Skip to content
Carmelics
TopicsThinkersChangesContributorsLoading account…

    Carmelics

    A reasoning platform. Break down any belief into clear reasons, explore both sides, and weigh the evidence honestly.

    Navigate

    • Topics
    • Search
    • Recent Changes
    • Contribute
    • How It Works
    • Glossary
    • Thinkers
    • Contributors
    • About
    • Statistics
    • Terms
    • Privacy

    Database

    Statements
    —
    Perspectives
    —
    Topics
    —

    Press ? for keyboard shortcuts

    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Descriptive complexity results are sensitive to the choic... — Carmelics
    Home
    HistoryEditSee Inverse

    Part of a larger discussion

    Challenges→First-order logic FO captures only the very weak complexity class AC^0 and cannot express properties in stronger classes such as P without extensions.

    Descriptive complexity results are sensitive to the choice of encoding, so the boundary between FO and stronger classes is not a fact about logic alone but about logic-plus-structure.

    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    1 reason against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.The same finite structure can be encoded as ordered or unordered, with/without successor, yielding different FO-expressibility boundaries.
      ?

      Think about whether this reason is strong or weak

    • 2.Encoding choices are conventional tools, not intrinsic to logical power, so descriptive complexity reflects the tool-structure pairing.
      ?

      Think about whether this reason is strong or weak

    • 3.If boundaries were purely logical facts, they'd hold across all encodings. Since they don't, encoding is essential to the boundary.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Different encodings of the same structure are informationally equivalent; relabeling vertices doesn't change what FO can express about it.
      ?

      Think about whether this reason is strong or weak

    • 2.The expressive gap between FO and second-order logic persists across all reasonable encodings, suggesting the boundary is logically robust.
      ?

      Think about whether this reason is strong or weak

    • 3.Sensitivity to encoding may reflect artifact of proof technique, not a deep dependence of logical power on representation choice.
      ?

      Think about whether this reason is strong or weak

    Sign in or register to share your perspective on this statement.

    Next step

    Based on where you are in your exploration

    Strongest counterpoint
    Explore the most compelling reason on the other side.

    Key Terms

    FO (First-Order Logic)(the main subject being discussed in the statement)
    A formal system for reasoning that can make statements about individual things and their properties, but cannot directly talk about properties themselves the way higher-level systems can.
    Logic-plus-structure(as used in mathematical and philosophical logic)
    The idea that what's true or possible depends not just on the rules of reasoning themselves, but also on the specific way things are organized or arranged in reality.
    descriptive complexity(Descriptive complexity theory, contrasted with computational complexity)
    A measure of a problem's complexity in proportion to the logical resources required to describe its instances, defined using formulas that characterize the problem's instances relative to an appropriate background class of finitary structures.
    encoding(Contrasted with exemplification; characterized as 'internal' predication. E.g., the winged horse encodes the property winged without exemplifying it.)
    The mode of predication attributed to non-existent objects, by which such objects bear a property without instantiating it in the ordinary sense.

    Connections

    2 topics

    Truth & Knowledge1 linkedPhilosophy of Language1 linked

    Related

    Different encodings of the same structure are informationally equivalent; relabe...Encoding choices are conventional tools, not intrinsic to logical power, so desc...First-order logic FO captures only the very weak complexity class AC^0 and canno...If boundaries were purely logical facts, they'd hold across all encodings. Since...
    +3 moreShow less
    Sensitivity to encoding may reflect artifact of proof technique, not a deep depe...The expressive gap between FO and second-order logic persists across all reasona...

    Details

    Type
    claim
    Perspectives
    2 (1 for, 1 against)
    Edits
    1 edit
    The same finite structure can be encoded as ordered or unordered, with/without s...