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An Image-Schematic Account of Spatial Categories



How we categorize certain objects depends on the processes they afford: something is a vehicle because it affords transportation, a house because it offers shelter or a watercourse because water can flow in it. The hypothesis explored here is that image schemas (such as LINK, CONTAINER, SUPPORT, and PATH) capture abstractions that are essential to model affordances and, by implication, categories. To test the idea, I develop an algebraic theory formalizing image schemas and accounting for the role of affordances in categorizing spatial entities.
An Image-Schematic Account of Spatial Categories
Werner Kuhn
Institute for Geoinformatics, University of Münster
Robert-Koch-Str. 26-28, D-48149 Münster
Abstract. How we categorize certain objects depends on the processes they afford: something is a
vehicle because it affords transportation, a house because it offers shelter or a watercourse because
water can flow in it. The hypothesis explored here is that image schemas (such as LINK,
CONTAINER, SUPPORT, and PATH) capture the necessary abstractions to model affordances for
spatio-temporal processes. To test the idea, I develop an algebraic theory formalizing image schemas
and accounting for the role of affordances in categorizing spatial entities.
1 Introduction
An ontology, according to (Guarino 1998), is "a logical theory accounting for the intended meaning of a
vocabulary". Ontologies map terms in a vocabulary to symbols and their relations in a theory. For example,
an ontology might specify that a house is a building, using symbols for the two universals (house, building)
and applying the taxonomic (is-a) relation to them; or it might explain what it means for a roof to be part of
a building, using a mereological (part-of) relation (Bittner and Smith 2004). What relations should account
for the meaning expressed by the italicized phrases in the following glosses from WordNet
a house is a "building in which something is sheltered or located"
a boathouse is a "house at edge of river or lake; used to store boats"
a vehicle is a “conveyance that transports people or objects
a boat is a "small vessel for travel on water"
a ferry is a “boat that transports people or vehicles across a body of water and operates on a
regular schedule”
a houseboat is a "barge that is designed and equipped for use as a dwelling"
a road is an “open way (generally public) for travel or transportation.”
These phrases express potentials for actions or processes, rather than actual processes. For such potentials,
I will use the term affordances in a very broad sense. This should not be taken as overextending Gibson’s
notion (Gibson 1979), but as a shorthand covering a collection of phenomena whose prototype is an
affordance in Gibson’s sense (and which have also been called telic relations in (Pustejovsky 1998)). They
all pose the same ontological challenge: how can a logical theory capture the categorizing effect of
potential processes? For example, how can it express that a vehicle is a vehicle because it affords
transportation, even though it may never be used to transport anything, or that a house is a house because it
can shelter something? The goal is not to capture peripheral cases of a category (e.g., the vehicle which
never transports), but conversely, properties which are central (e.g., the transportation and sheltering
affordances). As the above WordNet examples illustrate, affordances are indeed often central for
What is the relevance of affordances to ontologies of spatial information? Ordnance Survey (the
mapping agency of Great Britain) has identified affordance as one of five basic ontological relations to
make their geographic information more explicitly meaningful (together with taxonomic, synonym,
topological, and mereological relations) (Hart, Temple et al. 2004). Feature-attribute catalogues for
geographic information in fact abound with object definitions listing affordances as key characteristics
(Rugg, Egenhofer et al. 1997). Recent challenges to semantic interoperability have revolved around use
cases that involved affordances. For example, the OGC geospatial semantic web Interoperability
Experiment worked on a query retrieving airfields based on their affordance of supporting certain aircrafts
( Addressing the ontological challenge posed by
affordances is necessary to capture the meaning of vocabularies in spatio-temporal information and enable
semantic interoperability.
Affordances are also increasingly recognized as having a role to play in formal ontology (Bateman and
Farrar 2004). The enduring core of objects, particularly of artifacts or organisms, has been sought in
“bundles of essential functions” (Masolo, Borgo et al. 2004). Frank sees affordances as the properties that
define object categories in general and uses this idea in his ontological tiers (Frank 2003). Smith has
proposed an ontology of the environment, and in particular the notion of niche (Smith and Varzi 1999),
founded on Gibson’s affordances and underlying his Basic Formal Ontology. The Descriptive Ontology for
Linguistic and Cognitive Engineering (DOLCE) introduces categories “as cognitive artifacts ultimately
depending on human perception, cultural imprints and social conventions” and acknowledges a Gibsonian
influence (Gangemi, Guarino et al. 2003).
