Gödel Logic

Origin. Gödel introduced Gödel logics (1932) studying intuitionistic logic. Infinite-valued with min/max operations. Also called Gödel-Dummett logic. Characterized by linearity axiom: (A→B)∨(B→A). Intermediate between intuitionistic and classical.

Models. Linear order of truth values. Łukasiewicz: truth + arithmetical operations. Gödel: truth with min/max only. Values in [0,1] or finite chains. Conjunction = min, disjunction = max. Implication: A→B = 1 if A≤B, else B.

Formalism.

Truth values: [0,1] linearly ordered, or {0, 1/n, 2/n, ..., 1} for Gₙ₊₁.

Connectives:

A ∧ B = min(A, B)
A ∨ B = max(A, B)
¬A = A → 0
A → B = 1 if A ≤ B, else B

Implication property: A → B ∈ {1, B} (either fully true or takes B's value) No intermediate implication values.

Axioms (Gödel-Dummett): Intuitionistic logic + linearity: (A → B) ∨ (B → A) Also called LC (logic of linear chains).

Finite Gödel logics: Gₙ: n truth values {0, 1/(n-1), ..., 1}. G₂ = classical logic. G₃: three values, intermediate. G_ω: infinitely many values.

Key properties:

(A → B) ∨ (B → A) valid (linearity)
A ∨ ¬A not valid (not classical)
¬¬A → A not valid (not classical)

Symbols.

Symbol	Unicode	Name	Meaning
∧	U+2227	Min	Minimum
∨	U+2228	Max	Maximum
→	U+2192	Gödel implication	Residuum
¬	U+00AC	Negation	A → 0
0, 1	—	Bounds	False, true
Gₙ	—	Finite Gödel	n-valued

Metatheory. Gödel logic decidable (coNP-complete). Complete for linear Kripke frames. Intermediate logic: intuitionistic < Gödel < classical. First-order Gödel logic: undecidable, axiomatizable. Algebraically: Gödel algebras (Heyting + prelinearity).

Applies to. Fuzzy logic (qualitative comparison). Intermediate logics study. Constructive mathematics. Comparison-based reasoning. Vagueness with ordering. Decision procedures.

Limitations. Limited expressiveness for fuzzy. No Łukasiewicz-style arithmetic. Philosophical status debated. Less common in applications than Łukasiewicz. Specific niche in fuzzy logic family.