Completeness of certain bimodal logics for subset spaces
type: Item
instance of: scholarly article
title: Completeness of certain bimodal logics for subset spaces
zbMATH Open document ID: 1009.03015
DOI: 10.1023/A:1016372523344
author: M. Angela Weiss
author: Rohit Parikh
published in: Studia Logica
publication date: 2002-08-21T00:00:00Z
review text: The bimodal logic of subset spaces was introduced in a paper by \textit{A. Dabrowski, L. Moss} and \textit{R.~Parikh} [Ann. Pure Appl. Logic 78, No.~1-3, 73-110 (1996; Zbl 0861.68092)]. Several possible axioms for the logic of intersection-closed subset spaces were discussed there, too, but no complete axiomatization could be provided. A partial solution to that problem is given in the present paper. The authors call a subset frame \((X,{\mathcal O}),\) where \(X\) is a non-empty set and \({\mathcal O}\) a set of subsets of \(X,\) directed iff for all \(x\in X\) and \(U,V\in{\mathcal O}\) such that \(x\in U\cap V\) there exists \(W\in{\mathcal O}\) such that \(x\in W\subseteq (U\cap V).\) Then, a corresponding logical system is proposed and proved to be sound and complete for the class of directed spaces. The system contains a non-finitary axiom schema. This cannot be avoided, as is additionally shown in the paper. A theorem whose proof is contained in a forthcoming article by the first author will give a complete solution to the problem mentioned above.
reviewed by: Bernhard Heinemann
Mathematics Subject Classification ID: 03B45
Mathematics Subject Classification ID: 03B42
zbMATH DE Number: 1785606
zbMATH Keywords: multi-modal logics
zbMATH Keywords: logic of knowledge
zbMATH Keywords: topological reasoning
MaRDI profile type: Publication
full work available at URL: a:1016372523344
OpenAlex ID: W1482882201
description: scientific article; zbMATH DE number 1785606