The absence of strong completeness is a feature shared by virtually all normal modal logics with the Barcan formula or analogous infinitary commitments, not a deficiency unique to PDL.
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(Distinguished from weak completeness, which only concerns tautologies)
If φ is a semantic consequence of Γ (Γ ⊨ φ), then φ is provable from Γ (Γ ⊢ φ)
The Barcan formula(an example of what causes the absence of strong completeness)
A rule in modal logic (named after philosopher Ruth Barcan Marcus) that says if something possibly exists, then it's possible that something exists—it deals with how existence and possibility interact.