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	Logical and / Conjunction	Minimum; the meet of the chain
∨	U+2228	Logical or / Disjunction	Maximum; the join of the chain
→	U+2192	Rightwards arrow / Gödel implication	Residuum of min: 1 if A ≤ B, else B
¬	U+00AC	Not sign / Negation	Defined as A → 0
≤	U+2264	Less-than or equal to	The linear order on truth values
∈	U+2208	Element of	Membership in the value set, as in A → B ∈ {1, B}
0, 1	—	Bounds of the chain	Absolute falsity and absolute truth
[0,1]	—	Unit interval	The standard truth-value set
Gₙ	—	Finite Gödel logic	The logic of the n-element chain; G₂ is classical
G_ω	—	Infinite-valued Gödel logic	The logic of an infinite chain
LC	—	Logic of linear chains	Dummett's name for Gödel–Dummett logic
A, B	—	Formula metavariables	Range over formulas
min, max	—	Lattice operations	The interpretations of ∧ and ∨

Metatheory. Dummett (1959) axiomatized LC as IPC plus prelinearity and proved it complete for linearly ordered Heyting algebras, equivalently for linear Kripke frames, equivalently for the standard chain [0,1]; the same paper showed LC is the intersection of the finite Gödel logics Gₙ, which gives the finite model property. The consistent extensions of LC are exactly the Gₙ, so they form a chain LC ⊂ ... ⊂ G₄ ⊂ G₃ ⊂ G₂ = CPC — a completely classified interval of the lattice of intermediate logics, which is unusual and is why LC is the standard worked example. Validity is coNP-complete, so LC is strictly easier than IPC unless PSPACE = coNP: linearity collapses the branching that makes intuitionistic proof search hard. Algebraically the Gödel algebras are the prelinear Heyting algebras, the variety generated by [0,1], and in Hájek's classification they are the algebras of the minimum t-norm, the only one of the three fundamental fuzzy logics whose conjunction is idempotent and therefore also its lattice meet. First-order Gödel logic over [0,1] is recursively axiomatizable (Horn 1969) and undecidable; Baaz, Preining and Zach showed that for other closed truth-value sets recursive axiomatizability depends on the order topology of the set, so it is not a property of Gödel logic as such.

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

Limitations. The truth functions depend only on the order of the values, not their magnitude: every order-isomorphism of [0,1] preserves every valuation, so a degree carries comparative information and nothing else, and 0.9 against 0.1 is indistinguishable from 0.51 against 0.5. That is fatal for any fuzzy application requiring accumulation, because min is idempotent — A ∧ A = A — so there is no strong conjunction distinct from the meet and no arithmetic on degrees of the kind Łukasiewicz and product logic provide. Negation is worse than the other connectives: ¬A is 1 when A = 0 and 0 otherwise, so it discards all intermediate information and ¬¬A is 1 for every positive A, which is why the Gödel negation of a vague statement is never vague. Implication jumps discontinuously from B to 1 as A crosses B, so arbitrarily small changes in the inputs produce maximal changes in the output. And LC lacks the disjunction property despite being intuitionistic in origin: prelinearity is a theorem neither of whose disjuncts is.