Routley Star in Information-Based Semantics

TL;DR

This paper explores the integration of Routley star into information-based semantics, revealing a restricted yet elegant logical framework that aligns with known logics like Kalman logic and R-mingle.

Contribution

It extends Routley star semantics to information-based models, demonstrating that this integration yields a non-trivial, well-characterized logical system.

Findings

Double negation law holds only in involutive linear frames.

The logic of all linear frames is characterized axiomatically.

The logic of involutive linear frames matches Kalman logic.

Abstract

It is common in various non-classical logics, especially in relevant logics, to characterize negation semantically via the operation known as Routley star. This operation works well within relational semantic frameworks based on prime theories. We study this operation in the context of "information-based" semantics for which it is characteristic that sets of formulas supported by individual information states are theories that do not have to be prime. We will show that, somewhat surprisingly, the incorporation of Routley star into the information-based semantics does not lead to a collapse or a trivialization of the whole semantic system. On the contrary, it leads to a technically elegant though quite restricted semantic framework that determines a particular logic. We study some basic properties of this semantics. For example, we show that within this framework double negation law is valid only in involutive linear frames. We characterize axiomatically the logic of all linear frames and show that the logic of involutive linear frames coincides with a system that Mike Dunn coined Kalman logic. This logic is the fragment (for the language restricted to conjunction, disjunction and negation) of the "semi-relevant" logic known as R-mingle. Finally, we characterize by a deductive system the logic of all information frames equipped with Routley star.