R1. Introduction
Analogy as relational priming (henceforth, ARP) views the analogical abilities of young children as developing out of priming – a simple, ubiquitous, cognitive mechanism. Central to the theory is the importance of development as a fundamental theoretical constraint for any account of analogical reasoning (and by extension, other high-level cognitive processes). It is our view that development must constrain all theories of analogy (even those just focusing on adult competence) because, to paraphrase Stephen Jay Gould quoting Francis Crick, “you can't identify the value in something until you know how it is made” (Gould Reference Gould1992, p. 137). Consequently, we are happy to note that many commentators were highly sympathetic to this goal of our work (Cheshire, Ball, & Lewis[Cheshire et al.]; Goswami; Markman & Laux; Morrison & Cho; Schwering & Kühnberger; Wasserman). Indeed, several commentators provided interesting additional evidence supporting central tenets of our account (e.g., Estes & Jones; Raffray,Pickering,& Branigan [Raffray et al.]), and the converging support provided by the comparative angle of Greene and Wasserman on the evolution of high-level reasoning and relational knowledge is of particular interest. However, as made clear by other commentaries, much research into how we view analogical reasoning remains polarized (e.g., French, Holyoak & Hummel, Markman & Laux).
In this response, we begin by reconsidering why we believe development is so central to understanding analogical reasoning and why ignoring it (as some commentators do) inevitably distorts any theoretical account of analogy. We also reiterate our understanding of analogy as an emergent property, recycling existing cognitive processes to build up complex cognitive skills. We then consider the two main assumptions underlying the current implementation of the ARP account: relational priming and representing relations as transformations between semantic states, all the time paying particular attention to development. We then turn to issues arising from more difficult analogies that, we agree, require mechanisms or processes beyond those countenanced by ARP. We close by discussing other models of analogical completion and some meta-theoretical issues raised by some commentators.
R2. One doth not fly into flying
A key principle of the target article is the centrality of ontogeny to our account of analogy. While some commentators appear to have overlooked this issue (most notably Holyoak & Hummel), others clearly appreciate the importance of an ontogenetic (and phylogenetic) perspective on cognition and the consequences of such a perspective for theories of analogy. Wasserman is perhaps the most articulate of the commentators in this respect, when he notes that, quoting Nietzsche (Reference Nietzsche1891), “one doth not fly into flying.”
In our view, there are precursors to the kind of analogical reasoning and problem solving that is typically examined in experimental work with adult humans, just as there are precursors to flying. This is demonstrated by the developmental literature reviewed in the target article, which provides strong evidence that even very young children are able to apply analogical principles to solve simple proportional analogies, and that this ability improves with age as children move from basing their responses on surface similarity to relational similarity. As an account of the development of early analogical ability, ARP therefore provides more than just an account of a set of behavioral findings. It also provides an explanation of why behavior changes as children acquire greater knowledge of the world, and more generally of how analogical abilities relate to other cognitive mechanisms (i.e., priming).
Two points here require emphasis. First, in presenting this argument we do not mean to imply that ARP is intended as a precursor of a greater or more complex theory that at some point during adolescence replaces or supplants it. Rather, our view is that the developmental findings on which ARP is based provide fundamental insights into basic cognitive mechanisms that are preserved and refined as successive mechanisms (e.g., relating to inhibitory control and working memory) come into play.
The second point to note concerns the nature of explanation. There are many constraints that theorists might adopt in theory development. As an empirical science, psychological theories are generally constrained by or evaluated against empirical findings. But it is generally agreed that for a theory to have an explanatory role it must do more than merely match behavioral findings. It should not, for example, posit special purpose mechanisms or entities that have no support beyond the range of situations for which they were introduced. Equally, theoretical explanations are generally preferred if they are continuous with theories in related areas, accounting for behavior in one area using existing, well-supported concepts from other areas. ARP offers just such a theory of the analogical abilities of young children.Footnote 1
At the same time, if a theory cannot reach a particular level of cognitive complexity proposed as an explanation of adult competence, then this is not an explanation of the adult competence. Building a bridge on the basis of principles that cannot be implemented would never happen in engineering. Our approach therefore is extending (or in the case of Piaget et al. Reference Piaget, Montangero, Billeter and Piaget1977, re-extending) to analogical reasoning the developmental principles of decades of work in other areas of developmental biology and psychology (see Mareschal et al. Reference Mareschal, Johnson, Sirois, Spratling, Thomas and Westermann2007).
