the second degree.* 

 Without speculating at all in this chapter upon the question, 

 whether that circumstance is necessary in its own nature, we 

 may venture to assert that if it had not existed, the whole pro- 

 cedure of the understanding would have been different from what 

 it is. Thus it is a consequence of the fact that the fundamental 

 equation of thought is of the second degree, that we perform the 

 operation of analysis and classification, by division into pairs of 



* Should it here be said that the existence of the equation a- 2 = x necessitates 

 also the existence of the equation x 3 = x, which is of the third degree, and then 

 inquired whether that equation does not indicate a process of trichotomy ; the 

 answer is, that the equation x 3 = x is not interpretable in the system of logic, 

 For writing it in either of the forms 



*(!-) (14*) =0, (2) 



*(!-) (-1-*) = 0, (3) 



we see that its interpretation, if possible at all, must involve that of the factor 

 1 4 x, or of the factor 1 ar. The former is not interpretable, because we 

 cannot conceive of the addition of any class x to the universe 1 ; the latter is not 

 interpretable, because the symbol 1 is not subject to the law * (1 x} = 0, to 

 which all class symbols are subject. Hence the equation x 3 = x admits of no in- 

 terpretation analogous to that of the equation # 2 = x. Were the former equation, 

 however, true independently of the latter, i. e. were that act of the mind which 

 is denoted by the symbol j, such that its second repetition should reproduce the 

 result of a single operation, but not its first or mere repetition, it is presumable 

 that we should be able to interpret one of the forms (2), (3), which under the 

 actual conditions of thought we cannot do. There exist operations, known to 

 the mathematician, the law of which may be adequately expressed by the equa- 

 tion a: 3 = x. But they are of a nature altogether foreign to the province of 

 general reasoning. 



In saying that it is conceivable that the law of thought might have been dif- 

 ferent from what it is, I mean only that we can frame such an hypothesis, and 

 study its consequences. The possibility of doing this involves no such doctrine 

 as that the actual law of human reason is the product either of chance or of arbi- 

 trary will. 



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