One Modal Logic to Rule Them All? - Holliday, Wesley; Litak, Tadeusz Michal (Advances in Modal Logic 2018, 2018)
In this paper, we introduce an extension of the modal language with what we call the global quantificational modality [∀p]. In essence, this modality combines the propositional quantifier ∀p with the global modality A: [∀p] plays the same role as the compound modality ∀pA. Unlike the propositional quantifier by itself, the global quantificational modality can be straightforwardly interpreted in any Boolean Algebra Expansion (BAE). We present a logic GQM for this language and prove that it is complete with respect to the intended algebraic semantics. This logic enables a conceptual shift, as what have traditionally been called different “modal logics” now become [∀p]-universal theories over the base logic GQM: instead of defining a new logic with an axiom schema such as Pφ → PPφ, one reasons in GQM about what follows from the globally quantified formula [∀p](Pp → PPp).
Keywords: global quantificational modalities; propositional quantifiers; Boolean algebra expansions; Boolean algebras with operators; modal consequence