Modal Logic With Non-deterministic Semantics: Part II -- Quantified Case

TL;DR

This paper extends non-deterministic matrix semantics to quantified non-normal modal logics, providing new insights into longstanding issues like the identity predicate, Barcan's formulas, and de dicto/de re modalities.

Contribution

It introduces a formal non-deterministic semantic framework for quantified non-normal modal logics, addressing controversial issues in the field.

Findings

Framework successfully models quantified modal logic issues

Offers new perspectives on Barcan's formulas and identity

Enhances understanding of de dicto and de re modalities

Abstract

In the first part of this paper we analyzed finite non-deterministic matrix semantics for propositional non-normal modal logics as an alternative to the standard Kripke's possible world semantics. This kind of modal systems characterized by finite non-deterministic matrices was originally proposed by Ju. Ivlev in the 70's. The aim of this second paper is to introduce a formal non-deterministic semantical framework for the quantified versions of some Ivlev-like non-normal modal logics. It will be shown that several well-known controversial issues of quantified modal logics, relative to the identity predicate, Barcan's formulas, and de dicto and de re modalities, can be tackled from a new angle within the present framework.