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An Algebraic Approach to Modeling Creativity of Metaphor Bipin Indurkhya Department of Computer Science Tokyo University of Agriculture and Technology 2-24-16 Nakacho, Koganei, Tokyo 184-8588, Japan [email protected]

Abstract. In this article we consider the problem of creative metaphors — that is, those metaphors that induce new ontologies and new structures on an object or a situation, thereby creating new perspectives — and how they might be modeled formally. We argue that to address this problem we need to fix the model, and study how different theories organize the model differently. We briefly present some algebraic mechanisms that can be used to formalize this intuition, and discuss some of their implications. Then we provide a few examples to illustrate our approach. Finally, we emphasize that our proposed mechanisms are meant to supplement the existing algebraic approaches to formalizing metaphor, and are not suggested as a replacement.

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Introduction: Creativity in Metaphor

The creative aspect of metaphor that we focus on here concerns the phenomenon of gaining a new perspective on or an insight into an object or a situation. This kind of creativity in problem solving has been studied by Gordon (1961, 1965), Koestler (1964) and Sch¨ on (1963; 1979), among others. For example, Sch¨ on recounts how the idea that a paintbrush might be viewed as a pump led to a new ontology and new structure for the painting process, which in turn led to an improved synthetic-fiber paintbrush. More recent psychological research has also demonstrated this aspect of creativity in understanding metaphorical juxtaposition in poetry (Gineste, Indurkhya & Scart-Lhomme 1997; Nueckles & Jantezko 1997; Tourangeau & Rips 1991). The key point here is that a metaphor involved in this kind of creativity is not based on some existing similarities between its two objects or situations, but, if the metaphor is successful, creates the similarities. For example, people usually do not see any similarity between the ocean and a harp, but Stephen Spender’s beautiful poem Seascape draws on a compelling imagery, where the sunlight playing on the ocean waves is compared to the strumming of harp strings. In an explanatory model of this process that we have articulated elsewhere (Indurkhya 1992, 1997a), it is argued that such metaphors work by changing the representation of the object or situation that is the topic of the metaphor. C. Nehaniv (Ed.): Computation for Metaphors, Analogy, and Agents, LNCS 1562, pp. 292–306, 1999. c Springer-Verlag Berlin Heidelberg 1999

An Algebraic Approach to Modeling Creativity of Metaphor

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Moreover, this process is constrained by the intrinsic nature of the object or the situation, which resists arbitrary changes of representation. In our model, this intrinsic nature of the object is taken to be the either the sensory-motor data set corresponding to the object, if the object is perceptually available; or the imagery and episodic data (retrieved from memory) corresponding to the object, when the object is not perceptually available. Indeed, imagery and episodic memory have been known to play a key role in understanding certain metaphors (Marschark, Katz & Paivio 1983; Paivio 1979) — a claim that has been strengthened by recent neurolinguistic research (Bottini 1994; Burgess and Chiarello 1996) — to the extent that some researchers argue that metaphors are essentially grounded in perception (Dent-Read and Szokolszky 1993). In this article we outline an approach to formalizing these ideas using algebraic notions. The article is organized as follows. In the next section we motivate the need to introduce certain non-standard algebraic mechanisms to formalize our intuitions, and describe these mechanisms briefly. In Section 3, we discuss how we apply these mechanisms to approach the creativity of metaphor, and in Section 4 we present some examples to illustrate our approach. Finally, in Section 5, we remark on how our ideas relate to the existing research, and in Section 6, we conclude by summarizing the main points of the paper. We assume familiarity with some elementary algebraic notions. The discussion throughout is kept focused on the motivation for certain formal mechanisms, and so it has an informal tone, and definitions, theorems, etc. have been left out.

2

Outline of an Algebraic Approach

Classical model theory studies the properties of and relations between different models of a given theory. A similar approach is used in most other formalizations of semiotics (Goguen 1997). This situation is depicted in Figure 1 below. However, to understand creativity of metaphor, we need to reverse our standpoint and consider different theories of the same model. For example, in the painting-as-pumping metaphor mentioned above, one would like to see how the pumping theory restructures the painting model. In the Seascape example, we would like to be able to describe how the harp and its related concepts (which could be considered a theory) restructure the experiential datum (the model) of the ocean. This situation is depicted in Figure 2. To avoid the confusion between two senses of ‘model’: one referring to modeling creativity in metaphor, and the other to the model of a theory, we will henceforth use the term environment to refer to the model of a theory. Thus, Figure 1 should be read as ‘Focus on multiple environments of a theory’ and Figure 2 as ‘Focus on multiple theories of an environment’. We believe that in order to model creativity of metaphor we must focus on Figure 2, and study how different theories can conceptualize the same environment differently.

