Aspects of a Typology of Direction

Although direction plays an important role in the semantics of prepositions and verbs, there are 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. (locative) b. Alex went into/through the forest. (directional) 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 semantic categories are of central importance for the grammar of adpositions in languages across the world, as well as for case systems (Van Riemsdijk and Huijbregts 2001, Kracht 2002). 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. (atelic) b. ... into the forest/to the station/around the barrier. (telic) 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 to show 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). 1 Paths and prepositions Building 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. A path can be taken as a directed curve, corresponding to a sequence of positions in space. I will assume here that a path is formally defined as a continuous function p from the real interval [0,1] to some domain S of places, which is a common mathematical way to define a path. Such a mathematical definition has its limitations, of course, when it is applied to linguistic phenomena, and alternatives are easily conceivable. However, this formalization is convenient and it does serve our purposes well here. For some further discussion, see Zwarts (2005). A path has a starting point, that we indicate with p(0), an endpoint p(1), the two extremes of the path, and for every i between 0 and 1, p(i) is an intermediary point of the path, between the extremes. Paths have to be integrated in the interpretations of the sentence in one way or another. I will not go into that here, but assume that there is a thematic relation TRACE between a (motion) event e and the path p = TRACE(e) that represents the spatial trace of that event, an common enough assumption in work on aspect and prepositions (e.g. Krifka 1998). With this in place we can give definitions of directional prepositions (or rather, the PPs they project) and a rough first classification, based on the way the prepositions are defined. Most of the prepositions can be defined in terms of locative conditions they impose on particular parts of the path. There is a class of source prepositions, for instance, that impose a locative condition on the initial part of the path. Examples are from, out of, off, from under, and away from. The definition of from under the bridge is as follows: (3) [[ from under the bridge ]] = { p: there is a proper subinterval I of [0,1] that includes 0 and that consists of all the i ∈ [0,1] for which p(i) is under the bridge } Interval I corresponds to the portion of the path that is under the bridge. This portion includes the starting point 0, but it excludes the endpoint 1, because it picks out a proper part of the path. As a result, the definition divides the domain [0,1] of the path in two parts (phases in the terminology of Löbner 1987 and Fong 1997), a ‘positive’ part that is mapped to positions under the bridge and a ‘negative’ part mapped to positions that are not under the bridge. Schematically, with I corresponding to the + part: