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
    In Gentzen systems, if a connective does not appear in th... — Carmelics
    Home/Philosophy of Language
    HistoryEditSee Inverse

    In Gentzen systems, if a connective does not appear in the conclusion of a proof, it will not appear in the proof at all.

    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.In Gentzen systems, connectives are always introduced in a proof when read from top to bottom.
      ?

      Think about whether this reason is strong or weak

    • 2.In Gentzen systems, connectives cannot be eliminated once introduced.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Cut elimination is a metatheorem that must be proved for each Gentzen system, not a universal structural feature of all such systems.
      ?

      Think about whether this reason is strong or weak

    • 2.Before cut elimination is established, the cut rule permits a formula with arbitrary connectives to appear in a proof without appearing in the conclusion.
      ?

      Think about whether this reason is strong or weak

    • 3.Gentzen's Hauptsatz therefore presupposes successful cut elimination, making the claim conditional rather than definitional of Gentzen systems generally.
      ?

      Think about whether this reason is strong or weak

    Reason against 2 of 2
    ?
    • 1.In display calculi and hypersequent systems, which are recognized extensions of Gentzen's framework, structural rules can reintroduce connective complexity not present in the conclusion.
      ?

      Think about whether this reason is strong or weak

    • 2.The subformula property—on which the claim depends—fails for systems with non-standard structural rules such as those modeling modal or relevant logics.
      ?

      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.

    Topics

    Philosophy of LanguageTruth & Knowledge

    Related

    Before cut elimination is established, the cut rule permits a formula with arbit...Cut elimination is a metatheorem that must be proved for each Gentzen system, no...Gentzen's Hauptsatz therefore presupposes successful cut elimination, making the...In Gentzen systems, connectives are always introduced in a proof when read from ...
    +3 moreShow less
    In Gentzen systems, connectives cannot be eliminated once introduced.In display calculi and hypersequent systems, which are recognized extensions of ...The subformula property—on which the claim depends—fails for systems with non-st...

    Similar

    In Gentzen systems, proofs never lose structure because connectives ar...89%In Gentzen systems, connectives are always introduced in a proof when ...88%In Gentzen systems, connectives cannot be eliminated once introduced.85%Polynomial proof systems (polynomially bounded proof systems) likely d...75%

    Source

    AI-extracted1/3 agreementValid
    SEP: logic-substructural
    View source passageHide passage
    Gentzen systems, with their introduction rules on the left and the right, have very special properties which are useful in studying logics. Since connectives are always introduced in a proof (read from top to bottom) proofs never lose structure. If a connective does not appear in the conclusion of a proof, it will not appear in the proof at all, since connectives cannot be eliminated.
    Extraction notes

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

    Details

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