Recent formal ontology work has attempted to formalize image schemas as abstract descriptions
(Gangemi and Mika 2003). While this gives an ontological account of the mental abstraction of image
schemas, it does not yet allow for using it in category specifications. This would require an integration
with work on roles, particularly thematic roles, to address the categorizing power of objects participating in
processes. Thematic roles are essentially a set of types of participation for objects in processes (Sowa
2000). For example, a vehicle can play the thematic role of an instrument for transportation. The
formalization of image schemas given in this paper does capture such thematic roles, but at a more generic
level that has not yet been related to the thematic role types discussed in the linguistics literature.
In the context of spatial information theory, affordances and image schemas have been the object of several
formalization attempts (Raubal, Egenhofer et al. 1997; Raubal, Gartrell et al. 1998; Egenhofer and Andrea
Rodríguez 1999; Frank and Raubal 1999; Raubal and Worboys 1999; Kuhn and Raubal 2003; Raubal and
Kuhn 2004). At least one of them also made the relationship between the two notions explicit (Raubal and
Worboys 1999). But none has developed an image-schema based formalization of affordances to the point
where it can account for categorizations. The work presented here is building on these attempts and
developing them further in the context of spatial ontologies.
What are the basic ontological modeling options for affordances? Unary relations (e.g., “affords-
transportation”) leave the meaning of their predicates unspecified, produce an unlimited number of them,
and suppress the object to which a process is afforded. To model affordances as a binary or ternary relation
(“affords”), would require a kind of process as argument (e.g., “transportation”). For example, an individual
vehicle (say, your car) affords a kind of transportation (i.e., of people), which subcategorizes the vehicle
(e.g., as a car). However, first order theories do not permit process types as arguments, as long as these are
not treated as individuals. The Process Specification Language PSL (Gruninger and Kopena 2005) takes the
latter approach, but cannot state algebraic properties for the process types.
Due to this expressiveness issue, formal ontology has not yet approached affordance relations (see also
(Degen, Heller et al. 2001; Bittner and Smith 2004; Masolo, Borgo et al. 2004) for discussions of formal
ontological relations). The participation relation comes closest, but relates individual objects to actual
processes (Masolo, Borgo et al. 2004). It is also much looser than affordance. For example, a boat might
participate in a painting process, as well as a transportation event. Both conform to the usual intuition about
participation, that there are objects involved in processes, but one is an “accidental participation” and the
other an instance of a potential, afforded participation that categorizes the object.
Ontological relations between classes have been proposed for capturing spatial relations (Donnelly and
Bittner). However, they cover only classes of objects (not processes), and the approach would need to be
extended to modal logic in order to cope with potential processes, not only with existence and universal
quantification. Before taking this route of first order axiomatizations of affordance relations, one has to
analyze what relations are necessary. This is the goal pursued here. The main novelty is the combined
formalization of affordances and image schemas in a second order algebraic theory, accounting for spatial
categories and establishing a framework for ontology mappings. In the remainder of the paper, I will state
how image schemas form the basis for the theory (section 2), introduce the formalization method (section
3), present the results of the analysis (section 4), and conclude with implications and directions for future
research (section 5).
2 Image Schemas as Theoretical Foundation
The primary source on image-schematic categorization is Lakoff’s Women, Fire, and Dangerous Things
(Lakoff 1987). It presents detailed case studies on image schemas and their role in language and cognition.
Its main impact lay in demonstrating the inadequacy of traditional ideas about categorization, which are
based on necessary and sufficient conditions. Formalizations, however, were not Lakoff’s goal, due to the
perceived limitations of formal semantics at the time. Since then, a lot of work in cognitive semantics,
including applications to ontology engineering (Kuhn, Raubal et al. 2007) has built on Lakoff’s empirical
evidence and informal models for image schemas. This section reviews the widely accepted characteristics
and attempts to identify “ontological properties” of image schemas.
