severalvalued

Several-valued logic, also known as many-valued logic, refers to logical systems that use more than two truth values to evaluate propositions. These logics extend classical true/false semantics to handle partial knowledge, indeterminacy, or inconsistency. The term covers a broad spectrum, from three-valued systems to logics with infinitely many values. The development of many-valued logics began in the early 20th century with the work of Jan Łukasiewicz and Emil Post, who introduced formal frameworks for statements that are neither strictly true nor strictly false. Many-valued logics are distinct from probabilistic or fuzzy logics, which model degrees of belief or degrees of truth rather than fixed logical values.

Semantics in a many-valued system begin with a predefined set of truth values, which can be finite or infinite. Each proposition is assigned a value from this set, and the logical connectives (such as negation, conjunction, disjunction, and implication) are interpreted by truth-functional operations, often specified by truth tables or algebraic definitions. A designated subset of truth values identifies which valuations count as true or valid, shaping entailment and consequence relations in the logic.

Common examples include Kleene’s three-valued logic with values true, false, and unknown; Łukasiewicz’s three-valued logic with values 0, 1/2, and 1; and Post’s various three-valued logics. Four-valued systems, such as Dunn–Belnap logic, model incomplete or contradictory information using four values such as true, false, both, and neither. Higher-valued and infinite-valued logics also exist and are used in different theoretical and applied contexts.

Applications of many-valued logics appear in databases, knowledge representation, nonmonotonic reasoning, and aspects of artificial intelligence where information may be partial or inconsistent.

See also: many-valued logic, three-valued logic, four-valued logic, paraconsistent logic.