THEORIES CONCERNING AXIOMS. 307 



us the mysterious world of Things in themselves, are the two 

 principles, which he terms, after the schoolmen, the Principle 

 of Contradiction, and the Principle of Excluded Middle : the 

 first, that two contradictory propositions cannot both he true ; 

 the second, that they cannot both be false. Armed with these 

 logical weapons, we may boldly face Things in themselves, and 

 tender to them the double alternative, sure that they must 

 absolutely elect one or the other side, though we may be for 

 ever precluded from discovering which. To take his favourite 

 example, we cannot conceive the infinite divisibility of matter, 

 and we cannot conceive a minimum, or end to divisibility : yet 

 one or the other must be true. 



As I have hitherto said nothing of the two axioms in ques- 

 tion, those of Contradiction and of Excluded Middle, it is not 

 unseasonable to consider them here. The former asserts that 

 an affirmative proposition and the corresponding negative pro- 

 position cannot both be true ; which has generally been held 

 to be intuitively evident. Sir William Hamilton and the 

 Germans consider it to be the statement in words of a form 

 or law of our thinking faculty. Other philosophers, not less 

 deserving of consideration, deem it to be an identical proposi- 

 tion ; an assertion involved in the meaning of terms ; a mode 

 of defining Negation, and the word Not. 



I am able to go one step with these last. An affirmative 

 assertion and its negative are not two independent assertions, 

 connected with each other only as mutually incompatible. 

 That if the negative be true, the affirmative must be false, 

 really is a mere identical proposition ; for the negative pro- 

 position asserts nothing but the falsity of the affirmative, and 

 has no other sense or meaning whatever. The Principium 

 Contradictionis should therefore put off the ambitious phrase- 

 ology which gives it the air of a fundamental antithesis per- 

 vading nature, and should be enunciated in the simpler form, 

 that the same proposition cannot at the same time be false 

 and true. But I can go no farther with the Nominalists ; for 

 I cannot look upon this last as a merely verbal proposition. 

 I consider it to be, like other axioms, one of our first and most 

 familiar generalizations from experience. The original foun- 



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