Negation as a modality in a quantified setting

Journal of Logic and Computation. 2021. Vol. 31. No. 5. P. 1330–1355.

Speranski S. O.

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 in [Cabalar et al. 2006], while the latter has recently been advocated in [Leitgeb 2019], but was first introduced in [Moisil 1942]. 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.

Keywords: quantification, intuitionistic modal logic, modal negation, Heyting–Ockham logic, Hype, Routley star