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    Russell's paradox can be derived without appealing to the... — Carmelics
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    Russell's paradox can be derived without appealing to the principle of Excluded Middle

    Philosophy of Language
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    2 reasons for
    1 reason against

    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.Brouwer's intuitionism accepts the derivation of contradiction from R ∈ R ≡ ~(R ∈ R) without invoking Excluded Middle.
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    • 2.Intuitionistic logic validates ex contradictione sequitur quodlibet, so the paradox's explosive force survives without LEM.
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    • 3.Dummett's proof-theoretic semantics confirms that Russell's contradiction is constructively derivable from Comprehension alone.
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    Reason for 2 of 2
    ?
    • 1.Thierry Coquand showed in constructive type theory that the Burali-Forti and Russell paradoxes arise without classical logic.
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    • 2.The biconditional R ∈ R ≡ ~(R ∈ R) entails absurdity in minimal logic, which is strictly weaker than intuitionistic logic.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.The paradox can be reformulated using the Law of Non-contradiction instead of Excluded Middle
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    • 2.From the definition of R, it follows that R ∈ R ≡ ~(R ∈ R)
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    • 3.From R ∈ R ≡ ~(R ∈ R), we can derive R ∈ R ⊃ ~(R ∈ R)
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    Philosophy of LanguageTruth & Knowledge

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    2 topics

    Proof of definition segments7 linkedModality & Possibility2 linked

    Related

    Both R ∈ R and its negation are deduced using only intuitionistically acceptable...Brouwer's intuitionism accepts the derivation of contradiction from R ∈ R ≡ ~(R ...By modus tollens, we conclude ~(R ∈ R)Combined with R ∈ R ⊃ R ∈ R, this yields R ∈ R ⊃ (R ∈ R ∧ ~(R ∈ R))
    +9 moreShow less
    Dummett's proof-theoretic semantics confirms that Russell's contradiction is con...From R ∈ R ≡ ~(R ∈ R), it also follows that ~(R ∈ R) ⊃ R ∈ R, yielding R ∈ RFrom R ∈ R ≡ ~(R ∈ R), we can derive R ∈ R ⊃ ~(R ∈ R)

    Similar

    Russell's paradox can be formulated without relying on Excluded Middle92%Rejecting the principle of Excluded Middle does not resolve Russell's ...89%The problems with Naive Comprehension are not confined to Russell's pa...87%Bertrand's paradox provides no conclusive reason against the Indiffere...86%

    Source

    AI-extracted1/3 agreementValid
    SEP: russell-paradox
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    Another suggestion might be to conclude that the paradox depends upon an instance of the principle of Excluded Middle, that either \(R\) is a member of \(R\) or it is not. This is a principle that is rejected by some non-classical approaches to logic, including intuitionism. However it is possible to formulate the paradox without appealing to Excluded Middle by relying instead upon the Law of Non-contradiction. We do so as follows: Given the definition of \(R\) it follows that \(R \in R \equiv{\
    Extraction notes

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

    Details

    From the definition of R, it follows that R ∈ R ≡ ~(R ∈ R)
    Intuitionistic logic validates ex contradictione sequitur quodlibet, so the para...
    The Law of Non-contradiction gives us ~(R ∈ R ∧ ~(R ∈ R))
    The biconditional R ∈ R ≡ ~(R ∈ R) entails absurdity in minimal logic, which is ...
    The paradox can be reformulated using the Law of Non-contradiction instead of Ex...
    Thierry Coquand showed in constructive type theory that the Burali-Forti and Rus...
    Type
    claim
    Perspectives
    3 (2 for, 1 against)
    Edits
    1 edit