THEORIES CONCERNING AXIOMS. 



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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 question, those of 

 Contradiction andof Excluded Middle, 

 it is not unseasonable to consider them 

 here. The former asserts that an 

 affirmative proposition and the cor- 

 responding negative proposition can- 

 not both be true ; which has generally 

 been held to be intuitively evident. 

 Sir William Hamilton and the Ger- 

 mans 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 proposition, 

 an assertion involved in the meaning 

 of the 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 identi- 

 cal 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 fun- 

 damental antithesis pervading nature, 

 and should be enunciated in the 

 simpler form, that the same proposi- 

 tion 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 generalisations from experi- 

 ence. The original foundation of it 

 I take to be, that Belief and Disbelief 

 are two different mental states, ex- 

 cluding one another. This we know 

 by the simplest observation of our own 



minds. And if we carry our observa- 

 tion outwards, we also find that light 

 and darkness, sound and silence, mo- 

 tion and quiescence, equality and in- 

 equality, preceding and following, suc- 

 cession and simultaneousness, any 

 positive phenomenoi^ whatever and 

 its negative, are distinct phenomena, 

 pointedly contrasted, and the one 

 always absent where the other is pre- 

 sent. I consider the maxim in ques- 

 tion to be a generalisation from all 

 these facts. 



In like manner as the Principle of 

 Contradiction (that one of two contra- 

 dictories must be false) means that an 

 assertion cannot be both true and false, 

 so the Principle of Excluded Middle, 

 or that one of two contradictories 

 must be true, means that an assertion 

 must be either true or false : either 

 the affirmative is true, or otherwise 

 the negative is true, which means that 

 the affirmative is false. I cannot help 

 thinking this principle a surprising 

 specimen of a so called necessity of 

 Thought, since it is not even true, un- 

 less with a large qualification. A pro- 

 position must be either true or false, 

 provided that the predicate be one 

 which can in any intelligible sense be 

 attributed to the subject (and as this 

 is always assumed to be the case in 

 treatises on logic, the axiom is always 

 laid down there as of absolute truth). 

 " Abracadabra is a second intention " 

 is neither true nor false. Between 

 the true and the false there is a third 

 possibility, the Unmeaning ; and this 

 alternative is fatal to Sir William 

 Hamilton's extension of the maxim 

 to Noumena. That Matter must 

 either have a minimum of divisibility 

 or be infinitely divisible, is more than 

 we can ever know. For in the first 

 place, Matter, in any other than the 

 phenomenal sense of the term, may 

 not exist ; and it will scarcely be said 

 that a non-entity must be either in- 

 finitely or finitely divisible. In the 

 second place, though matter, con- 

 sidered as the occult cause of oun 

 sensations, do really exist, yet what 

 we caU divisibility may be an attri- 



