Metaphysically, this law asserts:: "Nothing can be both A and not-A." For propositions: "A proposition, P, can not be both true and false."

The Law of Noncontradiction is defended through retortion: any attempt to contradict the concept must rely on the acceptance that contradictions are false.

It can be proven using Propositional Logic:

Proof (by reductio):

1) (A & ~A) [Proposition]
2) A [Conjunction elimination from 1]
3) ~A [Conjunction elimination from 1]
4) ~(A & ~A) [Reductio, 1 - 3]

QED

The Law of Noncontradiction, while appearing prima facie and necessarily true, is questioned by modern logicians. See Paraconsistent Logic and Dialetheism.

The Principle of contradiction (principium contradictionis in Latin) is the second of the so-called three classic laws of thought. The oldest statement of the law is that contradictory statements cannot both at the same time be true, e.g. the two propositions A is B and A is not B are mutually exclusive. A may be B at one time, and not at another; A may be partly B and partly not B at the same time; but it is impossible to predicate of the same thing, at the same time, and in the same sense, the absence and the presence of the same quality. This is the statement of the law given by Aristotle. It takes no account of the truth of either proposition; if one is true, the other is not; one of the two must be true.

In the symbolism of propositional logic, the principle is expressed as:

Failed to parse (Can't write to or create math temp directory): neg (P wedge neg P).!

Interpretations

According to Allan Bloom, "the earliest-known explicit statement of the principle of contradiction - the premise of philosophy and the foundation of rational discourse" - is given in Plato's Politeia (The Republic) where the character Socrates states, "It's plain that the same thing won't be willing at the same time to do or suffer opposites with respect to the same part and in relation to the same thing" (436B).

The principle is also found in ancient Indian logic as a meta-rule in the Shrauta Sutras, the grammar of [[PÄ�á¹�ini]],<ref>Template:Citation (cf. Template:Citation)</ref> and the Brahma Sutras attributed to Vyasa. It was later elaborated on by medieval commentators such as Madhvacharya.<ref>Template:Citation</ref>

The law of non-contradiction is often used as a test of "absolute truth." For example, Christianity, and other religions, are based on the belief there is but one true God of the universe. Other religious beliefs may claim that truth is relativistic. The defenders of the Principle of Contradiction would argue that in order for the statement "there is no absolute truth" to be true, absolute must be true, thus making the statement self-refuting.Template:Fact

Aristotle's attempt at proof

In chapter 4, book IV of the Metaphysics, Aristotle attempts several proofs of this principle. He first argues that every expression has a single meaning (otherwise we could not communicate with one another). This rules out the possibility that by 'to be a man', 'not to be a man' is meant. But 'man' means 'two-footed animal' (for example), and so if anything is a man, it is necessary (by virtue of the meaning of 'man') that it must be a two-footed animal, and so it is impossible at the same time for it not to be a two-footed animal. Thus '"it is not possible to say truly at the same time that the same thing is and is not a man" (Metaphysics 1006b 35). Another argument is that anyone who believes something cannot believe its contradiction (1008b). Why should someone walk to Megara, rather than merely "twiddle his toes"?

Why does he not just get up first thing and walk into a well or, if he finds one, over a cliff? In fact, he seems rather careful about cliffs and wells <ref>1008b, trans. Lawson-Tancred</ref>.

Avicenna gives a similar argument:

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. <ref>Avicenna, Metaphysics, I; commenting on Aristotle, Topics I.11.105a4â��5.</ref>

Leibniz and Kant

Leibniz and Kant adopted a different statement, by which the law assumes an essentially different meaning. Their formula is A is not not-A; in other words it is impossible to predicate of a thing a quality which is its contradictory. Unlike Aristotle's law this law deals with the necessary relation between subject and predicate in a single judgment. For example, in Gottlob Ernst Schulze's Aenesidemus, it is asserted, "â�¦ nothing supposed capable of being thought may contain contradictory characteristics." Whereas Aristotle states that one or other of two contradictory propositions must be false, the Kantian law states that a particular kind of proposition is in itself necessarily false. On the other hand there is a real connection between the two laws. The denial of the statement A is not-A presupposes some knowledge of what A is, i.e. the statement A is A. In other words a judgment about A is implied.

Kant's analytical judgments of propositions depend on presupposed concepts which are the same for all people. His statement, regarded as a logical principle purely and apart from material facts, does not therefore amount to more than that of Aristotle, which deals simply with the significance of negationTemplate:Fact.

Alleged impossibility of its proof or denial

The law of non-contradiction is alleged to be neither verifiable nor falsifiable, on the grounds that any proof or disproof must use the law itself prior to reaching the conclusion, and thus beg the question.<ref>Contradiction (Stanford Encyclopedia of Philosophy)</ref> Since the early 20th century, however, numerous logicians have proposed logics that either weaken or deny the law. Collectively, these logics are known as "paraconsistent" or "inconsistency-tolerant" logics. Graham Priest advances the strongest thesis of this sort, which he calls "dialetheism".

In several axiomatic derivations of logic<ref>Steven Wolfram, A New Kind Of Science, ISBN 1579550088</ref>, this is effectively resolved by showing that (P â�¨ Â¬P) and its negation are constants, and simply defining TRUE as (P â�¨ Â¬P) and FALSE as Â¬(P â�¨ Â¬P), without taking a position as to the Principle of bivalence or Law of excluded middle.

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References

• Copi, I. M, Cohen, C., (2001), "Introduction to Logic", 11th Edition.
• Aristotle's Metaphysics translated with an introduction by H. Lawson-Tancred. Penguin 1998
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