Relational Models for the Lambek Calculus with Intersection and Constants

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...

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Bibliographic Details
Published in:	Logical methods in computer science Vol. 19, Issue 4
Main Author:	Kuznetsov, Stepan L.
Format:	Journal Article
Language:	English
Published:	Logical Methods in Computer Science e.V 01-01-2023
Subjects:	03g15 computer science - logic in computer science f.4.1 mathematics - logic
Online Access:	Get full text
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Summary:	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.
ISSN:	1860-5974 1860-5974
DOI:	10.46298/lmcs-19(4:32)2023