A Certain Version of Preservationism

A certain approach to paraconsistency was initiated by works of R. Jennings and P. Schotch. In their “Inference and necessity” [4] they proposed a notion of a level of inconsistency (incoherence) of a given set of premises. This level is a measure that assigns to a given set of premises X, the least number of elements of covers of X that consist of consistent subsets of X. The idea of the level of inconsistency allows to formulate a paraconsistent inference relation called by the authors forcing, while the obtained approach  preservationism. Similarly as classical inference relation is truth-preserving, the obtained inference relation is preserving the level of inconsistency. We will discuss some examples of inferences that are valid in the sense of Jennings-Schotch inference relation and rise some questions on them. Based on that we formulate an inference relation as an answer to the mentioned doubts. As regards forcing inference relation, the set of premises needed to derive a given conclusion can vary when changing covers from one to another. Our proposal is to stipulate to have some common set of relevant premises.