Towards a model theory of diagrams
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.
scientific article; zbMATH DE number 944228