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theory (sign system)

model 1

model 2

theory 1

model N

Fig. 1: Focus on multiple models of a theory.

theory 2

theory N

model (environment)

Fig. 2: Focus on multiple theories of a model.

Now in formalizing the environment (model), we need to keep in mind the following two points: (1) it should have an autonomous structure (that resists arbitrary restructuring); and (2) it should allow multiple restructurings. The second point has two further implications: (a) the structure should be rich enough so that any object has many potential structures; and (b) it should not be structured too strongly a priori — meaning that we should not predetermine the set of primitives, the sorts, the levels of abstractions, and so on. Intuitively, the motivation behind these requirements is as follows. Different languages and cultures — different semiotic systems — have different ways of describing (structuring) any given experience or situation. The sorts, categories, even which objects are considered as primitives and which as composites can vary considerably from one semiotic system to another. So if all these choices have already been predetermined in an environment, then there will be little possibility of restructuring it in novel ways. With all these factors in mind, the approach we propose is to formalize the environment as an algebra: that is, a set of objects and a set of operators over it. Now the term ‘structure’ here refers to how an object can be decomposed into its parts; or, to put it in other words, how an object can be composed from its components by applying certain operators. This sense of ‘structure’ is quite similar to the way it is used in most AI knowledge representation languages, KLOne, for example (see also Brachman & Levesque 1985). Notice, however, that in this sense a structure becomes a term of the algebra, and the term algebra contains all possible structures in the environment. A few other comments seem to be in order here. First of all, for the reasons mentioned above, we choose not to put any sorts in the algebra. Though, obviously, all the operators need not be defined on all the objects, still we can take care of that by having one or more ‘undefined’ or ‘error’ objects in the algebra. Secondly, the objects are not assigned any predetermined level of complexity. In fact, we expect circularity: meaning that situations where, for example, an object A generates B, B generates C, and C generates A are allowed. In these cases, there is no fixed set of primitives. If A is taken as a primitive, then B takes


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level-2 complexity, and C level-3. But if C is taken as a primitive, then A takes level-2 complexity and B level-3. This characteristic allows us to model cognitive interactions as “closed systems that are, at the same time, open to exchanges with the environment.” (Piaget 1967, pp. 154–58.) Another point worth emphasizing is that we deliberately choose to not put any predicates in the algebra for the environment because we feel that operators (corresponding to actions of an agent) are more primitive than relations (Piaget 1953), and all relations can be broken down to some sequence of operators. However, in some applications, as we will see in the example of legal reasoning in Section 4, it may be more convenient to allow predicates and relations in the environment algebra. Having formalized the environment like this, a theory can be formalized similarly as an algebra. Here, however, we allow sorts, complexity-levels, a predetermined set of primitives, ordering, and other structures or restrictions as may seem appropriate: perhaps similar to a sign system of Goguen (1997). Now a cognitive (or semantic) relation is formed by connecting the objects and operators of the theory algebra to the objects and operators of the environment algebra. As the environment algebra does not have any sorts, complexitylevels, etc., only the arity of the operators needs to be preserved. Notice first of all that we allow the two algebras to have different signatures. Secondly, we allow a cognitive relation to be a many-to-many relation, but it can be turned into a function by grouping the environment algebra appropriately. (See Indurkhya 1992, Chap. 6, for details.) Finally, though structure-preserving property, which we refer to as coherency, is the ideal for cognitive relations, a more useful notion for cognitive modeling is that of local coherency, that is, coherency within some restricted subalgebras of the theory and the environment. A cognitive relation induces a structure in the environment that reflects the structure of the theory: we can say that the environment is structured by the theory. A different theory would structure the environment differently. Though both these structures may look very different, they are both, nonetheless, constrained by the autonomous structure of the environment. Any incoherency that is detected by the agent must be countered by either modifying the cognitive relation (thereby changing the ontology of the environment as seen from within the theory) or by changing the structure of the theory. We emphasize again that the autonomous structure of the environment cannot be changed by the agent, though it can be organized in different ways by different theories.

