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    The equivalence of Hintikka's game-theoretic definition o... — Carmelics
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    The equivalence of Hintikka's game-theoretic definition of truth and Tarski's definition of truth is itself equivalent to the axiom of choice (given the other axioms of Zermelo-Fraenkel set theory).

    Truth & Knowledge
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.The argument that player ∃'s winning strategies for G(∀x φ(x)) can be assembled from strategies for G(φ(a)) requires the axiom of choice.
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    • 2.The axiom of choice is necessary and sufficient for the equivalence between game-theoretic and Tarskian truth definitions within ZF set theory.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Hodges (1985) demonstrated that the equivalence holds for first-order logic over well-orderable domains without invoking full AC.
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    • 2.The claim conflates the set-theoretic machinery needed for arbitrary structures with the logical equivalence itself, which is domain-relative.
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    • 3.Restricting quantifier domains to sets admitting a choice function is sufficient for GTS-Tarski equivalence in standard model-theoretic practice.
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    Reason against 2 of 2
    ?
    • 1.Constructive and intuitionistic reformulations of GTS by Felscher (1985) achieve Tarski-equivalence without classical AC by replacing winning strategies with computable functions.
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    • 2.If a constructively valid proof of equivalence exists that avoids AC, then AC cannot be strictly necessary for the equivalence, only for one classical formulation of it.
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    Topics

    Philosophy of LanguageTruth & Knowledge

    Key Terms

    Axiom of Choice(Foundations of mathematics; arises in second-order logic as the statement that every total binary relation has a choice function)
    Given a set A of non-empty pairwise disjoint sets, there exists a set B containing exactly one element from each set in A. When A is infinite, forming B requires making infinitely many simultaneous choices.
    Equivalence(what classical and intuitionistic logic disagree about)
    Two statements are equivalent when they mean exactly the same thing and always have the same truth value.
    Game-theoretic definition(Hintikka's approach to defining truth)
    An explanation of how something works using the logic of games, where different players make strategic moves to try to win or prove a point.
    Hintikka, Jaakko(philosopher being referenced for his definition of truth)
    A Finnish philosopher who developed a way to think about truth using game theory—imagining truth as something that can be verified through a strategic game between two players.
    Tarski, Alfred(philosopher being referenced for his definition of truth)
    A Polish-American logician who created an influential mathematical definition of what it means for a statement to be true.
    Zermelo-Fraenkel set theory(as used in mathematics and logic)
    The most common mathematical system for talking about collections of things (called 'sets') and the rules for how they work together.

    Related

    Constructive and intuitionistic reformulations of GTS by Felscher (1985) achieve...Hodges (1985) demonstrated that the equivalence holds for first-order logic over...If a constructively valid proof of equivalence exists that avoids AC, then AC ca...Restricting quantifier domains to sets admitting a choice function is sufficient...
    +3 moreShow less
    The argument that player ∃'s winning strategies for G(∀x φ(x)) can be assembled ...The axiom of choice is necessary and sufficient for the equivalence between game...

    Source

    AI-extracted1/3 agreementValid
    SEP: logic-games
    View source passageHide passage
    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

    The claim conflates the set-theoretic machinery needed for arbitrary structures ...

    Similar

    The axiom of choice is necessary and sufficient for the equivalence be...94%The non-equivalence of the Tarski definition and the Hintikka definiti...88%For any first-order sentence φ interpreted in a fixed structure A, one...84%The axiom of choice is vindicated by the correct explication of truth.81%
    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit