Splitting Meanings: A Unified View on Paraconsistency and Inconsistency Measurement
TL;DR
A modular logical framework for both reasoning with and measuring inconsistency in propositional knowledge bases that captures several existing forms of inconsistency-tolerant reasoning, including entailment based on maximal satisfiable subsets and the minimally inconsistent Logic of Paradox.
Abstract
We propose a modular logical framework for both reasoning with and measuring inconsistency in propositional knowledge bases. The framework extends propositional logic with markers attached to occurrences of atoms and interpreted as pointers to worlds. This allows different occurrences of the same atom to be evaluated in different contexts unless they share a marker. We define paraconsistent entailment relations via abnormality functions that associate each model with a set of deviations from a preferred behavior. Each entailment relation is then defined with respect to models that are minimal under set inclusion. We show that suitable markings and abnormality functions capture several existing forms of inconsistency-tolerant reasoning, including entailment based on maximal satisfiable subsets and the minimally inconsistent Logic of Paradox. We also show how a range of inconsistency measures can be expressed in the same setting. Thus our framework provides a uniform basis for diverse approaches to inconsistency handling and measurement.