In logic, the semantic principle (or law) of bivalence states that every declarative sentence expressing a proposition (of a theory under inspection) has exactly one truth value, either true or false.
[2] The principle of bivalence is studied in philosophical logic to address the question of which natural-language statements have a well-defined truth value.
[2] Many-valued logics formalize ideas that a realistic characterization of the notion of consequence requires the admissibility of premises that, owing to vagueness, temporal or quantum indeterminacy, or reference-failure, cannot be considered classically bivalent.
[5] The principle of bivalence is related to the law of excluded middle though the latter is a syntactic expression of the language of a logic of the form "P ∨ ¬P".
Assigning Boolean semantics to classical predicate calculus requires that the model be a complete Boolean algebra because the universal quantifier maps to the infimum operation, and the existential quantifier maps to the supremum;[7] this is called a Boolean-valued model.
[citation needed] One of the early motivations for the study of many-valued logics has been precisely this issue.
In the early 20th century, the Polish formal logician Jan Łukasiewicz proposed three truth-values: the true, the false and the as-yet-undetermined.
Issues such as this have also been addressed in various temporal logics, where one can assert that "Eventually, either there will be a sea battle tomorrow, or there won't be."
332–340) offers a 3-valued logic for the cases when algorithms involving partial recursive functions may not return values, but rather end up with circumstances "u" = undecided.
Consideration of its status will show that we are limited to a special kind of truth table".The following are his "strong tables":[12] For example, if a determination cannot be made as to whether an apple is red or not-red, then the truth value of the assertion Q: " This apple is red " is " u ".