Gödel logic

Propositional

Given a propositional Gödel logic, an interpretation of it is defined as follows:

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

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

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 ${\displaystyle M}$, the universe of the interpretation.
• For each variable symbol ${\displaystyle v}$, there is an element ${\displaystyle v^{I}\in M}$.
• For each k-ary function symbol ${\displaystyle f}$, there is a function ${\displaystyle f^{I}:M^{k}\to M}$.
• For each k-ary relation symbol ${\displaystyle R}$, there is a function ${\displaystyle R^{I}:M^{k}\to V}$.
• For each term ${\displaystyle f(t_{1},\dots ,t_{k})}$, its interpretation is ${\displaystyle f(t_{1},\dots ,t_{k})^{I}:=f^{I}(t_{1}^{I},\dots ,t_{k}^{I})}$.
• For each atomic formula ${\displaystyle R(t_{1},\dots ,t_{k})}$, its interpreted truth value is ${\displaystyle I(R(t_{1},\dots ,t_{k})):=R^{I}(t_{1}^{I},\dots ,t_{k}^{I})}$.
• The propositional logic connectives works the same as before.
• For each ${\displaystyle \forall x,A}$, its interpreted truth value is ${\displaystyle I(\forall x,A):=\inf _{m\in M}I[m/x](A)}$, where ${\displaystyle I[m/x]}$ is defined as the interpretation generated by setting ${\displaystyle x^{I}}$ to ${\displaystyle m}$ instead.
• For each ${\displaystyle \exists x,A}$, its interpreted truth value is ${\displaystyle I(\exists x,A):=\sup _{m\in M}I[m/x](A)}$,

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

Entailment

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

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