Abstract
The purpose of this paper is to present two kinds of analogical representational change, both occurring early in the analogy-making process, and then, using these two kinds of change, to present a model unifying one sort of analogy-making and categorization. The proposed unification rests on three key claims: (1) a certain type of rapid representational abstraction is crucial to making the relevant analogies (this is the first kind of representational change; a computer model is presented that demonstrates this kind of abstraction), (2) rapid abstractions are induced by retrieval across large psychological distances, and (3) both categorizations and analogies supply understandings of perceptual input via construing, which is a proposed type of categorization (this is the second kind of representational change). It is construing that finalizes the unification.
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Notes
Others also have the goal of unifying analogy and categorization; see, e.g., Atkins (2004) and Kuehne et al. (2000). Several researchers are interested in unifying analogy with other cognitive processes. Three examples are unifying analogy with reasoning [e.g., Hummel and Holyoak 2003 (in this paper, inference is added to the capacities of their analogy model, LISA 1997); Hummel and Choplin 2000; Kokinov and Petrov 2001], unifying analogy with perception (e.g., Mitchell 1993), and unifying analogy with schema induction and learning (e.g., Hummel and Holyoak 1997, 2003). In several of these projects, analogy is not so much unified with some other cognitive process, rather that cognitive process is shown to require or depend on analogy (usually analogical mapping) in order to work properly. Good examples of such an approach can be found in Doumas et al. (2008) and Hummel and Holyoak (2003).
For example, in the case of MAC/FAC (“many are called, but few are chosen”), rapid abstraction could be added between the MAC and FAC stages; i.e., between the computationally cheap matcher/retriever (MAC) and the more expensive and sophisticated final filterer and selector (FAC—which is basically SME, the Structure Mapping Engine, see Falkenheiner et al. 1989). Alternatively and more efficiently (especially given Liberman’s and Trope’s experimental data, see “Abstraction and psychological distance”, below), with appropriate modifications, rapid abstraction could just be incorporated into the MAC stage.
The actual experiments were run by Hans Geiger and Ernest Marsden in 1909, and they discovered the scattering alpha particles.
There are other important models of analogy which hypothesize various kinds of representational change, some of which are interpretable as structural changes, others are interpretable as abstractions similar to the kind considered here. AMBR2 is one such model [although Kokinov and Petrov seem to downplay within their theory the importance of structure in analogy; see Kokinov and Petrov (2001, especially, tables 3.1 and 3.4)]. Also, AMBR2 seems to assume a pre-existing semantic similarity. This might be theoretically fine, depending on the details of "semantic similarity." In any case, AMBR2 can map together quite non-similar predicates, but this is not due so much to representational changes in the predicates, but rather to special interpretations of the "semantics" of the relevant predicates. Other representational change analogy theories include COPYCAT (Mitchell 1993) and TABLETOP (French 1995). In these two, the representational changes occur during the making of the analogy and are roughly similar to the kind proposed here—rapid abstraction (COPYCAT has been criticized as not being an adequate model of analogy for other reasons; see Forbus et al. 1998. Their criticisms also apply to TABLETOP, for COPYCAT and TABLETOP are deeply similar, even analogous). Indurkhya has an interesting analogy model using representational change (see his 1992, 1997, 1998). Finally, some robust and highly suggestive computer models of analogy and representational change are based on psychological data collected in experiments. The performance of these models has also been favorably compared with human performance. In many ways, these programs and their ties to psychological experiments represent one of cognitive science's real success stories. See ACME and its associated model of memory access, ARCS (Holyoak and Thagard 1989; Thagard et al. 1990), IAM (Keane and Brayshaw 1988; Keane et al. 1994); Phineas (Falkenhainer 1990a, b), MACFAC (Forbus et al. 1995), LISA (Hummel and Holyoak 1997, 2003) and DORA (Doumas et al. 2008).
