Hitoshi Omori (Kyoto University)
One of the oldest systems of paraconsistent logic is the set of so-called C-systems of Newton da Costa, and this has been generalized into a family of systems now known as Logics of Formal Inconsistencies (LFIs) by Walter Carnielli, Marcelo Coniglio and Joao Marcos. The characteristic notion in these systems is the so-called consistency operator which, roughly speaking, indicates how gluts are behaving. One natural question then is to ask if we can let not only gluts but also gaps be around and generalize the notion of consistency into classicality. This is already considered by Andrea Loparic and da Costa in the style of C-systems. The aim of this paper is to develop a family of systems that generalizes the system of Loparic and da Costa which may be called Logics of Formal Classicality (LFCs). In developing the systems, we take the classic criticism of Graham Priest and Richard Routley saying that C-systems lack negation into account, and argue that we can restore negation by making a small change.