Contradiction

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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.

Contradiction in formal logic

In formal logic, particularly in propositional and first-order logic, a proposition Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi

is a contradiction if and only if Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi\vdash\bot

. Since for contradictory Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi

it is true that Failed to parse (Missing texvc executable; please see math/README to configure.):  \vdash\varphi\rightarrow\psi
for all Failed to parse (Missing texvc executable; please see math/README to configure.): \psi
(because Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi\rightarrow\bot\rightarrow\psi

), one may prove any proposition from a set of axioms which contains contradictions.

Proof by contradiction

Main article: reductio ad absurdum

For a proposition Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi

it is true that Failed to parse (Missing texvc executable; please see math/README to configure.): \vdash\varphi

, i. e. that Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi

is a tautology, i. e. that it is always true, if and only if Failed to parse (Missing texvc executable; please see math/README to configure.): \neg\varphi \vdash \bot

, i. e. if the negation of Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi

is a contradiction. Therefore, a proof that Failed to parse (Missing texvc executable; please see math/README to configure.): \neg\varphi \vdash \bot
also proves that Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi
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 Failed to parse (Missing texvc executable; please see math/README to configure.): A\vee\neg A
as an axiom.

In mathematics, the symbol used to represent a contradiction within a proof varies. [1] Some symbols that may be used to represent a contradiction include ↯, ⇒⇐ , ⊥, ↮, and ※. It is not uncommon to see Q.E.D. or some variant immediately after a contradiction symbol; this occurs in a proof by contradiction, to indicate that the original assumption was false and that the theorem must therefore be true.

Example: To fly is impossible, but I could possibly achieve it.

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

It often occurs in philosophy that the very presence of the argument contradicts the claims of the argument; An inconsistency arising because of the normal implications of saying something, rather than because of the content of what is said. [2] Examples include: 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

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.

In dialectics

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]}

Mao Zedong's most important philosophical essay furthered Marx and Lenin's thesis and suggested that all existence is the result of contradiction.

See also

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

External links

Look up contradiction, although in Wiktionary, the free dictionary.

• Stanford Encyclopedia of Philosophy, Contradiction

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