In this paper, we shall consider the so-called cancellation view of negation and the inferential role of contradictions. We will discuss some of the problematic aspects of negation as cancellation, such as its original presentation by Richard and Valery Routley and its role in motivating connexive logic. Furthermore, we will show that the idea of inferential ineffectiveness of contradictions can be conceptually separated from the cancellation model of negation by developing a system we call qLPm, a combination of Graham Priest’s minimally inconsistent Logic of Paradox with q-entailment (quasi-entailment) as introduced by Grzegorz Malinowski.
In his article "Reassurance via Translation" Marcel Crabbe proposed a formalism to obtain reassurance and classical recapture in the setting of minimal FDE. His formalism proved to be general enough to be extended in order to formalize other forms of non-monotonic systems based on preference relations. It is the aim of this article to show how his result can be extended in a natural way by combining two different reasoning systems, namely minimal FDE and circumscription, in order to get a paraconsistent and paracomplete version of circumscription, which we will call paracomplistent circumscription, which has the advantages of FDE and circumscription but is neither explosive nor lacks modus ponens in consistent contexts. Furthermore, we will complete a proof Crabbe left unfinished.
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