Paranormal modal logic – Part I. The system K? and the foundations of the logic of skeptical and credulous plausibility

Abstract

In this two-parts paper we present paranormal modal logic: a modal logic which is both paraconsistent and paracomplete. Besides using a general framework in which a wide range of logics  including normal modal logics, paranormal modal logics and classical logic  can be defined and proving some key theorems about paranormal modal logic (including that it is inferentially equivalent to classical normal modal logic), we also provide a philosophical justification for the view that paranormal modal logic is a formalization of the notions of skeptical and credulous plausibility.

Paranormal modal logic – Part II: K?, K and Classical Logic and other paranormal modal systems

“…For instance, since formulas resulting from the application of △ can be seen as abbreviations inside L ♦ , it might be said that there is a formal paraconsistent and paracomplete inferential relation (in addition to a conceptual one) based on a true paranormal modality-dependent negation inside normal modal logic. This we think strengths the thesis we have mentioned in Section 1 of [4] about normal modal logic being paranormal. This result make give room for a sort of objection that questions the whole worthiness of our endeavor: if all the expressive and inferential power of paranormal modal logic is already contained in normal modal logic, what is the point of developing and studying it?…”

“…In the course of laying down definitions and proving theorems in this paper we shall use many of the concepts introduced in [4]. 1 Where there is a need for referring to those definitions we shall prefix the corresponding label given in [4] with "I.…”

“…1 Where there is a need for referring to those definitions we shall prefix the corresponding label given in [4] with "I. "; I.4.9 for example means definition 4.9 of [4].…”

“…1 Where there is a need for referring to those definitions we shall prefix the corresponding label given in [4] with "I. "; I.4.9 for example means definition 4.9 of [4]. In this and the next sections, we have skipped the proof of those results which can be easily worked out with the help of elementary logical proof techniques.…”

“…The present paper is the second part of a two-part paper intent to further elaborate on several philosophical and technical aspects of paranormal modal logic. Besides giving a philosophical justification to paranormal modal logic as a logic of skeptical and credulous plausibility, we have in the first part of the paper [4] introduced a general framework in which a wide range of logics, including classical logic, traditional (normal) modal logics and the most basic of all paranormal modal logics, propositional system K ? , can be defined.…”