We show that vector space semantics and functional semantics in two- sorted first order logic are equivalent for pregroup grammars. An algo- rithm translates functional expressions to vector expressions and vice- versa. The semantics is compositional, variable free and invariant under change of order or multiplicity. It includes the semantic vector models of Information Retrieval Systems and has an interior logic admitting a comprehension schema. A sentence is true in the interior logic if and only if the 'usual' first order formula translating the sentence holds. The examples include negation, universal quantifiers and relative pronouns.