Relational Models for the Lambek Calculus with Intersection and Constants
Stepan L. Kuznetsov
TL;DR
This work analyzes relational (R-model) semantics for the Lambek calculus extended with intersection and constants, revealing nuanced completeness phenomena. It shows that standard interpretations of the constants $\mathbf{0}$ and $\mathbf{1}$ destroy completeness, but non-standard, relativised R-models can restore weak completeness, while strong completeness fails in general. The paper also extends to infinitary reasoning with iterative divisions ($A^*\backslash B$, $B/ A^*$) and demonstrates weak completeness in this setting, though strong completeness fails there as well. A non-standard exponential modality is introduced to achieve strong conservativity and product-free strong completeness, with a general result: strong completeness can be recovered for the product-free fragment, both with and without constants, under appropriate model/theoretic adjustments. Overall, the results delineate how Lambek's restriction, product, constants, and infinitary operations interact with relational semantics and what model refinements are needed to regain completeness.
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.