CHAP. III.] DERIVATION OF THE LAWS. 49 



PROPOSITION IV. 



That axiom of metaphysicians which is termed the principle of 

 contradiction, and which affirms that it is impossible for any being to 

 possess a quality, and at the same time not to possess it, is a conse- 

 quence of the fundamental law of thought, whose expression is x* = x. 



Let us write this equation in the form 



x - x* = 0, 

 whence we have 



x(l-x) = 0; (1) 



both these transformations being justified by the axiomatic laws 

 of combination and transposition (II. 13). Let us, for simplicity 

 of conception, give to the symbol x the particular interpretation 

 of men, then 1 - x will represent the class of " not-men" 

 (Prop, in.) Now the formal product of the expressions of two 

 classes represents that class of individuals which is common to 

 them both (II. 6). Hence #(1 - x) will represent the class 

 whose members are at once " men," and " not men," and the 

 equation (1) thus express the principle, that a class whose mem- 

 bers are at the same time men and not men does not exist. In 

 other words, that it is impossible for the same individual to be at 

 the same time a man and not a man. Now let the meaning of 

 the symbol x be extended from the representing of " men," to 

 that of any class of beings characterized by the possession of any 

 quality whatever ; and the equation (1) will then express that it 

 is impossible for a being to possess a quality and not to possess 

 that quality at the same time. But this is identically that 

 66 principle of contradiction" which Aristotle has described as the 

 fundamental axiom of all philosophy. " It is impossible that the 

 same quality should both belong and not belong to the same 

 thing. . . This is the most certain of all principles. . . Wherefore 

 they who demonstrate refer to this as an ultimate opinion. For 

 it is by nature the source of all the other axioms."* 



* To yap avrb apa vrcap^iv rt KOI /i?) virdp^iv advvarov r< airy /cat 

 TO avro. . . AVTT) Sri iraauiv tori (3((3aiOTaTij TWV ap^Stv. . . Ato 7raJT ol cnr 



vvvreg ti'e TCLVTI\V dvdyovffiv iffxdTrjv S6%av <j>v<rei yap dpxn 

 iravTtav Metapkysica, in. 3. 

 E 



