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    The translation from singular reference to 'the' singleto... — Carmelics
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    Home/Philosophy of Language
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    The translation from singular reference to 'the' singleton function into universal quantification over singleton functions introduces no inconsistency into mathematics.

    Modality & PossibilityPhilosophy of Language
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
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    • 1.There exist values for the singleton function variable s, so the quantification is non-vacuous.
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    • 2.It is provable that the embedded sentence has the same truth value for any eligible value of s, eliminating indeterminacy.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.Universal quantification over singleton functions presupposes a determinate domain of such functions, which itself requires prior set-theoretic commitments.
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    • 2.If the domain of singleton functions is indeterminate or subject to paradox (e.g., via Cantorian diagonalization on function spaces), the quantification inherits that indeterminacy.
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    • 3.Thus the translation defers rather than eliminates the foundational inconsistency, relocating it from singular reference to domain specification.
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    Reason against 2 of 2
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    • 1.Benacerraf's identification problem shows that mathematical objects like singletons lack determinate identity conditions across set-theoretic constructions.
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    • 2.If no privileged singleton function exists, universal quantification over all eligible candidates conflates structurally distinct but formally isomorphic objects, violating referential determinacy.
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    • 3.A translation that preserves truth value across indeterminate referents does not establish consistency but merely hides the ontological arbitrariness Benacerraf diagnosed.
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    Philosophy of LanguageModality & Possibility

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    Related

    A translation that preserves truth value across indeterminate referents does not...Benacerraf's identification problem shows that mathematical objects like singlet...If no privileged singleton function exists, universal quantification over all el...If the domain of singleton functions is indeterminate or subject to paradox (e.g...
    +4 moreShow less
    It is provable that the embedded sentence has the same truth value for any eligi...There exist values for the singleton function variable s, so the quantification ...Thus the translation defers rather than eliminates the foundational inconsistenc...Universal quantification over singleton functions presupposes a determinate doma...

    Similar

    Any claim previously using 'the' singleton function can be recast as a...88%Given plural quantification, one can specify exactly what constraints ...85%Mathematical claims made in terms of sets/classes can be understood as...80%There exist values for the singleton function variable s, so the quant...80%

    Source

    AI-extracted1/3 agreementValid
    SEP: david-lewis
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    One of the technical advances of (1991a) and (1993d) was that they showed how a structuralist account of set theory was even possible. This part of the work was co-authored with John P. Burgess and A. P. Hazen. Given a large enough universe (i.e., that the cardinality of the mereological atoms is an inaccessible cardinal), and given plural quantification, we can say exactly what constraints a function must satisfy for it to do the work we want the singleton function to do. (By ‘the singleton f
    Extraction notes

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

    Details

    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit