Negation as a modality in a quantified setting

The idea of treating negation as a modality manifests itself in various logical systems, especially in Došen's propositional logic N, whose negation is weaker than that of Johansson's minimal logic. Among the interesting extensions of N are the propositional logics N* and Hype; the former was proposed by Cabalar, Odintsov and Pearce as a framework for studying foundations of well-founded semantics for logic programs with negation, while the latter has recently been advocated by Leitgeb as a basic system for dealing with hyperintensional contexts, but was first described by Moisil in 1942. I shall develop predicate versions of N and N*, and provide a simple Routley-style semantics for Leitgeb's predicate version of Hype. The corresponding strong completeness results will be presented. Also, Leitgeb mistakenly claimed that Hype has the disjunction property, so I shall briefly discuss related issues. In particular, the predicate version of Hype has neither the disjunction property nor the existential property. Finally, it should be remarked that this work can be seen as a starting point for the investigation of predicate intuitionistic modal logics.