Finitevalued

Finite-valued describes systems in which the set of potential values is finite. In logic and semantics, a finite-valued logic uses a finite set of truth values, and the truth of complex expressions is determined by connectives defined on that finite set. A valuation assigns each formula a value from the value set, and logical consequence is defined with respect to designated values within that set.

Classical propositional logic is two-valued, using true and false. More generally, finite-valued logics encompass three-valued, four-valued, or higher-valued systems. Common examples include Kleene’s and Łukasiewicz’s three-valued logics, as well as other logics with four, five, or more truth values. Many finite-valued logics are truth-functional, meaning the value of a compound statement depends only on the values of its components via fixed truth tables.

Semantics for finite-valued logics often employ matrix semantics, specifying a finite set of truth values and a subset designated as the “true” or acceptable values. Algebraic approaches use finite structures such as lattices or specialized algebras (e.g., Kleene algebras or MV-algebras) to capture the behavior of logical connectives and to study properties like soundness, completeness, and decidability.

Applications and implications: finite-valued logics are typically decidable by finite truth-table evaluation and are used in areas such as reasoning under uncertainty, digital circuit design (including tri-state or multi-value logic), and certain branches of paraconsistent or many-valued logic research. They stand in contrast to infinite-valued logics, such as some forms of fuzzy logic, where the set of truth values is infinite or uncountable.