# Paraconsistency, resolution and relevance

## TL;DR

This paper explores a novel logical framework using digraphs and semikernels to handle inconsistent theories, providing sound and complete resolution-based reasoning that avoids classical fallacies and Ex Falso.

## Contribution

It introduces a subset of well-behaved semikernels for nontrivial models of inconsistent theories and adapts classical resolution for this semantics with specific weakenings.

## Key findings

- Semikernels yield models for inconsistent theories.
- Resolution with weakening is sound and complete for the new semantics.
- Reasoning without weakening avoids relevance fallacies.

## Abstract

Digraphs provide an alternative syntax for propositional logic, with digraph kernels corresponding to classical models. Semikernels generalize kernels and we identify a subset of well-behaved semikernels that provides nontrivial models for inconsistent theories, specializing to the classical semantics for the consistent ones. Direct (instead of refutational) reasoning with classical resolution is sound and complete for this semantics, when augmented with a specific weakening which, in particular, excludes Ex Falso. Dropping all forms of weakening yields reasoning which also avoids typical fallacies of relevance.