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Intraditional logic, acontradiction involves aproposition conflicting either with itself or establishedfact. It is often used as a tool to detectdisingenuous beliefs andbias. Illustrating a general tendency in applied logic,Aristotle'slaw 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."[1]
In modernformal logic andtype theory, the term is mainly used instead for asingle proposition, often denoted by thefalsum symbol; a proposition is a contradiction iffalse 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).[2][3] This can be generalized to a collection of propositions, which is then said to "contain" a contradiction.
By creation of aparadox,Plato'sEuthydemus dialogue demonstrates the need for the notion ofcontradiction. In the ensuing dialogue,Dionysodorus denies the existence of "contradiction", all the while thatSocrates is contradicting him:
... I in my astonishment said: What do you mean Dionysodorus? I have often heard, and have been amazed to hear, this thesis of yours, which is maintained and employed by the disciples of Protagoras and others before them, and which to me appears to be quite wonderful, and suicidal as well as destructive, and I think that I am most likely to hear the truth about it from you. The dictum is that there is no such thing as a falsehood; a man must either say what is true or say nothing. Is not that your position?
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?".[4]
In classical logic, particularly inpropositional andfirst-order logic, a proposition is a contradictionif 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").[5]
In acomplete logic, a formula is contradictory if and only if it isunsatisfiable.
For a set of consistent premises and a proposition, it is true inclassical logic that (i.e., proves) if and only if (i.e., and leads to a contradiction). Therefore, aproof that also proves that is true under the premises. The use of this fact forms the basis of aproof technique calledproof by contradiction, which mathematicians use extensively to establish the validity of a wide range of theorems. This applies only in a logic where thelaw of excluded middle is accepted as an axiom.
Usingminimal logic, a logic with similar axioms to classical logic but withoutex 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.[6] Each of these extensions leads to anintermediate logic:
In mathematics, the symbol used to represent a contradiction within a proof varies.[7] 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 inBoolean algebra). It is not uncommon to seeQ.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.
In general, aconsistency proof requires the following two things:
But by whatever method one goes about it, all consistency proofs wouldseem to necessitate the primitive notion ofcontradiction. Moreover, itseems as if this notion would simultaneously have to be "outside" the formal system in the definition of tautology.
WhenEmil Post, in his 1921 "Introduction to a General Theory of Elementary Propositions", extended his proof of the consistency of thepropositional calculus (i.e. the logic) beyond that ofPrincipia Mathematica (PM), he observed that with respect to ageneralized 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:
The prime requisite of a set of postulates is that it be consistent. Since the ordinary notion of consistency involves that of contradiction, which again involves negation, and since this function does not appear in general as a primitive in [thegeneralized set of postulates] a new definition must be given.[8]
Post's solution to the problem is described in the demonstration "An Example of a Successful Absolute Proof of Consistency", offered byErnest Nagel andJames R. Newman in their 1958Gödel'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:
The property of being a tautology has been defined in notions of truth and falsity. Yet these notions obviously involve a reference to somethingoutside the formula calculus. Therefore, the procedure mentioned in the text in effect offers aninterpretation of the calculus, by supplying a model for the system. This being so, the authors have not done what they promised, namely, "to define a property of formulas in terms of purely structural features of the formulas themselves". [Indeed] ... proofs of consistency which are based on models, and which argue from the truth of axioms to their consistency, merely shift the problem.[9]
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 oftautologous – as yet to be defined – is "inherited": if one begins with a set of tautologous axioms (postulates) and adeduction system that containssubstitution andmodus ponens, then aconsistent system will yield only tautologous formulas.
On the topic of the definition oftautologous, Nagel and Newman create twomutually exclusive andexhaustive 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".[10]
Hence Nagel and Newman can now define the notion oftautologous: "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".[11] This way, the property of "being tautologous" is described—without reference to a model or an interpretation.
For example, given a formula such as ~S1 V S2 and an assignment of K1 to S1 and K2 to S2 one can evaluate the formula and place its outcome in one or the other of the classes. The assignment of K1 to S1 places ~S1 in K2, and now we can see that our assignment causes the formula to fall into class K2. Thus by definition our formula is not a tautology.
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:
Definition.A system will be said to be inconsistent if it yields the assertion of the unmodified variable p [S in the Newman and Nagel examples].
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".[12]: 177
Adherents of theepistemological theory ofcoherentism typically claim that as a necessary condition of thejustification of abelief, that belief must form a part of a logically non-contradictorysystem of beliefs. Somedialetheists, includingGraham Priest, have argued that coherence may not require consistency.[13]
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.[14]
Indialectical materialism: Contradiction—as derived fromHegelianism—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 toMarxist theory, such a contradiction can be found, for example, in the fact that:
Hegelian and Marxist theories stipulate that thedialectic nature of history will lead to thesublation, orsynthesis, of its contradictions. Marx therefore postulated that history would logically makecapitalism evolve into asocialist society where themeans of production would equally serve theworking and producing class of society, thus resolving the prior contradiction between (a) and (b).[15]
Colloquial usage can label actions or statements as contradicting each other when due (or perceived as due) topresuppositions which are contradictory in the logical sense.
Proof by contradiction is used inmathematics to constructproofs.
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