A dissimilarity-based framework for generating inconsistency-tolerant logics

Many commonly used logics, including classical logic and intuitionistic logic, are trivialized in the presence of inconsistency, in the sense that inconsistent premises cause the derivation of any formula. It is thus often useful to define inconsistency-tolerant variants of such logics, which are faithful to the original logic with respect to consistent theories but also allow for nontrivial inconsistent theories. A common way of doing so is by incorporating distance-based considerations for concrete logics. So far this has been done mostly in the context of two-valued semantics. Our purpose in this paper is to show that inconsistency-tolerance can be achieved for any logic that is based on a denotational semantics. For this, we need to trade distances for the more general notion of dissimilarities. We then examine the basic properties of the entailment relations that are obtained and exemplify dissimilarity-based reasoning in various forms of denotational semantics, including multi-valued semantics, non-deterministic semantics, and possible-worlds (Kripke-style) semantics. Moreover, we show that our approach can be viewed as an extension of several well-studied forms of reasoning in the context of belief revision, database integration, consistent query answering, and inconsistency maintenance in knowledge-based systems.