# Contradiction, Law of

## Contradiction, Law of

(principle of contradiction), one of the fundamental general logical principles, according to which no contradiction is “admissible” (“acceptable”) as a formally logical criterion of a “text” (a statement, reasoning, or entire theory) or as an objective characteristic of the reality of which the text may be a description. The second, or “ontological,” aspect of the law of contradiction is historically the older aspect. It dates from the Sophists and was known to Socrates, who used it frequently, according to Plato. Aristotle formulated the law of contradiction as follows: “The same can and cannot belong to the same in the same reference” *(Metafizika,* Moscow-Leningrad, 1934). Even in Aristotle, however, the law of contradiction is encountered as a logical thesis (more precisely, as a methodological or, in contemporary terminology, a metalogical thesis). According to this thesis, every word (and therefore every sentence or every assertion) has a unique meaning in every case and in every specific context.

The first complete modern formulation of the law of contradiction was provided by G. W. von Leibniz: no proposition can be at the same time true and false *(Nouveaux Essais sur l’entendement humain;* Russian translation, *Novye opyty o chelovecheskom razume,* Moscow-Leningrad, 1936). Therefore, if an argument leads to a contradiction, the premises of the argument are inconsistent (contradictory), or errors have been admitted to the argument, or the logical system providing a framework for the argument is not useful (or is inadmissible).

The clearest and simplest formulation and explanation of the law of contradiction is found in mathematical logic. In the propositional calculus (or, at the semantic level, in sentential, or propositional logic) this formulation is expressed in the provable (identically true) formula ┐ *(A &* ┐ *A),* in which *A* is a propositional variable that designates an arbitrary proposition. On the methodological level, the law of contradiction is expressed in the assertion that this formula is provable or true, or that it is a tautology. In the predicate calculus the law of contradiction has an infinite number of formulations, which differ depending on the number of argument places used in the formulation of the predicates. For example, for one-place predicates the law of contradiction is formulated in the following way: no object can simultaneously possess and not possess the same property, or ∀x ┐ *(A(x)* & ┐ *A(x)).* For two-place predicates, it is formulated as follows: no two objects can simultaneously be and not be in the same relation or ∀_{x}∀_{y} ┐ *(B(x, y) &* ┐ *B(x, y)).*

These purely logical formulations of the law of contradiction also have self-evident “ontological” (relating to material reality) interpretations. The reason for all of these formulations is quite simple. In the overwhelming majority of logical and mathematical logical calculi, it is possible to deduce (prove) either the principle *A*& ┐ *A* ⊃ *B* (from a contradiction anything follows) or at least the weaker principle *A*& ┐ *A* ⊃ ┐ *B* (from a contradiction follows the negation of any assertion). Logical systems in which the law of contradiction is violated are, therefore, clearly unacceptable from the intuitive standpoint, since they do not correspond to material reality, in relation to which the ontological formulation of the law of contradiction is clearly true. Moreover, systems that violate the law of contradiction are of no logical value. The presence of contradictions (antinomies, paradoxes) in a system automatically leads to the provability (or at least, the unprovability) of *any* proposition that can be formulated in the language of that system. For this reason, the consistency (the validity of the law of contradiction) of a logical theory (or in general, a scientific theory) is a very important and crucial criterion of its usefulness. The law of contradiction is therefore of permanent importance.

### REFERENCES

Kolmogorov, A. N. “O printsipe*tertium non datur.” Matematieheskii sbornik,*vol. 32, issue 4, 1925.

Tarski, A.

*Vvedenie v logiku i metodologiiu deduktivnykh nauk.*Moscow, 1948. (Translated from English.)

Kleene, S. C.

*Vvedenie v metamatematiku.*Moscow, 1957. Chapter 3. (Translated from English.)

Church, A.

*Vvedenie v matematicheskuiu logiku,*vol. 1, pars. 17, 32.

Moscow, 1960. (Translated from English.)