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    The full ramified theory of types is not needed to resolv... — Carmelics
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    The full ramified theory of types is not needed to resolve mathematical or set-theoretical paradoxes.

    Proof of definition segmentsTruth & Knowledge
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 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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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 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 against 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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    Truth & KnowledgeProof of definition segments

    Related

    Chwistek (1921) and Ramsey (1931) independently proposed that only the simple th...Ramsey's own separation of logical from semantic paradoxes presupposes a distinc...Russell's substitutional theory and the no-class theory require ramification to ...Simple type theory alone cannot resolve semantic paradoxes because it lacks the ...
    +3 moreShow less
    The simple theory of types — distinguishing individuals, functions of individual...The vicious circle principle, which motivated ramification, is needed to block i...Without ramified orders, the axiom of reducibility must be introduced, which Qui...

    Similar

    The ramified theory of types is in tension with classical real analysi...85%Our theories in many fields of inquiry are at least apparently committ...80%The typical ambiguity of terms and formulas in the theory of types mea...80%Chwistek (1921) and Ramsey (1931) independently proposed that only the...80%

    Source

    AI-extracted1/3 agreementValid
    SEP: principia-mathematica
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    The paradoxes of the theory of sets are resolved by reducing assertions about sets to assertions about propositional functions. The restriction that a function of one type cannot apply to a function of the same type is enough to block the paradoxes. Thus the distinction between individuals, functions of individuals, and functions of such functions, categorized by what came to be called “simple theory of types” is enough for the purposes of reducing mathematics to classes, and so to logic. The id
    Extraction notes

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

    Details

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