A Compositional Semantics for Directional Modifiers 
- Locative Case Reopened - 
Erhard W. Hinrichs 
Bolt Beranek & Newman Laboratories 
I. Abstract 
This paper pre,'Jents a model-theoretic semantics for directional 
modifiers in English. The semantic theory presupposed for the 
analysis is that of Montague Grammar (cf. Montague 1970, 1973) 
which makes it possible to develop a strongly compositional 
treatment of directional modifiers, Such a treatment has 
significant computational advantages over case-based treatments 
of directional modifiers that are advocated in the A! literature. 
2. Case-based Treatments 
Among natural language processing systems which attempt to 
incorporate spatial information, the following strategy seems to 
prevail. Directional or locative modifiers are treated either as 
corresponding to slots in case "/rames in the canonical lexical 
representations of verbs (cf. Celce 1972, Hendrix, Thompson and 
Slocum 1973), or as corresponding to conceptual cases in the 
(meta-linguistic) conceptualization of actions (Schank 1975). 
Case based approaches to the semantics of directional modifiers 
can be characterized as weakly compositional in the following 
sense: In a verb phrase such as fly to Chicago the prepositional 
phrase contributes semantically the meaning of the NP Chicago 
' as the directional or locative goal of the action associated with 
~.he verb phrase. However, the directional preposition to itself 
does not make a semantic contribution at all to the meaning of 
the verb phrase as a whole. Instead, to merely serves as a 
syntactic marker for a semantic entity, namely a locative or 
directional case whose meaning cannot be separated from, but 
rather is an integral part of a given verb frame or conceptual 
structure. By contrast, the semantics of directional modifiers 
that I will be advocating in this paper is strongly csmpositional 
in the sense that directional prepositions serve as autonomous 
syntactic and semantic units. Consequently, each word in a 
phrase such as fly to Chicago contributes its own, independent 
meaning to the meaning of the phrase as a whole. 
This strongly compositional analysis of directional modifiers has a 
number of crucial computational advantages over case-based 
approaches. Consider how inferences between sentences such as 
(1) and (2) can be handled by the two types of approaches. 
(I) John went to New York. 
(2) John was in New York. 
In Schank (1975, p.53) sentence (I) corresponds to the 
conceptual structure in (3). 
(3) \[~New York 
John ~-%-~ PIRAte ~ John e-~-~ 
L .X 
(3) be should read as: "John is at some time in the past (p) 
engaged in an act of physical transfer (PTRANS) whose object (o) 
is John and whose direction (D) is from some location X to New 
York." The fact that (1) implies (2) is expressed by attaching to 
the bi-directional arrow in (3).the structure in (4). (of. Schank 
1975, p. 54) 
(4) John~ LOC(N.Y.) 
Schank calls the r-lin/c (v for result) between structures (3) and 
(4) an inference. However, the term inference is really a 
misnomer because the association between structures such as (3) 
and (4) is merely a matter of stipulation but does not follow from 
any general principles or axioms that would constrain the 
language of conceptual structures. For that matter, there is 
nothing in Schank's system that prevents a link between (3) end 
a structure which expresses that John does not reach the 
location New York. In the analysis we will develop below, on the 
other hand, the inference between (1) and (2) follows logically 
from the semantics of motion verbs such as go in conjunction 
with the semantics of directional modifiers, 
Consider next the issue of how easy or difficult it is to upscale 
natural language systems whose treatment of directional 
modifiers is case-based. Assume a case-based system in which 
only those verbal frames or conceptual structures are 
implemented that relate locative or directional case to verbs of 
motion. Now imagine that we! want to extend coverage to verbs 
such as wave which, as illustrated in (5), allow directional 
modifiers such as to, 
(5) The President waved to the reporters. 
Since wave, unlike verbs of motion, does not entail a change of 
location for the agent involved, a new verbal frame or conceptual 
structure would have to be introduced into a system which only 
covers motion verbs. Moreover, locative or directional case 
would have to be reintroduced into the system as well because in 
a case-based system the specific effect of a given semantic case 
has to be determined for each mdividual frame or conceptual 
structure. This is a direct consequence of the weakly 
compositional semantics of suctl systems and in turn leads to an 
highly redundant method of upscaling. Since my analysis of 
directional modifiers is, by contrast, strongly compositional, 
upscaling becomes much easier. In the case of extending 
coverage to a verb like wave, all that needs to be added is the 
lexical semantics for the w.~rb itself, while the semantics of 
directional modifiers can remain untouched. 
Finally, consider how a case-based approach to directional 
modifiers fares with respect to phrases such as the ones given m 
(6). 
