Lambek() proposed a categorial achitecture for natural language grammars, whereby syntax and semantics are modelled by a
biclosed monoidal category (bmc)
and a cartesian closed category (ccc) respectively, and semantic interpretation by a functor from syntax to semantics that preserves the biclosed monoidal structure; essentially this same architecture underlies the framework of abstract categorial grammar (ACG, de Groote ), except that the bmc is now symmetric, in keeping with the collapsing of Lambek’s directional implications / and \ into the linear implication
. At the same time, Lambek proposed that the semantic ccc bears the additional structure of a
, and that the meanings of declarative sentences—linguist’s propositions—can be identified with propositions in the sense of topos theory, i.e. morphisms from the terminal object 1 to the subobject classifier Ω. Here we show (1) that this proposal as it stands is untenable, and (2) that a serviceable framework results if a
preboolean algebra object
distinct from Ω is employed instead. Additionally we show that the resulting categorial structure provides ‘for free’, via Stone duality, an account of the relationship between fine-grained ‘hyperintensional’ semantics (,,,) and the familiar coarse-grained intensional semantics of Carnap () and Montague ().