Spatial adpositions such as opon can often be used in two different contexts. This is clear from the fact that example (165a) is ambiguous between two readings: in the first reading, Jan is on the table and jumps there; in the second reading, Jan performs the action of jumping, as a result of which he ends up in a different location, namely on the table. The ambiguity of (165a) is resolved if the clause is in the perfect tense: if the auxiliary hebben is used, as in (165b), only the first reading survives; if the auxiliary zijn is used, as in (165c), only the second reading survives.

It is sometimes claimed that the two different readings are due to the adposition op itself, i.e. that we are dealing with two homonymous adpositions denoting a location and a change of location, respectively. We will not adopt this position, but assume instead that the specific interpretation of the adposition is due to the verb used: if springen selects the auxiliary hebben, it is used as an atelic verb, and thus no change of location is implied; if springen selects zijn, it is used as a telic verb, and thus a change of location is implied. This would be consistent with the fact, illustrated in (166), that most spatial PPs can be used as complements of location verbs, like liggento lie and staanto stand, and of change-of-location verbs, like leggen and zettento put. This shows that adpositions of the type in (166) are compatible with both the location and the change-of-location reading.

A small number of spatial adpositions are special in that they cannot occur as complements of verbs of (change of) location. Such adpositions intrinsically denote a path (cf. Subsection V); since this notion plays a crucial role in our definition of the notion of direction as “movement along a path”, we will refer to such adpositions as directional adpositions. Directional PPs typically occur as the complement of verbs of traversing (which denote movement along a certain path), such as rijdento drive. Example (167) illustrates this for the preposition naarto.

Since the examples in (166) show that the difference between the location and change-of-location readings of the spatial adpositions is due to their syntactic environment (here: the verb), we can assume that spatial PPs headed by non-directional spatial adpositions simply refer to some point or area in space. Therefore, we will simply use the term locational adposition for such non-directional spatial adpositions.

The basic semantic contribution of the locational adpositions is that they establish a spatial relation between two entities. For example, the locational prepositional phrase in het huisin the house in (168) locates the subject of the clause Jan in space, which we can therefore call the located object (other terms found in the literature on spatial relations are theme, figure, and trajectory). More precisely, the located object is situated in space with respect to the complement of the preposition het huisthe house, which we can therefore call the reference object (other terms found in the literature are ground and landmark). The exact nature of the spatial relation is determined by the lexical meaning of the locational preposition inin; this relation would have been different if we had used voorin front of instead of in. Therefore, as a first approximation, it seems reasonable to consider the preposition in as a two-place predicate, and to assign the meaning in (168b) to the clause in (168a).

The assignment of meaning to the clause in (168a) seems rather straightforward: Jan is located in the house. In some cases, however, things are not so simple. There are actually three different ways of interpreting locational prepositions; they will be discussed in the following subsections.

Locational adpositions locate an object in space with respect to the reference object. However, the interpretation of locational PPs is often rather vague. This is illustrated by the sentences in (176).

Example (176a) is usually interpreted to mean that the photographer is standing in the position shown in Figure 5. Example (176b), on the other hand, need not be interpreted to mean that all lamps are positioned like lamp 3; the positions of the other lamps are included, i.e. the only requirement is that they remain within the gray area.

Figure 5 shows that locational PPs may sometimes refer to rather large areas. As a result of this, the areas denoted by different locational adpositions can overlap. For example, if we want to refer specifically to lamp 1 in Figure 5, we could also say that it stands naastnext to or links vanto the left of the camera. We can even express the appropriateness of the chosen adpositions by means of examples such as (177), which makes it explicit that achterbehind and naast “next to’ are both applicable, but that naast is the more appropriate preposition for expressing the intended spatial relation.

The positions in the area can be made even more specific by modifying the prepositional phrase. If we interpret the preposition achter deictically from the point of view of the photographed object, the statements in (178) are true. Note in passing that according to the inherent interpretation of achter, which takes the camera as its anchoring point, lamp 2 would be to the right of the camera.

The examples in (178) give an indication of the direction we have to look in order to find the located object in question. As is shown in (179), the distance between the reference and the located object can also be indicated.

This means that each point in the area referred to by the locational PP achter de camera can be defined in terms of a direction and a distance. In other words, the preposition achter can be thought of as a set of vectors that project the position of the reference object onto a potential position of the located object. Consider Figure 6, where the reference object occupies the (0,0) position. The preposition achter denotes those vectors V‹x,y› that originate at (0,0), and for which y > 0. Since the vectors V‹-4,3›, V‹0,5› and V‹8,1› map the position of the reference object onto the positions of lamps 2, 3 and 5, respectively, the latter are indeed in the area referred to by the PP achter de camera.

