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
    If there exist sentences φ such that Γ ⊨ φ yet Γ ⊬ φ with... — Carmelics
    Home
    HistoryEditSee Inverse

    Part of a larger discussion

    Challenges→Γ ⊢ φ (Γ proves φ)

    If there exist sentences φ such that Γ ⊨ φ yet Γ ⊬ φ within sufficiently expressive systems, the universal claim Γ ⊢ φ is false as a general principle.

    ?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.Semantic entailment (⊨) captures what must be true across all models; syntactic derivability (⊢) is bounded by proof systems and their axioms.
      ?

      Think about whether this reason is strong or weak

    • 2.Gödel's incompleteness shows sufficiently expressive systems have truths unprovable within them, demonstrating the gap between ⊨ and ⊢.
      ?

      Think about whether this reason is strong or weak

    • 3.If Γ ⊨ φ but Γ ⊬ φ exists as a concrete case, claiming Γ ⊢ φ universally contradicts that evidence and overstates deductive power.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.The distinction between ⊨ and ⊢ is system-relative; a claim true in one proof system may be derivable in a stronger one, so no universal failure exists.
      ?

      Think about whether this reason is strong or weak

    • 2.Gödel's results apply to *specific* unprovable sentences, not all entailed sentences—most logically valid inferences remain provable in standard systems.
      ?

      Think about whether this reason is strong or weak

    • 3.Rejecting 'Γ ⊢ φ universally' based on edge cases conflates 'not always provable' with 'the principle is false,' when it may just need qualification.
      ?

      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 incompleteness shows sufficiently expressive systems have truths unprova...Gödel's results apply to *specific* unprovable sentences, not all entailed sente...If Γ ⊨ φ but Γ ⊬ φ exists as a concrete case, claiming Γ ⊢ φ universally contrad...Rejecting 'Γ ⊢ φ universally' based on edge cases conflates 'not always provable...
    +3 moreShow less
    Semantic entailment (⊨) captures what must be true across all models; syntactic ...The distinction between ⊨ and ⊢ is system-relative; a claim true in one proof sy...Γ ⊢ φ (Γ proves φ)

    Details

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