2.1 Key Characteristics of Image Schemas
Image schemas are patterns abstracting from spatio-temporal experiences. For example, they capture the
basic common structures from our repeated experience of containment, support, linkage, motion, or
contact. The idea emerged from work in cognitive linguistics in the 1970’s and 1980’s, mostly by Len
Talmy, Ron Langacker, Mark Johnson, and George Lakoff. A recent survey is (Oakley). The idea gained
popularity through Lakoff and Johnson’s book on metaphors (Lakoff and Johnson 1980), Lakoff’s book on
categorization (Lakoff 1987), and Johnson’s book on embodied cognition (Johnson 1987). The latter
characterizes image schemas as follows (p. 29): “In order for us to have meaningful, connected experiences
that we can comprehend and reason about, there must be pattern and order to our actions, perceptions, and
conceptions. A schema is a recurrent pattern, shape, and regularity in, or of, these ongoing ordering
activities. These patterns emerge as meaningful structures for us chiefly at the level of our bodily
movements through space, our manipulation of objects, and our perceptual interactions”. Johnson’s
characterization highlights the spatial aspects of image schemas, their link to activities, and their power to
generate meaning and support reasoning. I will briefly discuss each of these aspects and then extract those
properties of image schemas that suggest their potential for ontology.
Image schemas are often spatial, typically topological (e.g., CONTAINMENT, LINK, PATH,
COUNTERFORCE). This makes them obvious candidates for structuring spatial categories in ontologies.
Since foundational ontologies still take a rather simplistic view of space, based on abstract geometric
locations (for a survey, see (Bateman and Farrar 2004)), any candidates for more powerful spatial patterns
should be tested for their ontological potential. This is what I am doing here with image schemas for the
purpose of building ontologies of spatio-temporal phenomena. As Lakoff and Johnson have shown, the
spatial nature of image schemas also gives them a role in structuring abstract categories and in constraining
conceptual mappings (Lakoff 1990). The scope of this paper, however, is limited to the role of image
schemas in categorizing spatial entities.
The second aspect in Johnson’s characterization closely relates to the first: image schemas capture
regularities in activities and processes. Examples are CONTAINMENT, SUPPORT or PATH. Since my
goal is to account for meaning grounded in affordances (in the sense of process potentials offered by the
environment), any regularity observed in processes is interesting. The structuring of processes in
foundational ontologies is as weak as that of space. Separate branches of foundational ontologies have been
devoted to processes, but their internal structure and the participation relations between processes and
objects remain underdeveloped (Masolo, Borgo et al. 2004). Image schemas provide a special form of such
Thirdly, image schemas support our understanding of and reasoning about experiences. Thus, they are
likely to shape the meaning of vocabularies we use to describe these experiences. After all, information
system terminology, like natural language, encodes human experiences in communicable form and as a
basis for reasoning. An obvious example are navigation systems (Raubal, Egenhofer et al. 1997), guiding
human movement through space. Most GIS applications, from cadastral systems and land use databases to
tourism services involve categorizations based on human activities and experiences (Kuhn 2001).
Exploiting abstract patterns of experiences for the design of spatial (and other) ontologies seems therefore
2.2 Ontological Properties of Image Schemas
A generally accepted precise definition or even formalization of image schemas is still missing, and the
evidence for their existence (in whatever form) comes mainly from linguistics. For the purpose of this
work, however, I require neither a precise definition nor a broader empirical basis for image schemas. I
simply take the notion as a useful theoretical construct with the following ontological properties:
1. Image schemas generalize over concepts (e.g., the CONTAINMENT schema abstracts
container behavior from concepts like cups, boxes, or rooms);
2. they are internally structured (e.g., the CONTAINMENT schema involves behavior associated
with an inside, an outside, a contained entity, and possibly a boundary);
3. they can be nested and combined (e.g., transportation combines behavior from the SUPPORT
and PATH schemas);
4. they are preserved in conceptual mappings (e.g., the LINK and PATH schemas are common to
all transportation links).
