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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    PDL's calculus is sound and complete only in the weak sense. — Carmelics
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    PDL's calculus is sound and complete only in the weak sense.

    Philosophy of LanguageTruth & Knowledge
    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.PDL includes the induction axiom (Axiom 5)
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    • 2.The induction axiom is responsible for PDL being uncompact
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    • 3.Compactness fails for PDL
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Weak completeness (every valid formula is provable) is the standard notion of completeness in propositional modal logic, making 'only' misleading.
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    • 2.Segerberg's original completeness proof for PDL establishes exactly what standard Hilbert-style axiomatizations are expected to establish.
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    • 3.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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    Reason against 2 of 2
    ?
    • 1.Uncompactness entails failure of strong completeness but leaves weak completeness fully intact, so the two properties operate at distinct levels of logical strength.
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    • 2.Kozen and Parikh's 1981 completeness result for PDL is standardly cited in the literature as a genuine completeness theorem without qualification, undermining the rhetorical force of 'only'.
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    Philosophy of LanguageTruth & Knowledge

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    Related

    Compactness fails for PDLKozen and Parikh's 1981 completeness result for PDL is standardly cited in the l...PDL includes the induction axiom (Axiom 5)Segerberg's original completeness proof for PDL establishes exactly what standar...
    +4 moreShow less
    The absence of strong completeness is a feature shared by virtually all normal m...The induction axiom is responsible for PDL being uncompactUncompactness entails failure of strong completeness but leaves weak completenes...Weak completeness (every valid formula is provable) is the standard notion of co...

    Similar

    XL can have a strongly complete calculus85%Soundness is an essential requirement of a calculus, while completenes...82%Without soundness, a calculus may derive false conclusions from true h...81%A deductive calculus must be sound to be useful.79%

    Source

    AI-extracted1/3 agreementValid
    SEP: logic-many-sorted
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    Given a Kripke structure \[\mathcal{A}=\langle \mathbf{W},\mathbf{R},\langle P^{\mathcal{A}}\rangle _{P\in \Atom}\rangle\] we say that \(\mathcal{AG}\) is a general structure built on \(\mathcal{A}\) if and only if \[\mathcal{AG}=\langle \mathbf{W},\mathbf{W}^{\prime },\mathbf{R},\epsilon _{1}^{\mathcal{A}},\langle P^{\mathcal{A}}\rangle _{P\in \Atom}\rangle\] where \(\Def \subseteq \mathbf{W}^{\prime }\subseteq \wp (\mathbf{W})\). [22] It can be proved that the set of worlds where a moda
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit