Law of noncontradiction

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:


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

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 can only be proved by showing the opponents of the principle to be themselves committed to it. 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", particularly in finding common or dissimilar beliefs in a religious context. For example, Christianity, and other religions, is based on the belief there is but one true God of the universe. Other religious beliefs, such as Hinduism, 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.

The law of noncontradiction is undeniable
The law of non-contradiction is indemonstrable (neither verifiable nor falsifiable) in that anyone who attempts to disprove it must use the law itself, and thus beg the question. In this way it can be said to be undeniable, that is, literally impossible to deny. The law is impossible to prove for the same reason, since one has to use the law to prove the law, and this is a circular argument.

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)