It is important to note that this issue is independent of the nature versus nurture debate that rages though developmental psychology. Regardless of whether one believes that the essential components of cognition are somehow encoded throughout the genome, acquired through interaction with the environment, or some combination of the two, the weight of proof remains on researchers to show how the adult state comes into being (Marcus Reference Marcus2004).
R3. Could analogy boil down to priming?
R3.1. Empirical evidence of relational priming
Several commentators critique ARP by questioning the existence of relational priming (e.g., Morisson & Cho), citing, in particular, the negative findings of Spellman et al. (Reference Spellman, Holyoak and Morrison2001). However, as argued by Estes & Jones, Greene, and Raffray et al., there is in fact ample evidence of relational semantic priming in a broad range of domains (including memory retrieval, noun-noun comprehension, and syntactic text processing). Much of this evidence has appeared since Spellman et al.'s (2001) article (for a review, see Estes & Jones Reference Estes and Jones2006). Furthermore, as Estes & Jones make clear in their commentary, the ARP model captures many aspects of the relational priming literature, while at the same time accounting for many aspects of the development of analogy. In addition, Raffray et al. point out recent evidence of a “lexical boost” in relational priming. This is consistent with the ARP account and further supports the concept of relational priming.
Despite this growing body of empirical findings, it is true that no studies have yet shown relational priming in children using verbal stimuli. Goswami suggests that behavioral evidence of relational priming in children is not critical, and that it would in fact be sufficient to provide evidence at the neural level of repetition suppression (Dehaene et al. Reference Dehaene, Naccache, Cohen, LeBihan, Mangin, Poline and Rivière2001). We agree, but recent work by Duscherer et al. (in press) suggests that it is plausible to expect behavioral evidence. Duscherer et al. found, using a visual priming paradigm, that the prior perception of an action pantomime facilitates the recognition of the corresponding image of a tool associated with the action, but not of an image of a tool unassociated with the action, in 5- to 12-year-olds. This provides initial evidence that components of a causal relation can be primed in young children. Moreover, it is reasonable to expect verbal relational priming in younger age groups, since semantic priming effects are normally larger in younger, less fluent participants (Chapman et al. Reference Chapman, Chapman, Curran and Miller1994).
Regardless of these empirical findings, there is currently something of a debate over what exactly relational priming consists of; however, we are agnostic with regards to many of the details of relational priming. Although we implement a specific type of relational priming in ARP (pattern completion in a dynamic recurrent network), more broadly, any kind of semantic priming would carry out the same effect. Indeed, when making an analogy, all sorts of different priming mechanisms could be at play – semantic priming, lexical priming in addition to “pure” relational priming – all or any of which would lead to the completion of the analogy through mechanisms that do not involve special architectures or processes such as explicit structural alignment.
R3.2. An associationist straitjacket?
Given the evidence for relational priming, is the mechanism sufficient for reasoning by analogy? Holyoak & Hummel argue that “the target article illustrates an all-too-common approach to connectionist modeling of those cognitive processes most central to human intelligence: suck the essence out, then force-fit what's left into an associationist straitjacket.” If by “sucking the essence out” they mean de-mystifying (i.e., explaining cognitive processes in terms of simpler processes such as memory that we have a better handle on), then yes, we absolutely agree with them. The aim of psychological science is to explain phenomena in terms of well-defined causal mechanisms whose effects are reliably measurable (see Smith [2000] for an extended discussion of this point). Priming in a semantic memory is an example of such a phenomenon.
