On syntactic concept lattice models for the Lambek calculus and infinitary action logic
Stepan L. Kuznetsov
TL;DR
This work investigates semantic models based on syntactic concept lattices to achieve strong completeness for the infinitary extension of the Lambek calculus, $ACT^{+}_{\omega}$. It develops PSCL-/SCL-models and a Lindenbaum-style embedding to translate abstract algebraic truth into lattice-based semantics, including a two-letter alphabet reduction and non-standard interpretation of the bottom constant $\bot$. It extends the framework to the full Lambek calculus without the non-emptiness restriction in fragments, using upper-cone constructions to obtain strong completeness results for certain variants, while highlighting open questions for the constants-inclusive full system. Overall, the paper clarifies how SCL-models overcome incompleteness issues that challenge L-models when adding iteration and constants, with implications for linguistically motivated extensions of Lambek grammars.
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.