We view the vectors of a semantic distributional model as vectors of a semi-module over the semi-ring of a real interval with its canonical lattice structure. In this approach, the quantum logic of subspaces is distributive and Boolean on the sublattice of subspaces generated by sets of basis vectors. If the basis vectors correspond to strings of words, the semi-module classifies words like a thesaurus. Its Boolean sublattice includes the concepts corresponding to single words. If the basis vectors identify with single words the connectives of quantum logic lift from the basis vectors to the whole space as linear maps. Concept vectors are linked to the vectors assigned by a pregroup grammar through the classes they determine. The functional composition of pregroup semantics corresponds to a composition of concepts by logical connectives. Examples including compound noun-phrases like \pi{no bank} and \pi{red herring} illustrate the equivalence of the quantum logic of subspaces to the compositional logic of pregroup grammars.