truthvalued

Truth-valued semantics and logics describe a framework in which the truth of declarative sentences is assigned a value from a fixed set of truth values. In the classical version, this set has two values: true and false, and the truth value of any compound sentence is determined by truth-functional connectives via truth tables. This two-valued or bivalent framework underpins classical logic and rests on the principle of bivalence, which holds that every statement is either true or false.

Many-valued truth-valued logics extend the idea by allowing more than two values, such as undefined, both true and false, or other designated values. Prominent examples include Kleene and Łukasiewicz logics for three-valued systems, and Belnap four-valued logic for handling inconsistent or incomplete information. Such logics model partial knowledge, uncertainty, or conflicting data, and they challenge classical principles like the law of excluded middle and the principle of non-contradiction.

Formally, a truth-valued logic consists of a set V of truth values and a collection of truth functions that interpret the logical connectives. A valuation assigns to each atomic sentence a value in V, and the value of any compound sentence is computed by applying the respective truth functions. Truth-valued semantics are contrasted with nontruth-valued approaches, such as truth-conditional theories or possible-worlds semantics, in which truth is analyzed relative to contexts or models.

Applications appear in computer science, database theory, and linguistics, where handling partial, uncertain, or conflicting information is essential.