I call these properties ontological, because they relate image schemas to essential questions in ontology,
which impact applications to spatial information: what entities should form the upper levels of ontologies?
can these have internal structure or should they be atomic? how can ontologies reconcile process and object
views of the world? how can ontological primitives be combined? what are useful properties of ontological
It is far beyond the scope of this paper to address any of these questions in detail. However, the fact that
image schemas suggest some answers to them encourages a research program that explores their potential
to (1) ground ontologies in human sensory-motor experience, and (2) provide invariants for ontology
mappings. In pursuing this longer term goal, I claim here that an algebraic theory built on image schemas
can account for those aspects of meaning that are grounded in affordances. I test this hypothesis by
developing a new style of formalization for image schemas, and applying it to account for empirical data
about spatial conceptualizations taken from WordNet.
3 Formalization Method
The relation between objects and process types, though missing from formal ontology, is a central pillar of
algebra and computing. Algebra groups values into sorts based on the kinds of operations they admit.
Programming languages model these sorts as data types and classify values or objects based on the kinds of
computational processes they offer. These processes are collectively referred to as behavior. For example, a
data type Integer offers operations (behavior) of addition, subtraction, and multiplication, but not division
or concatenation.
The software engineering technique of algebraic specification exploits this commonality between
programs and algebra (Ehrig and Mahr 1985). It models programs as many-sorted algebras, consisting of
sets of values with associated operations. Logical axioms, in the form of equations over terms formed from
these operations, define the semantics of the symbols used. For example, an axiom might say that an item is
in a container after having been put in. There exists a variety of flavors and environments for algebraic
specifications. In the context of ontology engineering, OBJ (Goguen 2005) and CASL (Lüttich 2006) are
the most popular environments.
For reasons of simplicity, expressiveness, and ease of testing, I use the functional language Haskell as an
algebraic specification language. It offers a powerful development and testing environment for ontologies,
without the restriction to subsets of first order logic and binary relations typical for ontology languages. It
also imposes less overhead for theorem proving than most specification environments and imposes the
healthy need to supply a constructive model for each specification. While we have previously published
ontological specifications in Haskell (e.g., in (Frank and Kuhn 1995), (Kuhn 2002), (Kuhn and Raubal
2003), (Raubal and Kuhn 2004)), the style has now been refined, clarified, and stabilized. The remainder of
this section introduces this style and the associated design decisions, as far as their knowledge is necessary
to understand the subsequent formalizations.
This is not a syntax-driven introduction to Haskell, but one that explains how the language can be used
to model individuals, universals, affordances, and image schemas. Introductions to Haskell as a
programming language can be found at, together with interpreters and compilers.
For beginners, the light-weight Hugs interpreter is highly recommended. It has been used to develop and
test all the specifications in this paper.
3.1 Universals as Data Types
Universals (a.k.a. categories, classes or concepts in ontology) will be modeled here as Haskell data types,
and individuals (a.k.a. instances) as values. The standard computing notion of what it means to belong to a
type captures the ontological relationship of instantiation between individuals and universals. Note that
universals are then not just flat sets of individuals, but are structured by the operations (a.k.a. methods)
defined on them.
The Haskell syntax for type declarations simply uses the keyword data followed by a name for the type
and a right-hand side introducing a constructor function for values, possibly taking arguments. Here is a
simple example, declaring a type for the universal Medium with two (constant) constructor functions
Water and Air (Haskell keywords will be boldfaced throughout the paper):
data Medium = Water | Air
Type synonyms can be declared using the keyword type. For example, one defines a synonym House
based on a previously declared type Building as follows:
type House = Building
By considering types as theories (Goguen 1991), where operation signatures define the syntax and
equational axioms the semantics for some vocabulary, one can now write theories of intended meanings,
i.e., ontologies, in Haskell. They introduce a type symbol for each universal (e.g., House), a function type
for each kind of process (e.g., enter, not shown here), and equations on them (e.g., stating that entering a
House results in being in it).
3.2 Subsumption (and more) through Type Classes
Organizing categories by a subsumption relation permits the transfer of behavior from super- to sub-
categories. For example, houses might inherit their ability to support a roof from buildings. Standard
ontology languages define subsumption in terms of instantiation: houses are buildings, if every house is
also a building. Haskell does not permit this instantiation of a single individual to multiple types, but offers
a more powerful form of subsumption, using type classes.
