From rational Gödel logic to ultrametric logic

Abstract

This paper is devoted to systematic studies of some extensions of firstorder Gödel logic. The first extension is the first-order rational Gödel logic which is an extension of first-order Gödel logic, enriched by countably many nullary logical connectives. By introducing some suitable semantics and proof theory, it is shown that the first-order rational Gödel logic has the completeness property, that is any (strongly) consistent theory is satisfiable. Furthermore, two notions of entailment and strong entailment are defined and their relations with the corresponding notion of proof is studied. In particular, an approximate entailment-compactness is shown. Next, by adding a binary predicate symbol d to the first-order rational Gödel logic, the ultrametric logic is introduced. This serves as a suitable framework for analyzing structures which carry an ultrametric function d together with some functions and predicates which are uniformly continuous with respect to the ultrametric d. Some model theory is developed and to justify the relevance of this model theory, the Robinson joint consistency theorem is proven.

Some Model Theoretic Properties for Pavelka-Style Gödel Logic, RGL* and Gödel Logic with Δ

“…Thus, according to Remark 2.10 in [26], if a theory T is satisfiable in an L-structure M, then T is strongly consistent. Now, the definition of a maximally strongly consistent theory and one of its characterizations are presented [26]. Definition 4.…”

“…The soundness theorem can be proved in RGL*, i.e., if T ϕ, then T |= ϕ, for every L-theory T and L-sentence ϕ. Thus, according to Remark 2.10 in [26], if a theory T is satisfiable in an L-structure M, then T is strongly consistent. Now, the definition of a maximally strongly consistent theory and one of its characterizations are presented [26].…”

“…Despite the standard semantics of Gödel logic, the set of truth values is I = [0, 1] 2 \{(0, r) : r > 0} with lexicographical ordering to have the compactness theorem in rational Gödel logic, RGL* [26].…”

“…The concepts of an L-structure, an assignment, L-terms, interpretations of terms, and L-formulas are defined as usual; see Definitions 2.1 and 2.2 in [26]. The interpretation of L-formulas is defined as follows:…”

“…The proof system for RGL* was expressed in Section 2.1 in [26]. This proof system has some axioms for propositional Gödel logic, quantifiers, and some book-keeping axioms:…”