Unfortunately, some of Holyoak & Hummel's criticisms seem to stem from a failure to grasp key elements of the model's functioning, leading them to characterize it as a “glorified paired associates task.” So, to reiterate in case there is any confusion, the ARP model consists of a network learning about temporally contiguous states of the world. It is designed to learn about the causal semantics of the world and not to draw analogies. However, analogical completion falls naturally out of its dynamic pattern completion properties. To assess analogical completion, the a and b terms of an analogy (e.g., bread and cut bread) are first presented to the network. These values are clamped until the network has settled. This results in an internal state of the network that is consistent with its previous experience of bread and cut bread. At this point, the a and b inputs are unclamped. The network is then presented with the c term of the analogy (e.g., lemon) and allowed to settle again. Because it was in a state consistent with cut and it now receives a new input lemon, it settles into the output state of cut lemon. Note that (contrary to Holyoak & Hummel's claim) “knife” is never presented and thus never clamped during the formation of the analogy. It follows that the network's ability to complete this analogy is not simply an example of “glorified paired associates” in which the presence of bread paired with knife leads to cut bread. The model's ability to complete analogies comes instead from the recurrent network's dynamic pattern completion abilities. Different prior events (i.e., different a:b probes) will situate the network in different locations within a continuous state space. Its response to the current input (e.g., lemon) is a function of that new input and its location within that state space.
R3.3. Systematicity
A second concern over the sufficiency of relational priming for reasoning by analogy concerns systematicity. Holyoak & Hummel, Schwering & Kühnberger, and Markman & Laux all flag the phenomenon of systematicity as a selection constraint in analogical reasoning that, these commentators claim, the ARP model fails to capture. We fully agree that a bias towards systems of relations is a desirable feature of any model of analogical reasoning; but, furthermore, we also believe that a bias towards systematicity is a natural feature of many artificial neural networks with hidden layers and nonlinear activation rules, that arises without the need to use more complex structural representations. The point (made in the target article) is that systems of relations will vary coherently across multiple domains reflecting an underlying statistical structure, whereas superfluous or irrelevant relations will often be restricted to individual domains. As such, the hidden units in a network will extract this underlying statistical structure across multiple domains. Relations that frequently co-occur will be represented closer together in the trained network's internal semantic space. Subsequent complex, multi-relation analogies using these relations will key into this proximity between representations of relations constituting a system, ensuring that the network has a bias towards picking systematic relations. We refer the reader to similar phenomena involving semantic cognition (Rogers & McClelland 2004) and metaphor (Thomas & Mareschal Reference Thomas and Mareschal2001).
R3.4. Analogical inference
Markman & Laux make the valid point that one of the main functions of analogical reasoning is to draw useful inferences that extend our knowledge. They also point out that ARP at present does not model how people draw these inferences. Although we have to date not focused on analogical inference, the possibility of drawing novel inferences is consistent with the ARP account. Under ARP we envisage analogical inferences as proceeding in a somewhat different way from more traditional mapping accounts (e.g., the Structure Mapping Engine) that explicitly choose features and relations to map from a base to a target domain. Instead, we suggest that analogical inferences might best be understood as novel generalizations governed by the distributional information about which input features and relations co-vary across the base and target domains. (The Metaphor as Pattern Completion account provides a template as to how this might proceed; see Thomas & Mareschal Reference Thomas and Mareschal2001.)
R4. Are relations simply transformations in a semantic space?
R4.1.#Reversible relations
Several commentators (Costello, Petrov, Sloutsky) point out that although some relations (particularly simple causal relations such as cutting) could usefully be viewed as additive transformations in a semantic space, this cannot be the case for many of the relations habitually used in analogies, particularly adult analogies. In fact, Costello clearly demonstrates the impossibility that a bidirectional relation (e.g., the relation kicks that can be reversed from John kicks Mary to Mary kicks John) could be represented as an additive transformation. However, this misapprehension arises not from an inherent limitation of our account but from an overly narrow reading of the notion of relations as transformations. To clarify, we may distinguish between a specific and a general theory of relations as transformation. The specific theory is that there is a one-to-one (or close to one-to-one) correspondence between a given relation and a given transformation and, consequently, that relations are additive transformations in semantic space. In our first simulation, we followed this approach because it makes sense in the context of simple causal relations of the kind easily used in analogies by young children. For a relation such as cutting it is plausible that cutting transforms everyday objects in somewhat similar ways. Furthermore, these simple causal relations tend to be unidirectional in that we see apples being transformed into cut apples but not the other way around. However, as Goswami points out, we need not be wedded to this specific theory of relations as additive transformations –instead what is important to the ARP account is a much more general theory of relations as transformations.