3

Formalizing Creativity of Metaphors

Many theories and their cognitive relations are inherited, biologically or culturally, or learned as we grow up. We can dub them as conventional cognitive relations. These cognitive relations structure our environment in various ways, and it is this structured environment that we live in and interact with. However, in certain situations, it becomes necessary to form new cognitive relations. A prime example of such situations is metaphor. In metaphor, a new cognitive

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relation is created between a theory and an environment. Usually, the vehicle theory interacts with the topic environment, but often the process is mediated by the topic theory. Not all metaphors result in a new perspective and a new representation. Actually, many metaphors can be understood by constructing some mapping between the topic and the vehicle theories, as in semiotic morphisms of Goguen (1997). However, for some metaphors, no such mappings could be found (there are no existing similarities.) In such cases, it becomes necessary to conceptualize the topic environment anew — as if it were encountered for the first time — using the concepts from the vehicle theory. In this interaction — and we must emphasize that the result of the interaction is determined in part by the structure of the topic environment, and in part by the structure of the vehicle theory — a new structure of the topic environment emerges (if the process is successful.) For example, in projecting the pumping theory onto painting process, a new ontology for paintbrush emerged, in which the space between the fibres played a key role. Thus, the process underlying creative metaphor becomes that of instantiating a new cognitive relation between a theory and an environment, such that it preserves the structure of each. This new cognitive relation restructures the environment, and as a result, new attributes of the environment may emerge and new information about the environment may become apparent. For example, in restructuring the painting environment by pumping theory, the part of the paintbrush where it bends away from the surface being painted becomes very crucial, and the part of the paintbrush which is already in contact with the surface fades into irrelevance. Or in understanding the ocean-as-a-harp metaphor, new perceptual similarities between the ocean and the harp emerge — similarities that were lost when the two were viewed from the conceptual level via their respective conventional theories — and one gets a glimpse of an alternative semiotic system in which the two would be semantically very close, and even be assigned the same category. Here, an interesting result can be obtained by generalizing the first isomorphism theorem (Cohn 1981, p. 60; Mal’cev 1973, pp. 47–8) for certain cognitive relations by taking into account the change of signature (see Indurkhya 1992, Chap. 6). The first isomorphism theorem essentially says that any homomorphism from a source algebra to a target algebra can be factored into a unique isomorphism. The trick is to first take the kernel of the source algebra, which means grouping the elements of the source algebra as follows: if any two elements map to the same element of the target algebra then they are put in the same group. Secondly, we limit the target algebra to its subalgebra that is the range of the homomorphism. That is, if a certain element of the target algebra is such that no element of the source algebra maps into it, then that element is not included in the subalgebra. After these two steps one finds that there exists an isomorphism between the kernel of the source algebra and the ‘range’ subalgebra of the target. Moreover, this isomorphism is unique, so that different homomorphisms factor into different isomorphisms. In other words, every iso-


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morphism factored by this process carries a unique stamp of the homomorphism from which it was derived. The mechanism corresponding to the first isomorphism theorem corresponds to a frequently used cognitive process, and failing to realize it has resulted in some needless controversy over whether metaphors ought to be formalized as a relation, a homomorphism or an isomorphism. For example, Max Black (1962, 1979) proposed that underlying every metaphor is an isomorphism between its topic and its vehicle, and many scholars have chided him for positing too strong a requirement. To realize the cognitive correlate of the first isomorphism theorem, consider how we use the map of a city. Obviously, the map does not represent everything in the city. (There is a charming story by Borges precisely on this theme.) Yet, in using the map, one gives the city an ontology or a representation where parts of the city are grouped together and are seen as primitives: two lanes of a street, the sidewalks, and the shops and building along the street are all seen as a unit and correspond to a line on the map. In using the map, one acts as if it were isomorphic to the city, even though the street is not painted orange, but the line on the map is, and the vehicles and the people on the street are nowhere to be found on the map. Thus, the operations of taking a subalgebra and forming groupings (as in taking the kernel) play an important role in modeling cognitive interaction. If we assume that a cognitive agent can be aware of its environment only as far as it is represented in a theory, then we can also provide an explanatory model of how new features can be created by metaphor (Indurkhya 1998). The approach outlined here has some other applications as well, and we would like to mention one of them briefly. Consider the prototype effect, which is demonstrated by Eleanor Rosch in her prolific work on human categorization (Rosch 1977). According to it, categories have a radial structure, with certain members occupying a more central position than others. (See also Lakoff 1987). To model this phenomenon, we have to realize that the environment does not have a preassigned set of primitives. Which objects are considered as primitives depends on the structure given to it by the cognitive relation. As the objects in an algebra are structured by its operators, if we deem a certain subset of objects of the algebra to be primitive (prototype), and assign a measure function that assigns a ‘distance’ to every other object depending on the length of the shortest description of that object using only the primitives, then a kind of radial structure (Lakoff 1987, Chap. 6) emerges. For example, in the Dyirbal classification system discussed by Lakoff (1987, p. 100), the category Balan includes women, fire and dangerous things. If women are considered as primitives, then dangerous things become distant members of the category, because the derivation from women to dangerous things is a long one: going from women to sun, then to fire, finally arriving at dangerous things. On the other hand, if fire is considered a primitive, then dangerous things become more central members of the category but women become more distant members.

