Principle of bivalence

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In logic, the semantic principle of bivalence states that every declarative sentence expressing a proposition (of a theory under inspection) is either true or false. The dual semantic principle, the principle of contravalence, states that no proposition is both true and false. The principle of bivalence is related to the excluded middle though the latter is a syntactic expression of the language of a logic of the form "P or ¬P". The difference between the principle and the law is important because there are logics which validate the law but which do not validate the principle, and vice versa. For example, the "Logic of Paradox" (LP) of Graham Priest validates the law of excluded middle though its intended semantics is not bivalent.

The intended semantics of classical logic is bivalent, but this is not true of every semantics for classical logic. Classical logic may be characterized by the class of boolean algebras and many-valued matrices in which propositions (sentences) may take one of more than two truth values. The two-valued semantics has a special status, however, besides being the intended one. E.g. every finite boolean algebra is isomorphic to a power of the two-valued boolean algebra.


Bivalence and non-contradiction

The principle of bivalence is a property of the semantics of logic, one that finds itself dual to the principle of contradiction:

  • Principle of bivalence:

Parallel to these semantic principles, there are laws that may be axioms or theorems of a logic.

  • Law of the excluded middle, parallel to bivalence:
  • The law parallel to non-contradiction:

Typically the logical law is validated by a semantics where the principle holds.

In second-order propositional logic, second-order quantifers are available to bind the propositional variables, allowing the formula scheme to be replaced by

  • Excluded middle: ∀P(P ∨ ¬P)
  • Non-contradiction: ∀P¬(P ∧ ¬P)

The law of bivalence itself has no analogue in either of these logics: on pain of paradox, it can be stated only in the metalanguage used to study the aforementioned formal logics.

These analogues of the law of the excluded middle are not valid in intuitionistic logic, although the weaker ∀P¬¬(P ∨ ¬P) is an intuitionistic theorem; the rejection of excluded middle is founded in intuitionists' constructivist as opposed to Platonist conception of truth and falsity.

On the other hand, in linear logic, formal analogues of both excluded middle and non-contradiction are valid, though using linear logic's semantically different "multiplicative" negation and conjunction/disjunction.

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