Excluded Middle, Law of the

(in Latin, tertium non datur), the principle of traditional formal logic asserting that any proposition is either true or false (symbolically this is expressed by the formula Av⌉A, where v designates “or,” A is the assertion “A is true,” and ⌉A is the assertion “A is false”).

In such a formulation, the law of the excluded middle is identical with the principle of bivalence. In the context of propo-sitional (sentential) calculus, the formula Av⌉A can be read differently: for any statement A, either A itself or its negation is true (here A is the arbitrary statement, and ⌉ A is the negation of A). A second formulation of the law of the excluded middle, in conjunction with the Aristotelian interpretation of this principle—that is, either A(x) is true for each x, or there is at least one x for which A(x) is not true—clearly expresses the content of the law of the excluded middle in the context of the set-theoretical logic of predicates, namely the equivalence of the negation of the universal quantifier and the existential quantifier. The equivalence, generally speaking, cannot be proved without applying the law of removal of double negation, which is equivalent to the law of the excluded middle. This leads to a vicious circle (petitio principii) in attempting to view its proof as a basis of the law of the excluded middle.

The generally ineffective nature of statements on existence obtained on the basis of the law of the excluded middle serves as a natural reason for abandoning this principle in intuitionistic and constructivist programs for the foundations of mathematics. Since neither the exclusion of the law of the excluded middle from the initial principles of theory nor, conversely, its inclusion among such principles leads to a contradiction, the law of the excluded middle, from the methodological viewpoint, is now viewed only as a postulate of traditional logic.