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    The empty sequent is not provable — Carmelics
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    Perspectives
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    The empty sequent is not provable

    Proof of definition segmentsTruth & 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.If the empty sequent were provable, it would have a cut-free derivation by the Hauptsatz
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    • 2.A cut-free derivation of the empty sequent can contain only empty sequents
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    • 3.No valid proof can consist solely of empty sequents, since every proof must contain axioms
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.In paraconsistent logics (e.g., Priest's LP), the explosion principle fails, so derivability from contradictions does not trivialize proof systems.
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    • 2.A proof system admitting true contradictions could license the empty sequent without collapsing into inconsistency, undermining P3's universality.
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    • 3.The unprovability of the empty sequent is thus a feature of classical and intuitionistic systems, not a logical necessity across all coherent proof systems.
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    Reason against 2 of 2
    ?
    • 1.Wittgenstein's rule-following considerations suggest that what counts as a 'valid axiom' depends on communal practice, not intrinsic formal properties.
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    • 2.If axiomatic status is practice-relative, P3's claim that every proof must contain axioms begs the question against alternative foundational frameworks.
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    • 3.A formalist who treats the empty sequent as a primitive stipulation would deny P3 without incoherence, making the unprovability claim system-relative rather than absolute.
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    Topics

    Truth & KnowledgeProof of definition segments

    Related

    A cut-free derivation of the empty sequent can contain only empty sequentsA formalist who treats the empty sequent as a primitive stipulation would deny P...A proof system admitting true contradictions could license the empty sequent wit...If axiomatic status is practice-relative, P3's claim that every proof must conta...
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    If the empty sequent were provable, it would have a cut-free derivation by the H...In paraconsistent logics (e.g., Priest's LP), the explosion principle fails, so ...No valid proof can consist solely of empty sequents, since every proof must cont...The unprovability of the empty sequent is thus a feature of classical and intuit...Wittgenstein's rule-following considerations suggest that what counts as a 'vali...

    Similar

    If the empty sequent were provable, it would have a cut-free derivatio...86%If the empty sequent is not deducible, then Z_2 is consistent82%No valid proof can consist solely of empty sequents, since every proof...82%A cut-free derivation of the empty sequent can contain only empty sequ...81%

    Source

    AI-extracted1/3 agreementValid
    SEP: proof-theory
    View source passageHide passage
    Proof: Assume that the empty sequent is provable; then, according to the Hauptsatz it has a cut-free derivation \(\cD\). The previous corollary assures us that only empty sequents can occur in \(\cD\); but such a \(\cD\) does not exist since every proof must contain axioms. \(\qed\)
    Extraction notes

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

    Details

    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit