Strong Kleene Logic
Origin. Kleene's strong three-valued logic (1938, 1952). Third value: unknown/undefined. Some operations short-circuit: F∧U=F, T∨U=T. Foundation for logic programming semantics. Connection to fixed-point semantics and partial predicates.
Models. Partial information with short-circuit. Three values: T (true), F (false), U (unknown). Strong: some operations determined by one input. F ∧ anything = F. T ∨ anything = T. Models computation with undefined subexpressions.
Formalism.
Truth values: {T, F, U} — true, false, unknown.
Strong Kleene tables:
| ∧ | T | F | U | ∨ | T | F | U | ¬ | |||
|---|---|---|---|---|---|---|---|---|---|---|---|
| T | T | F | U | T | T | T | T | T | F | ||
| F | F | F | F | F | T | F | U | F | T | ||
| U | U | F | U | U | T | U | U | U | U |
Key property: F ∧ U = F (false dominates) T ∨ U = T (true dominates) Short-circuit evaluation.
Implication: A → B = ¬A ∨ B T → U = U, F → U = T, U → T = T, U → F = U, U → U = U.
Designated value: T is designated (for validity). Valid: true for all valuations. A ∨ ¬A not valid (U when A = U).
Kripke-Kleene fixed point: Least fixed point of truth definition. Three-valued models for logic programs.
Symbols.
| Symbol | Unicode | Name | Meaning |
|---|---|---|---|
| T | — | True | Known true |
| F | — | False | Known false |
| U | — | Unknown | Undefined/unknown |
| ∧ | U+2227 | And | Strong conjunction |
| ∨ | U+2228 | Or | Strong disjunction |
| ¬ | U+00AC | Not | Negation |
| → | U+2192 | Implies | Material |
Metatheory. Strong Kleene decidable. Some tautologies preserved from classical. FDE is Strong Kleene with fourth value (both). Fixed-point semantics for logic programs. De Morgan lattice algebra. Relates to supervaluations.
Applies to. Logic programming (3-valued semantics). Partial functions. Database nulls. Vagueness. Truth-value gaps. Recursion semantics. Liar paradox treatments.
Limitations. Law of excluded middle fails. Implication weak. Multiple 3-valued logics. U interpretation varies. Classical reasoning compromised. Designated value choice matters.
© 2026 Lingenic LLC