The category of semi-modules over the lattice of a real interval $I$ serves as a common frame for extensional logical semantics and conceptual vector semantics of natural language. The vector lattice of an $I$-space is embedded in its projector lattice. Words are represented by an extensional vector given in a pregroup lexicon and a conceptual vector that lives in a tensor product of 2-dimensional spaces where each 2-dimensional space stand for a 'basic' concept. Syntactical analysis defines the extensional vector of a grammatical string compositionally. The latter is translated to the concept vector of the string such that the truth of a sentence is equivalent to the truth of a concept. Examples include compound noun phrases with adjectives, quantifiers, negation and relative pronouns.