Using the Universal Modality: Gains and Questions

Abstract

The paper investigates a simple and natural enrichment of the usual modal language ℒ = ℒ(□) with an auxiliary ‘universal’ modality u⃞ having Kripke semantics: u⃞ϕ is true at a world of a model iff Φ is valid in the model The enriched language ℒu⃞ = ℒ(□,⃞) turns out to be fairly different from the classical one. In particular the notions of satisfiability, validity and consequence in models become interreducible. Section 2 is devoted to modal definability in ℒu⃞. Model-theoretic characterizations of this definability are obtained. ℒu⃞-definability is proved to be equivalent with sequential definability in Lintroduced by Kapron. In section 3 the minimal normal ℒu⃞-logic Ku⃞ is axiomatized and a general model-completeness theroem for the family of normal extensions of Ku⃞ is proved. Section 4 deals with minimal extensions ℒu⃞ logics axiomatized with schemata of ℒ over Ku⃞. A general study to transfer of properties of ℒ-logics to their minimal extensions is initiated. Transfer of incompleteness, strong completeness, compactness and filtration is proved. The problems of transferring completeness, finite completeness and decidabilit are investigated and several general results are obtained. Uniform reductions of these properties of ℒu⃞-logics to corresponding natural properties in thier classical fragments are established. For a large class of ℒ-logics, completeness is shown to be inherited in thier minimal extensions. However, the general transferring problems remain still open.In section 5 several concrete completeness and decidability results for logics with essentially ℒu⃞-axiomatics are stated and some other applications of u⃞ are sketched. In an appendix independent join of ℒu⃞-logics is introduced and proved to preserve completeness when applied to minimal extensions. Besides the technical results, the paper pursues two main purposes: first, to advertise the universal modality as a natural and helpful tool, providing a better medium for the mission of modality; and second, to illustrate the typical problems arising when enrichments of modal languages are investigated.