In mathematical logic, Gödel logics, sometimes referred to as Dummett logics or Gödel–Dummett logics, is a family of finite- or infinite-valued logics in which the sets of truth values V are closed subsets of the unit interval [0,1] containing both 0 and 1. Different such sets V in general determine different Gödel logics. Gödel logics have several alternative definitions. Specifically, Gödel logics are:
logics of linearly-ordered Heyting algebras logics of (classes of) linearly ordered and countable intuitionistic Kripke structures with constant domains logics of relative comparison, in contrast to Łukasiewicz logic, which is a logic of absolute comparison or metric comparison The concept is named after Kurt Gödel and Michael Dummett.
Semantics
Propositional Given a propositional Gödel logic, an interpretation of it is defined as follows:
Each propositional variable p {\displaystyle p} is assigned a truth value I [ p ] ∈ V {\displaystyle I[p]\in V} .
I [ A ∨ B ] = max ( I [ A ] , I [ B ] ) {\displaystyle I[A\lor B]=\max(I[A],I[B])} .
I [ A ∧ B ] = min ( I [ A ] , I [ B ] ) {\displaystyle I[A\land B]=\min(I[A],I[B])} .
I [ A → B ] = { 1 if I [ A ] ≤ I [ B ] , I [ B ] if I [ A ] > I [ B ] {\displaystyle I[A\to B]={\begin{cases}1&{\text{ if }}I[A]\leq I[B],\\I[B]&{\text{ if }}I[A]>I[B]\end{cases}}}
I [ ⊥ ] = 0 {\displaystyle I[\bot ]=0}
I [ ¬ A ] := I [ A → ⊥ ] = { 1 if I [ A ] = 0 , 0 else. {\displaystyle I[\neg A]:=I[A\to \bot ]={\begin{cases}1&{\text{ if }}I[A]=0,\\0&{\text{ else.}}\end{cases}}}
I [ ⊤ ] := I [ ¬ ⊥ ] := I [ ⊥ → ⊥ ] = 1 {\displaystyle I[\top ]:=I[\neg \bot ]:=I[\bot \to \bot ]=1}
For this logic, there is usually also another unary logical connective Δ {\displaystyle \Delta } , such that a model of it must satisfy I ( Δ A ) = { 1 if I [ A ] = 1 , 0 else. {\displaystyle I(\Delta A)={\begin{cases}1&{\text{ if }}I[A]=1,\\0&{\text{ else.}}\end{cases}}} and a binary logical connective ≺ {\displaystyle \prec } defined by A ≺ B := ( B → A ) → B {\displaystyle A\prec B:=(B\to A)\to B} , which implies I [ A ≺ B ] = { 1 if I [ A ] < I [ B ] , I [ B ] if I [ A ] ≥ I [ B ] {\displaystyle I[A\prec B]={\begin{cases}1&{\text{ if }}I[A]<I[B],\\I[B]&{\text{ if }}I[A]\geq I[B]\end{cases}}} Note that for this, we do not need V {\displaystyle V} to be a closed set, only that V {\displaystyle V} contain 0 , 1 {\displaystyle 0,1} .
First-order Given a first-order logic, it corresponds to a first-order Gödel logic. An interpretation of it is defined essentially the same as the first-order logic:
There is a nonempty set M {\displaystyle M} , the universe of the interpretation. For each variable symbol v {\displaystyle v} , there is an element v I ∈ M {\displaystyle v^{I}\in M} . For each k-ary function symbol f {\displaystyle f} , there is a function f I : M k → M {\displaystyle f^{I}:M^{k}\to M} . For each k-ary relation symbol R {\displaystyle R} , there is a function R I : M k → V {\displaystyle R^{I}:M^{k}\to V} . For each term f ( t 1 , … , t k ) {\displaystyle f(t_{1},\dots ,t_{k})} , its interpretation is f ( t 1 , … , t k ) I := f I ( t 1 I , … , t k I ) {\displaystyle f(t_{1},\dots ,t_{k})^{I}:=f^{I}(t_{1}^{I},\dots ,t_{k}^{I})} . For each atomic formula R ( t 1 , … , t k ) {\displaystyle R(t_{1},\dots ,t_{k})} , its interpreted truth value is I ( R ( t 1 , … , t k ) ) := R I ( t 1 I , … , t k I ) {\displaystyle I(R(t_{1},\dots ,t_{k})):=R^{I}(t_{1}^{I},\dots ,t_{k}^{I})} . The propositional logic connectives works the same as before. For each ∀ x , A {\displaystyle \forall x,A} , its interpreted truth value is I ( ∀ x , A ) := inf m ∈ M I [ m / x ] ( A ) {\displaystyle I(\forall x,A):=\inf _{m\in M}I[m/x](A)} , where I [ m / x ] {\displaystyle I[m/x]} is defined as the interpretation generated by setting x I {\displaystyle x^{I}} to m {\displaystyle m} instead. For each ∃ x , A {\displaystyle \exists x,A} , its interpreted truth value is I ( ∃ x , A ) := sup m ∈ M I [ m / x ] ( A ) {\displaystyle I(\exists x,A):=\sup _{m\in M}I[m/x](A)} , Note that for this, we do need V {\displaystyle V} to be a closed set, since otherwise the quantified formulas is not guaranteed to have a truth value.
Entailment For any set Γ {\displaystyle \Gamma } of formulas, and any interpretation I {\displaystyle I} , define I ( Γ ) := inf A ∈ Γ I ( A ) {\displaystyle I(\Gamma ):=\inf _{A\in \Gamma }I(A)} , with the special case that I ( ∅ ) = 1 {\displaystyle I(\emptyset )=1} . We say that Γ ⊨ V A {\displaystyle \Gamma \models _{V}A} iff for any interpretation I {\displaystyle I} into V {\displaystyle V} , I ( Γ ) ≤ I ( A ) {\displaystyle I(\Gamma )\leq I(A)} . In particular, ⊨ V A {\displaystyle \models _{V}A} iff for any interpretation I {\displaystyle I} into V {\displaystyle V} , I ( A ) = 1 {\displaystyle I(A)=1} .
Syntax In 1959, Michael Dummett showed that infinite-valued propositional Gödel logic can be axiomatised by adding the axiom schema
( A → B ) ∨ ( B → A ) {\displaystyle (A\rightarrow B)\lor (B\rightarrow A)}
to intuitionistic propositional logic.
See also Intermediate logic
References