For example, compare Structure-Mapping Theory (e.g., Gentner 1983, 1989) to the developmental, connectionist relational priming model of Leech et al. (2008). The former well-known theory and accompanying computer model use classical propositional-style representations. Analogous representations are said to be in structural alignment. This alignment is the basis for the analogical mapping, and it assumes that the aligned elements already exist. Note that the three key constraints in SMT governing structural alignment—parallel connectivity, one-to-one mapping, and systematicity—are based on the assumption of pre-existing identical relational structures: the three constraints govern the best match between such representations. Parallel connectivity ensures that when two elements are placed in correspondence, the arguments of those elements are also placed in correspondence. The one-to-one mapping principle states that each element in a base domain can be mapped to at most one element in the corresponding target domain. The systematicity constraint requires that, all else being equal, correspondences between systems of elements in the domains are preferred to matches between isolated elements. The mapping process is local-to-global. At the beginning of the comparison, individual entities, attributes, and relations are matched, and then the constraints are utilized to determine more global systems of relations. In contrast, Leech et al.'s model of analogy uses a recurrent connectionist net that relies on relational priming and pattern completion. It's main focus is on developmental and learning issues. Relations are represented as transformations between states and are stored in the weights and hidden layer. In spite of these differences between the two models, Leech et al.'s model explicitly uses pre-existing, identical relations to fund the analogies their network makes. Their analogies are just as discovered (and not created) as those in SMT (see Dietrich 2008).
Turney (2008), provides an enlightening discussion of the problems of pre-existing and handcoded representations. His Latent Relational Mapping Engine is his proposed solution.
The fact that “orbits” and “deflects” are not identical would not be a problem for models such as LISA, but the fact that they are semantically so dissimilar would be.
COPYCAT (Mitchell 1993) does something roughly like this.
All trees are binary and are represented as lists thusly: [root (left-subtree) (right-subtree)]. Also, the nomenclature used here is simplified from that used in Dietrich et al. (2003).
In STRANG, mapping is constructing a link between the mapped items. This is implemented usually as coupling in a list. It is important to emphasize here that STRANG’s packing trees can violate the rules of its grammar. Of course, STRANG can do this because of the algebraic operation it implements, namely, packing: T(B) = A*. Packing implements implicit rules that can override the rules of the grammar. These implicit rules are derived from each operator string, T, as it is applied to its base string, B. When the grammar is overridden, STRANG usually marks this explicitly by a specific new node type in its packing tree. There is only one such new node type, but it can be employed in a wide variety of cases. This new node type represents that STRANG has a new parsing category for which it does not have a name and which is not integrated into the grammar.
A detailed explanation for why psychological distance covaries with and induces abstraction is missing from Liberman’s and Trope’s theory (but see their 2008, p. 1205, and also Liberman and Förster 2009, p. 1337, for some interesting speculation). Adding such an explanation to their theory would be a major advance. Also missing is an actual mechanism for producing the abstraction. Adding this would also be a major advance.
I’m leaving the notion of semantic distance intuitively defined because a full definition would require a theory of mental semantics or mental content (including intentionality). In Liberman’s and Trope’s theory, their notion of psychological distance resolves into many kinds of specific distances from self (origin), including distances of time, space, probability of occurring, social relations, etc. Each different kind of distance has a different metric; yet, all the different kinds of distances are related. See their 2008.
We know from Liberman and Trope's research that retrieval across large psychological distances either quickly produces or merely retrieves abstractions. It seems unlikely that the relevant abstractions are already made and are just lying around (for one thing, each concept would have to be abstracted relative to all other concepts in a given mind; that's a lot of abstractions, to put it mildly). So, we can conclude that large distance retrieval produces the abstractions. How? As mentioned above, Liberman and Trope do not speculate on the actual mechanism by which abstractions are created by psychological distance; they are concerned primarily to argue that such abstraction occurs and is due to psychological distance. Perhaps the STRANG algorithm could be the needed mechanism. Briefly, given some cognitive agent and any two of its mental representations, R and S, these two differ as a function of their semantics. Hence, representations that differ a lot are going to be semantically further apart than representations that differ only a little. Ergo, if R is an operator on S in the way described in this paper, the resultant representation, U, R(S) = U, will be more and more abstract as the semantic distance between R and S increases (compare the packing tree representations of abab and efef versus ababccc and mnopqrhijhijhij). The problem lies in making the STRANG algorithm fast enough in an architecture such as the brain. This might be easy or hard. At this point, we do not know. In fact, we do not in detail know how the brain implements any high-level cognitive processing, analogy included.
