B14. SPACE
1. Space and Spatial Analogies
Many "top-level" ontologies begin with a distinction between physical
objects and abstract entities. By contrast, we have made it through
twelve background theories without ever mentioning the distinction.
The reason for this is that the core of language doesn't seem to care
much about this distinction. We can be "in" a building, and we can be
"in" politics and "in" trouble. We can "move" a chair from the desk
to the door, and we can "move" the debate from politics to religion
and "move" money from one bank account to another. Ontologies that
begin with this distinction, or similar ones like Cyc's
tangible-intangible distinction (Lenat and Guha, 1990), fail to
capture important generalizations in language and as a result very
nearly make themselves irrelevant in linguistic applications at the
outset.
It has frequently been observed that we understand many abstract
domains by analogy with spatial relations (e.g., Vico, 1744 [1968];
Richards, 1936). A relatively recent rediscovery of this old truth
was by Lakoff and Johnson (1980). We operate in space from the moment
we are born and thus build up very rich spatial models. By setting up
a mapping between a new domain and space, we are able to commandeer
these rich models for the new domain and think about it in more
complex ways.
This would seem to indicate that the very first theory one should
develop in an enterprise such as ours is a very rich theory of space,
and then set up some mechanism for analogical reasoning. But what
does analogical reasoning involve? It is a matter of identifying
common properties of the things being compared, reasoning about them
in the familiar domain, and then transferring the results to the new
domain. But what is typically transferred from space to new domains
in spatial analogies is not just any property. Very rarely, for
example, do we transfer the properties of hardness or color or precise
distance. The common properties that are transferred are usually
topological properties.
In these background theories we have identified some of the most
important underlying properties of the spatial domain that are most
frequently utilized in analogies -- such things as complex structure,
scalar concepts, change of state, and so on. We have constructed
theories of these abstract properties. The content of these theories
is very similar to proposals in work in cognitive linguistics (e.g.,
Croft and Cruse, 2004) on the "image schemas" that seem to underlie
much of our thought.
If one were inclined to make innateness arguments, one position would
be that we are born with a instinctive ability to operate in spatial
environments. We begin to use this immediately when we are born, and
when we encounter abstract domains, we tap into its rich models. The
alternative, more in line with our development here, is that we are
born with at least a predisposition towards instinctive abstract
patterns -- composite entities, scales, change, and so on -- which we
first apply in making sense of our spatial environment, and then apply
to other, more abstract domains as we encounter them. This has the
advantage over the first position that it is specific about exactly
what properties of space might be in our innate repetoire. For
example, the scalar notions of "closer" and "farther" are in it; exact
measures of distance are not. A nicely paradoxical coda for summing
up this position is that we understand space by means of a spatial
metaphor.
It is also a curious and perhaps surprising fact about our efforts to
construct the cognitive theories of Part C that very few purely
spatial concepts turned out to be necessary. We occasionally need to
make reference to physical objects, to one physical object being "at"
another, and to a physical object being "near" another. And that is
all.
2. Spatial Systems and Distance
We will use the predicate "physobj" -- (physobj x) -- to say that
something is a physical object. We will not explicate this notion,
because it would take us deeply into commonsense physics rather than
commonsense psychology. We would have to talk about properties of
color, weight, and malleability. Instead we will simply stipulate
that something is a physical object, and not attempt to constrain the
possible interpretations of the predicate "physobj" by means of
further axioms.
The key notion in our treatment of space will be "spatialSystem". A
spatial system is a composite entity whose components are physical
objects and among whose relations are a "distance" relation.
(forall (s) (1)
(iff (spatialSystem s)
(and (compositeEntity s)
(exists (s1)
(and (componentsOf s1 s)
(forall (x) (if (member x s1)(physobj x)))))
(exists (s2 s3)
(and (relationsOf s2 s)(subset s3 s2)
(forall (e)
(if (member e s3)
(exists (d x1 x2 u)
(distance' e d x1 x2 u s)))))))))
Line 3 says that a spatial system s is a composite entity. Lines 4-6
say that its components are physical objects. Lines 7-12 say that
some (s3) of the relations (s2) of s are distance relations e between
two components x1 and x2 of s with respect to some spatial unit u.
In the predicate "distance", we take d to be a non-negative number
dependent upon the spatial unit u that is used, rather than reifying
distances as distinct entities. We won't say what a spatial unit is,
but it can be characterized in the same way temporal units were
characterized in Chapter B12. The constraints on the arguments of
"distance" are as follows:
(forall (d x1 x2 u s) (2)
(if (distance d x1 x2 u s)
(and (nonNegativeNumber d)(spatialSystem s)
(componentOf x1 s)(componentOf x2 s)
(spatialUnit u s))))
We will constrain the predicate "distance" by the usual mathematical
properties. The distance between an entity and itself is zero.
(forall (x u s) (distance 0 x x u s)) (3)
The distance between two entities is symmetric.
