Қарсылықтардың логикалық үйлесімсіздігі

Logical incompatibility between two or more propositions

In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's law of noncontradiction states that "It is impossible that the same thing can at the same time both belong and not belong to the same object and in the same respect." In modern formal logic and type theory, the term is mainly used instead for a single proposition, often denoted by the falsum symbol ; a proposition is a contradiction if false can be derived from it, using the rules of the logic. It is a proposition that is unconditionally false (i. e., a self contradictory proposition). This can be generalized to a collection of propositions, which is then said to "contain" a contradiction.

By creation of a paradox, Plato's Euthydemus dialogue demonstrates the need for the notion of contradiction. In the ensuing dialogue, Dionysodorus denies the existence of "contradiction", all the while that Socrates is contradicting him:

Indeed, Dionysodorus agrees that "there is no such thing as false opinion there is no such thing as ignorance", and demands of Socrates to "Refute me." Socrates responds "But how can I refute you, if, as you say, to tell a falsehood is impossible? ".

In classical logic, particularly in propositional and first order logic, a proposition is a contradiction if and only if Since for contradictory it is true that for all (because ), 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, anything follows"). In a complete logic, a formula is contradictory if and only if it is unsatisfiable.

For a set of consistent premises and a proposition , it is true in classical logic that (i. e., proves ) if and only if (i. e., and leads to a contradiction). Therefore, a proof that also proves that is true under the premises The use of this fact forms the basis of a proof technique called proof by contradiction, which mathematicians use extensively to establish the validity of a wide range of theorems. This applies only in a logic where the law of excluded middle is accepted as an axiom. Using minimal logic, a logic with similar axioms to classical logic but without ex falso quodlibet and proof by contradiction, we can investigate the axiomatic strength and properties of various rules that treat contradiction by considering theorems of classical logic that are not theorems of minimal logic. Each of these extensions leads to an intermediate logic:
Double negation elimination (DNE) is the strongest principle, axiomatized , and when it is added to minimal logic yields classical logic. Ex falso quodlibet (EFQ), axiomatized , licenses many consequences of negations, but typically does not help to infer propositions that do not involve absurdity from consistent propositions that do. When added to minimal logic, EFQ yields intuitionistic logic. EFQ is equivalent to ex contradiction quodlibet, axiomatized , over minimal logic. Peirce's rule (PR) is an axiom that captures proof by contradiction without explicitly referring to absurdity. Minimal logic + PR + EFQ yields classical logic. The Gödel Dummett (GD) axiom , whose most simple reading is that there is a linear order on truth values. Minimal logic + GD yields Gödel Dummett logic. Peirce's rule entails but is not entailed by GD over minimal logic. Law of the excluded middle (LEM), axiomatised , is the most often cited formulation of the principle of bivalence, but in the absence of EFQ it does not yield full classical logic. Minimal logic + LEM + EFQ yields classical logic. PR entails but is not entailed by LEM in minimal logic. If the formula B in Peirce's rule is restricted to absurdity, giving the axiom schema , the scheme is equivalent to LEM over minimal logic. Weak law of the excluded middle (WLEM) is axiomatised and yields a system where disjunction behaves more like in classical logic than intuitionistic logic, i. e. the disjunction and existence properties don't hold, but where use of non intuitionistic reasoning is marked by occurrences of double negation in the conclusion. LEM entails but is not entailed by WLEM in minimal logic. WLEM is equivalent to the instance of De Morgan's law that distributes negation over conjunction: .

In mathematics, the symbol used to represent a contradiction within a proof varies. Some symbols that may be used to represent a contradiction include ↯, Opq, , ⊥, / , and ※; in any symbolism, a contradiction may be substituted for the truth value "false", as symbolized, for instance, by "0" (as is common in Boolean algebra). It is not uncommon to see Q. E. D., or some of its variants, immediately after a contradiction symbol. In fact, this often occurs in a proof by contradiction to indicate that the original assumption was proved false—and hence that its negation must be true.

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 system of beliefs. Some dialetheists, including Graham Priest, have argued that coherence may not require consistency.

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.

Colloquial usage can label actions 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. The scientific method uses contradiction to falsify bad theory.