(6) From Russia with Love 
To New York and then to Atlanta 
Since in case-based systems locative or directional case is a 
relational notion and is crucially dependent on a verbal frarae or 
conceptual structure, it becomes impossible to assign an 
interpretation to verbless phrases as in (6). One strategy for 
extending case-based systems to such verbless phrases would 
consist in supplementing the relational notion of directional or 
locative case by a non-relational counterpart which does not 
depend on some verbal frame or conceptual structure. But the 
resulting account of locative or directional case would once 
again be highly redundant since essentially all of the cases in 
the system wouhi have to be split into a relational and a non- 
relational version. 
2, Motion Verbs as Location Predicates 
In their literal sense, locative use to and toZvavd typically modify 
motion verbs such as wal~, ru~, drive, slither, move etc. An 
adequate treatment of the directional modifiers themselves is, 
therefore, closely connected to a semantic account of such 
motion verbs. In Hinrichs (1985) I argue that motion verbs 
should be treated as stage level predicates in the sense of 
Carlson (1977), namely as predicates whose arguments refer to 
stages of individuals. Stages are connected to individuals in 
Carlson's ontology by a realization relation R, which associates a 
given individual with all of the (spoils-temporal) stages at which 
that individual is present. 
Motion verbs such as move can be understood as prototypical 
examples of stage-level predicates, since such verbs predicate 
something about the spatio-.-temporal location of one or more 
objects. Following Hiurichs (1985), I interpret a motion verb like 
move in terms of a three-place stage level predicate move + , 
whose first two argument positions range over individual stages 
realizing th~ referents of the object and subject NPs, 
respectively. Fallo~ing Davidson (1977), the rightmcst argument 
position r ~n~,es over events, or more specifically over evenl 
347 
stages which realize the event that the referents of the subject 
and object NPs are engaged in. Thus, move+(xs)(ys)(e s) should 
be read as: "the referents of x s and ys are engaged in an event 
stage e s realizing an event of moving." As is customary in 
Montague Grammar, I express constraints on lexical meaning in 
terms of meaning postulates that constrain the set of possible 
models of semantic interpretation? The meaning postulate in (7) 
states that an event stage e s which realizes a moving event 
spatio-temporally includes (symbolized as K) at least the location 
of the referent denoted by the object argument, i.e. yS K e s. 
This does not exclude the possibility that the location of the 
referent of the subject NP can be contained in the event stage 
as well, but this is not required for move, as (8) shows. 
(7) V xS,yS,e" \[ move+(xS)(yS)(e s) ---> yS ~ es\] 
(8) John moved the troops. 
Of course, different motion verbs will have different properties 
with respect to how the locations of the event stages relate to 
the stages that realize the individuals involved in these event 
stages. Consider verbs like slither, walk, and run which in my 
framework are analyzed as two-place stage level predicates. For 
these predicates the location of the event stage is equal to the 
location of the agent, i.e. the referent of the subject NP. This 
can be enforced by a meaning postulate as in (9). 
(9) V xS,e s \[8+(xS)(e s) --> xS=e s \], where ~ translates 
slither, walk, run, etc.. 
The lexical entailment associated with the verb move to the effect 
that the location of the referent of the object NP changes can 
be captured by the meaning postulate in (10). (The symbols < 
and #~ used in (10) stand for temporal precedence and spatial 
inequality, respectively.) 
(10) V eS,x~,y",x ° \[R(x~,x °) & move+(x~)(y~)(e~) --> a x~ 
\[R(x~,x °) & x~ < x~ & x~ ~. x~\] 
3, The Semantics of to and toward 
Now I let a t0-phrase, as a modifier of untensed verb phrases 
(IV*), operate semantically on the event stages in the denotation 
of the unmodified verb phrase in such way that the event stages 
in the denotation of the resulting IV* phrase constitute a 
spatio-temporal path (in the sense of Cresswell 1978) between 
some specified point of origin to the location of the term 
combining with to. The translation of to is given in (11). 
(11) to translates as XPXPXllXXiP \[kyIS12\[R(12,Yi ) & 
PATH(il,lr,12) & P(xi)(ll)\]\] 
The formula following the lambda abstractions in (11) introduces 
an individual stage /2 realizing an individual object yi which is 
the one bound by the noun phrase (NP) combining with to to 
form the IV* modifier. The second conjunct in the formula 
asserts that the denotation of the event stage located at l 1, 
which is to be bound by the translation of the IV* phrase that 
the to-phrase combines with, qualifies as a spatio-temporal path 
(a notion formally defined in Hinrichs 1985) between some point 
of origin I r and the spatio-temporal location of the point of 
destination. Finally, the third conjunct asserts the truth of the 
unmodified IV* phrase that the to-phrase combines with. It is 
this last conjunct that automatically guarantees the inference 
from sentences such as (12) to sentences such as (13). 
