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
    Home/Original/inverse
    See Original
    Inverse View

    It is not the case that The above proof of the intermediate value result can be read either as a syntactic derivation from the axioms or as a semantic argument

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    2 perspectives
    Reason for 1 of 2
    ?
    • 1.Syntactic derivability in second-order logic is not recursively enumerable, so no finite proof system can capture all second-order validities.
      ?

      Think about whether this reason is strong or weak

    • 2.A proof that cannot be mechanically verified as syntactically complete conflates semantic entailment with derivability, undermining the syntactic reading.
      ?

      Think about whether this reason is strong or weak

    Reason for 2 of 2
    ?
    • 1.Kreisel and Boolos demonstrated that second-order consequence is not axiomatizable, meaning 'derivation from axioms' in second-order logic is inherently incomplete relative to standard semantics.
      ?

      Think about whether this reason is strong or weak

    • 2.If the proof's validity depends on the intended interpretation of quantifiers over all subsets, then the semantic reading is doing essential work that the syntactic reading cannot replicate.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.On the surface the proof looks like a semantic argument
      ?

      Think about whether this reason is strong or weak

    • 2.Every step of the proof can be derived from the axioms
      ?

      Think about whether this reason is strong or weak

    • 3.Syntactic derivations in second-order logic based on the Comprehension Axiom Schema and Axioms of Choice are very much like syntactic derivations in set theory, and working mathematicians write both in shorthand
      ?

      Think about whether this reason is strong or weak

    Next step

    Based on where you are in your exploration

    Strongest counterpoint
    Explore the most compelling reason on the other side.
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42