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In logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical inversions of each other. 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.”

By extension, outside of formal logic, one can speak of contradictions between actions when one presumes that their motives contradict each other.

Contents 1 Contradiction in formal logic 1.1 Proof by contradiction 2 Contradictions and philosophy 2.1 Meta-contradiction 3 Contradiction outside formal logic 3.1 In colloquial speech 3.2 In dialectics 3.2.1 Marxism 3.2.2 Liberalism 4 Notes 5 See also 6 External links

[edit] Contradiction in formal logic

In formal logic, particularly in propositional and first-order logic, a proposition <math>\varphi</math> is a contradiction if and only if <math>\varphi\vdash\bot</math>. Since for contradictory <math>\varphi</math> it is true that <math> \vdash\varphi\rightarrow\psi</math> for all <math>\psi</math>, one may prove any proposition from a set of axioms which contains contradictions.

[edit] Proof by contradiction

Main article: reductio ad absurdum

For a proposition <math>\varphi</math> it is true that <math>\varphi \vdash \top</math>, i. e. that <math>\varphi</math> is a tautology, i. e. that it is always true, if and only if <math>\neg\varphi \vdash \bot</math>, i. e. if the negation of <math>\varphi</math> is a contradiction. Therefore, a proof that <math>\neg\varphi \vdash \bot</math> also proves that <math>\varphi</math> is true. The use of this fact constitutes the technique of the proof by contradiction, which mathematicians use extensively. This applies only in a logic using the excluded middle <math>A\vee\neg A</math> as an axiom.

[edit] Contradictions and 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 (consistent) system of beliefs. Some dialetheists, including Graham Priest, have argued that coherence may not require consistency.

[edit] Meta-contradiction

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

[edit] Contradiction outside formal logic

[edit] In colloquial speech

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

[edit] In dialectics

[edit] Marxism

In dialectical materialism, contradiction, as 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 stem from the social structure of society and inherently lead to class conflict, economic crisis, and eventually revolution, the existing order’s overthrow and the formerly oppressed classes’ ascension to political power.^{[citation needed]}

[edit] Liberalism

Marxist thought does not have a monopoly on the idea of a contradiction as a conflict based in a social structure. Liberal thinkers may interpret the problem of public goods as a contradiction in that conflict arises between what benefits society (such as the production of a public good) and what benefits individual free riders who refuse to pay the costs of the public good. This case provides another interpretation of the Hegelian contradiction.<ref name="deising">Deising, Paul. Hegel’s Dialectical Political Economy. ISBN 0-8133-9131-8.</ref>

[edit] Notes

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[edit] See also

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

[edit] External links

• Stanford Encyclopedia of Philosophy, Contradictionde:Widerspruch

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