(12) Fangs slithered to the rock. 
(13) Fangs slithered. 
Using the translation for to suggested in (11), sentence (12) 
receives the reduced translation in (14) according to my 
analysis. 
1All the meaning postulates appearing in this paper ore formulated 
In the language of extensional logic developed in Hinrichs (1985), 
(14) ~Iea,e i \[R(eS,e i) & PAST(e 5) & 3x e \[R(xS,f) & 2x°Vz ° \[ 
roek'(z °) & ~z a \[R(zS,z °) & slither'(xe)(e s) & 
PXTH(eS,lr,ZS)\] ~ x ° = z°\]\] 
Paraphrasing (14), it says that there is an event stage realizing 
some individual event of Fangs' slithering such that that event 
stage lies in the past and the spatio-temporal location of the 
event stage constitutes a path between some implicit point of 
reference l r and the location of some unique rock object. The 
point of reference l r occurs as a free variable in the formula in 
(14); l r is to be understood as an indexical parameter similar to 
the notion of a reference point proposed by Reichenbach (1947) 
for the interpretation of tenses in English. 
Notice that the notion of a path in the translation of to in (11) 
and hence also in the translation for (12) given in (14) is defined 
to hold of the process making up a particular event. Moreover, 
due to the postulate in (9), the referent of the subject NP, when 
it combines with a motion verb such as slither to the rock, is 
realized by a stage spatio-temporally co-extensive to the path 
denoted by the to phrase. This fact guarantees the inference 
between sentences such as (12) and (15). 
(15) Fangs was at the rock. 
For other classes of verbs the same type of inference, namely 
identifying the path with the position(s) of the referent of the 
subject NP, cannot be drawn. For sentences such as (16) we do 
not want to claim that the stages realizing John make up a path 
to Boston. Rather it is the object NP, in this case an event 
term, that constitutes the path. The same is true of (17); it is 
the ball whose locations constitute a path to the location 
specified in the to-phrase. 
(16) John made a phone call to Boston. 
(17) Carol set the ball to Lucy. 
Let us now turn to the treatment of the preposition toward 
whose lexical translation rule is given in (18). 
(18) toward translates as kPkPkeakxlP(ky I 31 \[R(l,y i) & 31' 
\[PATH(I',lr,\])) & e s _<, 1' & I r < e s & P(xi)(eS)\]\]) 
The translation for toward constrains the value of the event 
stage variable s s in such a way that e s has to be spatio- 
temporally contained in some initial segment of a path l' from 
some implicit point of origin I r to the location I of the referent . 
of the NP with which toward. The requirement that the value of 
e s has to be an initial segment of such a path follows from the 
condition that the implicit point of origin I r has to be properly 
contained in e s. Proper containment is necessary in order to 
avoid that the value of e s could be equal to the point of origin, 
in which case an object could count as moving toward another 
object if the spatial location of the first object remains 
unchanged. 
Using (18), sentence (19) is translated as in (20). 
(19) Fangs slithered toward the rock. 
(20) 3eS,e i \[R(ee,e i) & PAST(e s) & 3x ~ \[R(xa,f) & ~x°\[Vz ° 
\[rock'(z °) <--> xO=z o\] & 3z a \[R(ze,z °) & slither'(xS,e s) 
& 31 \[PATH(1,1r,Z s) & e s _~ 1 & 1 r < eS\]\]\]\]\] 
The translation in (E0) says that there is an event stage 
realizing some individual event of Fangs' slithering such that 
that event stage lies in the past and the spatio-temporal 
location of the event stage constitutes the initial part of a path 
between some implicit point of reference I r and the location of 
some unique rock object. Since es in (20) is an initial part of a 
complete path to the rock, the truth of a sentence such as (12) 
entails the truth of (19), but not vice versa. Moreover, (12), but 
not (19), entails (15). 
4. The AspeetualEffect of to and toward 
Apart from supporting the relevant inference patterns between 
sentences such as (12), (15) and (19), an adequate analysis of to 
and toward should also account for a systematic difference in 
348 
the aspectual behavior of these two directional modifiers. 
Sentences such as (21a) which involve the preposition to describe 
atelis events or, in the terminology of Vendler (1967), activities. 
Sentences such as (21b), on the other hand, refer to telic events 
or to accomplishments in Vendler's classification, 
(21) a. John walked to the library. 
b. John walked toward the library. 
These aspeetual properties can be demonstrated by examining the 
cooccurrenee restrictions of the sentences in (21) with temporal 
modifiers such as in an hour as in (22) and with for an hour as 
in (23). 
