Are (Linguists’) Propositions (Topos) Propositions?

  • Carl Pollard
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6736)


Lambek([22]) 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 [12]), except that the bmc is now symmetric, in keeping with the collapsing of Lambek’s directional implications / and \ into the linear implication \(\multimap\). At the same time, Lambek proposed that the semantic ccc bears the additional structure of a topos, 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 ([6],[33],[27],[28]) and the familiar coarse-grained intensional semantics of Carnap ([2]) and Montague ([26]).


proposition hyperintension intension topos Stone duality preboolean algebra 


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Authors and Affiliations

  • Carl Pollard
    • 1
  1. 1.Department of LinguisticsThe Ohio State UniversityColumbusUSA

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