Principle of bivalence - Definition and Overview

In logic, the principle of bivalence states that for any proposition P, either P is true or P is false.

This is not to be confused with the law of excluded middle and the law of noncontradiction. See bivalence and related laws for a summary of the differences.

In classical logic, the principle of bivalence is equivalent to the result that there are no propositions that are neither true nor false. A proposition P that is neither true nor false is undecidable. In intuitionistic logic, sometimes the truth-value of a proposition P cannot be determined (i.e. P cannot be proved nor disproved). In such a case, P simply does not have a truth-value. Other logics, e.g. multi-valued logic, may assign P an indeterminate truth-value.

The principle of bivalence is intuitionistically provable.

Define ¬A as (A → contradiction). I.e., a false statement is one from which one can derive a contradiction. This is the standard intuitionistic definition of what it is for a statement to be false.

So using this definition, if we have (A ∧ ¬A) this can be written as (A ∧ (A → contradiction)) → contradiction.

So (A ∧ ¬A) → contradiction.

So ¬ (A ∧ ¬A)