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1.3.1.1. The semantics of spatial adpositions
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This section discusses the semantics of spatial adpositions. Subsection I starts by briefly discussing the central semantic notions of (change of) location and direction. Subsection II shows that spatial adpositions can be interpreted in three different ways, which we will refer to as deictic, inherent and absolute, subsections III and IV argue that the mathematical notion of a vector is very useful to properly describe the semantics of spatial adpositions. Subsection V, finally, goes somewhat deeper into the notion of a path that enters the definition of directional PPs.

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[+]  I.  (Change of) location and direction

Spatial adpositions like op'on' in (160) can normally be used in two distinct contexts. Example (160a) is ambiguous between two readings: according to the first reading, Jan is situated on the table and he is jumping there; according to the second reading, Jan is performing the action of jumping as the result of which he ends up in a different location, viz., on the table. The ambiguity of (160) is resolved if the clause is put in the perfect tense: if the auxiliary hebben is used, as in (160b), only the first reading survives; if the auxiliary zijn is used, as in (160c), only the second reading survives.

Example 160
a. Jan springt op de tafel.
location or change of location
  Jan jumps  on(to)  the table
b. Jan heeft op de tafel gesprongen.
location
  Jan has  on  the table  jumped
c. Jan is op de tafel gesprongen.
change of location
  Jan has  onto  the table  jumped

It sometimes claimed that these two different readings are due to the adposition op itself, that is, that we are dealing with two homonymic adpositions which denote, respectively, a location and a change of location. We will not adopt this position, but instead assume that the specific interpretation of the adposition is due to the verb that is used: if springen selects the auxiliary hebben it is used as an atelic verb, and hence no change of location is implied; if springen selects zijn it is used as a telic verb, and hence a change of location is implied. This would be consistent with the fact, illustrated in (161), that most spatial PPs can be used as complements both of locational verbs like liggen'to lie' and staan'to stand' and of verbs of change of location like leggen and zetten'to put'. This shows that adpositions of the sort in (161) are compatible both with a location and with a change of location reading.

Example 161
a. De lamp staat bij/naast/onder/op de tafel.
location
  the lamp stands  near/next.to/under/on  the table
  'The lamp is standing near/next to/under/on the table.'
a'. De ladder ligt achter/langs/tegen de muur.
  the ladder  lies  behind/along/against  the wall
  'The ladder is lying behind/along/against the wall.'
b. Jan zet de lamp bij/naast/onder/op de tafel.
change of location
  Jan puts  the lamp  near/next.to/under/on  the table
  'Jan is putting the lamp near/next to/under/on the table.'
b'. Jan legt de ladder achter/langs/tegen de muur.
  Jan puts  the ladder  behind/along/against  the wall
  'Jan is putting the ladder behind/along/against the wall.'

      A small number of spatial adpositions are exceptional in that they cannot occur as complements of verbs of (change of) location. These adpositions intrinsically denote a path (cf. Subsection V), and since we define 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 (verbs that denote movement along a certain path), such as rijden'to drive'. In (162) an example is given with the preposition naar'to'.

Example 162
Jan rijdt naar Groningen.
direction
  Jan  drives  to Groningen
'Jan is driving to Groningen.'

Since the examples in (161) show that the difference between the location and change of location reading of the spatial adpositions is due to the syntactic environment (here: the verb), we may assume that spatial PPs headed by non-directional spatial adpositions simply refer to some point or region in space. We will therefore simply use the term locational adposition for such non-directional spatial adpositions.

[+]  II.  Deictic, inherent and absolute interpretations

The basic meaning contribution of the locational adpositions is that they establish a spatial relation between two entities. The locational prepositional phrase in het huis'in the house' in (163), for example, situates the subject of the clause Jan in space, which we may 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 huis'the house', which we may therefore call the reference object (other terms found in the literature are ground and landmark). The precise nature of the spatial relation is determined by the lexical meaning of the locational preposition in'in'; the relation would have been different if we had used voor'in front of' instead of in. Therefore, as a first approximation, it seems reasonable to consider the preposition in as a two-place predicate, and assign the clause in (163a) the meaning in (163b).

Example 163
a. Jan is in het huis.
  Jan is in the house
b. in (Jan, het huis)

The meaning assignment to the clause in (163a) seems rather straightforward: Jan is situated inside the house. In some cases, however, things are not that simple. There are actually three different ways in which locational prepositions can be interpreted.

