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    Carmelics

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    Made withinDC&Austin
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    Home/Original/inverse
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    Inverse View

    It is not the case that Simple Type Theory (STT) gives rise to Russell's Appendix B paradox when supplemented with the principle that propositions differing by a constituent are distinct propositions and a correlation of propositions with classes they mention.

    ?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 'Structure principle' that propositions differing by a constituent are distinct conflates syntactic individuation with intensional identity of structured propositions.
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    • 2.Church's own Logic of Sense and Denotation (Alternative (0)) individuates propositions by synonymous isomorphism, not mere constituent difference, blocking the diagonal construction.
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    • 3.If propositions are individuated by sense-equivalence rather than syntactic structure, the correlation W cannot be well-defined, since 'mentioning' a class is not a purely extensional relation.
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    Reason for 2 of 2
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    • 1.Quine argued in 'Set Theory and Its Logic' that Russell-style paradoxes generated by type theory reveal the instability of treating propositions as set-like objects subject to membership conditions.
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    • 2.If propositions are not genuine objects in the domain over which quantifiers range—as deflationary nominalists like Prior urged—then no well-formed correlation between propositions and classes exists to generate the diagonal.
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    • 3.Prior's substitutional account of propositional quantification in 'Objects of Thought' shows STT's paradox depends on a realist ontology of propositions that is independently contestable.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.If two propositions differ by a constituent, then they are different propositions (the Structure principle).
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    • 2.If propositions are correlated with the classes they mention, then diagonalization is possible.
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    • 3.Let W be the class of propositions not in the classes with which they are correlated; the proposition 'Every proposition in W is true' is in W if and only if it is not in W.
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