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
    Recursive enumerability of validities is necessary but no... — Carmelics
    Home
    HistoryEditSee Inverse

    Part of a larger discussion

    Challenges→XL can have a strongly complete calculus

    Recursive enumerability of validities is necessary but not sufficient for strong completeness without a complete proof system being explicitly constructible.

    ?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.Recursive enumerability guarantees that all valid formulas can be systematically listed, which is logically prior to any completeness result.
      ?

      Think about whether this reason is strong or weak

    • 2.Strong completeness requires semantic entailment to match provability, but this equivalence need not depend on an explicit, constructive proof system.
      ?

      Think about whether this reason is strong or weak

    • 3.Gödel's completeness theorem shows validity enumeration suffices for first-order logic without requiring proof system construction details.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Strong completeness inherently demands an explicit proof system: if no derivation method exists, the notion of 'provability' becomes semantically empty.
      ?

      Think about whether this reason is strong or weak

    • 2.Recursive enumerability of validities alone cannot establish the semantic-syntactic equivalence that strong completeness requires to be meaningful.
      ?

      Think about whether this reason is strong or weak

    • 3.Non-constructive existence proofs fail to ground completeness claims; showing a system must exist differs fundamentally from showing it suffices.
      ?

      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.

    Connections

    2 topics

    Truth & Knowledge1 linkedPhilosophy of Language1 linked

    Related

    Gödel's completeness theorem shows validity enumeration suffices for first-order...Non-constructive existence proofs fail to ground completeness claims; showing a ...Recursive enumerability guarantees that all valid formulas can be systematically...Recursive enumerability of validities alone cannot establish the semantic-syntac...
    +3 moreShow less
    Strong completeness inherently demands an explicit proof system: if no derivatio...Strong completeness requires semantic entailment to match provability, but this ...XL can have a strongly complete calculus

    Details

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