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We introduce a sequent calculus FL' for non-commutative substructural logic.It has at most one formula on the right side of sequent, and excludes threestructural inference rules, i.e. contraction, weakening and exchange. (FL' isbased on our investigations of the Gentzen-style natural deduction fornon-commutative substructural logics.) FL' has the same proof strength as thestandard sequent calculus FL (Full Lambek), which is the basic sequent calculusfor all other substructural logics. For the standard FL, we use Ono'sformulation. Although FL' and the standard FL are equivalent, there is a subtledifference in the left rule of implication. In the standard FL, two parameters$\Gamma_1$ and $\Gamma_2$(resp.), each of which is just an finite sequence ofarbitrary formulas, appear on the left and right side (resp.) of a formulawhich is placed on the left side of the sequent on the upper left side of theleft rule $\imply$ (which corresponds to $\imply'$ in FL'). On the other hand,there is no such parameter on the left side of the sequent on the upper leftside in the left rule for $\imply'$ of FL'. In FL', $\Gamma_1$ is always empty,and only $\Gamma_2$ is allowed to occur in the left rule for $\imply'$.(Similar differences occur in multiplicative and additive conjunctions, and inadditive disjunction.) This difference between FL' and FL matters deeply, forwe are led to a construction of proof-figures in FL', which show how theassociativity of multiplicative conjunction and the distributivity ofmultiplicative conjunction over additive disjunction are involved in theeliminations of the cut rule in those proof-figures. We specify howassociativity and distributivity are related to the non-eliminability of anapplication of the cut rule in those proof-figures of FL'.

On non-eliminability of the cut rule and the roles of associativity and distributivity in non-commutative substructural logics