(22) a. John walked to tile library in an hour. 
b. * John walked toward the library in an hour. 
(23) a. Johu walked to the library for an hour. 
b. Johu walked toward the library for an hour. 
As first pointed out by Vendler, only relic events or 
accomplishments can occur with temporal modifiers such as in an 
hour. Modifiers such as for an hour can occur with both 
activities and accomplishments. However, when modified by 
temporal for, only activities as in (23a) can be interpreted as 
describing a single event. If temporal for occurs with sentences 
that describe accomplishments as in (23b), such sentences have 
to be interpreted in some special fashion to make them 
semantically acceptable, (23b), for example, can best be 
understood as referring to an iterative event, namely of John's 
repeatedly walking to the library during the period of one hour, 
Since doing something for x amount of time means doing 
something during most if not all subintervals of the interval x, 
sentences such as (24), which refer to atelic events or activities, 
can be characterized as being temporally homogeneous. 
(24) Fangs slithered toward the rock. 
To do something in x amount of time, on the other hand, means 
to do something at some unique interval within x. Since relic 
events or accomplishments can be modified by temporal in, they, 
in contrast to activities or atelic events, can be described as 
being temporally heterogeneous: telic events such as (25) come 
about over tile course of some unique time interval I', i,e. not at 
some subinterval of I' or at some interval properly containing I'. 
(25) Fangs slithered to the rock. 
If my analysis of directional toward and to is an adequate one, it 
should predict that verb phrases formed with directional toward 
refer to temporally homogeneous events, while verb phrases 
formed with to refer to temporally hcterogenous events, Due to 
the way in which I have defined toward as an initial subpart of a 
path to the projected point of destination, the reference 
property of temporal homogeneity associated with toward can, in 
fact, be reconstructed in the following way. Let us assume that 
there is a location I t which qualifies as an initial segment of a 
path from a putative point of origin r t to a destination d. 
Moreover, let us assume that r 2, the temporally final bound of l/, 
is in turn the temporally initial bound for a location 12 which 
forms the intial part of a path from r 2 to d. Then it follows that 
lf+l 2, the spatio-temporal sum of l t and l 2, is also an initial 
segment of a path from r I to d. This is precisely what is 
required to stake the semantics of toward homogeneous. 
Since my account of motion verbs and directional toward does 
predict that sentences such as (26) correspond to atelic and 
semantically homogeneous events, my analysis can support 
inferences from sentences such as (26) to sentences such as 
(27). 
(26) United Flight 342 has moved toward Logan Airport 
for the last fifteen minutes. 
(27) United Flight 342 moved toward Logan Airport ten 
minutes ago. 
Inference patterns between sentences such as (26) and (27) are, 
in fact, highly relevant for data base interface systems that 
process spatial information. Imagine that sentence (26) is 
presented to a database that monitors plane movements. If the 
system does not have the capability to infer that the event 
described in (26) is true at any subinterval of the fifteen 
minutes mentioned in (26), the United flight in question would 
erroneously not be counted when the answer to a subsequent 
query such as (28) is computed. 
(28) How many planes moved toward Logan Airport ten 
minutes ago? 
If we compare the semantics of toward with the semantics of to 
as defined in (11), it turns out that to is heterogeneous in its 
reference in the same way as accomplishments. Recall that the 
semantics of to is defined in terms of a complete path between a 
point of origin and a point of destination, Since for any given 
path there do not exist any sublocations within that path that 
themselves would qualify as a path between the same two 
locations, the heterogeneons reference property of to follows 
automatically. 
5. Conclusion 
In order to make an even stronger case in favor of my analysis 
of directional modifiers, I would have to demonstrate how it can 
be generalized to locative prepositions other than to and toward. 
Even though I cannot discuss this issue in detail in the present 
paper, I should like to point out in conclusion that the notion of 
a PATH plays an important role in the treatment of other 
directional prepositions such as between, along and across. In 
the case of across the path seems to be bounded by two 
locations on tile peripherie of the referent of the NP across is 
combined with; i.e. across the meadow specifies some path 
extending from one end of the meadow to the other, Notice, 
however, that the two locations that mark the two endpoints of 
such a path cannot be chosen arbitrarily but in some sense have 
to be "opposite each other". Undoubtedly, various pragmatic 
considerations enter the picture if one wants to make this 
requirement of oppositeness formally more precise. Thus, it 
appears that the notion of a path has to be complemented by 
additional constraints, if one wants to account for semantically 
store complex prepositions such as across, Even though I will 
have to leave the formulation of such additional constraints to 
future research, it should be obvious from these brief remarks 
that the notion of a path is a central notion for the semantics of 
directional modifiers in general, 

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