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| In ], the '''Principle of contradiction''' (''principium contradictionis'' in ]) is the second of the so-called ]. The oldest statement of the law is that contradictory statements cannot both at the same time be true, e.g. the two propositions ''A'' is ''B'' and ''A'' is not ''B'' are mutually exclusive. ''A'' may be ''B'' at one time, and not at another; ''A'' may be partly ''B'' and partly not ''B'' at the same time; but it is impossible to predicate of the same thing, at the same time, and in the same sense, the absence and the presence of the same quality. This is the statement of the law given by ]. It takes no account of the truth of either proposition; if one is true, the other is not; one of the two must be false. | |||
| In the symbolism of ], the principle is expressed as: | |||
| : <math> \neg (P \wedge \neg P).\!</math> | |||
| == Interpretations == | |||
| According to ], "the earliest-known explicit statement of the principle of contradiction — the premise of philosophy and the foundation of rational discourse" — is given in ]'s '']'' ('']'') where the character ] states, "It's plain that the same thing won't be willing at the same time to do or suffer opposites with respect to the same part and in relation to the same thing" (436B). | |||
| The principle is also found in ancient ] as a meta-rule in the '']'', the grammar of ],<ref>{{citation|author=]|title=Universals: Studies in Indian Logic and Linguistics|publisher=]|year=1988|pages=109–28}} (] {{citation|title=Seeing Things Hidden|first=Malcolm|last=Bull|publisher=Verso|year=1999|isbn=1859842631|page=53}})</ref> and the '']'' attributed to ]. It was later elaborated on by medieval commentators such as ].<ref>{{citation|title=A History of Indian Philosophy|first=Surendranath|last=Dasgupta|publisher=]|year=1991|isbn=8120804155|page=110}}</ref> | |||
| The traditional source of the Principle of Contradiction is ]’s ] where he gives three different versions.<ref>{{Harvtxt|Łukasiewicz|1971|p=487}}</ref> | |||
| :]: “It is impossible that the same thing belong and not belong to the same thing at the same time and in the same respect.” (1005b19-20) | |||
| :]: “No one can believe that the same thing can (at the same time) be and not be.” (1005b23-24) | |||
| :]: “The most certain of all basic principles is that contradictory propositions are not true simultaneously.” (1011b13-14) | |||
| == Aristotle's attempt at proof == | |||
| In chapter 4, book IV of the '']'', Aristotle attempts several proofs of this principle. He first argues that every expression has a single meaning (otherwise we could not communicate with one another). This rules out the possibility that by 'to be a man', 'not to be a man' is meant. But 'man' means 'two-footed animal' (for example), and so if anything is a man, it is necessary (by virtue of the meaning of 'man') that it must be a two-footed animal, and so it is impossible at the same time for it ''not'' to be a two-footed animal. Thus '"it is not possible to say truly at the same time that the same thing is and is not a man" (''Metaphysics'' 1006b 35). Another argument is that anyone who believes something cannot believe its contradiction (1008b). | |||
| :Why does he not just get up first thing and walk into a well or, if he finds one, over a cliff? In fact, he seems rather careful about cliffs and wells <ref>1008b, trans. Lawson-Tancred</ref>. | |||
| ] gives a similar argument: | |||
| :Anyone who denies the law of non-contradiction should be beaten and burned until he admits that to be beaten is not the same as not to be beaten, and to be burned is not the same as not to be burned. <ref>Avicenna, Metaphysics, I; commenting on Aristotle, Topics I.11.105a4–5.</ref> | |||
| == Leibniz and Kant == | |||
| ] and ] adopted a different statement, by which the law assumes an essentially different meaning. Their formula is A is not not-A; in other words it is impossible to predicate of a thing a quality which is its contradictory. Unlike Aristotle's law this law deals with the necessary relation between subject and predicate in a single judgment. For example, in ]'s '']'', it is asserted, "… nothing supposed capable of being thought may contain contradictory characteristics." Whereas Aristotle states that one or other of two contradictory propositions must be false, the Kantian law states that a particular kind of proposition is in itself necessarily false. On the other hand there is a real connection between the two laws. The denial of the statement A is not-A presupposes some knowledge of what A is, i.e. the statement A is A. In other words a judgment about A is implied. | |||
| Kant's analytical judgments of propositions depend on presupposed concepts which are the same for all people. His statement, regarded as a logical principle purely and apart from material facts, does not therefore amount to more than that of Aristotle, which deals simply with the significance of negation{{Fact|April 2009|date=April 2009}}. | |||
| == Alleged impossibility of its proof or denial == | |||
| As is true of all ], the law of non-contradiction is alleged to be neither verifiable nor falsifiable, on the grounds that any proof or disproof must use the law itself prior to reaching the conclusion. In other words, in order to verify or falsify the laws of logic one must resort to logic as a weapon, an act which would essentially be ]. Since the early 20th century, certain logicians have proposed logics that denies the validity of the law. Collectively, these logics are known as "]" or "inconsistency-tolerant" logics. ] advances the strongest thesis of this sort, which he calls "]". | |||
| In several axiomatic derivations of logic<ref>Steven Wolfram, A New Kind Of Science, ISBN 1579550088</ref>, this is effectively resolved by showing that (P ∨ ¬P) and its negation are constants, and simply defining TRUE as (P ∨ ¬P) and FALSE as ¬(P ∨ ¬P), without taking a position as to the ] or ]. | |||
| ==See also== | |||
| * ] | |||
| * ] | |||
| * ] | |||
| * ] | |||
| * ] | |||
| * ] | |||
| * ] | |||
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| * ] | |||
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| * ] | |||
| ==Notes== | |||
| <references/> | |||
| == References == | |||
| *''Aristotle's Metaphysics'' translated with an introduction by H. Lawson-Tancred. Penguin 1998 | |||
| * ], J.: 1971, 'On the Principle of Contradiction in Aristotle', ''Review of Metaphysics'' '''24''', 485-509 (original in Polish in 1910). | |||
| *{{Wikisource1911Enc Citation|Principle of Contradiction}} '''(dead link)''' | |||
| ==External links== | |||
| * ], "" (]) | |||
| * ], "" (]) | |||
| * Graham Priest and Koji Tanaka, "" (]) | |||
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