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.

Symbol	Unicode	Name	Meaning
⊤	U+22A4	True	Designated/true value
⊥	U+22A5	False	Anti-designated
½, u, ⊙	—	Intermediate	Third value
[0,1]	—	Unit interval	Continuous truth degrees
μ	U+03BC	Membership	Fuzzy membership function
N, B	—	Neither, Both	Belnap values
⊗	U+2297	T-norm	Fuzzy conjunction
⊕	U+2295	T-conorm	Fuzzy disjunction

Metatheory. Many-valued logics form a spectrum. Łukasiewicz logics have algebraic semantics (MV-algebras). Fuzzy logic connects to continuous t-norms. Three-valued logics often lack certain classical tautologies. Functional completeness varies: which functions can be defined from the primitives? Some many-valued logics are decidable; complexity varies. Gödel-Dummett logic (infinite-valued) is intermediate between classical and intuitionistic.

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. No single "correct" many-valued logic — different applications need different systems. Sorites paradox (heap) isn't fully resolved by any finite-valued logic. Fuzzy logic criticized for ad hoc membership functions. The meaning of intermediate values is often unclear. Many-valued logics sacrifice some useful classical properties (like A ∨ ¬A). Inference procedures are more complex.