The general account of relations as transformations is that relations are transformations from one state to another in semantic/conceptual space, but there is no need for equivalence between a given relation and a specific additive transformation. This allows the network to represent relations that are reversible. Indeed, the second simulation considering the Gulf War/World War II analogies (sect. 4 of the target article) is implemented using this more general form of relation as transformations. What is important is that a given exemplar of a relation is a transformation between states (i.e., a mapping from one state to another) and the network learns different mappings (i.e., transformations) for different exemplars of the same relations. This approach, as we point out in the target article, is not unique to us but similar to what is envisaged in other transformational accounts of cognitive abilities (e.g., Rogers & McClelland 2004).
In general, therefore, we contend that relations should be understood as transformations in semantic space (a conceptual system), but, importantly, that a given relation is not necessarily represented by a specific single transformation. Instead, a given relation covers a set of learned transformations that are triggered by the specific interaction of objects with the available context.
Within the model, relations are represented implicitly in the matrix of connection weights, so there is nothing in the model per se that corresponds directly to the relation cutting. These implicit representations lead Dietrich to question whether the ARP model could account for explicit (i.e., conscious) awareness of relational knowledge. We have two responses to this point: first, (as made clear by the animal literature referred to by Greene and Wasserman; see also Oden et al. [2001] for a discussion of analogy in nonhuman species) there is no necessity for the relations underlying analogies to be explicitly known. Humans and some animal species appear to be capable of solving analogies using implicit or close-to-implicit knowledge and without much conscious awareness. Second, we accept that more complex, adult-like analogies will make considerable use of explicit strategies (see sect. R5 of this response). Note finally that by associating verbal labels with relations, for example, by augmenting the context layer of the network (see sect. 3.3.5 of the target article), it is easy to see how the implicit representations of relations could be linked to explicit (or at least verbalizable) representations in a manner that makes the knowledge consciously accessible.
As the previous point makes clear, we are not committed to any particular type of context representation driving analogies, such as causal agents that correspond to physical objects in the world (see Sloutsky). We introduced causal agents in the first simulations because we were modeling a specific task involving causal agents. However, as pointed out in the second simulation on more complex analogies, the context could take the form of many different types of mental representation, for example, something as concrete as a knife but also something more abstract such as heat or a verbal label.
R4.2. Multiple possible relations
A second objection to relations as transformations, raised most directly by Akman but also underlying Bouissac's uneasiness with ARP and central to one of French's concerns, is that multiple relations may hold between two objects. Consider Akman's example: puppy:dog::kitten:?. Is the relation of concern offspring, or is it any one of a number of other possibilities (is_younger_than, is_cuter_than)? Bouissac argues that the multiplicity of relations that might hold between the given terms in a proportional analogy undermines any possibility that there might be any single “correct” answer. His specific concern is with the socializing process that leads our society to prioritize one relation over another (and hence to suggest that there is just one right answer). In fact, this is wholly consistent with the ARP approach: when multiple relations hold between the a and b terms of a proportional analogy, the one that the model predicts will be applied is the one that is favored by the model's training history and current context. In other words, the model does not predict that there is a single “correct” answer to an analogy. As Bouissac notes, correctness is a function of the child's/ system's knowledge, which in turn is a function of culture.