(forall (d x1 x2 u s) (4)
(iff (distance d x1 x2 u s)(distance d x2 x1 u s)))
The triangle inequality holds.
(forall (d1 d2 d3 d4 x1 x2 x3 u s) (5)
(if (and (distance d1 x1 x2 u s)(distance d2 x2 x3 u s)
(distance d3 x1 x3 u s)(sum d4 d1 d2))
(leq d3 d4)))
We will get to the definition of "near" by three steps: define a
"shorterDistance" relation, define a "near-ness" scale, and define
"near" as in the Hi region of that scale.
A distance d1 is a shorter distance than d2 in a spatial system s
under the obvious conditions.
(forall (d1 d2 s) (6)
(iff (shorterDistance d1 d2 s)
(exists (u x1 x2 x3 x4)
(and (distance d1 x1 x2 u s)(distance d2 x3 x4 u s)
(lt d1 d2)))))
A scale for "near-ness" for a spatial system is the reverse of a scale
whose elements are the distances between elements of s and whose
ordering function is "shorterDistance". We reverse the scale so that
shorter distances end up in the Hi region.
(forall (s1 s) (7)
(iff (nearnessScale s1 s)
(exists (u s2 s3 e d1 d2)
(and (scaleFor s2 s3 e)
(forall (d)
(if (member d s3)
(exists (x1 x2)(distance d x1 x2 u s))))
(shorterDistance' e d1 d2 s)(reverse s1 s2)))))
In this definition, s3 is a set of distances in s, s2 is the scale of
these distances, and s1 is the reverse of that scale. The eventuality
type e is a "shorterDistance" relation in s, d1 and d2 are its
parameters, and u is an arbitrary spatial unit.
Finally, for x1 to be near x2 is for the distance between them to be
in the Hi region of a near-ness scale.
(forall (x1 x2 s) (8)
(if (and (componentOf x1 s)(componentOf x2 s))
(iff (near x1 x2 s)
(exists (s1 s2 d u)
(and (nearnessScale s1 s)(Hi s2 s1)
(distance d x1 x2 u s)(inScale d s2))))))
Since we have defined "near" between two entities in terms of the Hi
region of a scale, we can draw inferences about the distributional and
functional properties of the distance between them, as indicated in
Section B8.4.
3. Location
In section B7.3 we introduced the figure-ground relation "(at x y s)",
meaning that external entity x is at component y of composite entity
s. A spatial system is a possible ground by virtue of the fact that
all its components are physical objects. That is the common property
they have that makes a unitary interpretation of the "at" relation
possible. (cf. Axiom 6.14).
(forall (s) (if (spatialSystem s)(ground s))) (9)
We can now characterize the predicate "atLoc", for "at a location"
as "at" where s is specialized to spatial systems.
(forall (x y s) (10)
(if (atLoc x y s)
(and (spatialSystem s)(at x y s))))
Since s is a spatial system and x is at y in s, y must be a phyiscal
object. We are silent about what kinds of things x can be. Certainly
physical objects can be at another physical object -- Jill is at her
desk. Events involving physical objects can also be at physical
objects -- Jill is typing at her desk. Not all the arguments of an
event need to be at the location. We can say that a telecon is going
on at a point in Jill's office, but not all the participants of the
telecon need to be at that location. We do not say anything about
whether abstract eventualities or abstract entities can be at physical
locations.
Two entities that are at locations in a spatial system are near if
their locations are near.
(forall (s y1 y2 x1 x2) (11)
(if (and (componentOf y1 s)(componentOf y2 s)
(atLoc x1 y1 s)(atLoc x2 y2 s))
(iff (near x1 x2)(near y1 y2))))
Predicates Introduced in this Chapter
(physobj x): x is a physical object.
(spatialSystem s): s is a composite entity whose components
are physical objects related by
distance.
(distance d x1 x2 u s): d is the distance in units u between x1
and x2 in spatial system s.
(spatialUnit u): u is a spatial unit.
(shorterDistance d1 d2 s): d1 and d2 are distances between components
of spatial system s, and d1 is less
than d2.
(nearnessScale s1 s): s1 is a scale whose components are
distances between pairs of entities in
spatial system s.
(near x1 x2 s): x1 is near x2 in s.
(atLoc x y s): x is at y in spatial system s.
References:
Croft, William, and D.A. Cruse, 2004. Cognitive Linguistics,
(Cambridge Textbooks in Linguistics.) Cambridge: Cambridge University
Press
Lakoff, George, and Mark Johnson, 1980. {\it Metaphors We Live By},
Chicago: University of Chicago Press.
Lenat, Douglas B., and R. V. Guha, 1990. {\it Building Large
Knowledge-Based Systems}, Addison-Wesley, Reading, Massachusetts.
Richards, I. A., 1936. {\it The Philosophy of Rhetoric}, Oxford:
Oxford University Press.
Vico, Giambattista, 1744. {\it The New Science of Giambattista Vico},
T. Bergin and M. Frisch, trans. Ithaca NY: Cornell University Press.
1968.