Qualitative reasoning in a two-layered framework

TL;DR

This paper develops a formal two-layered logical framework for qualitative reasoning about uncertainty, including belief measures and handling contradictory information, with proofs of completeness.

Contribution

It introduces novel two-layered logics for qualitative uncertainty, formalising belief reasoning and paraconsistent extensions within a unified framework.

Findings

Formalisation of qualitative belief reasoning

Design and proof of completeness for the logics

Extension to paraconsistent logic for contradictory info

Abstract

The reasoning with qualitative uncertainty measures involves comparative statements about events in terms of their likeliness without necessarily assigning an exact numerical value to these events. The paper is divided into two parts. In the first part, we formalise reasoning with the qualitative counterparts of capacities, belief functions, and probabilities, within the framework of two-layered logics. Namely, we provide two-layered logics built over the classical propositional logic using a unary belief modality $\Be$ that connects the inner layer to the outer one where the reasoning is formalised by means of G\"{o}del logic. We design their Hilbert-style axiomatisations and prove their completeness. In the second part, we discuss the paraconsistent generalisations of the logics for qualitative uncertainty that take into account the case of the available information being contradictory or inconclusive.