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Some Examples

We now present a few examples to illustrate our approach. The first example is from the Copycat domain pioneered by Hofstadter (1984), which concerns proportional analogy problems between letter strings, as in: abc : abd :: pqr : ??

(1)

This domain may seem rather simple at first but in fact, as Hofstadter has shown, a number of rich and complex analogies can be drawn in it. In particular, the Copycat domain is quite suitable for demonstrating the context effect, according to which an object needs to be represented differently depending on the context, thereby revealing the limitations of fixed-representation approaches. For instance, in the analogy problems (2) and (3) below, the first term of the analogy (abba) is the same, but it needs to be given a different representation to solve each problem: for analogy (2), abba needs to be represented as a symmetrical object, with the string ab, reflected and appended to itself; and for analogy (3) it needs to be seen as an iterative structure, namely two copies of b, flanked by the same object, namely a, on either side. abba : abab :: pqrrqp : ??

(2)

abba : abbbbba :: pqrrpq : ??

(3)

In order to model this context effect in our approach, we take Leeuwenberg’s Structural Information Theory [SIT henceforth] (Leeuwenberg 1971) as the starting point. In SIT, a certain way of expressing different representations (also known as ‘gestalts’) of a pattern in terms of iteration, symmetry and alternation operators is defined. Then a measure called ‘information load’ is defined on every representation. According to SIT, for any given pattern, the representation with the minimum information load is the preferred gestalt. (See also Van der Helm and Leeuwenberg 1991.) In integrating SIT within our algebraic approach, we extend SIT in two significant ways. One is to allow domain-dependent operators to participate in the gestalt representations. For example, in the Copycat domain, the operators ‘successor’ and ‘predecessor’ play a key role, so that an object like ‘abcd’ can be seen to have an iterative structure where the operator ‘successor’ is applied at each iteration. Secondly, whereas SIT only accounts for the preferred gestalts of patterns in isolation, we incorporate context effect by taking into consideration the complexity of representation algebras also, which can be simply measured by counting the number of elements and the number of operators in it. For example, applying the information load criterion, the preferred gestalt for ‘abba’ is the one that sees a symmetry structure in it. However, when this object is considered together with ‘abbbbba’, as in the analogy (3) above, we must also take into account the complexity of the representation algebra that generates the gestalts for both. Though ‘abbbbba’ can also be written as a symmetry structure — albeit with


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odd symmetry, for it has a pivot point in the middle ‘b’ — the representation algebras that generate the minimum information load gestalts for each of these terms individually have mostly different elements, and so when we combine them to get the representation algebra that can generate both the terms, the complexity of the resulting algebra is almost cumulative. However, if we represent ‘abba’ and ‘abbbbba’ as iterative structures, then their individual representation algebra have a high degree of overlap, so that the complexity of the combined representation algebra remains almost the same. Fuller details of our approach can be found in Dastani, Indurkhya and Scha (1997), and Dastani (1998).

A

:

::

:

:

::

:

:

B

::

C

:

D

Fig. 3: Two examples of proportional analogy relations A is to B as C is to D involving geomteric figures. Notice that the terms A and B are the same in each example, yet different figures for the C term forces a different way of decomposing figures A and B.