To declare that houses inherit the behavior of buildings, one first introduces type classes for both. Class
names have only upper case letters here, to distinguish them visually from type names. The sub-category is
then derived (=>) from the super-category, and the types are declared instances of their classes:
class BUILDING building where
<specify the behavior of buildings here, if any>
class BUILDING house => HOUSE house where
<specify additional behavior of houses here, if any>
instance BUILDING Building where
<specify how the type Building implements the behavior of type class BUILDING>
instance HOUSE House where
<specify how the type House implements the behavior of type class HOUSE>
Note that Haskell’s instance relation is one between a type and a type class. The symbols building
and house are just variables for the types that can be instances (we could also call them x or a).
While this may look like a lot of overhead for the conceptually simple subsumption relation, it is not
only mathematically cleaner and more transparent (every value has exactly one type), but also more
flexible than subsumption in standard ontology languages. Type classes allow for behavior inheritance
from multiple classes (with the same => syntax, denoting a so-called context), without creating dubious
cases of subsumption. For example, a class can model the combination of behavior inherited by houseboats
from housings and boats, without requiring houseboats to be both, houses and boats. Type classes also
allow for partial inheritance, so that penguins can be birds which do not implement the flying behavior.
3.3 Image Schemas as Multi-Parameter Type Classes
The previous section has shown that Haskell type classes generalize over types (which model universals or
concepts), get structured by behavior (in the form of computable functions), and combine behavior from
multiple classes. They also define theory morphisms, i.e., mappings from one theory (type) to another. For
example, an instantiation of a type Boathouse to the HOUSE class defined above would map the basic
behavior of sheltering (not shown) from the prototypical House type to the sheltering of a boat. With this
mapping capacity, type classes exhibit all the ontological properties required to represent image schemas
listed in section 2.
To make the specification method more powerful, we now extend it to capture generalizations over types
belonging to different classes. In a popular extension of Haskell (and possibly in the next Haskell standard),
type classes can have multiple parameters. For example, a class representing the CONTAINMENT image
schema might have two parameters (type variables), one for the containing type (called container here)
and one for the type of the containee (called for):
class CONTAINMENT container for where
isIn :: for -> container -> Bool
The methods of this class use these parameters in their signatures. Thus, the second line of the example
states that an implementation of the isIn query requires a containee (for) type and a container type as
inputs and returns a Boolean.
Multi-parameter type classes represent relations between types. For example, a container and a containee
type are in the relation CONTAINMENT, if an isIn method exists that takes a value of each and returns
a Boolean. Thus, the device that models subsumption can in fact model any other relations between
categories. Image schemas are such a relation, but probably not the only interesting one for ontology.
Relations between types are more abstract than relations between individuals. For example, the
CONTAINMENT schema provides a characteristic containment relation for object types (relating, for
example, boats and boathouses). As a schema, it relates individual objects more weakly than the
topological contains relation, but more generally. The contains relation affirms, for example, that a
particular boathouse contains a particular boat. The CONTAINMENT schema, by contrast, affirms that a
boathouse (i.e., any instance of type boathouse) is a container for boats, no matter whether a particular
boathouse contains a particular boat at any point in time.
3.4 Affordances as Behavior Inherited from Image Schemas
The link between objects and process types that characterizes affordances has already been captured in the
algebraic notion of a type: The processes afforded by an individual are modeled as the methods offered by
the data type representing the universal. For example, the affordance of a house to shelter people is
expressed as a set of methods applicable to objects of type house and person (such as the methods to enter
and leave a house and to ask whether a person is inside or outside).
Since a lot of this behavior generalizes to other universals (in the example, to all “people containers”,
including rooms and cars), it makes sense to model it at the most generic level. This level, I claim here,
corresponds to image schemas. For example, the behavior required to ask whether somebody is in a house
is the isIn method of the CONTAINMENT class above. Houses get this affordance from the
CONTAINMENT image schema by instantiating the House type to the CONTAINMENT type class:
instance CONTAINMENT House Person where
isIn person house =
(container person = house) || isIn (container person) house
The axiom for isIn uses variables for individuals of type Person (person) and House (house) and
states that the person is in the house if the house contains her or, recursively, if the house contains a
container (e.g., a room) containing her. To state this, I use a representation of state as a labeled field
container, here assumed to be defined for person types.