French uses the puppy:dog semi-analogy to make a related point. His argument is that the relation between the a and b terms that is appropriate within a proportional analogy depends on the c term. If the c term is kitten then offspring is the appropriate relation, but if the c term is watch then the appropriate relation is something else, maybe small_version_of. In fact, there is substantial semantic overlap between offspring and small_version_of. We therefore see no difficulty in accounting for this analogy within ARP, provided of course that the system is exposed to exemplar watches and clocks. A more critical example would be a case where multiple very different relations hold between the a and b terms of the analogy and when, given c 1, the appropriate response is d 1, but when, given c 2, an alternate response d2 is appropriate. It remains to be demonstrated that young children can respond appropriately in such a situation. ARP would predict that, prior to the mastery of executive control, they could not. That is, within ARP the successful completion of such analogies is likely to require the kind of additional machinery for inhibiting responses included in our model of the Gulf War/World War II analogy, as described in the target article and further discussed in section R5 here.
R4.3. Abstract relations
A final issue concerning relations as transformations that several commentators brought up (Dietrich, Petrov, Phillips, Schwering & Kühnberger) is how the model can handle abstract relations. Again, in part this issue arises because of the confusion between the specific and general theories of relations as transformations. The general account of relations as transformations allows for abstract relations to be represented. For example, “pen left_of pencil,” although not involving a meaningful transformation in the world, could be implemented as a mapping between the representation of pen and the representation of pencil. However, we acknowledge that there is still the important question of how relations implemented implicitly as transformations can be used abstractly and flexibly in novel contexts. For example, in the situation “pen left_of pencil” this is a temporary configuration that we may never have seen before (or at least never analogized about before). As Phillips makes clear, we have not had the chance to be exposed to hundreds of epochs of learning about the relative location of pens and pencils; and even if we had, it would not be any use if we then saw a pencil to the left of a pen. This is a legitimate concern, and again we have two responses to this issue. First, we maintain that some of the complex and flexible usage of relations observed in the world relies on complex cognitive control strategies involving working memory and inhibition. Second, and perhaps more important, we contend, however, that there really are no examples of completely abstract and novel analogical or relational reasoning. Although we may never have seen a given pen and pencil in a specific and completely temporary configuration (or even two novel objects, e.g., greebles), we have seen and thought about a vast array of other objects in similar types of configurations.
R4.4. Empirical evidence for relations as transformations
As discussed in the target article, Thomas and Mareschal (Reference Thomas, Mareschal, Shafto and Langley1997) and Hahn et al. (Reference Hahn, Chater and Richardson2003) have both stressed the importance of transformations for judging similarity and, by extension, the importance of transformations for analogical reasoning. In addition, as discussed in section 5.1.2 of the target article, Leech et al. (Reference Leech, Mareschal and Cooper2007) directly tested a strong prediction of the current implementation of causal relations as transformation. They found that the size of a transformation instantiating a relation between two states of the world (e.g., shrinks) was a performance factor affecting analogical reasoning, in both adults and adolescents, using a range of different analogical reasoning tasks. Such similarity effects fall naturally out of the implicit distributed nature of relations as transformation and are difficult to interpret in terms of explicit propositional representations of relations.
R5. ARP and the bigger picture
R5.1. The challenge of complex analogies
Researchers of adult analogy often make use of very complex analogies requiring far more than just the mechanisms that they propose in order to explain analogy. Take, for example, the analogy described by French in the beginning of his commentary relating baseball to academia. It is (on the surface at least) far more complex than a:b::c:d. But what has increased its complexity? Such analogies require additional components, such as advanced semantic knowledge of baseball and the world of academia, as well as sentence and text processing skills, working memory, and pragmatic conversational skills. Are these intended to be central to analogy? It seems entirely plausible that children's failure on these more involved analogies is at least in part due to shortcomings in world knowledge, working memory (see sect. R5.2), sentence processing, and pragmatic skills, all of which develop substantially throughout childhood.
We use proportional analogies in presenting and exploring ARP because they embody what is essential to analogies (Sternberg Reference Sternberg1977a). Could French's example be recast as a proportional analogy? Does “Ty Cobb:Home runs::Academic:Journal articles” make sense? Absolutely, but only if the relevant background knowledge is primed (i.e., if the reader is located in the appropriate part of their semantic space) – as French does, by providing the reader with the background context so as to highlight the intended relation between Ty Cobb and home runs that French wishes us to transfer to the domain of academia.
