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    Carmelics

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    Inverse View

    It is not the case that Player ∃ has a winning strategy in G(¬φ) if and only if player ∃ does not have a winning strategy in G(φ).

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    2 perspectives
    Reason for 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.
      ?

      Think about whether this reason is strong or weak

    • 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.
      ?

      Think about whether this reason is strong or weak

    Reason for 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)'.
      ?

      Think about whether this reason is strong or weak

    • 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.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Hintikka's game G(φ) has finite length for any first-order sentence φ.
      ?

      Think about whether this reason is strong or weak

    • 2.The Gale-Stewart theorem entails that G(φ) is determined.
      ?

      Think about whether this reason is strong or weak

    • 3.Determinacy means player ∃ has a winning strategy in exactly one of G(φ) and its dual game.
      ?

      Think about whether this reason is strong or weak

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    Strongest counterpoint
    Explore the most compelling reason on the other side.
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