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In logic, the Principle of contradiction (principium contradictionis in Latin) is the second of the so-called three classic laws of thought. 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. This is the statement of the law given by Aristotle. 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 propositional logic, the principle is expressed as:

${\displaystyle \neg (P\wedge \neg P).\!}$

The Principle of Contradiction, along with its complement, the Law of Excluded Middle, are correlates of the Law of Identity; the first principle of thought (reason). Because the principle of identity intellectually partitions the Universe into exactly two parts: ‘self’ and ‘other’, it creates a dichotomy wherein the two parts are ‘mutually exclusive’ and ‘jointly exhaustive’. The principle of contradiction is merely an expression of the mutually exclusive aspect of that dichotomy, and the principle of excluded middle, an expression of its jointly exhaustive aspect.

Interpretations

One difficulty in applying the principle of contradiction is ambiguity in the propositions. For instance, if time is not explicitly specified as part of the propositions A and B, then A may be B at one time, and not at another. A and B may in some cases be made to sound mutually exclusive linguistically even though A may be partly B and partly not B at the same time. However, 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.

According to Allan Bloom, "the earliest-known explicit statement of the principle of contradiction — the premise of philosophy and the foundation of rational discourse" — is given in Plato's Politeia (The Republic) where the character Socrates 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 Indian logic as a meta-rule in the Shrauta Sutras, the grammar of Pāṇini, and the Brahma Sutras attributed to Vyasa. It was later elaborated on by medieval commentators such as Madhvacharya.

The traditional source of the Principle of Contradiction is Aristotle’s Metaphysics where he gives three different versions.

ontological: “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)

psychological: “No one can believe that the same thing can (at the same time) be and not be.” (1005b23-24)

logical: “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 Metaphysics, 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 .

Avicenna 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.

Leibniz and Kant

Leibniz and Kant 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 Gottlob Ernst Schulze's Aenesidemus, 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.

Alleged impossibility of its proof or denial

As is true of all axioms, 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 self-defeating. Since the early 20th century, certain logicians have proposed logics that deny the validity of the law. Collectively, these logics are known as "paraconsistent" or "inconsistency-tolerant" logics. Graham Priest advances the strongest thesis of this sort, which he calls "dialetheism".

In several axiomatic derivations of logic, 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 Principle of bivalence or Law of excluded middle.

Some, such as David Lewis, have objected to paraconsistent logic on the ground that it is simply impossible for a statement and its negation to be jointly true. A related objection is that “negation” in paraconsistent logic is not really negation; it is merely a subcontrary-forming operator.

See also

• Contradiction
• First principle
• Identity (philosophy)
• Law of excluded middle
• Law of identity
• Laws of thought
• Peirce's law
• Principle of bivalence
• Principle of explosion
• Reductio ad absurdum
• Oxymoron

Notes

1. Frits Staal (1988), Universals: Studies in Indian Logic and Linguistics, Chicago, pp. 109–28 (cf. Bull, Malcolm (1999), Seeing Things Hidden, Verso, p. 53, ISBN 1859842631)
2. Dasgupta, Surendranath (1991), A History of Indian Philosophy, Motilal Banarsidass, p. 110, ISBN 8120804155
3. Łukasiewicz (1971) p.487
4. 1008b, trans. Lawson-Tancred
5. Avicenna, Metaphysics, I; commenting on Aristotle, Topics I.11.105a4–5.
6. Steven Wolfram, A New Kind Of Science, ISBN 1579550088
7. See Lewis (1982).
8. See Slater (1995), Béziau (2000).

References

• Aristotle's Metaphysics translated with an introduction by H. Lawson-Tancred. Penguin 1998
• Łukasiewicz, Jan (1971) , "On the Principle of Contradiction in Aristotle", Review of Metaphysics, 24: 485–509
•  Chisholm, Hugh, ed. (1911). Encyclopædia Britannica (11th ed.). Cambridge University Press. {{cite encyclopedia}}: Missing or empty |title= (help) (dead link)

External links

• Laurence Horn, "Contradiction" (Stanford Encyclopedia of Philosophy)
• Graham Priest, "Dialetheism" (Stanford Encyclopedia of Philosophy)
• Graham Priest and Koji Tanaka, "Paraconsistent logic" (Stanford Encyclopedia of Philosophy)

• Logic
• Rules of inference