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$LE^{t}_{rightarrow}$ , $LR^{circ}_{wedgesim}$ , LK and Cutfree Proofs

Katalin Bimbó
Journal of Philosophical Logic
Vol. 36, No. 5 (Oct., 2007), pp. 557-570
Published by: Springer
Stable URL: http://www.jstor.org/stable/30226905
Page Count: 14
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$LE^{t}_{rightarrow}$
          ,
          $LR^{circ}_{wedgesim}$
          , LK and Cutfree Proofs
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Abstract

Two consecution calculi are introduced: one for the implicational fragment of the logic of entailment with truth and another one for the disjunction free logic of nondistributive relevant implication. The proof technique-attributable to Gentzen-that uses a double induction on the degree and on the rank of the cut formula is shown to be insufficient to prove admissible various forms of cut and mix in these calculi. The elimination theorem is proven, however, by augmenting the earlier double inductive proof with additional inductions. We also give a new purely inductive proof of the cut theorem for the original single cut rule in Gentzen's sequent calculus LK without any use of mix.

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