Strong completeness and limited canonicity for PDL

Abstract

Propositional dynamic logic (PDL) is complete but not compact. As a consequence, strong completeness (the property Γtrue signΦ⇒ Γ⊢Φ) requires an infinitary proof system. In this paper, we present a short proof for strong completeness of PDL relative to an infinitary proof system containing the rule from [α; β n ]φ for all n N conclude αβ*]Phi;. The proof uses a universal canonical model, and it is generalized to other modal logics with infinitary proof rules, such as epistemic knowledge with common knowledge. Also, we show that the universal canonical model of PDL lacks the property of modal harmony, the analogue of the Truth lemma for modal operators. © 2007 Springer Science+Business Media B.V.