MAXIMAL NEIGHBORHOOD UNSOUND CONGRUENTIAL MODAL LOGICS

Abstract

We construct an uncountable sequence of minimal varieties of modal algebras which do not contain any modal algebra whose Boolean reduct forms a powerset algebra. By algebraizabilty and duality between modal algebras and neighborhood frames, this yields a result stating that there exists an uncountable set of Post complete and neighborhood unsound (strongly incomplete) congruential modal logics. Our result resolves an open problem posed by Peter Fritz in his paper from 2016. Furthermore, it shows a sharp contrast with the lattice of normal modal logics (NMLs), since every NML is known to be Kripke (and therefore neighborhood) sound.