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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.

In other words, the notion of "contradiction" can be dispensed when constructing a proof of consistency; what replaces it is the notion of "mutually exclusive and exhaustive" classes. An axiomatic system need not include the notion of "contradiction".[citation needed]

## Philosophy

Adherents of the epistemological theory of coherentism typically claim that as a necessary condition of the justification of a belief, that belief must form a part of a logically non-contradictory system of beliefs. Some dialetheists, including Graham Priest, have argued that coherence may not require consistency.[14]

A pragmatic contradiction occurs when the very statement of the argument contradicts the claims it purports. An inconsistency arises, in this case, because the act of utterance, rather than the content of what is said, undermines its conclusion.[15]

### Dialectical materialism

In dialectical materialism: Contradiction—as derived from Hegelianism—usually refers to an opposition inherently existing within one realm, one unified force or object. This contradiction, as opposed to metaphysical thinking, is not an objectively impossible thing, because these contradicting forces exist in objective reality, not cancelling each other out, but actually defining each other's existence. According to Marxist theory, such a contradiction can be found, for example, in the fact that:

• (a) enormous wealth and productive powers coexist alongside:
• (b) extreme poverty and misery;
• (c) the existence of (a) being contrary to the existence of (b).

Hegelian and Marxist theory stipulates that the dialectic nature of history will lead to the sublation, or synthesis, of its contradictions. Marx therefore postulated that history would logically make capitalism evolve into a socialist society where the means of production would equally serve the exploited and suffering class of society, thus resolving the prior contradiction between (a) and (b).[16]

Mao Zedong's philosophical essay On Contradiction (1937) furthered Marx and Lenin's thesis and suggested that all existence is the result of contradiction.[17]

## Outside formal logic

In dialectical materialism: Contradiction—as derived from Hegelianism—usually refers to an opposition inherently existing within one realm, one unified force or object. This contradiction, as opposed to metaphysical thinking, is not an objectively impossible thing, because these contradicting forces exist in objective reality, not cancelling each other out, but actually defining each other's existence. According to Marxist theory, such a contradiction can be found, for example, in the fact that:

• (a) enormous wealth and productive powers coexist alongside:
• (b) extreme poverty and misery;
• (c) the existence of (a) being contrary to the existence of (b).

Hegelian and Marxist theory stipulates that the dialectic nature of history will lead

Hegelian and Marxist theory stipulates that the dialectic nature of history will lead to the sublation, or synthesis, of its contradictions. Marx therefore postulated that history would logically make capitalism evolve into a socialist society where the means of production would equally serve the exploited and suffering class of society, thus resolving the prior contradiction between (a) and (b).[16]

Mao Zedong's philosophical essay On Contradiction (1937) furthered Marx and L

Mao Zedong's philosophical essay On Contradiction (1937) furthered Marx and Lenin's thesis and suggested that all existence is the result of contradiction.[17]

Colloquial usage can label actions or statements as contradicting each other when due (or perceived as due) to presuppositions which are contradictory in the logical sense.

Proof by contradiction is used in mathematics to construct proofs.