Preferential and Preferential-discriminative Consequence relations

The present paper investigates consequence relations that are both non-monotonic and paraconsistent. More precisely, we put the focus on preferential consequence relations, i.e. those relations that can be defined by a binary preference relation on states labelled by valuations. We worked with a general notion of valuation that covers e.g. the classical valuations as well as certain kinds of many-valued valuations. In the many-valued cases, preferential consequence relations are paraconsistant (in addition to be non-monotonic), i.e. they are capable of drawing reasonable conclusions which contain contradictions. The first purpose of this paper is to provide in our general framework syntactic characterizations of several families of preferential relations. The second and main purpose is to provide, again in our general framework, characterizations of several families of preferential discriminative consequence relations. They are defined exactly as the plain version, but any conclusion such that its negation is also a conclusion is rejected (these relations bring something new essentially in the many-valued cases).Comment: team Logic and Complexity, written in 2004-200

Pivotal and Pivotal-discriminative Consequence Relations

In the present paper, we investigate consequence relations that are both paraconsistent and plausible (but still monotonic). More precisely, we put the focus on pivotal consequence relations, i.e. those relations that can be defined by a pivot (in the style of e.g. D.~Makinson). A pivot is a fixed subset of valuations which are considered to be the important ones in the absolute sense. We worked with a general notion of valuation that covers e.g. the classical valuations as well as certain kinds of many-valued valuations. In the many-valued cases, pivotal consequence relations are paraconsistant (in addition to be plausible), i.e. they are capable of drawing reasonable conclusions which contain contradictions. We will provide in our general framework syntactic characterizations of several families of pivotal relations. In addition, we will provide, again in our general framework, characterizations of several families of pivotal discriminative consequence relations. The latter are defined exactly as the plain version, but contradictory conclusions are rejected. We will also answer negatively a representation problem that was left open by Makinson. Finally, we will put in evidence a connexion with X-logics from Forget, Risch, and Siegel. The motivations and the framework of the present paper are very close to those of a previous paper of the author which is about preferential consequence relations