From Aristotle’s Metaphysics – the three laws of thought – logic.

Law of identity (A is A)

Now ‘why a thing is itself’ is a meaningless inquiry (for (to give meaning to the question ‘why’) the fact or the existence of the thing must already be evident–e.g. that the moon is eclipsed – but the fact that a thing is itself is the single reason and the single cause to be given in answer to all such questions as why the man is man, or the musician musical’, unless one were to answer ‘because each thing is inseparable from itself, and its being one just meant this’; this, however, is common to all things and is a short and easy way with the question). [Metaphysics, Book 7, Part 17]

Law of non-contradiction (Non-contradiction)

There is a principle in things, about which we cannot be deceived, but must always, on the contrary recognize the truth, – viz. that the same thing cannot at one and the same time be and not be, or admit any other similar pair of opposites. About such matters there is no proof in the full sense, though there is proof ad hominem. For it is not possible to infer this truth itself from a more certain principle, yet this is necessary if there is to be completed proof of it in the full sense. But he who wants to prove to the asserter of opposites that he is wrong must get from him an admission which shall be identical with the principle that the same thing cannot be and not be at one and the same time, but shall not seem to be identical; for thus alone can his thesis be demonstrated to the man who asserts that opposite statements can be truly made about the same subject. Those, then, who are to join in argument with one another must to some extent understand one another; for if this does not happen how are they to join in argument with one another? Therefore every word must be intelligible and indicate something, and not many things but only one; and if it signifies more than one thing, it must be made plain to which of these the word is being applied. He, then, who says ‘this is and is not’ denies what he affirms, so that what the word signifies, he says it does not signify; and this is impossible. Therefore if ‘this is’ signifies something, one cannot truly assert its contradictory.

Further, if the word signifies something and this is asserted truly, this connexion must be necessary; and it is not possible that that which necessarily is should ever not be; it is not possible therefore to make the opposed affirmations and negations truly of the same subject. Further, if the affirmation is no more true than the negation, he who says ‘man’ will be no more right than he who says ‘not-man’. It would seem also that in saying the man is not a horse one would be either more or not less right than in saying he is not a man, so that one will also be right in saying that the same person is a horse; for it was assumed to be possible to make opposite statements equally truly. It follows then that the same person is a man and a horse, or any other animal.

While, then, there is no proof of these things in the full sense, there is a proof which may suffice against one who will make these suppositions. And perhaps if one had questioned Heraclitus himself in this way one might have forced him to confess that opposite statements can never be true of the same subjects. But, as it is, he adopted this opinion without understanding what his statement involves. But in any case if what is said by him is true, not even this itself will be true – viz. that the same thing can at one and the same time both be and not be. For as, when the statements are separated, the affirmation is no more true than the negation, in the same way – the combined and complex statement being like a single affirmation – the whole taken as an affirmation will be no more true than the negation. Further, if it is not possible to affirm anything truly, this itself will be false – the assertion that there is no true affirmation. But if a true affirmation exists, this appears to refute what is said by those who raise such objections and utterly destroy rational discourse. [Metaphysics, Book 11, Part 5 which is Book 4, Part 4 summarized]

Law of the excluded middle (Either-or)

[T]here cannot be an intermediate between contradictories, but of one subject we must either affirm or deny any one predicate. This is clear, in the first place, if we define what the true and the false are. To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true; so that he who says of anything that it is, or that it is not, will say either what is true or what is false; but neither what is nor what is not is said to be or not to be. [Metaphysics, Book 4, Part 7]

Some interesting asides-

  • The phrases in brackets are the headings Rand used for the three parts of Atlas Shrugged.
  • The 10th century Islamic philosopher Ibn Sina wrote in his Metaphysics-

    Anyone who denies the law of non-contradiction should be beaten and burned until he admits that to be beaten is not the same as not to be beaten, and to be burned is not the same as not to be burned.

  • Brand Blanshard, while debating axioms, refers to the law of non-contradiction and says

    Take the law of contradiction: a thing cannot be A and not-A in the same sense at the same time. Now, that is often alleged, of course, to be a self-evident proposition. But I’m inclined to think that [Bernard] Bosanquet is right, in the second volume of his Logic [Logic, or the Morphology of Knowledge, Volume II, Chapter 7; Oxford University Press, 1911], when he includes the law of contradiction among what he calls the “postulates” of inference.

    Let me raise it in this way. Suppose someone just says, offhand, “The law of contradiction is absolutely certain within its own meaning.” And then I ask this question: “Do you not become more certain of the law when you see if you can say, “Either this or nothing.” Because if you deny the law of contradiction, then your whole world disappears in smoke. It vanishes, because it’d be impossible to say that anything was what it was, or anything was related to anything else, because the contradictory of that might be true. So your world would just dissolve. That law of contradiction is the bottleneck of all thinking.

    The determined intuitionist might reply: “I don’t see that I am the least more certain because of seeing that.” But I think he is misrepresenting himself. I think that Bosanquet is right when he says that if you see that this proposition, which at first seems to require nothing but itself for its own justification, is the bottleneck of the existence, or the truth, of everything, then you do have an element of certainty added to it. And he would say, then, that the real verification or justification of the law of contradiction is that it is the, I wouldn’t say the “foundation,” but the postulate upon which all thinking is based. So that your world of thought vanishes without it.

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