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.