On Nonmonotonic Consequence Relations

Abstract

We discuss nonmonotonic reasoning in terms of consequence relations and corresponding operators. Based on the matrix consequence that gives the monotonic case, we define a restricted matrix consequence that illustrates the nonmonotonic case. The latter is a generalization of the relation of logical friendliness introduced by D. Makinson. We prove that any restricted single matrix consequence, although it may be nonmonotonic, is always weakly monotonic and, in the case of a finite matrix, the restricted matrix consequence is very strongly finitary. Further, by modifying the definition of logical friendliness relation formulated specifically in a proof-theoretic manner, we show a possibility of obtaining other reflexive nonmonotonic consequence relations, for which a limited result towards finitariness is proved. This leads to numerous questions about nonmonotonic consequence relations in the segment between the monotonic consequence relation based on intuitionistic propositional logic and logical friendliness.