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 is assigned a truth value . . .
For this logic, there is usually also another unary logical connective , such that a model of it must satisfyand a binary logical connective defined by , which impliesNote that for this, we do not need to be a closed set, only that contain .
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 , the universe of the interpretation. For each variable symbol , there is an element . For each k-ary function symbol , there is a function . For each k-ary relation symbol , there is a function . For each term , its interpretation is . For each atomic formula , its interpreted truth value is . The propositional logic connectives works the same as before. For each , its interpreted truth value is , where is defined as the interpretation generated by setting to instead. For each , its interpreted truth value is ,
Note that for this, we do need to be a closed set, since otherwise the quantified formulas is not guaranteed to have a truth value.
Entailment For any set of formulas, and any interpretation , define , with the special case that .
We say that iff for any interpretation into , . In particular, iff for any interpretation into , .
Syntax In 1959, Michael Dummett showed that infinite-valued propositional Gödel logic can be axiomatised by adding the axiom schema
to intuitionistic propositional logic.
See also Intermediate logic
References