Implication connectives for logics with right weakening

Abstract

We study non-monotonic inference relations, focusing on the properties of the underlying monotonic consequence relation. This sheds new light on the general properties that may or may not be satisfied by a nonmonotonic relation. We let the underlying consequence relation to be any monotonic logic, indicating under what precise conditions it gives rise to a non-monotonic inference relation that obeys major principles such as right weakening and the rule of detachment. We begin to explore the border line that non-monotonic relations cannot cross if they are to enjoy these properties. This work may be considered either as a constructive opposition, or as a complement, to the previous work about the general properties of nonmonotonic inference relations.