Modal Logic With Non-deterministic Semantics: Part I - Propositional Case

TL;DR

This paper introduces weaker non-deterministic modal logic systems with eight values, proves their completeness, and demonstrates that finite deterministic matrices cannot characterize these systems.

Contribution

It extends Ivlev's non-deterministic semantics for modal logic by proposing weaker eight-valued systems and establishing their completeness, while also showing limitations of finite deterministic matrices.

Findings

Proposed new eight-valued modal logic systems.

Proved completeness of these systems.

Showed finite deterministic matrices cannot characterize these systems.

Abstract

In 1988, Ivlev proposed four-valued non-deterministic semantics for modal logics in which the alethic T axiom holds good. Unfortunately, no completeness was proved. In previous work, we proved completeness for some Ivlev systems and extended his hierarchy, proposing weaker six-valued systems in which the T axiom was replaced by the deontic D axiom. Here, we eliminate both axioms, proposing even weaker systems with eight values. Besides, we prove completeness for those new systems. It is natural to ask if a characterization by finite ordinary (deterministic) logical matrices would be possible for all those systems. We will show that finite deterministic matrices do not characterize any of them.