Law of noncontradiction

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In logic, the law of noncontradiction (also called the law of contradiction) states, in the words of Aristotle, that "one cannot say of something that it is and that it is not in the same respect and at the same time". In the symbolism of propositional logic, this is expressed as:

$\neg (P \wedge \neg P).\!$

It is one of the three classic laws of thought.

1 Interpretations
2 Alleged impossibility of its proof or denial
3 Quotes
4 See also
5 References
6 External links

[edit] 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).

According to Aristotle and Thomas Aquinas, this is a fundamental principle of thought, which is so basic that it can be successfully argued for merely by showing that any opponents of the principle must be using it (and thus be committed to it) themselves. Thus, Aristotle considers the case of someone who denies the principle in the strong way — holding that every proposition is both true and false — and asks why such a person goes on the Megara road to get to Megara from Athens, since on such a person's view it is just as true that any other road would get him to Megara.

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 there are many gods. As a principle of logic, the law of non-contradiction would allow one to conclude that there are many Gods, or there are not many gods, (or neither, if the principle of bivalence is not assumed), but not both.

[edit] Alleged impossibility of its proof or denial

The law of non-contradiction is alleged to be neither verifiable nor falsifiable, on the ground that any proof or disproof must use the law itself, and thus beg the question. ^[1] 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".

[edit] Quotes

"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." (Avicenna, Medieval Philosopher)^[2]

"It is impossible that the same thing can at the same time both belong and not belong to the same object and in the same respect." (Aristotle, Metaphysics)