3.5 Testable Models
As a logical theory, an ontology per se does not need a formal model. However, if the theory is expressed
in a programming language like Haskell, it does, for the code to become executable. The benefit of this is
to make the theory testable through its model. Our experience has been that this is enormously beneficial
for ontology engineering, as it reveals missing pieces, inconsistencies, and errors immediately,
incrementally, and without switching development environments (Frank and Kuhn 1999). Ideally, one uses
the best of both worlds, executability from a language like Haskell, and the greater freedom of expression
that comes with non-constructive languages like CASL or even just Description Logics. With tools like
Hets (,
spanning multiple logics, such heterogeneous specifications can now be integrated and tested for
For the case at hand here, the executability requirement was certainly more beneficial than limiting.
Constructing individuals of the categories under study often revealed errors in axioms, and it gave the
satisfaction (well known from programming) of seeing a solution “run” (though this is, of course, never a
proof of correctness or adequacy). To state the effect in simpler terms: this paper would have been
submitted to COSIT two years ago, if Haskell was as patient as paper (or Word) is. It would have had fewer
insights, more confused ideas, and certainly more errors than now.
A model of the kind of logical theory presented here consists of a way to record and change state
information. It has to satisfy the signatures and axioms of the theory. Haskell offers a simple device to keep
track of the state of an object and allow object updates, through its labeled fields (already used above). For
example, a parameterized type Conveyance can be constructed from two values (a conveying entity of
type Instrumentation and a conveyed entity of variable type) as follows:
data Conveyance for =
Conveyance {conveyor :: Instrumentation, conveyed :: for}
Values of the type can then be constructed and updated by supplying values for these labeled fields to the
constructor function Conveyance.
4 Formalization Results
The immediate results of applying the above formalization method are, besides testing and improving the
method itself, a set of seven image schemas specifications and around twenty category specifications
(including affordances). The current state of these specifications can be downloaded from http://musil.uni- (Resources -> Downloads). Rather than listing the specifications in full length, this section
will present results in the form of insights gained in the process, and illustrate these by excerpts from the
specifications. It starts with some observations on image schemas and their interactions in general and then
shows how an account for affordance-based categorizations is derived from the formalized image schemas.
4.1 Image Schemas and their Combinations
Image schemas are treated here as purely relational structures, regardless of whether they are also reified as
objects. What we consider containers, for example, are objects playing the role of containing something
else. This role is afforded by the CONTAINMENT schema, but it does not have to turn the object into a
container type. The same object may also take on other roles, such as supporting or covering and it might
stop to have them at any time. This distinction is a direct result from the formal difference between a type
class (as a relation on types, used to model a role) and a type (used to model rigid properties that an object
cannot stop to have (Guarino and Welty 2002)).
In formalizing particular image schemas, it became clear that some of them combine simpler patterns
which are themselves image-schematic. This generalizes the earlier observation that image schemas have
internal structure toward differentiating primitive from complex image schemas. For example, the usual
forms of the CONTAINMENT and SUPPORT schemas described in the literature involve a path (at least
implicitly), to allow for the contained or supported objects to access and leave containers or supports. It
seemed appropriate to factor out the PATH schema to obtain simpler and more powerful CONTAINMENT
and SUPPORT schemas. These can then also structure situations where something is contained or
supported without ever having (been) moved into or onto something else (e.g., a city in a country, a roof on
a building). The resulting specification, as seen previously, is rather simple:
class CONTAINMENT container for where
isIn :: for -> container -> Bool
class SUPPORT support for where
isOn :: for -> support -> Bool
The SUPPORT schema is used here for situations that are sometimes attributed to a surface schema instead.
The choice of name reflects the fact that it models support (e.g., against gravity) which may or may not
come from a surface. Objects like buildings (supporting a roof), parcels (on which buildings stand), and
vehicles (supporting a transported object) obtain this affordance from the schema.
