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    Russell's type hierarchy requires the axiom of reducibili... — Carmelics
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    Challenges→The paradoxes of set theory are resolved by reducing assertions about sets to assertions about propositional functions.

    Russell's type hierarchy requires the axiom of reducibility, which Ramsey and Wittgenstein argued is an empirical assumption smuggled into logic.

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    1 reason for
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    Reasons For

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    Reason for
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    • 1.Reducibility asserts that all propositional functions are equivalent to predicative ones, which is not derivable from logical axioms alone.
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    • 2.Ramsey and Wittgenstein correctly identified that this assumption goes beyond formal logic into claims about the structure of reality or language.
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    • 3.Without reducibility, Russell's system cannot prove that all well-formed sentences have truth-values, making it incomplete as a logical foundation.
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    Reasons Against

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    Reason against
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    • 1.Reducibility follows logically from the ramified hierarchy itself if one accepts that only predicative functions are genuinely definable.
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    • 2.Calling reducibility 'empirical' conflates metaphysical claims with empirical ones; it remains a formal axiom choice, not an observation about nature.
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    • 3.Ramsey's own later work abandoned the ramified hierarchy entirely, suggesting his critique targeted Russell's approach rather than proving reducibility empirical.
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    Proof of definition segments1 linkedTruth & Knowledge1 linked

    Related

    Calling reducibility 'empirical' conflates metaphysical claims with empirical on...Ramsey and Wittgenstein correctly identified that this assumption goes beyond fo...Ramsey's own later work abandoned the ramified hierarchy entirely, suggesting hi...Reducibility asserts that all propositional functions are equivalent to predicat...
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    Reducibility follows logically from the ramified hierarchy itself if one accepts...The paradoxes of set theory are resolved by reducing assertions about sets to as...Without reducibility, Russell's system cannot prove that all well-formed sentenc...

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