The phrase “see as” is meant to flag the distinction between seeing that something, X, has property F, versus seeing X as F. This distinction has a long tradition in philosophy of mind and is notoriously hard to make clear. Nevertheless, it can be useful. It is roughly the distinction between just seeing something and interpreting it a certain way. For example, one might just see two faces where someone else sees a family resemblance between the two faces. The latter case is a case of seeing-as. This is arguably a species of what Jerome Bruner called “going beyond the information given,”—see Bruner (1957). For a study of the relationship between Bruner’s notion and seeing-as, see Usborne and Lee (1997).
Another argument for construing theory is that it nicely fits with many current psychological and philosophical approaches of semantics in cognitive systems. These approaches divide semantics into two dimensions: an internal dimension and an external one. The internal one is based on the implicative or inferential power of mental representations. The external one is based on informational contact with the world. Both dimensions are needed for a full semantical connection to one’s environment (See Dietrich 2006, and Markman and Dietrich 2000, 1998, where this is argued in detail). Construing uses both dimensions and so can be said to be fully semantical. For a given construal, the internal dimension is realized by the generic concept’s inferential potential. The external dimension is realized by mapping the initial visual description (derived from perceptual input) onto the generic concept.
An objection to the rapid abstraction-construing theory of analogy is that it seems to imply that far analogies are easier to make than near analogies. It has been suggested that work such as Gick and Holyoak’s (1980) seems to tell against this. The quick answer to this is, Yes, the greater the psychological distance, the lower the analogical retrieval time, other things being equal, and assuming that abstractions are handled faster; but, No, Gick and Holyoak’s results are not a problem for this. Here is a longer answer, in three steps. First, field data indicate that spontaneous analogical remindings are rather common (Holyoak himself, together with Thagard, makes this point in their 1995). But they are rare in the psychology lab. Dunbar calls this the analogical paradox (2001). As I said at the beginning of “Semantically distant analogies and rapid abstraction”, I follow Dunbar (2001) in the view that natural spontaneous analogical remindings are quite common. Dunbar explains away this paradox by pointing out that lab settings for analogy production are actually quite a bit different from natural settings in several ways. One of those ways is that in natural settings, people generate analogies, rather than merely chose them, as they often do in lab settings. The research reported in this paper is geared toward analogical generation. Second, the research reported here does seem to suggest that given four concepts, C1, C2, C3, C4, if the psychological distance between C1 and C2 is greater than the psychological distance between C3 and C4, then C1(C2) [the rapid abstraction operation] will happen faster than C3(C4). But this is just a general, loose statement. It is not clear that rapid abstraction-construing theory predicts this because (1) factors like the difference in complexity of the operations C1(C2) and C3(C4) might swamp any distance effect [e.g., C1(C2) might be more complex on average or in certain cases than C3(C4)], and (2) construing times will not affect C1(C2), but will affect the time for final generation of the analogical insight, and rapid abstraction and construing are very tightly coupled; it is not clear how to separate them, experimentally. Given (1) and (2), it is not obvious how to set up an experiment to test this "distance-is-quicker" hypothesis, and it is not clear what a negative result would mean. Third, Gick and Holyoak never explored the role of relative distance between concepts in their famous experiments. Their tests were concerned only with some sort of "absolute" or fixed distance, namely that presupposed in Duncker's Radiation Problem. Also, Gick and Holyoak only found that, in the lab, a certain distant analogy was hard to make. Besides the problems Dunbar raises in his 2001 paper, in their experiments, Gick and Holyoak wanted their subjects to find a specific analogy. They did not experiment on open-ended, spontaneous analogical remindings.
For a nice picture of Rutherford's colliding alpha particles, see: http://www.chemsoc.org/timeline/pages/1911.html. For his 1911 paper, see http://www.chemteam.info/Chem-History/Rutherford-1911/Rutherford-1911.html.
The research for and work on this paper has benefitted from good conversations with Rick Dale and Clay Morrison. This version also benefitted from good comments made by two careful referees. Previous versions dating back to a distant time have benefitted from conversations with Dedre Gentner, Ken Forbus, Rob Goldstone, Michiharu Oshima, and Clay Morrison. I also thank Art Markman for many discussions on analogy, representation, cognitive science, and a host of other topics. And finally, as always, I thank Chris Fields for decades of exciting, learned, insightful, conversation about everything from analogy to zombies.
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Dietrich, E. Analogical insight: toward unifying categorization and analogy. Cogn Process 11, 331–345 (2010). https://doi.org/10.1007/s10339-010-0367-7
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DOI: https://doi.org/10.1007/s10339-010-0367-7