This is closely related to, though distinct from, the law of excluded middle and the law of noncontradiction. See bivalence and related laws for a summary of the differences.
In classical logic and fuzzy logic, the principle of bivalence is equivalent to the result that there are no propositions that are neither true nor false.
This follows because any statement has to have a truth value, such as true, false, or if neither of those hold, it has to have a third truth value (which is the fuzzy logic way out). But that is the classical way of thinking about logic, and no longer holds in intuitionistic logic.
In Intuitionistic logic, sometimes one can't determine a truth value, and in that situation, one just leaves it at that. One doesn't try to assign an indeterminate truth value.
The theorem that there are no propositions that are neither true nor false has a simple proof in intuitionistic logic:
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 )
In intuitionisitic logic, a conjecture is true if proved. It is false if shown to lead to a contradiction. One can say that it is either true or false if there is a method which is guaranteed to decide the question in a finite number of steps, even if it hasn't been decided yet. If none of those apply, one can't say anything at all about the truth/falsity. One then leaves it at that.
Also known as Tertium non datur (Latin).
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