Strong Completeness and Limited Canonicity for PDL

Authors:
Gerard Renardel De Lavalette;Barteld Kooi;Rineke Verbrugge
Affiliations:
Department of Computing Science, University of Groningen, Groningen, The Netherlands 9700 AK;Faculty of Philosophy, University of Groningen, Groningen, The Netherlands;Department of Artificial Intelligence, University of Groningen, Groningen, The Netherlands
Venue:
Journal of Logic, Language and Information
Year:
2008

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Strong Completeness and Limited Canonicity for PDL

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Abstract

Propositional dynamic logic ( $$\mathsf{PDL}$$ ) is complete but not compact. As a consequence, strong completeness (the property $$\Gamma \models \varphi \Rightarrow \Gamma \vdash \varphi$$ ) requires an infinitary proof system. In this paper, we present a short proof for strong completeness of $$\mathsf{PDL}$$ relative to an infinitary proof system containing the rule from [驴; β n ]驴 for all $$n \in {\mathbb{N}}$$ , conclude $$[\alpha;\beta^*] \varphi$$ . 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 $$\mathsf{PDL}$$ lacks the property of modal harmony, the analogue of the Truth lemma for modal operators.