The question remains open how CONTAINMENT and SUPPORT should be formally distinguished
(beyond the structurally equivalent class definitions above). The presence of a boundary (as suggested in
many characterizations of CONTAINMENT) does not seem to achieve this distinction, as not all containers
have explicit boundaries. A more likely candidate to make the distinction is transitivity: containment is
transitive (the person in the room is also in the house), but support is not, or at least not in the same direct
way (the person on the floor of the room is not supported in the same way by the ground supporting the
Though PATH is also a complex schema, the LINK schema cannot be factored out, as it is an essential
part, i.e., any path necessarily involves a link. PATH is specified here as a combination of the LINK and
SUPPORT schemas. It requires a link between the from and to locations and support for the moving
entity (for) at these locations. All movement furthermore involves a medium (Gibson 1979). The
specification is given by a class context involving the three schemas, and by the signature of the move
behavior afforded:
class (LINK link from to, SUPPORT from for,
SUPPORT to for, MEDIUM medium)
=> PATH for link from to medium where
move :: for -> link -> from -> to -> medium -> for
Surfaces, understood as separators of media (Gibson 1979), can be used to further constrain the motion to a
surface (water body, road) through a second combination with SUPPORT, where a surface supports the
Among the remaining schemas specified until now, the PART-WHOLE schema was modeled using the
relations defined in (Bittner and Smith 2004). The COLLECTION schema adds collections of objects of a
single type. It is useful to capture definite plurals ("the cars"), and it can be combined with the ORDER
schema to create ordered collections. The COVER schema is still underdeveloped and structurally not yet
distinguished from CONTAINMENT and SUPPORT.
Exploiting this algebra of image schemas further, one arrives at specifications for some fundamental
spatial processes. For example, the category of a conveyance (WordNet: "Something that serves as a means
of transportation") is a combination of the PATH and SUPPORT schemas. The specification becomes more
complex, adding constraints on the conveying and conveyed object types and putting the PATH context
into the transport method, but its basic structure is CONVEYANCE = PATH + SUPPORT:
class (INSTRUMENTATION conveyance, SUPPORT conveyance for,
=> CONVEYANCE conveyance for where
transport :: PATH conveyance link from to medium =>
for -> conveyance -> link -> from -> to -> medium -> conveyance
Conveyances afford support and transportation on paths. Moving a conveyance moves the object it
supports, which is captured by the transport method. This affordance structures all derived specifications
for transportation-related categories (vehicles, boats, ferries etc.). Note that conveyances may also contain
transported items (to be expressed by a combination with the CONTAINMENT schema), but in any case
need to support them, due to gravity.
4.2 Deriving Affordances from Image Schemas
The hypothesis posited for this paper was that image schemas capture the necessary abstractions to model
spatio-temporal affordances, or more specifically, that an algebraic theory built on the notion of image
schemas can account for those aspects of meaning that are grounded in affordances. The formalization
method discussion in section 3 has shown that the formal instrumentation is up to this task. As a proof of
concept and illustration for this claim, the above specification of the CONVEYANCE type class will now
be extended to specify vehicle as a “conveyance that transports people or objects” (see its WordNet
definition cited in the introduction).
Rather than listing all required instance declarations, the rest of this section focuses on the ones that are
essential to specify the transportation affordance. First, a data type representing conveyances is declared. It
is the parameterized type already introduced above, where the parameter stands for the type of (physical)
object to be transported on the conveyance:
data Conveyance for =
Conveyance {conveyor :: Instrumentation, conveyed :: for}
On the right-hand side of this declaration, the constructor function (Conveyance) takes two arguments in
the form of labeled fields. The first (conveyor) is of type Instrumentation and contains the state
information of the conveyance (e.g., its location). The second (conveyed) has the type of the transported
object and is represented by the parameter for this type (for). This parameterized specification allows for
having conveyances transporting any kinds of objects. It essentially wraps a conveyor type and adds a
parameter for what is conveyed.
Secondly, conveyances have to be declared instances of the PATH schema in order to obtain the
affordance to move. This is achieved by instantiating the necessary types of the PATH class. For example, a
conveyance for physical objects (generalizing people and objects, as indicated by the WordNet gloss), three
particulars (standing for a link, its source, and its goal) and a medium (in which the motion occurs) are
together instantiating the PATH relation. This produces an axiom stating that the result of moving a
conveyance is to move the conveyor:
instance PATH (Conveyance PhysicalObject) Particular Particular
Particular Medium where
move conveyance link start end med =
conveyance {conveyor = move (conveyor conveyance) link start end med}
The recursion on move is then resolved by instantiating PATH again for the conveyor, i.e., the
Instrumentation type. The resulting axiom specifies that the result of moving the conveyor is that
it will be supported by the end of the link:
instance PATH Instrumentation Particular Particular Particular Medium
where move conveyor link start end med = conveyor {support = end}
Finally, the transportation affordance results from instantiating conveyances for physical objects to the
CONVEYANCE class. The axiom specifies the effect of transport as a move of the conveyance.
instance CONVEYANCE (Conveyance PhysicalObject) PhysicalObject where
transport for con link from to med = move con link from to med
Since each object “knows” its location (in terms of another object supporting it), moving the conveyance
automatically moves the object it supports.
