A contradiction is a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which are the logical inversions of each other. By extension, outside of logic, contradictions are also said to occur between actions for which the motives are presumed to be contradictory. Illustrating a general tendency in applied logic, Aristotle’s law of noncontradiction states that “One cannot say of something that it is and that it is not in the same respect and at the same time.”

Contradiction outside of formal logic

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In colloquial speech

Colloquially, actions or statements, or both, are said to contradict each other when they are, or are perceived as being, due to presuppositions which are contradictory in the logical sense.

In Dialectics

Marxism

In dialectical materialism contradiction, derived by Karl Marx from Hegelianism, usually refers to an opposition of social forces. Most prominently, according to Marx, capitalism entails a social system that has contradictions because the social classes have conflicting collective goals. These contradictions are based in the social structure of society and inherently lead to class conflict, crisis, and eventually revolution, the existing order’s overthrow and the formerly oppressed classes’ ascension to political power.

Liberalism

The idea of a contradiction as a conflict based in a social structure is not unique to Marxist thought. For liberal thinkers, the problem of public goods may be interpreted as a contradiction in that there is a conflict between what is good for society, e. g., the production of a public good, and what is good for individual free riders who refuse to pay the costs of the public good. This is another interpretation of the Hegelian contradiction.

Contradiction in formal logic

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In formal logic, particularly in propositional and first-order logic, a proposition $\backslash varphi$ is a contradiction if and only if $\backslash varphi\backslash vdash\backslash bot$. Since for contradictory $\backslash varphi$ it is true that $\backslash vdash\; \backslash varphi\; \backslash rightarrow\; \backslash psi$ for all $\backslash psi$, any proposition may be proven from a set of axioms which contains contradictions.

Proof by contradiction

For a proposition $\backslash varphi$ it is true that $\backslash varphi\; \backslash vdash\; \backslash top$, i. e. that $\backslash varphi$ is a tautology, i. e. that it is always true, if and only if $\backslash neg\backslash varphi\; \backslash vdash\; \backslash bot$, i. e. if the negation of $\backslash varphi$ is a contradiction. Therefore, a proof that $\backslash neg\backslash varphi\; \backslash vdash\; \backslash bot$ also proves that $\backslash varphi$ is true. The use of this fact constitutes the technique of the proof by contradiction, which is used extensively in mathematics.

Contradictions and philosophy

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Coherentism is an epistemological theory whose adherents typically claim that a necessary condition for the justification of a belief is that it is part of a logically non-contradictory (consistent) system of beliefs. Some dialethists, including Graham Priest, have argued that coherence may not require consistency.

Meta-contradiction

It often occurs in philosophy that the presence of the argument contradicts with the claims of the argument. An example of this is Heraclitus’s proposition that knowledge is impossible, or, arguably, Nietzsche’s statement that one should not obey others.

Notes

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See also

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• Paradox
• Oxymoron
• Dialectical materialism
• Resolution (logic)
• TRIZ

External links

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• Stanford Encyclopedia of Philosophy, Contradiction

Logic | Mathematical logic | Marxist theory | Marxism | Conflict

Widerspruch | Contradiction | 모순 | Mótsögn | Prieštaravimas | Contradictie | 矛盾 | Contradição | Противоречие (философия) | Motsägelse | 矛盾