Relational Models for the Lambek Calculus with Intersection and Constants

TL;DR

This paper explores relational semantics for an extended Lambek calculus with intersection and constants, revealing limitations and possibilities of completeness under various interpretations and restrictions.

Contribution

It demonstrates the failure of strong and weak completeness in standard interpretations and establishes new completeness results for non-standard interpretations and product-free fragments.

Findings

Weak completeness holds for non-standard constant interpretations.

Strong completeness fails without restrictions and in standard interpretations.

Results extend to infinitary settings with iterative divisions.

Abstract

We consider relational semantics (R-models) for the Lambek calculus extended with intersection and explicit constants for zero and unit. For its variant without constants and a restriction which disallows empty antecedents, Andreka and Mikulas (1994) prove strong completeness. We show that it fails without this restriction, but, on the other hand, prove weak completeness for non-standard interpretation of constants. For the standard interpretation, even weak completeness fails. The weak completeness result extends to an infinitary setting, for so-called iterative divisions (Kleene star under division). We also prove strong completeness results for product-free fragments.