[+]  A.  The deictic interpretation

Consider example (164a). Assigning this example the semantic representation in (164b) seems to somehow miss the point; see the discussion in Subsection B for an account of this. This representation only establishes a relation between the located and the reference object, whereas a third participant seems to be involved: a speaker who utters (164a) seems to compute the position of Jan in relation to both the tree and himself. Example (164a) refers to the situation in Figure 1B; the situation in Figure 1A seems somehow undefined.

Example 164
a. Jan staat voor de boom.
  Jan stands  in.front.of  the tree
b. voor (Jan, de boom)

Figure 1: Jan staat voor de boom'Jan is standing in front of the tree'

The interpretation in Figure 1B, which is dependent on some additional anchoring point that also enters into the computation of the location of the located object, is called deictic. When an adpositional phrase is interpreted deictically, we will indicate this in the semantic representation by means of a D in superscript after the adposition: PD (x,y). That we are dealing with the anchoring point z can be indicated as follows: PD,z (x,y). Example (164a), that is, the situation depicted in Figure 1B, can therefore be represented as in (165), in which D,s indicates that we are dealing with a deictic interpretation of the preposition voor with the speaker s as its anchoring point.

Example 165
voorD,s (Jan, de boom)

The examples in (166) show that the anchoring point need not be the speaker. In (166a) the anchoring point is the addressee a, and in (166b) it is some other participant in the discourse, viz., de kerk'the church'.

Example 166
a. Vanuit jou gezien, staat Jan recht voor de boom.
  from  you  seen  stands  Jan straight  in.front.of the tree
  'Seen from your position, Jan is standing right in front of the tree.'
a'. voorD,a (Jan, de boom)
b. Vanuit de kerk gezien, staat Jan recht voor de boom.
  from  the church  seen  stands  Jan straight  in.front.of the tree
  'Seen from the church, Jan is standing right in front of the tree.'
b'. voorD,de kerk (Jan, de boom)
[+]  B.  The inherent interpretation

The anchoring point from which the location of the located object is computed may also be the reference object, that is, the complement of the adposition. This is only possible if the reference object is structured with respect to the relevant dimension(s). Consider example (167a), which can be depicted as in Figure 2A, in which the location of Jan is computed by taking the reference object as the anchoring point; the position of the speaker with respect to Jan and the car is not relevant. The difference between (164a) and (167a) has to do with the dimensional structuring of the reference object; a car can be seen as an object with a back and a front (indicated in Figure 2A by means of an arrow through the object), whereas we normally do not perceive trees in that way. Because of these dimensional properties of cars and trees, we can take the first but not the latter as the anchoring point for the preposition voor. The interpretation of locational adpositions in which we take the reference object as the anchoring point will be called inherent and from now on be indicated by means of an I in superscript: PI. On the inherent interpretation, example (167a) can therefore be represented as in (167b). Observe that the deictic interpretation is possible as well; (167a) can also refer to the situation depicted in Figure 2B, which can be assigned the semantic representation in (167c).

Example 167
a. Jan staat voor de auto.
  Jan stands  in.front.of  the car
  'Jan is standing in front of the car.'
b. voorI (Jan, de auto)
c. voorD,s (Jan, de auto)

Figure 2: Jan staat voor de auto'Jan is standing in front of the car'

      The deictic and the inherent interpretations of adpositional phrases differ in their logical properties. This can be illustrated by means of the examples in (168). When we interpret the adpositions deictically, the conclusion in (168c) follows from (168a&b), as can be seen in Figure 3A. However, if we interpret the adpositions inherently, the inference in (168) is no longer valid, as is shown in Figure 3B. The same difference in logical properties holds for other adpositions that can be used both deictically and inherently, like the simple prepositions achter'behind', naast'next to' and the phrasal prepositions links/rechts van'to the left/right of'.

Example 168
a. De auto staat voor de kerk. AND
  'The car is standing in front of the church.'
b. Jan staat voor de auto.
  'Jan is standing in front of the car.'
c. Jan staat voor de kerk.
  'Jan is standing in front of the church.'

Figure 3: Deictic vs. inherent interpretation of voor'in front of'

      The inference in (168) is also invalid if we switch from one perspective to another. The preposition in in (169a) is inherent, whereas the preposition voor in (169b) is interpreted deictically; cf. Figure 1. From these two examples, we cannot conclude that (169c) holds: for example, if Jan is situated in front of the trunk of the tree, and the bird is in the top of the tree, (169c) will clearly be false.

