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    Carmelics

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    It is not the case that The full ramified theory of types is not needed to resolve mathematical or set-theoretical paradoxes.

    ?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 vicious circle principle, which motivated ramification, is needed to block impredicative definitions that generate semantic paradoxes like Richard's and Berry's.
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    • 2.Simple type theory alone cannot resolve semantic paradoxes because it lacks the stratification of propositional functions by their definition order.
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    • 3.Ramsey's own separation of logical from semantic paradoxes presupposes a distinction that itself requires philosophical justification beyond simple typing.
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    Reason for 2 of 2
    ?
    • 1.Russell's substitutional theory and the no-class theory require ramification to avoid paradoxes arising from propositional functions defined over all propositions.
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    • 2.Without ramified orders, the axiom of reducibility must be introduced, which Quine and others argued undermines the logicist program's epistemic justification entirely.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.The simple theory of types — distinguishing individuals, functions of individuals, and functions of such functions — is sufficient for reducing mathematics to classes and logic.
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    • 2.Chwistek (1921) and Ramsey (1931) independently proposed that only the simple theory of types is required.
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