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 \bot
; 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).[1] [2] 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?".[3]
In classical logic, particularly in propositional and first-order logic, a proposition
\varphi
\varphi\vdash\bot
\varphi
\vdash\varphi → \psi
\psi
\bot\vdash\psi
In a complete logic, a formula is contradictory if and only if it is unsatisfiable.
See main article: Proof by contradiction. For a set of consistent premises
\Sigma
\varphi
\Sigma\vdash\varphi
\Sigma
\varphi
\Sigma\cup\{\neg\varphi\}\vdash\bot
\Sigma
\neg\varphi
\Sigma\cup\{\neg\varphi\}\vdash\bot
\varphi
\Sigma
A\vee\negA
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.[5] Each of these extensions leads to an intermediate logic:
\neg\negA\impliesA
\bot\impliesA
A\land\negA\impliesB
((A\impliesB)\impliesA)\impliesA
A\impliesB\veeB\impliesA
A\vee\negA
(\negA\impliesA)\impliesA
\negA\vee\neg\negA
\neg(A\landB)\iff(\negA)\vee(\negB)
In mathematics, the symbol used to represent a contradiction within a proof varies.[6] Some symbols that may be used to represent a contradiction include ↯, Opq,
⇒ \Leftarrow
\leftrightarrow
In general, a consistency proof requires the following two things:
But by whatever method one goes about it, all consistency proofs would seem to necessitate the primitive notion of contradiction. Moreover, it seems as if this notion would simultaneously have to be "outside" the formal system in the definition of tautology.
When Emil Post, in his 1921 "Introduction to a General Theory of Elementary Propositions", extended his proof of the consistency of the propositional calculus (i.e. the logic) beyond that of Principia Mathematica (PM), he observed that with respect to a generalized set of postulates (i.e. axioms), he would no longer be able to automatically invoke the notion of "contradiction"such a notion might not be contained in the postulates:
Post's solution to the problem is described in the demonstration "An Example of a Successful Absolute Proof of Consistency", offered by Ernest Nagel and James R. Newman in their 1958 Gödel's Proof. They too observed a problem with respect to the notion of "contradiction" with its usual "truth values" of "truth" and "falsity". They observed that:
Given some "primitive formulas" such as PM's primitives S1 V S2 [inclusive OR] and ~S (negation), one is forced to define the axioms in terms of these primitive notions. In a thorough manner, Post demonstrates in PM, and defines (as do Nagel and Newman, see below) that the property of tautologous – as yet to be defined – is "inherited": if one begins with a set of tautologous axioms (postulates) and a deduction system that contains substitution and modus ponens, then a consistent system will yield only tautologous formulas.
On the topic of the definition of tautologous, Nagel and Newman create two mutually exclusive and exhaustive classes K1 and K2, into which fall (the outcome of) the axioms when their variables (e.g. S1 and S2 are assigned from these classes). This also applies to the primitive formulas. For example: "A formula having the form S1 V S2 is placed into class K2, if both S1 and S2 are in K2; otherwise it is placed in K1", and "A formula having the form ~S is placed in K2, if S is in K1; otherwise it is placed in K1".[7]
Hence Nagel and Newman can now define the notion of tautologous: "a formula is a tautology if and only if it falls in the class K1, no matter in which of the two classes its elements are placed".[8] This way, the property of "being tautologous" is described—without reference to a model or an interpretation.
Post observed that, if the system were inconsistent, a deduction in it (that is, the last formula in a sequence of formulas derived from the tautologies) could ultimately yield S itself. As an assignment to variable S can come from either class K1 or K2, the deduction violates the inheritance characteristic of tautology (i.e., the derivation must yield an evaluation of a formula that will fall into class K1). From this, Post was able to derive the following definition of inconsistency—without the use of the notion of contradiction:
In other words, the notion of "contradiction" can be dispensed when constructing a proof of consistency; what replaces it is the notion of "mutually exclusive and exhaustive" classes. An axiomatic system need not include the notion of "contradiction".[9]
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.[10]
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.[11]
In dialectical materialism: Contradiction—as derived 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. According to Marxist theory, such a contradiction can be found, for example, in the fact that:
Hegelian and Marxist theories stipulate that the dialectic nature of history will lead to the sublation, 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 working and producing class of society, thus resolving the prior contradiction between (a) and (b).[12]
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.