# Proof Theory for Positive Logic with Weak Negation

## TL;DR

This paper develops proof-theoretic methods for positive logic with weak negation, establishing cut-free calculi, interpolation, and PSPACE-completeness of the systems.

## Contribution

It introduces novel sequent calculi for positive logic with weak negation, proving properties like cut-elimination, interpolation, and complexity results.

## Key findings

- Cut-free complete sequent calculi constructed
- Systems satisfy Craig interpolation property
- Logical systems are PSPACE-complete

## Abstract

Proof-theoretic methods are developed for subsystems of Johansson's logic obtained by extending the positive fragment of intuitionistic logic with weak negations. These methods are exploited to establish properties of the logical systems. In particular, cut-free complete sequent calculi are introduced and used to provide a proof of the fact that the systems satisfy the Craig interpolation property. Alternative versions of the calculi are later obtained by means of an appropriate loop-checking history mechanism. Termination of the new calculi is proved, and used to conclude that the considered logical systems are PSPACE-complete.