Distance-based non-deterministic semantics for reasoning with uncertainty

Non-deterministic matrices, a natural generalization of many-valued matrices, are semantic structures in which the value assigned to a complex formula may be chosen non-deterministically from a given set of options. We show that by combining non-deterministic matrices and distance-based considerations, one obtains a family of logics that are useful for reasoning with uncertainty. These logics are a conservative extension of those that are obtained by standard (i.e., deterministic) distance-based semantics, and so usual distance-based methods (in the context of, e.g., belief revision, information integration, and social choice theory) are easily simulated within our framework. We investigate the basic properties of the distance-preferential non-deterministic logics, consider their application for reasoning with incomplete and inconsistent information, and show the correspondence between some particular entailments in our framework and well-known problems like max-SAT.