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    If the axiom of reducibility is not a logical truth, the ... — Carmelics
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    Challenges→The paradoxes of set theory are resolved by reducing assertions about sets to assertions about propositional functions.

    If the axiom of reducibility is not a logical truth, the reduction of sets to propositional functions fails to resolve paradoxes on purely logical grounds.

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    1 reason for
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    Reasons For

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    Reason for
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    • 1.The axiom of reducibility is not derivable from standard logical axioms, making it a substantive mathematical assumption rather than a logical truth.
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    • 2.Without reducibility, type theory cannot guarantee all propositional functions correspond to legitimate sets, leaving impredicative definitions unresolved.
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    • 3.If paradox resolution requires only logical truths, then any non-logical assumption undermines claims of purely logical grounds for that resolution.
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    Reasons Against

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    • 1.The axiom of reducibility, though unprovable, may be a constitutive principle defining what counts as logical rather than an empirical claim needing proof.
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    • 2.Paradoxes are resolved by the type hierarchy structure itself; reducibility merely simplifies the system without being essential to paradox prevention.
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    • 3.Distinguishing 'purely logical' from 'mathematical' is ambiguous; foundational axioms supporting logic may not require independent justification as logical truths.
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    Related

    Distinguishing 'purely logical' from 'mathematical' is ambiguous; foundational a...If paradox resolution requires only logical truths, then any non-logical assumpt...Paradoxes are resolved by the type hierarchy structure itself; reducibility mere...The axiom of reducibility is not derivable from standard logical axioms, making ...
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    The axiom of reducibility, though unprovable, may be a constitutive principle de...The paradoxes of set theory are resolved by reducing assertions about sets to as...Without reducibility, type theory cannot guarantee all propositional functions c...

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