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    Carmelics

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

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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Original/inverse
    See Original
    Inverse View

    It is not the case that Γ ⊢ φ (Γ proves φ)

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

    Reasons For

    2 perspectives
    Reason for 1 of 2
    ?
    • 1.Henkin's completeness proof presupposes classical logic, including the law of excluded middle and double negation elimination.
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    • 2.Intuitionistic logic (Brouwer, Heyting) rejects these classical laws, severing the bridge from semantic consequence to syntactic provability.
      ?

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    • 3.Therefore, Γ ⊨ φ does not entail Γ ⊢ φ in constructive systems where proof requires explicit construction, not indirect refutation.
      ?

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    Reason for 2 of 2
    ?
    • 1.Gödel's first incompleteness theorem establishes that any consistent ω-complete formal system strong enough to express arithmetic contains true sentences with no proof.
      ?

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    • 2.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.
      ?

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    • 3.Completeness results like Henkin's hold only for first-order logic, and extending the claim beyond that domain conflates a restricted theorem with a universal one.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Γ ⊨ φ (φ is a semantic consequence of Γ)
      ?

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    • 2.If Γ ⊨ φ, then Γ ∪ {¬φ} has no model (is unsatisfiable)
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    • 3.Henkin's theorem: if a set of formulas is unsatisfiable, it is syntactically contradictory
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    Next step

    Based on where you are in your exploration

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