Contradiction

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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, usually opposite 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 classical logic, one can speak of contradictions between actions when one presumes that their motives contradict each other.

[edit] 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 unsatisfiable.

[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.^[1]

[edit] 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.^[2] 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.

[edit] Dialectical materialism

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 others 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, 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 resolution, 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).

Mao Zedong's philosophical essay furthered Marx and Lenin's thesis and suggested that all existence is the result of contradiction. ^[3]

[edit] 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.

Proof by contradiction is used in mathematics to construct proofs.

[edit] See also

[edit] References

^ In Contradiction: A Study of the Transconsistent By Graham Priest
^ Stoljar, Daniel (2006). Ignorance and Imagination. Oxford University Press - U.S.. pp. 87. ISBN 0195306589.
^ [1]