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    Carmelics

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    Inverse View

    It is not the case that A type-theoretical formal property theory can be constructed that avoids Russell's paradox

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    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.Type restrictions are stipulative and do not reflect the actual logical structure of predication as it occurs in natural language and thought.
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    • 2.If a formal system avoids paradox only by prohibiting grammatically well-formed and intuitively meaningful propositions, it has not solved Russell's paradox but merely quarantined it.
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    • 3.Ramsey and Wittgenstein showed that type distinctions, when applied to propositional functions, generate an infinite hierarchy that cannot itself be described within the system without violating its own constraints.
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    Reason for 2 of 2
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    • 1.Grelling's paradox and other semantic paradoxes arise independently of Russell's paradox and are not dissolved by type-theoretic restrictions on property self-predication.
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    • 2.A formal property theory that resolves Russell's paradox through typing while leaving semantic paradoxes intact has not achieved a principled solution but only a piecemeal one.
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    • 3.Kripke's work on truth demonstrated that paradox-avoidance requires grounding conditions that type theory does not supply, making typing insufficient as a foundational remedy.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.If a predicate can be predicated of another predicate only when the former is of a higher type than the latter, then self-predication is banished
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    • 2.Russell's paradox depends on the possibility of self-predication
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    • 3.If self-predication is banished, Russell's paradox cannot even be formulated
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