It ain't necessarily so: Basic sequent systems for negative modalities

TL;DR

This paper develops general proof systems for non-classical negations and negative modalities within modal logic frameworks, addressing inconsistency and indeterminacy across various frame types.

Contribution

It introduces a unified modal approach to non-classical negations, providing analytic proof systems applicable to multiple frame classes and exploring their relation to classical negation.

Findings

Established proof systems for negative modalities over various frames

Demonstrated how classical negation can be defined within these systems

Provided insights into handling inconsistency and indeterminacy in modal logic

Abstract

We look at non-classical negations and their corresponding adjustment connectives from a modal viewpoint, over complete distributive lattices, and apply a very general mechanism in order to offer adequate analytic proof systems to logics that are based on them. Defining non-classical negations within usual modal semantics automatically allows one to treat equivalent formulas as synonymous, and to have a natural justification for a global version of the contraposition rule. From that perspective, our study offers a particularly useful environment in which negative modalities and their companions may be used for dealing with inconsistency and indeterminacy. After investigating modal logics based on arbitrary frames, we extend the results to serial frames, reflexive frames, functional frames, and symmetric frames. In each case we also investigate when and how classical negation may thereby be defined.