this paper we describe a new, categorical approach to normalization in typed -

Abstract. We discuss different options for two-sided sequent systems of noncommutative linear logic and prove a restricted form of cut elimination. By “classical Lambek logic ” we denote a sequent system with sequences of propositional formulas on the right and left side of the sequent sign, which has no structural rule except cut. We credit this logic to J. Lambek since he was the first to investigate Gentzen-systems without structural rules — originally in an intuitionistic setting, i.e. with not more than one formula in the succedent of a sequent, and motivated by linguistic considerations (see [4]). From the point of view of linear logic classical Lambek logic can be considered as pure (i.e., without exponentials) noncommutative (i.e., without the structural rules of exchange) classical (i.e., multiple succedent) linear propositional logic. This is the starting point of Abrusci’s [1] paper. Abrusci presents a sequent calculus together with a semantics in terms of phase spaces. By proving completeness he gives a semantic justification of the sequent system. Independently, under the heading “bilinear logic ” Lambek himself has studied this system (see [6]) based

Let H be the term calculus of Howard's system of constructive ordinals. I use this to name iteration gadgets (generalized ordinal notations). I isolate a family of higher order ordinal functions which give a partial semantics of H. I indicate how H provides an intrinsic measure of each ordinal below the Howard ordinal.

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