On Matrix Consequence (Extended Abstract)

TL;DR

This paper provides a semantic characterization of uniform and couniform consequence relations in matrix model theory, establishing conditions under which these properties are equivalent and applying them to compare consequences across different languages.

Contribution

It introduces the notions of uniform and couniform bundles/atlases and shows their equivalence to certain classes of consequence relations, advancing the semantic understanding of matrix consequence.

Findings

Uniform bundles define uniform consequence relations.

Couniform classes of matrices define couniform consequence relations.

Uniform and couniform consequences coincide in certain language extensions.

Abstract

These results are a contribution to the model theory of matrix consequence. We give a semantic characterization of uniform and couniform consequence relations. These properties have never been treated individually, at least in a semantic manner. We consider these notions from a purely semantic point of view and separately, introducing the notion of a uniform bundle/atlas and that of a couniform class of logical matrices. Then, we show that any uniform bundle defines a uniform consequence; and if a structural consequence is uniform, then its Lindenbaum atlas is uniform. Thus, any structural consequence is uniform if, and only if, it is determined by a uniform bundle/atlas. On the other hand, any couniform set of matrices defines a couniform structural consequence. Also, the Lindenbaum atlas of a couniform structural consequence is couniform. Thus, any structural consequence is couniform if, and only if, it is determined by a couniform bundle/atlas. We then apply these observations to compare structural consequence relations that are defined in different languages when one language is a primitive extension of another. We obtain that for any structural consequence defined in a language having (at least) a denumerable set of sentential variables, if this consequence is uniform and couniform, then it and the \emph{ W\'{o}jcicki's consequence} corresponding to it, which is defined in any primitive extension of the given language, are determined by one and the same atlas which is both uniform and couniform.