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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.

*Although direction plays an important role in the semantics of prepositions and verbs, thereare not many precise and systematic treatments of this notion. This article offers a characterization and typology of directionality, based on two algebraic properties of spatial paths: cumulativity and reversability. This typology makes clear how directionality relates to the aspectual property of telicity, it generates implicational predictions about directional marking in systems of cases and adpositions around the world, and it suggests broad parallels between the realizations of directionality in prepositions and verbs.Like all parts of speech, prepositions can be classified in different ways. When we focus on the semantic side of spatial prepositions, we find a major division between locative prepositions (like in and under) and directional prepositions (like into and through):(1) a.Alex is in/under the car.b. Alex went into/through the forest.Locative prepositions correspond to places (where something is), directional prepositions to paths (where something is going) (see Jackendoff 1983 and many others). Further classifications can be made within both locative and directional expressions, like the distinction of goal ('to'), source ('from') and route ('via') prepositions, for example. Such We can also classify prepositions as telic or atelic, according to the contribution that they make to the aspectual structure of a sentence:(2) Alex walked … a. … in the forest/towards the station/along the beach.b. … into the forest/to the station/around the barrier.According to the usual aspectual tests (Dowty 1979), prepositions like in, towards and along lead to atelic, unbounded aspect, while into, to and around make the sentence they modify telic, bounded in aspect. All locative prepositions are atelic, but directional prepositions can be telic or atelic, depending on their particular lexical definition.The purpose of this paper is to take a closer look at classes of prepositions (section 1), building on the algebra of paths in Piñon (1993), Nam (2000), and Zwarts (2005), and toshow that this algebra allows a richer typology of direction than the one that has emerged in the linguistic literature (section 2). This typology does not only show us more clearly how the system of prepositions hangs together semantically and how the spatial and aspectual dimension relate (section 3 and section 4), but it also provides a basis for markedness patterns in the morphosyntax of directionality, whether expressed by adpositions or case markers (section 5). The directional typology that we find for prepositions can be extended to verb meanings, giving a partial typology of 'event shapes', similar to the more informal verb contours of Talmy (1978) and others (section 6). Paths and prepositionsBuilding on much earlier work on prepositions, I assume that the interpretation of directional prepositions is based on paths, more specifically, that a directional PP denotes a set of paths. 1A path can be taken as a directed curve, correspo...

Many languages use prepositions to qualify numerals, as in around ten or between ten and twenty. In this article we study the syntactic and semantic properties of these prepositions. We will argue that they take numerals as arguments and form full-fledged PPs that can function as numerals themselves. This leads us to reconsider the status of numerals. # When we think of numerals, the construction that usually comes to mind first is that of a bare numeral modifying a noun:(1) John speaks ten languages Theories about the syntax and semantics of numerals are usually based on this simple and common construction. However, the lack of agreement about the status of numerals suggests that this empirical basis might be too narrow (Section 1). We believe that it is necessary to study a wider range of constructions featuring numerals and in this article we will look at one construction in particular, in which a numeral is combined with a preposition (Section 2): www.elsevier.com/locate/lingua Lingua 116 (2006) 811-835