The syntactic calculus, as proposed by Lambek in 1961, is a logic
completely without structural rules: rules affecting multiplicity
(contraction, weakening) or structure (commutativity, associativity) of
the grammatical resources are not considered. Originally conceived with
linguistics in mind, Lambek’s calculus (both the ’61 and the
associative ’58 variant or its modern pregroup incarnation) have
found many models outside linguistics: as the logic for composition of
informational actions, for example, and in fields such as mathematical
morphology or quantum physics.
In terms of expressivity, Lambek’s calculi are strictly context-free.
The context-free limitation makes itself felt in situations
where syntactic and semantic composition seem to be out of sync:
long distance dependencies in syntax, or the dynamics of scoping
in semantics. Competing frameworks in the ‘mildly context-sensitive’
family (TAG, MG, MCFG, etc) handle such phenomena gracefully.
In the talk, I discuss the Lambek-Grishin calculus, a symmetric
generalization of the syntactic calculus allowing multiple conclusions.
I focus on two features that help resolve tensions at the syntax-semantics
-A continuation-passing-style interpretation, making contexts an
explicit part of the composition process. As a result of the richer
view on the mapping between syntax and semantics, the syntactic
source calculus itself can be kept very simple.
-Distributivity principles relating Lambek’s original type-forming
operations and their duals. These principles characterize syntactic
deformations under which interpretations are stable. They allow
a quite natural treatment of patterns beyond CF.
Moortgat 2009, Symmetric categorial grammar. J Philosophical Logic