Multi-type display calculus for dynamic epistemic logic

Abstract

In the present paper, we introduce a multi-type display calculus for dynamic epistemic logic, which we refer to as Dynamic Calculus. The displayapproach is suitable to modularly chart the space of dynamic epistemic logics on weaker-than-classical propositional base. The presence of types endows the language of the Dynamic Calculus with additional expressivity, allows for a smooth proof-theoretic treatment, and paves the way towards a general methodology for the design of proof systems for the generality of dynamic logics, and certainly beyond dynamic epistemic logic. We prove that the Dynamic Calculus adequately captures Baltag-Moss-Solecki's dynamic epistemic logic, and enjoys Belnap-style cut elimination.

Multi-type display calculus for propositional dynamic logic

“…This definition and theorem are more compact versions of the ones of [10], hence they will not be expanded on, but the Belnap-style cut elimination for the Dynamic Calculus for PDL (cf. Section 6) will be shown on the basis of this more compact definition.…”

“…Here we report on the environment of multi-type display calculi introduced in [10,Section 3]. Our starting point is a propositional language, the terms of which form n pairwise disjoint types T 1 .…”

“…The relativized display property is discussed in more details in [10 C' 8 : Eliminability of matching principal constituents. This condition requests a standard Gentzen-style checking, which is now limited to the case in which both cut formulas are principal, i.e.…”

“…See Section 2.3 for more on this). The cut elimination result for any such calculus follows straightforwardly from the corresponding Belnap-style metatheorem [10,Theorem 3.3].…”

“…The multi-type display calculus for PDL introduced in the present paper has a design similar to the one in [10], from which features analogous to (1) and (3) above will follow. This calculus is shown to be properly displayable (cf.…”