Note on witnessed Gödel logics with Delta
type: Item
instance of: scholarly article
title: Note on witnessed Gödel logics with Delta
zbMATH Open document ID: 1183.03022
author: Matthias Baaz
author: Oliver Fasching
published in: Annals of Pure and Applied Logic
publication date: 2009-11-04T00:00:00Z
review text: The paper deals with witnessed Gödel logics with or without the \(\Delta\)-operator (recall that \(\Delta(x) = 1\) if \(x = 1\), \(\Delta(x) = 0\) otherwise, for each \(x \in [0,1]\)). Witnessed Gödel logics are based on the interpretation of the universal (resp. existential) quantifier by minimum (resp. maximum) instead of infimum (resp. supremum). The main results of the paper are: -- An axiomatization of witnessed Gödel logic (with or without \(\Delta\)) over \([0,1]\), and the fact that each uncountable subset of the truth-values set \([0,1]\) determines the same witnessed Gödel logic. -- For each truth-value set \(V \subseteq [0,1]\) a translation from the witnessed Gödel logic with \(\Delta\) over \(V\) to the witnessed Gödel logic (without \(\Delta\)) over \(V\). -- The fact that the sets of all Gödel logics, witnessed Gödel logics, Gödel logics with \(\Delta\), witnessed Gödel logics with \(\Delta\) contain countably many elements.
reviewed by: Stefano Aguzzoli
Mathematics Subject Classification ID: 03B52
Mathematics Subject Classification ID: 03C80
zbMATH DE Number: 5624901
zbMATH Keywords: Gödel logics
zbMATH Keywords: disentangled fragment
zbMATH Keywords: witnessed interpretations
zbMATH Keywords: relative completeness
MaRDI profile type: Publication
full work available at URL: j.apal.2009.05.011
OpenAlex ID: W1975789524
cites work: Q4893133
cites work: Monadic Fragments of Gödel Logics: Decidability and Undecidability Results
cites work: First-order Gödel logics
cites work: Continuous Fraïssé conjecture
cites work: Linear Kripke frames and Gödel logics
cites work: On theories and models in fuzzy predicate logics
cites work: Another proof of the strong completeness of the intuitionistic fuzzy logic
cites work: Intuitionistic fuzzy logic and intuitionistic fuzzy set theory
cites work: Q3722491
cites work: Completeness of global intuitionistic set theory
DOI: 10.1016/J.APAL.2009.05.011
description: scientific article; zbMATH DE number 5624901