This approach can be further illustrated by considering the creation of similarity in proportional analogies involving geometric figures. In the two proportional analogy relations shown in Fig. 3, figures A and B are the same, yet they must be seen differently, or described differently, for understanding each example. People can comprehend them easily, but analogy systems based on mappings between fixed representation cannot account for them. The reason is that in fixed-representation systems, one must first choose how each figure is represented or described. If figure A is described as a triangle put on top of another inverted triangle, then the upper analogy relation in Fig. 3 can be comprehended but not the lower one. If, on the other hand, figure A is described as a hexagon with an outside facing equilateral triangle on each of its six sides, then the lower analogy relation in Fig. 3 can be understood, but not the upper one. Notice that if we describe figures A and B in terms of line segments/arcs (or pixels), then neither of the analogies can be comprehended, for the ontology

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of various closed figures, like ‘triangle’, and their structural configurations are essential to understanding the analogies. What seems necessary here is to provide a sufficiently low-level description of the figures (say, in terms of line segments and arcs), and a rich repertoire of operators and gestalts that allow one to build different higher-level structured representations from these low-level descriptions. For the examples in Fig. 3, we need the gestalts of ‘triangle’, ‘hexagon’, ‘ellipse’, etc.; and operators like ‘invert’ (turn upside down), ‘juxtapose’ ‘rotate-clockwise’, and so on. A structured representation using these gestalts and operators essentially shows how the figure can be constructed from the line segments and arcs.1 Needless to say, there are many ways to construct each figure, so there are many corresponding structured representations. Thus the heart of the problem, in this approach, lies in searching for a structured representation that is most appropriate in a given context. As representations correspond to algebraic terms, it means we must find suitable representation algebras for each of the figures — where ‘suitability’ must take into account complexity of representation algebras, complexity of representations, existence of an isomorphic mapping between representation algebras, and the complexity of this mapping. We must emphasize two somewhat unusual aspects of our approach here. One is that we require a mapping between representation algebras, and not between representations themselves, to capture the analogical relation. The reason for this is that a mapping between representation algebras is more robust with respect to trivial changes of representation — such as ones arising from symmetry or transitivity of operators. The second distinctive feature is that we require an isomorphism rather than a homomorphism. However, as explained above in Section 3, this by no means constitutes a limitation of our approach; on the contrary, it focuses attention on the isomorphism underlying each homomorphism. (See Indurkhya 1991 for a further elaboration of these issues and a formally worked out example.) The next example we would like to present, taken from Indurkhya (1997b), concerns modeling a certain kind of creative arguments in legal reasoning. Very briefly, the example is about a college professor, Weissman, who deducted the expenses of maintaining an office at home from his taxable income. A precedent that was helpful to Weissman’s arguments was the case of a concert violinist, Drucker, who was allowed to claim home-office deduction for keeping a studio at home where he practiced. However, the Revenue Service tried to distinguish Weissman from Drucker on the grounds that Drucker’s employer provided no space for practice, which is obviously required of a musician, whereas Weissman’s employer provided an office (a shared one). The judges, however, ruled that Weissman’s employer provided no suitable space for carrying out his required 1

It should be noted here that the algebra corresponding to this domain would be like the algebraic specification of any drawing or graphics program such as Superpaint. In any such graphics program, the user can create various objects on the screen, group them in certain ways to create different gestalts, and apply a variety of operations on them.


