Pivotal and Pivotal-discriminative Consequence Relations

TL;DR

This paper explores pivotal consequence relations that are paraconsistent and plausible, providing syntactic characterizations, addressing open representation problems, and connecting with X-logics within a general valuation framework.

Contribution

It introduces a general framework for pivotal and pivotal-discriminative consequence relations, including syntactic characterizations and solutions to open representation issues.

Findings

Pivotal consequence relations are paraconsistent and plausible.

Characterizations of various families of pivotal relations are provided.

Negative solution to an open representation problem by Makinson.

Abstract

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.