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    Carmelics

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    Made withinDC&Austin
    Statements
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    Perspectives
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    42
    Home/Original/inverse
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    Inverse View

    It is not the case that The fact that toposes support closed set logic as readily as open set logic is an argument that inconsistent theories are equally reasonable as items of mathematical study.

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

    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.The vindication of intuitionism from topos theory rests not merely on open set logic being supported, but on its deep connection to constructive proof and epistemic constraints on mathematical truth.
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    • 2.Closed set logic's category-theoretic availability does not supply paraconsistent mathematics with a corresponding epistemic or proof-theoretic motivation analogous to intuitionism's rejection of the law of excluded middle.
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    • 3.Structural parity in a semantic framework is insufficient for philosophical parity without an independent account of what cognitive or mathematical practice the logic is answerable to.
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    Reason for 2 of 2
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    • 1.Quine and Putnam's indispensability arguments ground mathematical ontology in the explanatory and predictive success of scientific theories, not in the mere formal consistency or categoricity of logical systems.
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    • 2.Inconsistent theories, lacking application in empirically successful science, fail the indispensability criterion that historically justifies treating classical and even intuitionistic mathematics as legitimate objects of study.
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    • 3.Category-theoretic generality that encompasses a formalism does not constitute the kind of scientific indispensability that warrants treating inconsistent theories as equally reasonable mathematical objects.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Toposes support open set logic, which has been taken as a vindication of mathematical intuitionism.
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    • 2.It can be proved that toposes support closed set logic as readily as they support open set logic.
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    • 3.Closed set logic provides the only category-theoretic semantics for a paraconsistent logic to date.
      ?

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