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We define the algorithm ALBA for the language of the same distributive modal logic (DML) for which a Sahlqvist theorem was proved by Gehrke, Nagahashi, and Venema. Successful executions of ALBA compute the local first-order correspondents of input DML inequalities, and also guarantee their canonicity. The class of inequalities on which ALBA is successful is… (More)

In this paper, we start studying epistemic updates using the standard toolkit of duality theory. We focus on public announcements, which are the simplest epistemic actions, and hence on single-agent 1 Public Announcement Logic (PAL) without the common knowledge operator. As is well known, the epistemic action of publicly announcing a given proposition is… (More)

In this paper we introduce canonical extensions of partially ordered sets and monotone maps and a corresponding discrete duality. We then use these to give a uniform treatment of completeness of re-lational semantics for various substructural logics with implication as the residual(s) of fusion.

abstract. We investigate an alternative presentation of classical and positive modal logic where the coalgebraic cover modality is taken as primitive. For each logic, we present a sound and complete Hilbert-style axiomatiza-tion. Moreover, we give a two-sided sound and complete sequent calculus for the negation-free language, and for the language with… (More)

This paper is a study into some properties and applications of Moss' coalgebraic or 'cover' modality ∇. First we present two axiomatizations of this operator, and we prove these axiomatizations to be sound and complete with respect to basic modal and positive modal logic, respectively. More precisely, we introduce the notions of a modal ∇-algebra and of a… (More)

Positive Modal Logic is the restriction of the modal local consequence relation defined by the class of all Kripke models to the propositional negation-free modal language. The class of positive modal algebras is the one canonically associated with PML according to the theory of the algebrization of logics [12]. A Priestley-style du-ality is established… (More)

We develop the mathematical theory of epistemic updates with the tools of duality theory. We focus on the Logic of Epistemic Actions and Knowledge (EAK), introduced by Baltag-Moss-Solecki, without the common knowledge operator. We dually characterize the product update construction of EAK as a certain construction transforming the complex algebras… (More)