Completeness of Second-Order Propositional S4 and H in Topological Semantics

Abstract

We add propositional quantifiers to the propositional modal logic S4 and to the propositional intuitionistic logic H, introducing axiom schemes that are the natural analogs to axiom schemes typically used for first-order quantifiers in classical and intuitionistic logic. We show that the resulting logics are sound and complete for a topological semantics extending, in a natural way, the topological semantics for S4 and for H.

Convergence, Continuity and Recurrence in Dynamic Epistemic Logic

“…With Theorem 1, a connection arises between dynamic epistemic logic and dynamic topological logic (see e.g. [23,24,34,35]): Each system (X d , f ) may be considered a dynamic topological model with atom set L Λ and the 'next' operator's semantics given by an application of f , equivalent to a f dynamic modality of DEL. The topological 'interior' operator has yet no DEL parallel.…”

On the Logic of Belief and Propositional Quantification

“…In a completely non-technical paper [20], we also saw perhaps the earliest proposal of treating ∀p as quantifying directly over objects in a lattice of propositions, a proposal perhaps inspired by the philosophical stance defended in that paper. Since then, there has been a steady stream of interest devoted to this topic, with general theoretical results focusing on expressive power under the standard possible-world semantics ( [21][22][23][24]), specific results mostly establishing non-axiomatizability ( [25][26][27][28][29][30][31][32][33]) with the exception of [31] and [33], and more application-oriented works: [1,[34][35][36][37][38][39].…”

Neighbourhood and Lattice Models of Second-Order Intuitionistic Propositional Logic

“…(2) Given a subset U of Idl X, we have that U is a completely prime element 7 of Ω * X if and only if U = ↑ (↓ a) for some a ∈ X.…”

“…To overcome it, one reasonable approach would be to weaken a requirement for the underlying structure of complete lattice so as to regard the Lindenbaum algebra induced from the propositions of NJ 2 as a model. In fact, this kind of generalization has been successful in various standpoints of semantics [6], and Kremer [7] actually gives a framework of models for which the completeness of NJ 2 is ensured. In contrast with these positive aspects, any structures appearing in these discussions would not be comparable with Sobolev's framework because in his definition the range of the valuation of bound variables sensitively depends on the possible world in which we have to define the interpretation.…”