Abstract
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 associated with the given model into the complex algebra associated with the updated model. This dual characterization naturally generalizes to much wider classes of algebras, which include, but are not limited to, arbitrary BAOs and arbitrary modal expansions of Heyting algebras (HAOs). As an application of this dual characterization, we axiomatize the intuitionistic analogue of the logic of epistemic knowledge and actions, which we refer to as IEAK, prove soundness and completeness of IEAK w.r.t. both algebraic and relational models, and illustrate how IEAK encodes the reasoning of agents in a concrete epistemic scenario.
Highlights
Duality theory is an established methodology in the mathematical theory of modal logic, and has been the driving engine of some of its core results, as well as of its generalizations, and of extensions of techniques and results from modal logic to other nonclassical logics (e.g. Sahlqvist correspondence for substructural logics)
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-MossSolecki, without the common knowledge operator
Summary
Duality theory is an established methodology in the mathematical theory of modal logic, and has been the driving engine of some of its core results (e.g. the theory of canonicity), as well as of its generalizations (e.g. coalgebraic logics), and of extensions of techniques and results from modal logic to other nonclassical logics (e.g. Sahlqvist correspondence for substructural logics). Together with [18], the present paper is concerned with applying duality theory to a close cognate of modal logic, namely Dynamic Epistemic Logic, and starting to take stock of the results of this application. The dynamic epistemic logic considered in the present paper is the Logic of Epistemic Actions and Knowledge due to Baltag-Moss-Solecki [2], and we refer to it as EAK. The main feature of the relational semantics of EAK is the so-called product update construction, which is grounded on a Kripke-style encoding of epistemic actions. Epistemic actions in this setting are formalized as action structures: finite pointed relational structures, each state of which is endowed with a formula (its precondition). Key words and phrases: Dynamic Epistemic Logic, duality, intuitionistic modal logic, algebraic models, pointfree semantics, Intuitionistic Dynamic Epistemic Logic
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