Constructive Negation and the Modal Logic of Consistency
The aim of this chapter is twofold. First of all, we shall take a closer look at the strong, constructive negation ~ introduced in Chapter 8. We shall consider ~ from the point of view of a proof-theoretic characterization of negation and argue that negation may be seen as a connecting link between provability and disprovability (refutability). This notion of negation as falsity will be developed against the background of N. Tennant’s  considerations of negation in intuitionistic relevant logic, where Tennant also attends to disproofs in addition to proofs. It is shown that negation in intuitionistic relevant logic is a negation as syntactical inconsistency in the sense of Gabbay , and that every such negation as inconsistency is a negation as falsity, while the converse is not true. Secondly, we shall consider semantics-based nonmonotonic reasoning as introduced by, again, Gabbay . In this approach, nonmonotonic inference is defined using a modal consistency operator that is interpreted as possibility with respect to the information order in semantical models of a monotonic base logic. It will be shown that certain anomalies of Gabbay’s approach can very naturally be avoided using David Nelson’s constructive three-valued system N3  as the monotonic base system instead of intuitionistic logic, IPL or Kleene’s three-valued logic, 3. The counterintuitive features of semantics-based nonmonotonic reasoning based on IPL or on 3 also disappear if certain properties of the information order in Kripke models for IPL and model structures for 3 are given up. In the case of IPL this leads to subintuitionistic logics. In this way, the present chapter prepares the ground for Chapter 10. Whereas Chapter 10 is devoted to display sequent calculi for subintuitionistic logics,  solves the problem of providing a sound and complete proof-system for the modal logic of consistency over Nelson’s four-valued logic N4.
KeywordsModal Logic Intuitionistic Logic Kripke Model Elimination Rule Strong Negation
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