Example 169
a. De vogel is in de boom. AND
  the bird  is  in the tree 
b. Jan staat voor de boom. ⇏
  Jan stands  in.front.of  the tree
c. Jan staat voor de vogel.
  Jan stands  in.front.of  the bird
[+]  C.  The absolute interpretation

Consider example (170a), which can be depicted as in Figure 4A. The fact that the computation of the location of the painting is independent of the speaker suggests that we are dealing with an inherent interpretation of the prepositional phrase boven die stoel'above that chair': bovenI (het schilderij, de stoel). This would be consistent with the fact that a chair can be considered to have a bottom and a top. There is reason to doubt, however, that the dimensions of the chair are really involved in the computation, because (170a) can also be felicitously used to refer to the situation depicted in Figure 4B, where the constellation between the chair and the painting has been changed. Moreover, (170a) cannot be used to refer to the situation in Figure 4C, where the constellation between the chair and the painting is essentially the same as in Figure 4C; note, for completenesssake that (170a) is also unable to refer to the situation in Figure 4C when we replace the verb hangen by liggen'to lie', which may provide a more appropriate description of the positioning of the painting. Interpretations of adpositional phrases that have neither an internal nor an external anchoring point will be called absolute (that is, depending only on the natural environment such as the surface of the earth); these interpretations will be indicated without a superscript, as in (170b).

Example 170
a. Het schilderij hangt boven die stoel.
  the painting  hangs  above  that chair
  'The painting is hanging above that chair.'
b. boven (het schilderij, de stoel)

Figure 4: Het schilderij hangt/ligt boven de stoel'The painting hangs/lies above the chair'

[+]  III.  Regions and vectors

Locational adpositions locate an object in space with respect to the reference object. It should be noted, however, that the interpretation of locational PPs is often rather vague. Consider the examples in (171). Example (171a) is most naturally interpreted in such a way that the photographer is standing in the position depicted in Figure 5. Example (171b), on the other hand, need not be interpreted in such a way that all the lamps are positioned like lamp 3; they may also occupy the positions of the other lamps as long as they remain in the grey area.

Example 171
a. De fotograaf staat achter de camera.
  the photographer  stands  behind the camera
  'The photographer is standing behind the camera.'
b. De lampen staan achter de camera.
  the lamps  stand  behind the camera
  'The lamps are behind the camera.'

Figure 5: Achter de camera'behind the camera'

Figure 5 shows that locational PPs sometimes refer to rather extensive regions. As a result of this the regions denoted by different locational adpositions may overlap. For example, if we specifically want to refer to lamp 1 in Figure 5, we could also say that it stands naast'next to' or links van'to the left of' the camera. We can even express the appropriateness of the chosen adpositions by means of examples such as (172), which makes explicit that achter'behind' and naast “next to' are both applicable, but that naast is the most appropriate preposition to express the intended spatial relation.

Example 172
Lamp 1 staat meer naast dan achter de camera.
  lamp 1  stands  more next.to  than  behind the camera
'Lamp 1 is situated more next to than behind the camera.'

      The positions in the region can also be made more specific by modifying the prepositional phrase. If we interpret the preposition achter deictically from the point of view of the object photographed, the statements in (173) 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.

Example 173
a. Lamp 3 staat recht achter de camera.
  lamp 3  stands  straight  behind the camera
  'Lamp 3 is positioned straight behind the camera.'
b. Lamp 2 staat links achter de camera.
  lamp 2  stands  left  behind the camera
  'Lamp 2 is positioned to the left behind the camera.'

The examples in (173) give an indication of the direction we have to look in order to find the located object in question. As is shown in (174), the distance between the reference and the located object can also be indicated.

Example 174
a. De fotograaf staat vlak achter de camera.
  the photographer  stands  right  behind the camera
  'The photographer is standing right behind the camera.'
b. Lamp 3 staat vijf meter achter de camera.
  lamp 3  stands  five meters  behind the camera
  'Lamp 3 is positioned five meters behind the camera.'

This means that each point in the region referred to by the locational PP achter de camera can be defined by means of a direction and a distance. In other words, the preposition achter can be considered to denote a set of vectors that project the position of the reference object onto a potential position of the located object. Consider Figure 6 in which the reference object occupies the (0,0) position. The preposition achter denotes those vectors V‹x,y› that originate in (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 region referred to by the PP achter de camera.

Figure 6: Achter de camera'behind the camera'

We can also denote subsets of the set of vectors denoted by the PP achter de camera'behind the camera': the modified PP recht achter de camera'straight behind the camera' denotes the subset of vectors denoted by achter in which x = 0, the modified PP links achter de camera denotes the subset of vectors in which x < 0, and the modified PP rechts achter de camera denotes the subset of vectors in which x > 0. Furthermore, the modified PP vlak achter de camera denotes the subset of vectors that are smaller than some contextually determined magnitude (length). These examples show that the assumption that locational prepositions denote a set of vectors will be very useful in the discussion of the modification properties of locational PPs in Section 3.1. For the moment, however, it suffices to observe that locational PPs refer to regions instead of fixed points in space.

