Negation as a modality in a quantified setting

The idea of treating negation as a modality manifests itself in various logical systems, especially in Dogen'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 in Cabalar et al. (2006, Proceedings of the 10th International Conference on Principles of Knowledge Representation and Reasoning, 25-36), while the latter has recently been advocated in Leitgeb (2019, J Philos. Logic, 48, 305-405), but was first introduced in Moisil (1942, Disquisitiones Math. et Phys., 2, 3-98). I shall develop predicate versions of N and N* and provide a simple Routley-style semantics for the predicate version of Hype. The corresponding strong completeness results will be proved by means of a useful general technique. It should be remarked that this work can also be seen as a starting point for the investigation of intuitionistic predicate modal logics.