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duties (the office, being a shared one, was not safe for keeping books and other research material), just as Drucker’s employer provided no suitable space for Drucker to practice. The key issue in modeling this argument is how to specialize category ‘no space provided by the employer’ to ‘no suitable space provided by the employer’, because the former distinguishes Weissman from Drucker, but the latter category allows Drucker to be applied to Weissman. We have argued that the new category can be obtained from other precedents. In this example, there was another precedent, Cousino, a high-school teacher who was denied home-office tax deduction, because the judges argued that his employer provided him a suitable space for each task for which he was responsible. A very interesting aspect of this example is that though Cousino and Drucker, when they are individually applied to Weissman, lead to a decision against Weissman; but when Cousino is used to reinterpret Drucker, and then reinterpreted Drucker is applied to Weissman, a decision in favor of Weissman can be obtained. In modeling this argument in our approach, the environment level is associated with the facts of a case, and the model or theory level is associated with the rationale for the decision of the case (Hunter and Indurkhya 1998). For example, facts of the Cousino case would include: ‘employer-of (Cousino) = XYZ’, ‘high-school (XYZ)’, ‘responsible (Cousino, teach)’, ‘responsible (Cousino, grade-papers)’, ‘provided (Cousino, XYZ, classroom)’ ‘provided (Cousino, XYZ, staff-room)’, ‘suitable-for (classroom, teaching), ‘suitable-for (staff-room, gradepapers) ’, etc. Notice that because the facts are themselves composed of linguistic and abstract categories, we need to allow predicates and relations in the environment algebra. The rationale of the case, in this example, would consist of a complex term (we mean algebraic term here) ‘employer provided suitable space for the tasks for which the employee is responsible’. As this is a precedent, that has already been decided, the terms of the rationale level would already be connected to the facts level (meaning that a cognitive relation exists). This already shows the grouping phenomenon, and how the facts level seems isomorphic to the rationale level. The object ‘tasks’ at the rationale level is connected to different objects at the facts level, including ‘teach’, ‘grade-papers’, ‘prepare-lessons’, ‘talk-to-parents’, etc. So all these activities are grouped together and are seen as a unit from the rationale level. Also, many facts at the facts level are not considered relevant, and so are not connected to anything at the rationale level. Nonetheless, it is necessary to keep these facts, for they may become necessary in reinterpreting the Cousino case, which is precisely what happens when Cousino is applied to reinterpret Drucker. In applying the rationale of Cousino — which contains the gestalt ‘suitable space’ — to the facts of Drucker, a new rationale and a new cognitive relation between the rationale and the facts levels of the Drucker case emerges. Using this new rationale, the facts of the Weissman case can also be organized in such a way that a decision favorable to Weissman can be obtained, and moreover, Drucker can be cited as a precedent to support this argument.

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Our final example concerns linguistic metaphor, and is taken from a certain translation of the Bible. As Stephen is persecuted for spreading the teachings of Jesus, he rebukes his persecutors: “You stiff-necked people, uncircumcised in heart and ears, you always resist the Holy Spirit. As your fathers did, so do you.” (Acts 7:51. The Oxford Annotated Bible with the Apocrypha. Revised Standard Version. Oxford University Press, 1965.) The phrase we would like to focus on is ‘uncircumcised in heart and ears’. Now several gestalt descriptions (or algebraic terms) can be associated with ‘circumcised’: for example ‘surgically removing prepuce’, ‘purify spiritually’, etc. Note that these descriptions themselves contain gestalts like ‘prepuce’, ‘purify’, which can be further decomposed into other gestalts. However, at some point, we have to try to interpret the gestalt descriptions by finding similar operations in the context of ears and heart. For example, ‘surgically remove’ is an operation applied to ‘prepuce’, so we have to find a similar operation that can be applied to some part of the ear. This process may require creating imagery for ear (and possibly for circumcision as well) using perceptual knowledge about it. Perhaps the gestalt that is easiest to interpret is ‘purify’ or ‘cleanse’, which means ‘uncircumcised’ would correspond to ‘unclean’ (negation operation is applied). But ‘unclean’ for ears could suggest ears plugged up by earwax, for example, so that the person cannot hear the message. Finding the right gestalt of ‘uncircumcised’ to interpret in the context of ‘heart’ is more complex, because ‘heart’ itself is used metaphorically, not for the physical organ that pumps blood, but for feelings and understanding. Here one can perhaps construct an image where something that is unclean cannot receive new ideas or impressions (e.g. adding a new tint to the dirty water), and the person with the unclean heart does not see what is the truth according to Stephen. There may also be the association that as circumcision requires a surgical procedure, something drastic needs to be done to purify the heart. We should add that all this analysis is done from a viewpoint that is outside of the Bible, for when viewed within the Bible, circumcision is a dead or a conventional metaphor (e.g. ‘Circumcise yourselves to the Lord’. Jeremiah 4:4.) Also, in some other translations a more literal approach is taken: “ ‘How stubborn you are!’ Stephen went on to say, ‘How heathen your hearts, how deaf you are to God’s message! You are just like your ancestors: you too have always resisted the Holy Spirit!’ ” (Acts 7:51. The Good News Bible. The Bible in Today’s English Version translated and published by the United Bible Societies, 1976.)

