Towards a model theory of diagrams
type: Item
instance of: scholarly article
title: Towards a model theory of diagrams
zbMATH Open document ID: 0860.03036
author: Norman Danner
author: Eric M. Hammer
published in: Journal of Philosophical Logic
publication date: 1996-11-12T00:00:00Z
review text: In his book: The logical status of diagrams (1994; Zbl 0829.03002), \textit{Sun-Joo Shin} constructs a formal system of Venn diagrams by making precise the syntax, the rules of inference and the notion of logical consequence. Next Shin proves a soundness theorem and a finite completeness theorem: if \(\Delta\) is a finite set of diagrams, \(D\) is a diagram and \(D\) is a logical consequence of \(\Delta\), then \(D\) is provable from \(\Delta\). In the present paper the authors extend this completeness theorem to the general case, i.e., for any set \(\Delta\) of diagrams. Section 1 presents the syntax: the well-formed diagrams are defined precisely. Section 2 presents the semantics in terms of sets and gives the definition of \(\Delta \models D\), where \(\Delta\) is a set of diagrams and \(D\) a diagram. Section 3 presents the rules of inference and gives the definition of \(\Delta \vdash D\). In section 4 soundness and completeness are shown. The completeness proof is by constructing maximally consistent sets of diagrams. The argument is complicated by the lack of a straightforward counterpart of atomic formulas for diagrams and by the lack of a counterpart of negation for most diagrams.
Mathematics Subject Classification ID: 03C99
Mathematics Subject Classification ID: 03B99
zbMATH DE Number: 944228
zbMATH Keywords: formal system of Venn diagrams
zbMATH Keywords: completeness
zbMATH Keywords: syntax
zbMATH Keywords: well-formed diagrams
zbMATH Keywords: semantics
zbMATH Keywords: rules of inference
zbMATH Keywords: soundness
zbMATH Keywords: maximally consistent sets of diagrams
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 944228