Infinitary propositional relevant languages with absurdity

TL;DR

This paper extends fundamental theorems and properties to infinitary propositional relevant logics, including isomorphism, Karp's theorem, and compactness, and explores their expressive power and model-theoretic characteristics.

Contribution

It establishes key model-theoretic results for infinitary relevant logics, including analogues of classical theorems and an interpolation theorem, advancing the understanding of their structure and expressiveness.

Findings

Proved analogues of Scott's isomorphism theorem and Karp's theorem for infinitary relevant logics.

Established a form of the interpolation theorem for $L_{ olinebreak} olinebreak_{ olinebreak} olinebreak fty ext{omega}$ logic.

Characterized the expressive power of infinitary relevant languages using relevant directed bisimulation.

Abstract

Analogues of Scott's isomorphism theorem, Karp's theorem as well as results on lack of compactness and strong completeness are established for infinitary propositional relevant logics. An "interpolation theorem" (of a particular sort introduced by Barwise and van Benthem) for the infinitary quantificational boolean logic $L_{\infty \omega}$ holds. This yields a preservation result characterizing the expressive power of infinitary relevant languages with absurdity using the model-theoretic relation of relevant directed bisimulation as well as a Beth definability property.