5

Related Research

In the last twenty years or so there has been much interest in metaphor, and many researchers from different disciplines have approached the problem from


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various angles. Our approach outlined here is based on the insights of Max Black (1962; 1979) and Nelson Goodman (1978), among others. However, because of not being spelled out precisely, these ideas have often been misunderstood. We already mentioned above that Black has been unfairly criticized for claiming that there is an isomorphism underlying every metaphor. Then Black has also been inconsistent on the symmetry of metaphor: at times suggesting that metaphors may be symmetrical, while in most places his account is clearly asymmetrical. This again has caused some needless misunderstanding (see, for example, Lakoff & Turner 1989, pp. 131–133). Our approach towards formalizing their insights and extending it further, we hope, dispels many of these misunderstandings. The research on metaphor and its role in organizing our conceptual system has received a huge impetus from the work of George Lakoff and his colleagues (Lakoff & Johnson 1980; Lakoff 1987). While the empirical data they have amassed to demonstrate how metaphors pervade our everyday life and discourse are indeed impressive, their attempts to explain how a metaphor can reorganize the topic and create new features in it are fraught with contradictions. In some places they claim that certain topic domains derive their structure primarily through metaphors, and they do not have a pre-metaphorical structure. At other places they imply that the topic constrains the possible metaphorical mappings and creation of feature slots. (See also Indurkhya 1992, pp. 78–84, pp. 124–127.) We believe that our formal approach clearly resolves this apparent paradox of how metaphor can restructure the topic, and yet it is not the case that anything goes.2 More recently, Gilles Fauconnier and Mark Turner have introduced a theory of conceptual blending (see, for example, Turner & Fauconnier 1995), which introduces a multiple space model. However, their theory works primarily with concepts, showing how concepts from many spaces blend together to produce metaphorical meanings. While we acknowledge that the multiple-space model does indeed come close to the way real-world metaphors work, we also feel that it is crucial to involve the object or the situation (what we have been calling the environment) in the interaction. Without incorporating this orthogonal component, we believe, the creativity of metaphor cannot be accounted for satisfactorily. Thus, in our view the approach presented here supplements the conceptual blending theory, and in the future we expect to broaden it by considering how multiple environments and multiple theories interact together to produce new meanings. 2

On the formal side, Goguen (1997) has embarked on an ambitious project to develop a formal framework for systems of signs and their representations. However, we believe that the mechanisms proposed here would have to be incorporated in the semiotic morphisms of Goguen in order to be able to account for creativity in metaphor. Though we must add that this kind of creative restructuring is neither always required, nor always desirable. Therefore, there may well be many situations where semiotic morphisms without allowing restructuring would work just fine. But a more comprehensive framework would have to allow the possibility of restructuring.

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Finally, a very different approach to modeling creativity of analogy and metaphor is taken by Doug Hofstadter and his colleagues (Hofstadter 1995). By cleverly designing a number of seemingly simple microdomains that capture the creative aspects of analogy and metaphor in their full complexity, they have focused right on the crux of the problem, and have built computational systems to model creativity of metaphor. Though they have deliberately eschewed any formalization of their ideas, their computational systems are a kind of formal system. Nonetheless, some of their underlying principles are not clear and it is difficult to glean them from their description of the systems. For example, a key concept used in many of Hofstadter’s systems is that of ‘temperature’. The lower the temperature, the better the analogy is supposed to be. However, it is not clear at all how the temperature is computed: its underlying principles are not made explicit. A formal approach such as the one outlined here allows such hidden principles to be articulated explicitly. For example, in our model of proportional analogy described in Section 4, we adapt Leeuwenberg’s concept of information load (Leeuwenberg 1971) to articulate the goodness of analogy. Thus, we feel that our formal approach fills an important niche left open by Hofstadter and his colleagues’ research.

6

Conclusions

In this paper we have focused on the problem of how metaphor can restructure an object or a situation, and create new perspectives on it. With this goal in mind, we outlined some algebraic mechanisms that can be used to model creativity and restructuring of metaphor. Needless to say, the approach presented here is merely a step towards a fuller understanding of the creativity of metaphor. First of all, the model, as it is, needs to be elaborated considerably, and computational mechanisms need to be developed to implement its different mechanisms. For example, elsewhere (Indurkhya 1997b) we have suggested a blackboard architecture for modeling interaction between a cognitive model and an environment in the domain of legal reasoning. Secondly, the approach needs to be expanded to incorporate language, communication between agents, and so on. Obviously, all these issues will keep us busy for years to come.

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