On the classification of first order Gödel logics
type: Item
instance of: scholarly article
title: On the classification of first order Gödel logics
zbMATH Open document ID: 1445.03034
author: Matthias Baaz
author: Norbert Preining
published in: Annals of Pure and Applied Logic
publication date: 2018-11-07T00:00:00Z
review text: The representation of Gödel logics via their set of truth values has the disadvantage that uncountably many truth value sets have indistinguishable logical properties. In this paper, the representation of Gödel logics via their set of valid sentences is considered. This has to deal with the insufficiencies with respect to satisfiability and entailment. It is not obvious how to describe the sets of 1-satisfiable sentences of these logics. Furthermore, while there are uncountably many entailment relations, there are only countably many sets of validities [\textit{A. Beckmann} et al., Order 25, No. 4, 281--298 (2008; Zbl 1171.03027)]. One of the corollaries of this article deals with the former insufficiency by providing a functional relation between the set of sets of valid formulas and the set of sets of 1-satisfiable formulas. This is achieved by giving a complete characterization of axiomatizability of validity and (the complements of) two types of satisfiabilities, based on the topological properties of the underlying truth value set. The absoluteness operator $\Delta$, introduced in [Interdisciplinary Syst. Res. 88, 291--301 (1986; Zbl 0593.03031)] by \textit{G. Takeuti} and \textit{S. Titani} as ``necessity operator'' $\square$ is considered. Recursively enumerable first order Gödel logics with and without $\Delta$ are characterized with respect to validity and non-satisfiability.
reviewed by: Albert Hoogewijs
Mathematics Subject Classification ID: 03B50
Mathematics Subject Classification ID: 03B52
Mathematics Subject Classification ID: 03B55
zbMATH DE Number: 6974478
zbMATH Keywords: satisfiability
zbMATH Keywords: validity
zbMATH Keywords: axiomatizability
zbMATH Keywords: delta operator
zbMATH Keywords: Gödel logics
MaRDI profile type: Publication
full work available at URL: j.apal.2018.08.010
OpenAlex ID: W2887494859
cites work: Q4893133
cites work: Q5406038
cites work: First-order Gödel logics
cites work: Continuous Fraïssé conjecture
cites work: Separating intermediate predicate logics of well-founded and dually well-founded structures by monadic sentences
cites work: Deciding logics of linear Kripke frames with scattered end pieces
cites work: Metamathematics of fuzzy logic
cites work: Q4323294
cites work: Descriptive set theory
cites work: Q4428309
DOI: 10.1016/J.APAL.2018.08.010
Recommended article: Q3131328
Recommended article: Note on witnessed Gödel logics with Delta
Recommended article: First-order Gödel logics
Recommended article: Gödel Logics – A Survey
Recommended article: The modal logic of Gödel sentences
Recommended article: Hypersequent Calculi for Godel Logics -- a Survey
Recommended article: Q4428309
Recommended article: Quantitative Logic Theory in Gödel System
Recommended article: Gödel Homomorphisms as Gödel Modal Operators
Recommended article: Two notions of compactness in Gödel logics
description: scientific article; zbMATH DE number 6974478