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Bipin Indurkhya - Google Scholar Citations
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Lakoff , George and Mark Johnson Metaphors we live by. Lakoff , George and Mark Turner More than cool reason: a field guide to poetic metaphor. Levin , Samuel R. Markman , Arthur B. American Journal of Psychology , 4 , — Burlington: The University of Vermont. Mio , Jeffery Scott and Arthur C. Metaphor and Symbolic Activity 6 , 2 , 87— These key components of the model will be illustrated through formalizing the two following proportional analogy problems borrowed from [ 8 ]:.
A K-algebra is a nonempty set C where it is defined a family of finitary operations indexed by K i. Example 0. Let C be the set of all non-null strings on the letters a, b , …. Definition 0. Let us consider the K-algebra from example 0. Notice that f t represents the append operation. In contrast, the operation f s takes two strings and appends to the first one, the result of applying symmetry to the second one. In Example 0.
To introduce our formalism, let us first consider a similar, widely known algebraic notion. Let A and B be two K -algebras. This algebraic definition formalizes the preservation of the structure of the K -algebra A into the K -algebra B. As mentioned in the introduction, notions such as quasi-homomorphisms [ 4 ] and local-homomorphisms [ 8 ] have been proposed as models for analogy. To preserve the potential usefulness of these constructions in modeling of cognitive phenomena, we propose a model based on homomorphisms.
But we need a strategy to achieve certain flexibility for formalizing at least some of the analogies with psychological interest. A first version of our model is presented now. Let us model the analogy problem abba : abab :: pqrrqp :? The example above formalizes a relation between the two key letter-strings in the source domain. This relation allows determining the missing item in the target domain via analogical transfer: abba and abab are built from the same element ab by applying the operations f t and f s imposed on the source domain see S2 Note for more details about this relation and its transfer.
These observations are consistent with empirical data suggesting that analogy is based on the mapping of relations [ 6 , 53 ] and is able to emphasize certain features of domains while downplaying certain others [ 54 ]. Clearly, our definition of domains is reminiscent of similar proposals in the literature see [ 5 , 8 , 40 ] where a domain is a mathematical set together with a collection of operations defined on it.
To see this, observe that under the perspective of the K -algebra C Example 0.
An Interactionist Approach
At the same values, the operation determined by t 2 returns the string abbbbba. This last string is the answer given by this particular model of the problem. We have been assuming that the source and the target of an analogy share the same context the same K -algebra. These assumptions are systematically violated by most analogies because, in general, the source and the target of an analogy differ radically.
To account for this observation, we need a way to relate terms of the two domains—because each one may have its own language. This is our core definition, and we refer to it as the mathematical model for analogy MMA. Notice that Definition 0. Each domain has two dimensions. And the semantic dimension is conformed by sets A and B that abstract possibly non-symbolic elements such as conceptual objects or spatial coordinates. The copycat model reports that the most preferred answer is the letter string ijll followed by ijkl and then by hjkk. The next example uses the MMA to provide us with insight about the algebraic nature of the solution hjkk.
Metaphor and Cognition
Let us endow the K-algebra C Example 0. They are defined in a way that agrees with the alphabet ordering i. The example above illustrates how the complete model is more flexible than the restricted model. However, the constraints imposed by the MMA on the modeling of analogies are still strong.