Investigations into a left-structural right-substructural sequent calculus

Abstract
We study a multiple-succedent sequent calculus with both of the structural rules Left Weakening and Left Contraction but neither of their counterparts on the right, for possible application to the treatment of multiplicative disjunction (fission, ‘cotensor’, par) against the background of intuitionistic logic. We find that, as Hirokawa dramatically showed in a 1996 paper with respect to the rules for implication, the rules for this connective render derivable some new structural rules, even though, unlike the rules for implication, these rules are what we call ipsilateral: applying such a rule does not make any (sub)formula change sides—from the left to the right of the sequent separator or vice versa. Some possibilities for a semantic characterization of the resulting logic are also explored. The paper concludes with three open questions.
Keywords Substructural logics  Multiplicative disjunction  Fission  Structural rules  Sequent calculus
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Lloyd Humberstone (2013). Replacement in Logic. Journal of Philosophical Logic 42 (1):49-89.
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