[+]  IV.  The null vector

Prepositions like achter'behind', and boven'above' seem to express that the magnitude of the vector is larger than zero, that is, that there is some distance between the reference object and the located object. Other prepositions, however, require there to be some physical contact between the two objects: they require that the magnitude of the vector be zero. This difference can be illustrated by means of the prepositions op'on' and boven'above' in (175); example (175a) implies that there is physical contact between the table and the lamp, as in Figure 7A, whereas (175b) suggests the absence of such contact, as in Figure 7B.

Example 175
a. De lamp staat op de tafel.
  the lamp  stands  on the table
  'The lamp is standing on the table.'
b. De lamp hangt boven de tafel.
  the lamp  hangs  above the table
  'The lamp is hanging above the table.'

Figure 7: Op'on' and boven'above'

Note that the choice of verb does not affect the interpretation of the preposition; substituting hangen for staan in (175b) is possible if we refer to a situation like Figure 7B, in which the lamp is hanging so low that it touches the table. Somewhat more common examples such as this are given in (176).

Example 176
a. De gordijnen hangen op de vensterbank.
  the curtains  hung  on the windowsill
  'The curtains are touching the windowsill.'
b. Je rok hangt op de grond.
  your skirt  hangs  on the floor
  'The hem of your skirt is touching the floor.'

      The fact that the preposition op denotes the null vector, whereas boven denotes a larger set of vectors with a magnitude greater than 0 accounts for the fact that (on 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 (177) is valid, whereas the one in (178) is not.

Example 177
a. Jan loopt op het plein. AND
  'Jan is walking on the square.'
b. Het plein is voor de kerk.
  'The square is in front of the church.'
c. Jan loopt voor de kerk.
  'Jan is walking in front of the church.'
Example 178
a. De luchtballon zweeft boven het plein. AND
  'The hot-air balloon is floating above the square.'
b. Het plein is voor de kerk.
  'The square is in front of the church.'
c. De luchtballon zweeft voor de kerk.
  'The hot-air balloon is floating in front of the church.'

There is, however, also a potential problem for 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 (179) is compositionally determined, we should conclude that achter and boven are compatible with the null vector since otherwise a contradiction would arise.

Example 179
a. De productiedatum staat achter op het blik.
  the manufacturing.date  stands  behind  on  the can
  'The manufacturing date can be found on the back of the can.'
b. De productiedatum staat boven op het blik.
  the manufacturing.date  stands  above  on  the can
  'The manufacturing date can be found on top of the can.'

We will not discuss this problem here, but simply claim that the underlying assumption that the meanings of achter/boven op is compositionally determined is false. Section 3.1.3 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. Given that the reference object het blik in (179) 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 situated at different positions on the surface of the reference object, and it is therefore correctly predicted that the compounds achterop and bovenop denote distinct subsets of this set of null vectors denoted by op.
      Although the discussion of the examples in (179) clearly shows that this is, strictly speaking, 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 discussion that will follow; this motivates why we will normally refer to “the null vector” instead of “the set of null vectors”. The three-dimensional extensions of the reference object will only be taken into account if this is necessary for the discussion.

[+]  V.  Paths

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 di rectional preposition cannot simply be considered a set of vectors. This does not imply, 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 certain position on the time line. The path denoted by van A naar B'from 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 by means of a blocked or dotted arrow.

Figure 8: Van A naar B'from A to B'

The situation depicted in Figure 8 is an appropriate characterization of the directional phrase in (a), in which the PPs act as complements of the verb of traversing rijden'to drive'. Besides this “core” directional reading, directional PPs can also have two slightly different readings. The PPs in (180b) do not denote a path that is traversed, but indicate the extent of the located object deze weg'the 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 (180b) as the extent reading (note that the verb lopen is also used as a verb of traversing meaning “to walk” if the subject is animate). The directional PP in (180c), finally, is used with an orientation reading.

Example 180
a. Jan rijdt van Utrecht naar Groningen.
directional reading
  Jan drives  from Utrecht  to Groningen
b. Deze weg loopt van Utrecht naar Groningen.
extent reading
  this road  goes  from Utrecht  to Groningen
c. De richtingaanwijzer wijst naar Groningen.
orientation reading
  the direction.sign  points  to Groningen
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