Journal of Logic, Language and Information

, Volume 17, Issue 1, pp 69–87

Strong Completeness and Limited Canonicity for PDL

  • Gerard Renardel de Lavalette
  • Barteld Kooi
  • Rineke Verbrugge
Open AccessArticle

DOI: 10.1007/s10849-007-9051-4

Cite this article as:
Renardel de Lavalette, G., Kooi, B. & Verbrugge, R. J of Log Lang and Inf (2008) 17: 69. doi:10.1007/s10849-007-9051-4


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 [α; βn]φ 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.


Propositional dynamic logic Strong completeness Canonical model Model disharmony 
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© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  • Gerard Renardel de Lavalette
    • 1
  • Barteld Kooi
    • 2
  • Rineke Verbrugge
    • 3
  1. 1.Department of Computing ScienceUniversity of GroningenGroningenThe Netherlands
  2. 2.Faculty of PhilosophyUniversity of GroningenGroningenThe Netherlands
  3. 3.Department of Artificial IntelligenceUniversity of GroningenGroningenThe Netherlands