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    Γ ⊢ φ (Γ proves φ) — Carmelics
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Philosophy of Language
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    Γ ⊢ φ (Γ proves φ)

    Philosophy of LanguageTruth & Knowledge
    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 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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    Reasons Against

    2 perspectives
    Reason against 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 against 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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    Topics

    Philosophy of LanguageTruth & Knowledge

    Connections

    2 topics

    All sources support it1 linkedModality & Possibility1 linked

    Related

    Calculus rules permit the elimination of ¬φ from the assumption set to yield Γ ⊢...Completeness results like Henkin's hold only for first-order logic, and extendin...Gödel's first incompleteness theorem establishes that any consistent ω-complete ...Henkin's completeness proof presupposes classical logic, including the law of ex...
    +7 moreShow less
    Henkin's theorem: if a set of formulas is unsatisfiable, it is syntactically con...If there exist sentences φ such that Γ ⊨ φ yet Γ ⊬ φ within sufficiently express...If Γ ∪ {¬φ} is contradictory, then Γ ∪ {¬φ} ⊢ φ

    Similar

    The Main Theorem establishes that Γ ⊨_MSL φ if and only if Trans(Γ)∪Π ...87%The Gale-Stewart theorem entails that G(φ) is determined.85%Parikh's theorem states that if IΔ_0 proves ∀x∃y φ(x,y) with φ being Σ...81%Γ ⊨ φ (φ is a semantic consequence of Γ)81%

    Source

    AI-extracted1/3 agreementValid
    SEP: logic-many-sorted
    View source passageHide passage
    To see that Henkin’s theorem implies Strong completeness, let us assume the antecedent, \(\Gamma \models \varphi\). Therefore, \(\Gamma \cup \{ \lnot \varphi \}\) is not satisfiable, it has no model. Using Henkin’s theorem we conclude that \(\Gamma \cup \{ \lnot \varphi \}\) is contradictory and so \(\Gamma \cup \{ \lnot \varphi \} \vdash \varphi\). The calculus rules allow us to get rid of \(\lnot \varphi\) and infer \(\Gamma \vdash \varphi\).
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

    If Γ ⊨ φ, then Γ ∪ {¬φ} has no model (is unsatisfiable)
    Intuitionistic logic (Brouwer, Heyting) rejects these classical laws, severing t...
    Therefore, Γ ⊨ φ does not entail Γ ⊢ φ in constructive systems where proof requi...
    Γ ⊨ φ (φ is a semantic consequence of Γ)
    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit