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 $I[p]\in V$.
• $I[A\lor B]=\max(I[A],I[B])$.
• $I[A\land B]=\min(I[A],I[B])$.
• $I[A\to B]={\begin{cases}1&{\text{ if }}I[A]\leq I[B],\\I[B]&{\text{ if }}I[A]>I[B]\end{cases}}$
• $I[\bot ]=0$
• $I[\neg A]:=I[A\to \bot ]={\begin{cases}1&{\text{ if }}I[A]=0,\\0&{\text{ else.}}\end{cases}}$
• $I[\top ]:=I[\neg \bot ]:=I[\bot \to \bot ]=1$

For this logic, there is usually also another unary logical connective $\Delta$, such that a model of it must satisfy$$I(\Delta A)={\begin{cases}1&{\text{ if }}I[A]=1,\\0&{\text{ else.}}\end{cases}}$$and a binary logical connective $\prec$ defined by $A\prec B:=(B\to A)\to B$, which implies$$I[A\prec B]={\begin{cases}1&{\text{ if }}I[A]<I[B],\\I[B]&{\text{ if }}I[A]\geq I[B]\end{cases}}$$Note that for this, we do not need $V$ 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 $\forall x,A$, its interpreted truth value is $I(\forall x,A):=\inf _{m\in M}I[m/x](A)$, where $I[m/x]$ is defined as the interpretation generated by setting $x^{I}$ to $m$ instead.
• For each $\exists x,A$, its interpreted truth value is $I(\exists x,A):=\sup _{m\in M}I[m/x](A)$,

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 $\Gamma$ of formulas, and any interpretation $I$, define $I(\Gamma ):=\inf _{A\in \Gamma }I(A)$, with the special case that $I(\emptyset )=1$.

We say that $\Gamma \models _{V}A$ iff for any interpretation $I$ into $V$, $I(\Gamma )\leq I(A)$. 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

$(A\rightarrow B)\lor (B\rightarrow A)$

to intuitionistic propositional logic.

See also

• Intermediate logic