The Hidden Structural Rules of the Discontinuous Lambek Calculus

The sequent calculus sL for the Lambek calculus L ([2]) has no structural rules. Interestingly, sL is equivalent to a multimodal calculus mL, which consists of the nonassociative Lambek calculus with the structural rule of associativity. This paper proves that the sequent calculus or hypersequent calculus hD of the discontinuous Lambek calculus1 ([7], [4] and [8]), which like sL has no structural rules, is also equivalent to an ω-sorted multimodal calculus mD. More concretely, we present a faithful embedding translation (·) between mD and hD in such a way that it can be said that hD absorbs the structural rules of mD. 1 The Discontinuous Lambek Calculus D and its Hypersequent Syntax D is model-theoretically motivated, and the key to its conception is the class FreeDisp of displacement algebras. We need some definitions: (1) Definition (Syntactical Algebra) A syntactical algebra is a free algebra (L,+, 0, 1) of arity (2, 0, 0) such that (L,+, 0) is a monoid and 1 is a prime. I.e. L is a set, 0 ∈ L and + is a binary operation on L such that for all s1, s2, s3, s ∈ L, s1+(s2+s3) = (s1+s2)+s3 associativity 0+s = s = s+0 identity The distinguished constant 1 is called a separator. (2) Definition (Sorts) The sorts of discontinuous Lambek calculus are the naturals 0, 1, . . .. The sort S (s) of an element s of a syntactical algebra (L,+, 0, 1) is defined by the morphism of monoids S to the additive monoid of naturals defined thus: S (1) = 1 S (a) = 0 for a prime a , 1 S (s1+s2) = S (s1) + S (s2) 1 In [5] and [8], the term displacement calculus is used instead of Discontinous Lambek Calculus as in [7] and [9]. I.e. the sort of a syntactical element is simply the number of separators it contains; we require the separator 1 to be a prime and the syntactical algebra to be free in order to ensure that this induction is well-defined. (3) Definition (Sort Domains) Where (L,+, 0, 1) is a syntactical algebra, the sort domains Li of sort i of generalized discontinuous Lambek calculus are defined as follows: Li = {s|S (s) = i}, i ≥ 0 (4) Definition (Displacement Algebra) The displacement algebra defined by a syntactical algebra (L,+, 0, 1) is the ωsorted algebra with the ω-sorted signature ΣD = (⊕, {⊗i+1}i∈ω, 0, 1) with sort functionality ((i, j → i + j)i, j∈ω, (i + 1, j → i + j)i, j∈ω, 0, 1): ({Li}i∈ω,+, {×i+1}i∈ω, 0, 1)