R5.2. Working memory and cognitive control
We agree with Doumas & Richland that there is more to solving multirelational analogies than relational knowledge. As Morrison & Cho point out, there are clear constraints from working memory and changes in inhibitory control on children's developing abilities to use and comprehend increasingly complex analogies. Again, this is entirely consistent with the role of executive control that we posit for interpreting complex multi-relational analogies. We chose, however, to focus on the kind of analogies that very young children (3 to 4 years old and younger) seem competent at, and we produced an emergentist account of how these types of analogy could emerge. The next, and by no means easy, step is to investigate the interplay of the emergence of analogy based on relational priming (with a central focus on knowledge accretion) with the other emerging abilities of working memory and executive or cognitive control (e.g., active and selective maintenance and updating of working memory) that presumably underlie very complex analogical reasoning. This research agenda will have to address some of the issues regarding working memory that Morrison & Cho note (i.e., integration of multiple relations and the suppression of distracting stimuli). However, from our perspective it seems likely that analogy is parasitic on the rest of cognition. So, as Petrov points out, quoting French (Reference French2002, p. 204), “it is probably safe to say that any program capable of doing analogy-making in a manner truly comparable to human beings would stand a very good chance of passing the Turing Test.” The implication of this statement is that to model the development of analogical reasoning, we must model the development of a huge raft of interrelated cognitive skills (semantic cognition, working memory, inhibition, number, mathematical and spatial reasoning skills, etc.). As discussed in section R6 of this response, these are all capabilities pre-assumed by other existing models of analogical development
R5.3. Reversibility and role assignment
A further point raised by multiple commentators is that people are able to reason about analogies with the terms reversed; as Akman says, “one apparent objection has to do with the ‘direction’ of the relationship.” So Holyoak & Hummel ask whether ARP, trained on apple:sliced apple and bread:sliced bread, could correctly complete the analogy “sliced apple:apple::sliced bread:?”. Estes & Jones similarly suggest that ARP could usefully be extended to have some mechanism for resolving roles, that is, discovering, for example, whether relational terms are in the order cause–effect, or effect–cause. The strength of this criticism, as pointed out by Akman, depends upon whether young children can process these types of analogies. If, following Akman's intuition, such analogies are harder for young children when they appear in a non-canonical form (e.g., sliced apple:apple::sliced bread:bread), what additional processes are occurring that make them harder? Tentatively we suggest that in the extended account of analogical reasoning (i.e., ARP embedded within additional cognitive control systems), the analogizer may learn to funnel information that is noted to be in a non-canonical order into the appropriate order for relational priming to work.
R5.4. ARP as part of a larger theory
We are very comfortable with Markman & Laux's comment that “ARP theory does not explain full competence, and cannot, in principle, be extended to do so without it becoming a part of a larger theory.” Where we differ from Markman & Laux is that we see ARP as a potential fundamental component of the wider theory, sufficient on its own for some analogies and necessary for many other more complex analogies. As illustrated by the second simulation regarding the Gulf War and World War II (sect. 4 of the target article), we show that, in principle, ARP could be embedded within other cognitive processes to construct the kind of complex extended analogies used to test models such as the Structure Mapping Engine (SME) and Learning and Inference with Schemas and Analogies (LISA). In essence, relational priming is iteratively applied in a controlled way to construct large multi-relation analogies. It may be helpful here to make an analogy with research into language development, which until relatively recently placed most weight on studying syntax as the fundamental aspect of human language (analogous to explicit structure mapping). It is, in fact, increasingly acknowledged that syntactic development and semantic development are fundamentally inseparable (Bates & Goodman Reference Bates and Goodman1997), especially when a truly developmental perspective is taken. Similarly, we believe that research on analogical development (and analogy in general) will only truly get anywhere if simple processes such as relational priming that are ontogenetically and phylogenetically (Wasserman) plausible can be put to good use in increasingly complex ways (i.e., with executive control building up to complex structure mapping).
