# A reconstruction of the multipreference closure

## TL;DR

This paper introduces the Multi Preference closure (MP-closure), a new reasoning approach that extends rational closure in preferential logics, offering a stronger and more nuanced way to handle exceptions and inheritance in propositional logic.

## Contribution

It reconstructs the MP-closure in propositional logic, showing it as a natural variant of lexicographic closure and demonstrating its semantic properties and strength over existing closures.

## Key findings

- MP-closure is a natural variant of lexicographic closure.
- It results in a rational consequence relation stronger than rational closure.
- MP-closure is also stronger than the Relevant Closure.

## Abstract

The paper describes a preferential approach for dealing with exceptions in KLM preferential logics, based on the rational closure. It is well known that the rational closure does not allow an independent handling of the inheritance of different defeasible properties of concepts. Several solutions have been proposed to face this problem and the lexicographic closure is the most notable one. In this work, we consider an alternative closure construction, called the Multi Preference closure (MP-closure), that has been first considered for reasoning with exceptions in DLs. Here, we reconstruct the notion of MP-closure in the propositional case and we show that it is a natural variant of Lehmann's lexicographic closure. Abandoning Maximal Entropy (an alternative route already considered but not explored by Lehmann) leads to a construction which exploits a different lexicographic ordering w.r.t. the lexicographic closure, and determines a preferential consequence relation rather than a rational consequence relation. We show that, building on the MP-closure semantics, rationality can be recovered, at least from the semantic point of view, resulting in a rational consequence relation which is stronger than the rational closure, but incomparable with the lexicographic closure. We also show that the MP-closure is stronger than the Relevant Closure.