Gödel logic

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 $p$ is assigned a truth value $I[p]\in V$.
• $I[A\vee B]=\max (I[A],I[B])$.
• $I[A\wedge B]=\min (I[A],I[B])$.
• $I[A\to B]=\{\begin{array}{ll}1& \text{ if }I[A]\le I[B],\\ I[B]& \text{ if }I[A]>I[B]\end{array}$
• $I[\mathrm{\perp }]=0$
• $I[\mathrm{\neg }A]:=I[A\to \mathrm{\perp }]=\{\begin{array}{ll}1& \text{ if }I[A]=0,\\ 0& \text{ else.}\end{array}$
• $I[\mathrm{\top }]:=I[\mathrm{\neg }\mathrm{\perp }]:=I[\mathrm{\perp }\to \mathrm{\perp }]=1$

For this logic, there is usually also another unary logical connective $\mathrm{\Delta }$, such that a model of it must satisfy$$I(\mathrm{\Delta }A)=\{\begin{array}{ll}1& \text{ if }I[A]=1,\\ 0& \text{ else.}\end{array}$$and a binary logical connective $\prec$ defined by $A\prec B:=(B\to A)\to B$, which implies$$I[A\prec B]=\{\begin{array}{ll}1& \text{ if }I[A]<I[B],\\ I[B]& \text{ if }I[A]\ge I[B]\end{array}$$Note that for this, we do not need $V$ to be a closed set, only that $V$ contain $0,1$.

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 $M$, the universe of the interpretation.
• For each variable symbol $v$, there is an element $v^I\in M$.
• For each k-ary function symbol $f$, there is a function $f^I:M^k\to M$.
• For each k-ary relation symbol $R$, there is a function $R^I:M^k\to V$.
• For each term $f(t_1,\dots ,t_k)$, its interpretation is $f(t_1,\dots ,t_k{)}^I:=f^I(t_1^I,\dots ,t_k^I)$.
• For each atomic formula $R(t_1,\dots ,t_k)$, its interpreted truth value is $I(R(t_1,\dots ,t_k)):=R^I(t_1^I,\dots ,t_k^I)$.
• The propositional logic connectives works the same as before.
• For each $\mathrm{\forall }x,A$, its interpreted truth value is $I(\mathrm{\forall }x,A):=\underset{m\in M}{\inf }I[m/x](A)$, where $I[m/x]$ is defined as the interpretation generated by setting $x^I$ to $m$ instead.
• For each $\mathrm{\exists }x,A$, its interpreted truth value is $I(\mathrm{\exists }x,A):=\underset{m\in M}{\sup }I[m/x](A)$,

Note that for this, we do need $V$ to be a closed set, since otherwise the quantified formulas is not guaranteed to have a truth value.

Entailment

For any set $\mathrm{\Gamma }$ of formulas, and any interpretation $I$, define $I(\mathrm{\Gamma }):=\underset{A\in \mathrm{\Gamma }}{\inf }I(A)$, with the special case that $I(\mathrm{\varnothing })=1$.

We say that $\mathrm{\Gamma }{\models }_{V}A$ iff for any interpretation $I$ into $V$, $I(\mathrm{\Gamma })\le I(A)$. In particular, ${\models }_{V}A$ iff for any interpretation $I$ into $V$, $I(A)=1$.

Syntax

In 1959, Michael Dummett showed that infinite-valued propositional Gödel logic can be axiomatised by adding the axiom schema

$(A\to B)\vee (B\to A)$

to intuitionistic propositional logic.

See also

• Intermediate logic