R5.5. Meta-cognitive processes
Providing children with feedback on their analogical performance has been shown to improve subsequent analogical performance (at least in the short term), as has asking them to explain their responses (Cheshire et al. Reference Cheshire, Ball, Lewis, Bara, Barsalou and Bucciarelli2005; Reference Cheshire, Muldoon, Francis, Lewis and Ball2007). Cheshire et al. argue in their commentary that such effects cannot be accounted for within ARP as it stands, and question how (or indeed whether) ARP might be extended to allow behavior to be influenced by feedback and reflection. On the one hand, we see this as a criticism of a specific connectionist implementation of ARP, rather than a criticism of ARP itself. On the other hand, we are wary about arbitrarily extending the implementation to incorporate feedback and self-explanation when, as noted in our target article, analogical abilities develop in the absence of explicit training on an analogy. To elaborate, Cheshire et al.'s data does not undermine either the mechanism of relational priming or the representation of relations as transformations. Rather, it suggests additional learning mechanisms that may influence performance on proportional analogies. Could the learning mechanism of our implementation be extended to account for the role of feedback and self-explanation? Possibly, but it is our view that the primary effects of feedback and self-explanation lie beyond the semantic system held to be involved in relational priming processes. They may, for example, be accounted for by learned attentional processes that differentially weight features of the stimuli, resulting in different inputs to the semantic system. Alternatively, they may affect post-semantic decision processes, resulting, for example, in the rejection (and subsequent inhibition) of targets that match only on surface similarity.
R6. Other models of analogy
Several commentators raise implicit objections to ARP by presenting alternative models of one or another aspect of analogy. Borges, d'Avila Garcez, & Lamb (Borges et al.) suggest that we have prematurely rejected hybrid “neural-symbolic” models, arguing that such models provide the great advantage of combining “trainable neural networks with symbolic representations,” whereas Schwering & Kühnberger question the expressive power of ARP in relation to analogies in the physical sciences involving quantified variables and logic operations.
We accept the potential advantages of hybrid systems (i.e., combining learning and symbolic representations) in some circumstances. However, we dispute the relevance of these properties to our target phenomena. Thus, in presenting their position, Borges et al. describe two neural-symbolic systems, Connectionist Inductive Learning and Logic Programming (CILP) and Sequential Connectionist Temporal Logic (SCTL). The latter certainly appears to be capable of performing analogical reasoning of the type addressed by ARP, but Borges et al. make no attempt to relate the account to empirical phenomena, let alone developmental evidence. In particular, there is no evidence that the SCTL approach might replicate any of our seven observations on the development of analogy, beyond perhaps Observation 1 (concerning knowledge accretion). Even in that case, however, it is unclear how the learning mechanism of SCTL might relate to a child's learning of relational knowledge, which, we claim, is the product of witnessing many cases of a relation in a variety of contexts, and not learning from a set of propositional clauses. Furthermore, as noted by Schwering & Kühnberger, our non-symbolic (or more precisely, non-atomic) representation of relations has the advantage that it allows for semantically similar relations to have similar representations. Thus, our encoding is in principle able to account for analogies involving similar, but non-identical, relations (such as “occupies” and “annexes,” to use Schwering & Kühnberger's example). At the same time ARP cannot be easily compared with the Heuristic Driven Theory Projection (HDTP) model (see Gust et al. Reference Gust, Krumnack, Kühnberger, Schwering, Geibel and Jain2007) cited by Schwering & Kühnberger – another model that is concerned solely with adult-level competence.
Doumas & Richland correctly point out that we overlooked one model that accounts for how explicit relational representations can be learned. The Discovery of Relations by Analogy (DORA) model (Doumas et al. Reference Doumas, Hummel and Sandhofer2008) proposes to do just this. We apologise for this oversight and, therefore, provide due coverage of this model in the present reply. At first glance, we are highly sympathetic to the aims of this work. As argued throughout our target article, answering the question of how and from where an ability emerges is essential to any understanding of that ability. Unfortunately, a more detailed look at the DORA model suggests that it fails on this vital task. In fact, the very name of the model, “Discovery of Relations by Analogy,” suggests that there is a presupposition that the mechanisms for analogical retrieval and inference are in place prior to the discovery of any form of relational representations.
