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In classical 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 non-contradiction 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 classical logic, one can speak of contradictions between actions when one presumes that their motives contradict each other.



Contradiction in formal logic


In classical logic, particularly in propositional and first-order logic, a proposition \varphi is a contradiction if and only if \varphi\vdash\bot.  Since for contradictory \varphi it is true that  \vdash\varphi\rightarrow\psi for all ψ (because \varphi\rightarrow\bot\rightarrow\psi), one may prove any proposition from a set of axioms which contains contradictions. This is called the "principle of explosion" or "ex falso quodlibet" ("from falsity, whatever you like").


In a complete logic, a formula is contradictory if and only if it is un-satisfiable.



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.


Pragmatic contradictions


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.  For examples, Heraclitus’s proposition that knowledge is impossible; or, arguably, Nietzsche’s statement that one should not obey others, or Moore's paradox. These are self-refuting statements and performative contradictions.



Contradiction outside formal logic


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


In dialectical materialism, contradiction, as derived by Karl Marx 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. Mao Zedong's philosophical essay furthered Marx and Lenin's thesis and suggested that all existence is the result of contradiction.


Proof by contradiction is used in mathematics to construct proofs.



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