Linear Kripke frames and Gödel logics

Abstract

We investigate the relation between intermediate predicate logics based on countable linear Kripke frames with constant domains and Gödel logics. We show that for any such Kripke frame there is a Gödel logic which coincides with the logic defined by this Kripke frame on constant domains and vice versa. This allows us to transfer several recent results on Gödel logics to logics based on countable linear Kripke frames with constant domains: We obtain a complete characterisation of axiomatisability of logics based on countable linear Kripke frames with constant domains.1 Furthermore, we obtain that the total number of logics defined by countable linear Kripke frames on constant domains is countable.

First-order Gödel logics

“…In the propositional case, infinite-valued Gödel logic can be axiomatized by the intuitionistic propositional calculus extended by the axiom schema (A → B) ∨ (B → A). This connection extends also to the Kripke semantics for intuitionistic logic: Gödel logics can also be characterized as logics of (classes of) linearly ordered and countable intuitionistic Kripke structures with constant domains [14].…”

Gentzen Calculi for the Existence Predicate

“…(A. Beckmann and N. Preining[2]) For every countable linear frame F there exists a Gödel set V such that G V |= A ⇔ A holds in all Kripke models on F with constant domains, (i) and vice versa: for every Gödel set V there exists a countable linear frame F such that (i).In[10] so-called Scott logics, structures S V are introduced which correspond to linear frames, but now for possibly non constant domains. That is, we haveTheorem 1.3.…”

Deciding logics of linear Kripke frames with scattered end pieces

“…This article combines and extends (and fixes some errors) of [3] giving non recursive enumerability results for Gödel logics, and [4], linking Gödel logics with logics defined by countable Kripke frames. The extensions presented here are two-fold: -extension to linear Kripke frames of arbitrary size, -extension to the case of increasing domains.…”