contradiction, law of

Also called the law (or principle) of non-contradiction. One of the traditional three laws of thought (the other two being the laws of identity and of excluded middle).

Variously formulated as saying that no proposition can be both true and not true; or that nothing can be – without qualification – the case and not the case at the same time; or that nothing can -without qualification – both have and lack a given property at the same time.

The law cannot be logically proved without begging the question, though arguments of a different kind (among those called transcendental arguments) have been offered in its defence since Aristotle (384-322 BC) in his Metaphysics (book 4, chapter 4).

However, recently a notion of dialetheism has been defended which allows breaches of the law in certain cases.

Also see: paraconsistency

G Priest, ‘Contradiction, Belief and Rationality’, Proceedings of the Aristotelian Society (1985-86)

In traditional logic, a contradiction consists of a logical incompatibility or incongruity between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other. Illustrating a general tendency in applied logic, Aristotle’s law of noncontradiction states that “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.”[1]

In modern formal logic, the term is mainly used instead for a single proposition, often denoted by the falsum symbol {\displaystyle \bot };[2] a proposition is a contradiction if false can be derived from it, using the rules of the logic. It is a proposition that is unconditionally false (i.e., a self-contradictory proposition).[3][4] This can be generalized to a collection of propositions, which is then said to “contain” a contradiction.

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