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    Player ∃ has a winning strategy in G(¬φ) if and only if p... — Carmelics
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    Player ∃ has a winning strategy in G(¬φ) if and only if player ∃ does not have a winning strategy in G(φ).

    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.Hintikka's game G(φ) has finite length for any first-order sentence φ.
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    • 2.The Gale-Stewart theorem entails that G(φ) is determined.
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    • 3.Determinacy means player ∃ has a winning strategy in exactly one of G(φ) and its dual game.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Gale-Stewart determinacy requires games of *countable* length, but extensions of GTS to infinitary logics like Lω₁ω yield games that may be non-determined.
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    • 2.If φ is an infinitary sentence, G(¬φ) need not be the strict dual of G(φ), so neither player may possess a winning strategy in either game.
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    Reason against 2 of 2
    ?
    • 1.Intuitionistic and constructivist semantics (Dummett, Martin-Löf) reject the equivalence of 'no winning strategy for ∃ in G(φ)' with 'a winning strategy for ∃ in G(¬φ)'.
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    • 2.Without classical negation, the absence of a proof of φ does not constructively yield a proof of ¬φ, so the biconditional holds only under classical assumptions the claim silently imports.
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    Philosophy of LanguageTruth & Knowledge

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    Related

    Determinacy means player ∃ has a winning strategy in exactly one of G(φ) and its...Gale-Stewart determinacy requires games of *countable* length, but extensions of...Hintikka's game G(φ) has finite length for any first-order sentence φ.If φ is an infinitary sentence, G(¬φ) need not be the strict dual of G(φ), so ne...
    +3 moreShow less
    Intuitionistic and constructivist semantics (Dummett, Martin-Löf) reject the equ...The Gale-Stewart theorem entails that G(φ) is determined.Without classical negation, the absence of a proof of φ does not constructively ...

    Similar

    Player ∃ has a winning strategy for Hintikka's game G(φ) if and only i...89%Determinacy means player ∃ has a winning strategy in exactly one of G(...87%If player ∃ has a winning strategy f_a for each game G(φ(a)), then the...85%The negation of the statement that player A has a winning strategy is ...84%

    Source

    AI-extracted1/3 agreementValid
    SEP: logic-games
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    One can prove that for any first-order sentence \(\phi\), interpreted in a fixed structure \(A\), player \(\exists\) has a winning strategy for Hintikka’s game \(G(\phi)\) if and only if \(\phi\) is true in \(A\) in the sense of Tarski. Two features of this proof are interesting. First, if \(\phi\) is any first-order sentence then the game \(G(\phi)\) has finite length, and so the Gale-Stewart theorem tells us that it is determined. We infer that \(\exists\) has a winning strategy in exactly one
    Extraction notes

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

    Details

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