Łukasiewicz Logic

Origin. Jan Łukasiewicz introduced three-valued logic (1920), then infinitely-valued (1930). Foundation for fuzzy logic. MV-algebras as algebraic semantics (Chang, 1958). One of the main many-valued logics. Motivated by future contingents.

Models. Truth values in [0,1]. Classical: {0, 1}. Łukasiewicz: [0,1] (or finite subsets). Connectives: ¬x = 1-x, x ⊕ y = min(1, x+y). Designated: value 1 (or > 0 for tautologies). Models vagueness and degrees of truth.

Formalism.

Connectives:

• ¬x = 1 - x (negation)
• x ⊕ y = min(1, x + y) (strong disjunction)
• x ⊗ y = max(0, x + y - 1) (strong conjunction)
• x → y = min(1, 1 - x + y) (implication)
• x ∧ y = min(x, y) (weak conjunction/lattice meet)
• x ∨ y = max(x, y) (weak disjunction/lattice join)

Three-valued: Values: 0, ½, 1. Third value: "indeterminate" or "possible."

Infinite-valued: Values: rational or real [0,1]. Tautology: evaluates to 1 for all valuations.

Axioms (Łukasiewicz):

1. φ → (ψ → φ)
2. (φ → ψ) → ((ψ → χ) → (φ → χ))
3. (¬φ → ¬ψ) → (ψ → φ)
4. ((φ → ψ) → ψ) → ((ψ → φ) → φ)

MV-algebras: Algebraic semantics: (A, ⊕, ¬, 0). Categorically equivalent to certain lattice-ordered groups.

Symbols.

Metatheory. Ł₃ (three-valued) decidable. Ł_ω (infinite-valued) decidable (coNP-complete). Complete for MV-algebra semantics. Functional completeness with ⊕, ¬. Standard completeness: [0,1] is the standard MV-algebra. Deduction theorem holds with ⊕.

Applies to. Fuzzy logic and control. Vagueness modeling. Approximate reasoning. Future contingents (philosophical). Hardware with undefined states. Default reasoning. Partial information.

Limitations. Not classical tautologies (p ∨ ¬p ≠ 1). Implication differs from classical. Multiple many-valued logics compete. Connection to probability unclear. Designated value conventions vary. Practical fuzzy systems often use simpler operations.

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