THEORIES CONCERNING AXIOMS. 205 



they must absolutely elect one or the other side, though we may be forever 

 precluded from discovering which. To take his favorite example, we can 

 not conceive the infinite divisibility of matter, and we can not 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 and of 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 mean- 

 ing 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 prop- 

 osition asserts nothing but the falsity of the affirmative, and has no other 

 sense or meaning whatever. The Principium Contradictionis should there- 

 fore put off the ambitious phraseology which gives it the air of a funda- 

 mental antithesis pervading nature, and should be enunciated in the simpler 

 form, that the same proposition can not at the same time be false and true. 

 But I can go no further with the Nominalists ; for I can not 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 foundation of it I take to be, that Belief and Disbelief are two dif- 

 ferent mental states, excluding one another. This we know by the simplest 

 observation of our own minds. And if we carry our observation outward, 

 we also find that light and darkness, sound and silence, motion and quies- 

 cence, equality and inequality, preceding and following, succession and si- 

 multaneousness, any positive phenomenon whatever and its negative, are 

 distinct phenomena, pointedly contrasted, and the one always absent where 

 the other is present. I consider the maxim in question to be a generaliza^ 

 tion 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 can not be both true and 

 false, so the Principle of Excluded Middle, or that one of two contradic- 

 tories 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 can not help thinking this principle a sur- 

 prising specimen of a so-called necessity of Thought, since it is not even 

 true, unless with a large qualification. A proposition must be either true 

 or isX'&Q, provided thixt 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. Be- 

 tween 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 max- 

 im 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 nonentity must be either infinite- . 

 ly or finitely divisible. In the second place, though matter, considered as 



