Model Theory and Proof Theory of Coalgebraic Predicate Logic

TL;DR

This paper introduces a coalgebraic extension of first-order logic, exploring its axiomatization, completeness, expressive power, and proof systems, with applications to various classes of structures like neighbourhood frames and topological spaces.

Contribution

It generalizes first-order logic for coalgebraic structures, providing new completeness results, expressive comparisons, and a sequent calculus with cut-elimination.

Findings

A general coalgebraic predicate logic framework is proposed.

Completeness results are established for certain classes of structures.

A sequent calculus with cut-elimination is developed.

Abstract

We propose a generalization of first-order logic originating in a neglected work by C.C. Chang: a natural and generic correspondence language for any types of structures which can be recast as Set-coalgebras. We discuss axiomatization and completeness results for several natural classes of such logics. Moreover, we show that an entirely general completeness result is not possible. We study the expressive power of our language, both in comparison with coalgebraic hybrid logics and with existing first-order proposals for special classes of Set-coalgebras (apart from relational structures, also neighbourhood frames and topological spaces). Basic model-theoretic constructions and results, in particular ultraproducts, obtain for the two classes that allow completeness---and in some cases beyond that. Finally, we discuss a basic sequent system, for which we establish a syntactic cut-elimination result.