3VL

Three-valued logic (3VL) is a non-classical logical framework that generalizes classical logic by introducing a third truth value, commonly interpreted as unknown or indeterminate. The standard values are true, false, and unknown. A valuation assigns one of these values to each proposition, and the truth value of compound statements is determined by extending the ordinary logical connectives to handle the extra value. The most influential versions are Kleene’s three-valued logic (K3) and Łukasiewicz’s three-valued logic (L3).

In Kleene’s K3, the connectives are defined with the following tables (values are T for true, F for false, and U for unknown). Negation: ¬T = F, ¬F = T, ¬U = U. Conjunction: T∧T = T, T∧U = U, T∧F = F, U∧T = U, U∧U = U, U∧F = F, F∧X = F. Disjunction: T∨T = T, T∨U = T, T∨F = T, U∨T = T, U∨U = U, U∨F = U, F∨F = F. Implication is usually defined as A→B := ¬A ∨ B, giving values: TT → T, TU → U, TF → F, UT → T, UU → U, UF → U, FT → T, FU → T, FF → T.

Łukasiewicz’s L3 uses numerical values 0, 1/2, and 1. Negation is ¬x = 1 − x, conjunction is min(x, y), disjunction is max(x, y), and implication is defined as min(1, 1 − x + y). This logic emphasizes graded truth with a smooth transition between values and is studied alongside K3 as a representative of finite-valued logics.

Applications of 3VL include reasoning with partial information, formalizing semantics of programming languages with undefined values, and modeling SQL databases where NULL introduces three-valued truth in query conditions. While related to paraconsistent logics, 3VL typically aims to manage partial knowledge rather than tolerate outright contradictions.