Goldblat-Thomason Theorems for Fundamental (Modal) Logic

TL;DR

This paper extends the semantics of Fundamental Logic by establishing a Goldblatt-Thomason theorem, characterizing axiomatic classes of frames and applying it to fundamental modal logic with standard modal operators.

Contribution

It provides a Goldblatt-Thomason theorem for Fundamental Logic and its modal extension, identifying conditions for classes of frames to correspond to axiomatic logics.

Findings

Characterization of axiomatic classes of fundamental frames

Extension of semantics to fundamental modal logic

Application of theorem to modal operators (Box and Diamond)

Abstract

Holliday recently introduced a non-classical logic called Fundamental Logic, which intends to capture exactly those properties of the connectives "and", "or" and "not" that hold in virtue of their introduction and elimination rules in Fitch's natural deduction system for propositional logic. Holliday provides an intuitive relational semantics for fundamental logic which generalizes both Goldblatt's semantics for orthologic and Kripke semantics for intuitionistic logic. In this paper, we further the analysis of this semantics by providing a Goldblatt-Thomason theorem for Fundamental Logic. We identify necessary and sufficient conditions on a class K of fundamental frames for it to be axiomatic, i.e., to be the class of frames satisfying some logic extending Fundamental Logic. As a straightforward application of our main result, we also obtain a Goldblatt-Thomason theorem for Fundamental Modal Logic, which extends Fundamental Logic with standard Box and Diamond operators.