The Lambek Calculus: Relational Semantics and the Method of Labelling
Natasha Kurtonina
Relational Semantics as it is introduced in [vB88], [vB91], [Bus.86]
and [Or89] provides a dynamic interpretation for the Lambek Calculus:
valuation of formulae is carried out on pairs of points, which can be
treated as informational states. Propositions can be understood as
programs of transformation of informational states. In [M92] Szabolcs
Mikulas presents an algebraic proof of the corresponding Completeness
Theorem.
The purpose of this paper is to propose a new, rather simple completeness
proof by using the method of labelling. The general theory of labelled
deductive systems is proposed by D.Gabbay. In the labelled version of
the Lambek Calculus a pair of labels is attached to every formula and
the way of labelling reflects corresponding semantical truth conditions.
Straighforward parallelism between syntax and semantics enables one to
obtain the completeness result.