Classical Lambek Logic

. 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 roles 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 [ I] 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 on categorical considerations. The following investigation is purely proof-theoretic. In the first part we will attempt to give a proof-theoretic motivation of the sequent-systems Abrusci and Lambek propose. This motivation is not obvious since the cut rule, the negation rules and the rules for the binary multiplicatives have to be formulated with restrictions (side conditions) which at first sight do not seem natural at all. We will consider different possible forms of these rules and show that essentially two systems, Jt (corresponding to the calculi proposed by Abrusci and Lambek) and B are possible, which by permutation of antecedents or succedents of sequents can be embedded into each other. It can be shown that the unrestricted rules one would expect from usual sequent systems make exchange derivable. So our criterion for the acceptance of a rule is that its side conditions preclude the derivation of exchange, but are as weak as possible in that respect. In the second part of this paper we deal with cut elimination. Abrusci has given an example which shows that even the restricted cut rule cannot be eliminated from the sequent system considered. We show that cut elimination holds if we confine ourselves to sequents in which in the scope of a negation sign no connective except the same negation occurs. Part I: Sequent Systems for Classical Lambek Logic We write sequents in the form Ft-A, where F and A stand for sequences of formulas. When no confusions can arise, we write these sequences without commas. Expressions like/'1,41 or A1ABA2 are then understood in the obvious way.