Many-Valued Logic

Origin. Jan Łukasiewicz introduced three-valued logic (1920) for future contingents ("there will be a sea battle tomorrow"). Emil Post explored n-valued and infinite-valued logics (1921). Stephen Kleene developed three-valued logic for partial recursive functions (1938). Lotfi Zadeh introduced fuzzy logic (1965). Multiple traditions with different motivations.

Models. Truth values beyond true and false. Reasons include: indeterminacy (future, vagueness), partiality (undefined computations), degrees of truth (fuzzy), quantum superposition, paradox accommodation. Different systems assign different meanings to intermediate values.

Formalism.

Łukasiewicz three-valued (Ł3): Values: 0 (false), ½ (indeterminate), 1 (true)

• ¬A = 1 - A
• A ∧ B = min(A, B)
• A ∨ B = max(A, B)
• A → B = min(1, 1 - A + B)

Tautologies differ from classical: A ∨ ¬A evaluates to ½ when A = ½.

Kleene strong three-valued (K3): Values: 0, u (unknown), 1

• Connectives as in Łukasiewicz for ¬, ∧, ∨
• A → B = ¬A ∨ B
• Designed for partial functions: u propagates

Kleene weak three-valued: u is "undefined" and infects all computations.

Łukasiewicz infinite-valued (Ł∞): Values: [0, 1] real interval Same connective definitions, continuous.

Fuzzy logic (Zadeh): Membership functions μ: X → [0,1]

• μ(A ∧ B) = min(μ(A), μ(B)) or product
• μ(A ∨ B) = max(μ(A), μ(B)) or probabilistic sum
• Fuzzy inference: rules with fuzzy antecedents and consequents

Belnap four-valued (FDE): Values: N (neither), F (false), T (true), B (both) Arranged in a lattice. Handles incomplete and inconsistent information.

Symbols.

Metatheory. The general framework is the logical matrix: an algebra of truth values with a designated subset, and consequence as preservation of designation. Łoś and Suszko (1958) proved that the consequence relations definable by classes of matrices are exactly the structural ones, so every substitution-invariant logic has a many-valued semantics and the content lies in which matrix, not whether one exists. Gödel (1932) showed intuitionistic propositional logic has no finite characteristic matrix, which is why the intermediate logics required algebraic rather than tabular semantics. Individual systems are settled: Chang (1959) proved completeness of infinite-valued Łukasiewicz logic for MV-algebras and for the standard [0,1] algebra; Dummett (1959) identified Gödel–Dummett logic LC as the logic of linearly ordered Heyting algebras; Cignoli, Esteva, Godo, and Torrens (2000) proved standard completeness of Hájek's BL for the continuous t-norms. Validity in any fixed finite matrix is decidable by evaluation, and Mundici (1987) showed tautologyhood in Ł∞ is coNP-complete. Suszko's reduction (1977) shows every such logic also admits a bivalent, non-truth-functional semantics.

Applies to. Vagueness and approximate reasoning. Fuzzy control systems. Database nulls (SQL three-valued logic). Partial evaluation and undefined values. Quantum logic. Expert systems. Modeling uncertainty distinct from probability.

Limitations. Truth-functionality is the binding constraint: the value of a compound is a function of the values of its parts, so a valuation cannot record that p and q are penumbrally connected, and v(p ∧ ¬p) is fixed by v(p) alone — no finite- or infinite-valued matrix resolves the sorites, because assigning borderline cases determinate intermediate degrees reproduces the problem one level up as higher-order vagueness. Suszko's objection cuts deeper: since designation is a two-way partition, every structural consequence relation is logically two-valued, and the extra values are algebraic apparatus rather than truth values. Non-truth-functional systems — supervaluationism, da Costa's C-systems — therefore fall outside the framework entirely. Individual systems have their own losses: K3 has no tautologies at all, LP's material conditional is not detachable, and the deduction theorem survives in Łukasiewicz logic only in a local form. First-order infinite-valued Łukasiewicz logic is not recursively axiomatizable.

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