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Using the universal modality: Gains and questions. (English) Zbl 0774.03003
The paper investigates a simple and natural enrichment of the usual modal language \({\mathcal L}={\mathcal L}(\square)\) with an auxiliary universal modality \(\square_ u\) having Kripke semantics; \(\square_ u\phi\) is true at a world of a model iff \(\phi\) is valid in the model. The enriched language \({\mathcal L}_{\square_ u}={\mathcal L}(\square,\square_ u)\) turns out to be fairly different from the classical one. In particular the notions of satisfiability, validity and consequence in models become interreducible.
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. (From the authors’ abstract).

03B45 Modal logic (including the logic of norms)
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