Vectors as Relative Positions: A Compositional Semantics of Modified PPs1

The interpretation of modified PPs, like one meter behind the desk, far outside the village, or right under the lamp has never received much attention in the literature about prepositions. However, as this paper shows, the modification of PPs presents an intriguing problem for a compositional semantics of PPs. This problem can be solved when the PP is interpreted, not as a set of points or mereological portions of space, but as a set of vectors, that represent positions relative to the reference object. Modifiers map a set of vectors to a subset. For example, if behind the desk denotes the set of vectors pointing backward from the desk, then one meter behind the desk denotes the subset consisting of those vectors that have a length of one meter. Given the familiar operations on vectors, the denotations of PPs can be studied in a systematic way, by formulating formal properties that characterize empirically relevant subclasses of locative PPs or that provide general constraints on their denotation. The examples are taken from Dutch, but the conclusions are valid for all languages that have the kind of PP modification that exists in Dutch and English.