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    Inverse View

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

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

    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.Cut elimination is a metatheorem that must be proved for each Gentzen system, not a universal structural feature of all such systems.
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    • 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.
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    • 3.Gentzen's Hauptsatz therefore presupposes successful cut elimination, making the claim conditional rather than definitional of Gentzen systems generally.
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    Reason for 2 of 2
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    • 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.
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    • 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.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.In Gentzen systems, connectives are always introduced in a proof when read from top to bottom.
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    • 2.In Gentzen systems, connectives cannot be eliminated once introduced.
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