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    If second-order logic is interpreted with full semantics ... — Carmelics
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    Challenges→There are structures of every infinite cardinality which are not second-order characterizable.

    If second-order logic is interpreted with full semantics relative to a given set-theoretic universe, what counts as 'all structures' shifts across universes, undermining the fixed comparison between sentence-count and structure-count.

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    Reasons For

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    Reason for
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    • 1.Full second-order semantics quantifies over all subsets relative to a universe V; different set-theoretic universes have different power sets.
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    • 2.Completeness theorems depend on fixing a background universe; changing universes changes which structures satisfy a sentence.
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    • 3.Without absolute universe-independence, 'all structures' is not an absolute notion, making cross-universe comparison of logical strength unstable.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.The invariant content of a sentence under full semantics persists across universes: satisfaction relations remain coherent under elementary embeddings.
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    • 2.Set-theoretic pluralism doesn't undermine logic; we can compare universes using a metatheory that relativizes quantifiers without loss of meaning.
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    • 3.Second-order expressiveness is captured by what sentences can distinguish structurally, which is independent of which universe counts as 'all'.
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    Key Terms

    Full semantics(as used in logic and philosophy of language)
    A way of interpreting logical symbols where they're understood to mean exactly what they appear to mean, without any restrictions or shortcuts.
    Second-order logic(as used in mathematical logic)
    A formal system that goes beyond basic logic by allowing you to quantify over (talk about) properties and relations themselves, not just individual objects.
    Sentence-count and structure-count(as used in mathematical logic)
    The number of logical sentences (statements) compared to the number of possible interpretations or models those sentences can describe.
    Structure (in logic)(as used in mathematical logic)
    A mathematical object that shows how symbols and rules of a logical system relate to and apply to real things.
    set-theoretic universe(mathematical foundations)
    The complete collection of all sets that mathematicians study; the foundational framework that most modern mathematics is built on.

    Connections

    2 topics

    Modality & Possibility1 linkedSkepticism1 linked

    Related

    Completeness theorems depend on fixing a background universe; changing universes...Full second-order semantics quantifies over all subsets relative to a universe V...Second-order expressiveness is captured by what sentences can distinguish struct...Set-theoretic pluralism doesn't undermine logic; we can compare universes using ...
    +3 moreShow less
    The invariant content of a sentence under full semantics persists across univers...There are structures of every infinite cardinality which are not second-order ch...

    Details

    Type
    claim
    Perspectives
    2 (1 for, 1 against)
    Edits
    1 edit
    Without absolute universe-independence, 'all structures' is not an absolute noti...