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First of all, let us define the language of LC. Given a denumerable set P of primitive symbols, we let the set of formulae FormLC be the smallest set containing every primitive symbol and closed under “\”, “/”, and “•”, i.e., if A,B ∈ FormLC, then A\B, A/B, A•B ∈ FormLC. The set of sequents is the set of all expressions of the form A1, . . . , An ⇒ A0 where n is a positive integer and Ai ∈ FormLC for each i ≤ n. LC is given by the following axiom and rules of inference, where A, B, C stand for formulae and x, y, z stand for finite sequences of formulae including the empty sequence unless the contrary is asserted. Axiom: (0) A⇒ A .

Strong completeness of the Lambek Calculus with respect to Relational Semantics