166 PROFKSSOR BOXILE S MATHEMATICAL THFORT 



but can be derived from it only by the aid of the principle of coDtra-- 

 diction. Not the ultimate foundation ; for what is the Dictum, but 

 a particular ease of a more eomprehensive, and (in this sense) more- 

 fundamental, law ? Aristotle saw this, and has expressed it aa clearlj 

 as any man that ever lived.. " It is manifest," he says, *' that no one 

 can conceive to himself that the same thing can at once be and not 

 be, for thus he would hold repugnant opinions, and subvert the 

 reality of truth. Wherefore, all who attempt to demonstrate, reduce 

 everything to this as the ultimate doctrine ; for this is by nature 

 the principle o-f all otlier axioms." 



Professor Boole's acceptance of the Leibnitzian maxim (though it 

 was much older than Leibnitz) that the true foundation of the sci- 

 ence of logic is the principle of coutradictionj. has the appearance of 

 being at variance with some extraordinary statements which he else- 

 where makes, to the effect that the principle of contradiction is a 

 consequence of the law of duality. We may remind our readers, 

 that the law of duality [see (4) and (7)] is substantially the prin- 

 ciple out of which all the details of Professor Boole's own doctrine 

 are evolved. Now, under the influence of what was, perhaps, not 

 an unnatural desire to vindicate for his system a peculiar depth of 

 foundation. Professor Boole has been betrayed into observations by 

 which his fame as a philosophic thinker must be seriously affected^. 

 Eor instance: "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 consequence of the fundamental law of thought, whose expres- 

 sion is x'^ = X." And again : " tlie above interpretation has been 

 introduced, not on account of its im-mediate value in the present 

 system, but as an illustration of a significant fact in the philosophy 

 of the intellectual powers, viz., that what has commonly been re- 

 garded as the fundamental axiom of metaphysics is but the conse- 

 quence of a law of thought, mathematical in its form." In thus 

 speaking of the principle of contradiction as a consequence of the 

 law of duality, Professor Boole seems to take away the fundamental 

 character of the principle of contradiction ; for, if that principle be, 

 in the proper sense of the term, a consequence of something else, it 

 cannot be itself truly fundamental. Yet, as we have seen. Professor 

 Boole admits that it is the real and deepest foundation of the science- 

 of logic. What,, then,, does be mean ? On the one handj he cer- 



