# Sequent systems for negative modalities

## TL;DR

This paper develops sequent systems for non-classical negations within modal logics over distributive lattices, addressing their semantic and proof-theoretic properties, and explores conditions for defining classical negation.

## Contribution

It introduces a general framework for sequent systems for non-classical negations in modal logics, including cut-free systems and conditions for classical negation definition.

## Key findings

- Sequent systems are established for various modal frame classes.
- Many systems are shown to admit cut-free proofs.
- Conditions for explicit classical negation are identified.

## Abstract

Non-classical negations may fail to be contradictory-forming operators in more than one way, and they often fail also to respect fundamental meta-logical properties such as the replacement property. Such drawbacks are witnessed by intricate semantics and proof systems, whose philosophical interpretations and computational properties are found wanting. In this paper we investigate congruential non-classical negations that live inside very natural systems of normal modal logics over complete distributive lattices; these logics are further enriched by adjustment connectives that may be used for handling reasoning under uncertainty caused by inconsistency or undeterminedness. Using such straightforward semantics, we study the classes of frames characterized by seriality, reflexivity, functionality, symmetry, transitivity, and some combinations thereof, and discuss what they reveal about sub-classical properties of negation. To the logics thereby characterized we apply a general mechanism that allows one to endow them with analytic ordinary sequent systems, most of which are even cut-free. We also investigate the exact circumstances that allow for classical negation to be explicitly defined inside our logics.