We can also denote subsets of the set of vectors denoted by the PP achter de camerabehind the camera: the modified PP recht achter de camerastraight behind the camera denotes the subset of vectors denoted by achter with x = 0, the modified PP links achter de camera denotes the subset of vectors with x < 0, and the modified PP rechts achter de camera denotes the subset of vectors with x > 0. Furthermore, the modified PP vlak achter de camera denotes the subset of vectors that are smaller than some contextually determined magnitude (here, length). These examples show that the assumption that locational prepositions denote a set of vectors will be very useful in the discussion of the use of locational PPs as modifiers in Section 34.1. For now, however, it is sufficient to note that locational PPs refer to areas rather than fixed points in space.

Prepositions like achterbehind and bovenabove seem to express that the magnitude of the vector is greater than zero, i.e. that there is some distance between the reference object and the located object. However, some prepositions require that there should be physical contact between the two objects, i.e. they require that the magnitude of the vector is zero. This difference can be illustrated by the prepositions opon and bovenabove in (180); example (180a) implies that there is physical contact between the table and the lamp, as in Figure 7A, whereas (180b) suggests that there is no such contact, as in Figure 7B.

Note that the choice of verb does not affect the interpretation of the preposition; it is possible to replace staan with hangen in (180a) if we are referring to a situation like Figure 7B, where the lamp hangs so low that it touches the table. More common examples of this kind are given in (181).

The fact that the preposition op denotes the null vector, while boven denotes a larger set of vectors with a magnitude greater than 0, accounts for the fact that (under the idealization that the length of the part of the square adjacent to the church does not exceed the length of the side of the church adjacent to the square) the inference in (182) is valid, while the one in (183) is not.

However, there is also a potential problem with the claim that the denotations of prepositions like achter and boven do not include the null vector. If we assume that the meaning of achter/boven op in (184) is compositionally determined, we should conclude that achter and boven are at least compatible with the null vector, since otherwise a contradiction would arise.

We will not discuss this problem here, but simply assume that the underlying hypothesis that the meanings of achter/boven op are compositionally determined is incorrect. Section 34.1.3 will show that formations like achterop and bovenop are compounds, and we therefore expect that the denotation of these formations consists of a subset of the denotation of the second member, the preposition op. Since the reference object het blik in (184) is not a point in space but a three-dimensional object, the preposition op can be assumed to denote a non-singleton set of null vectors which are located at different positions on the surface of the reference object. It is therefore correctly predicted that the compounds achterop and bovenop denote different subsets of the set of null vectors denoted by op.

Although the discussion of the examples in (184) clearly shows that, strictly speaking, this is not correct, we will often take the reference object to be a point in space instead of a physical object with three-dimensional extensions in order to simplify the following discussion. This is why we will usually refer to “the null vector” instead of “the set of null vectors”. The three-dimensional extensions of the reference object will be considered only when this is necessary for the discussion.

Directional adpositions differ from locational ones in that they do not situate the located object in a fixed position in space. As a result, the denotation of a directional preposition cannot simply be considered as a set of vectors. This does not mean, however, that the notion of vector is irrelevant in the case of directional PPs. Directional PPs express that the located object traverses a certain path. A path can be defined as an ordered set of vectors, each of which is associated with a particular position on the timeline. The path denoted by van A naar Bfrom A to B can then be represented as in Figure 8, which can be read as a cartoon. In our visual representations below, we will often indicate paths with a blocked or dotted arrow.

The situation depicted in Figure 8 is an appropriate characterization of the directional phrase in (185a), in which the PPs van Utrecht from Utrecht’ and naar Groningento Groningen act as complements of the verb of traversing rijdento drive. Besides this “core” directional reading, directional PPs can also have two slightly different readings. The PPs in (185b) do not denote a path that is traversed, but indicate the extent of the located object deze wegthe road; this example expresses that the road is situated on a path that goes from Utrecht to Groningen. We will refer to this use of the PP in (185b) as the extent reading (note that the verb lopen is also used as a verb of traversing meaning “to walk” when the subject is animate). Finally, the directional PP in (185c) is used with an orientation reading.

This subsection concludes our overview of the semantic concepts that will play a role in the semantic description of the various individual adpositions. The discussion is divided into four parts, corresponding to the syntactic classification based on the position of the nominal complement of the adposition.