On syntactic concept lattice models for the Lambek calculus and infinitary action logic

TL;DR

This paper extends the semantics of the Lambek calculus with Kleene iteration using syntactic concept lattices, proves strong completeness, and addresses issues with constants, enhancing its applicability in linguistic semantics.

Contribution

It introduces an extension of the SCL semantics to infinitary Lambek calculus with Kleene iteration and proves strong completeness, addressing previous limitations with constants.

Findings

Proved strong completeness for the extended calculus.

Extended semantics to include infinitary operations and constants.

Addressed issues with zero, unit, and top constants in the models.

Abstract

The linguistic applications of the Lambek calculus suggest its semantics over algebras of formal languages. A straightforward approach to construct such semantics indeed yields a brilliant completeness theorem (Pentus 1995). However, extending the calculus with extra operations ruins completeness. In order to mitigate this issue, Wurm (2017) introduced a modification of this semantics, namely, models over syntactic concept lattices (SCLs). We extend this semantics to the infinitary extension of the Lambek calculus with Kleene iteration (infinitary action logic), prove strong completeness and some interesting corollaries. We also discuss issues arising with constants - zero, unit, top - and provide some strengthenings of Wurm's results towards including these constants into the systems involved.