Modelling Sources of Inconsistent Information in Paraconsistent Modal Logic

Abstract

Epistemic logics based on normal modal logic are notoriously bad at handling inconsistent and yet non-trivial information. This fact motivates epistemic logics based on paraconsistent logic, examples of which can be traced back at least to the 1980s. These logics handle inconsistent and non-trivial information, but they usually do not articulate sources of the inconsistency. Yet, making the origin of an inconsistency present in a body of information explicit is important to assess the body—can we trace the mutually conflicting pieces of information to sources of information relevant to the body or is the inconsistency a result of an error unrelated to any outside sources? Is the inconsistency derived from various equally trustworthy sources or from a single inconsistent source? In this article we show that a paraconsistent modal logic, namely, the logic B K introduced by Odintsov and Wansing, is a first step toward a formalism capable of making these distinctions explicit. We interpret the accessibility relation between states in a model as a source relation—states accessible from a given state are seen as sources of potential justification of the information contained in the original state. This interpretation also motivates the study of a number of extensions of B K. We focus here on extensions of B K able to articulate the relation of compatibility between bodies of information and extensions working with labels explicitly differentiating between bodies of information. In the case of compatibility-based extensions a more detailed technical study including a completeness proof is provided; technical features of the simpler case of label-based extensions, on the other hand, are discussed without going into details.