The specifications for houses, boats, houseboats and boathouses take the same form. They combine the
image schemas required for their affordances into type classes, possibly subclass these further, and then
introduce data types inheriting the affordances and specifying the categories. The signatures of these logical
theories for categories are those of the inherited behavior. The axioms are the equations stating how the
types implement the classes.
5 Conclusions
The paper has presented a method to account for spatial categorizations using a second order algebraic
theory built on the notion of image schemas. The method posits image schemas as relations on types which
pass afforded behavior on to the types instantiating them. The type classes representing image schemas can
be combined to produce more complex behavior and pass this on in the same way. The case study material
contains an account for spatial categories from WordNet and has been illustrated here by the category of
The contributions of this work lie in several areas. The primary goal was an integrated formalization of
image schemas and affordances with the power to account for affordance-related aspects of meaning. The
choice of a functional (as opposed to logic) programming language and the use of its higher order
constructs (type classes) preclude an easy transfer of the results into standard ontology languages.
However, the results suggest a different route: Rather than recoding the image schematic relations into yet
another top level ontology, they can serve as a mapping structure to link domain ontologies. If standard
ontologies can be translated into Haskell data types, and these are declared to be instances of Haskell type
classes, one has obtained mappings among ontologies, without compromising the simplicity of the
ontologies or the expressiveness of the mapping environment. The equations generated by the type class
instantiations then serve as axioms bridging multiple conceptualizations.
With this goal in mind, this work prepares an important step in realizing semantic reference systems
(Kuhn 2003), i.e., to establish a transformation mechanism between multiple conceptualizations.
Consequently, future work will primarily study this ontology mapping capability of the approach. The
follow-up hypothesis could be that a grounding of ontologies in image schemas can take the form of image-
schematic mappings rather than that of some new ontological primitives. Lakoff’s invariance hypothesis
(Lakoff 1990) supports this strategy from a cognitive point of view.
Another direction in which to extend this work is toward a theory of (relative) location based on image-
schematic relations. With the two central image schemas of SUPPORT and CONTAINMENT, the essential
locating relations (isOn, isIn) have already been specified (see also (Hood and Galton 2006)). The approach
treats location as a role, allowing for multiple location descriptions for a single configuration, and it is
extensible with other image-schematic relations.
The idea of combining image schematic relations to specify complex behavior warrants further
exploration. Beyond the case of transportation shown here (as a combination of SUPPORT and PATH
behavior), other fundamental spatial processes may be modeled in this way, such as diffusion, as a
combination of CENTER-PERIPHERY and PATH.
The image schema specifications themselves will require some refinement. For example, a function from
time to a path position will model continuous motion, beyond the current discrete start-to-end motion.
Similarly (but not limited to image schemas), it remains to be seen how best to deal in this context with
physical and geometrical constraints occurring in category descriptions (such as sizes, weights, and
Discussions with Andrew Frank, Martin Raubal, Florian Probst, David Mark and others over many years
have shaped and sharpened my ideas on image schemas and affordances and their role in ontology. Partial
funding for this work came from the European SWING project (IST-4-026514). The paper was finished
while I was a resident Theme Leader on Spatial Semantics for Automating Geographic Information
Processes at the British e-Science Institute in Edinburgh.
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Specifications of software interfaces are essential for interoperability. If an interface is not clearly specified, software that exposes or uses the interface necessarily contains assumptions that can make it non-interoperable with other software based on different assumptions. Eliminating, or at least coordinating such assumptions requires a precise and complete specification language. If such a language also allows for prototyping of the specified components, it is a much more powerful tool in the specification process and can even support conformance testing. This chapter describes why and how functional languages serve all these purposes.