Vectors as relative positions: a compositional semantics of modified PPs

Abstract

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.

The Emergence of Axial Parts

“…The semantic contribution of Place is to specify how space is projected from a region; I will assume a modelling of space in terms of vectors along the lines proposed by Zwarts (1997), Zwarts and Winter (2000). Vectors are one-dimensional objects with direction and length which define points in a space when they are drawn from a region.…”

“…The exact conditions on measure and directional phrases are complex (see Zwarts 1997, Zwarts and Winter 2000, Winter 2001); I will assume that even non-measurable locative expressions include a component Place.…”

Locative and Directional Prepositions in Conceptual Spaces: The Role of Polar Convexity

“…We have been able to show that locative and directional prepositions support the general idea that concepts can be represented by convex regions in conceptual spaces (Gärdenfors 2000), if convexity is defined in terms of a new notion of betweenness over polar coordinates. Polar convexity generalizes and unifies the earlier constraints of continuity in Zwarts (1997). Several prepositions that seem to present counterexamples to polar convexity turn out to have special features.…”

“…The use of vectors in the semantics of prepositions has been advocated from different points of view: to account for their relation with cognitive maps (O'Keefe 1996(O'Keefe , 2003, for the way they are modified (e.g. three feet above, right behind, Zwarts 1997;Zwarts and Winter 2000), the role that attention plays in their use (Regier and Carlson 2001), for modeling their frames of reference and points of view (Bohnemeyer 2012), and their relation with syntactic structures (Svenonius 2006(Svenonius , 2008Kracht 2008). The notion of a vector is a simple but powerful way to represent the position of the trajector relative to the landmark.…”

The Degree Semantics Parameter and cross-linguistic variation

“…Since Washo lacks explicit comparisons, it follows that differential measure phrases are also not found in this language. Additionally, measure phrases are also commonly found with certain prepositional phrases (Zwarts 1997, Zwarts & Winter 2000. Attempts to elicit such structures in Washo, even with nonce measure phrases (e.g., Beverly is three people ahead of me in line), have met with some difficulty, and more thorough investigation must be left to future research.…”