So, what are the assumptions that, according to Doumas et al. (Reference Doumas, Hummel and Sandhofer2008), underlie the development of this ability in DORA? Fortunately, the authors are exceptionally clear in what they take as necessary requirements for the model to function. Thus, we read, “Armed with a basic vocabulary of perceptual and relational invariants … DORA discovers relations through general learning processes and develops as a result of experience” (Doumas et al. Reference Doumas, Hummel and Sandhofer2008, p. 30). We have nothing to quibble about concerning this proposition. Next come the assumptions: “The model assumes that memory and perception are present at the start of learning. It also requires a capacity for comparison, mapping, and SSL [Self Supervised Learning], as well as the ability to flexibly treat a feature as either a (holistic) feature of an object or as a feature of an explicit predicate” (Doumas et al. Reference Doumas, Hummel and Sandhofer2008, p. 30; emphasis added). Even putting aside further mysteries such as how the complex machinery involving multiple modules, multiple levels of representation, and multiple processes and control streams got put together in the first place, most if not all of the skills believed by many to underlie analogy are pre-assumed in the DORA model. To us, this is not an explanation of how analogical abilities emerge, but rather an explanation of how, once analogy is in place, it can be used to generate new explicitly accessible relational representations.
Doumas & Richland make the further claim that DORA can account for all of the phenomena accounted for by ARP and much more. At the moment, there is no published evidence of this. As the DORA model is not a model of analogical development (nor does it claim to be), it does not try to account for the same tasks as we do. Although it may be able to capture some of the performance characteristics of the ARP model on a:b::c:d analogies, we would be curious to see how easily it could capture specific developmental phenomena such as the profile of errors that children are found to make at different ages (see sect. 1.1.3 of the target article; see also Rattermann & Gentner Reference Rattermann and Gentner1998a) and the dynamic temporal markers associated with development on analogical completion tasks (sect. 1.1.4; see also Hosenfeld et al. Reference Hosenfeld, van der Maas and van den Boom1997).
There is one area in which the DORA and ARP models do overlap – they both suggest that there is often a shift in children's performance from appearing to rely on surface similarity in drawing analogical inferences to relying on relational similarity. The two accounts differ in that ARP suggests that it should be possible to find examples in which the opposite shift occurs. It turns out that this is in fact the case. Opfer and Bulloch (in press) tested 3-, 4-, and 5-year-olds on how they would generalize new information. On offspring problems, relational information yields more accurate generalizations, whereas in prey problems, surface similarity yields more accurate generalization. Opfer and Bulloch found that with offspring problems there was an increasing reliance on relational similarity with age (as previously reported), but on prey problems there was a decreasing reliance on relational similarity and an increasing reliance on surface similarity with age. These authors suggest that the relational trend commonly observed in analogical completion may, in fact, reflect increasing sensitivity to cue validity rather than an overall preference to generalize on the basis of relational similarity over perceptual similarity. These findings are entirely consistent with what is observed in the ARP model, and in fact validate a primary prediction made in section 5.1.1 of the target article. It is difficult to see how such a trend would fall naturally out of DORA or any of the associated structural alignment models.
A precursor to DORA coming from the same group of researchers is LISA, which Morrison & Cho claim can model many developmental findings. In particular, Richland et al. (Reference Richland, Morrison and Holyoak2006) demonstrate that by increasing an inhibition parameter, LISA is able to account for effects of decreasing sensitivity to distracter items and increasing ability with complex relations with age. As Richland and colleagues argue, the functional effect of increasing inhibition within LISA is to allow increased working memory capacity, and, as argued earlier in our response, we agree that such developmental changes are likely to promote improved performance in some analogical tasks. However, as in the case of DORA, the LISA model is not really a developmental model – it assumes that complex computational machinery is in place, and all that “develops” is better inhibitory control. This is also the case with the efforts to model development using SME that Markman & Laux direct us to (e.g., Gentner et al. Reference Gentner, Rattermann, Markman, Kotovsky, Simon and Halford1995). While it is a valid hypothesis to argue that performance improvements are caused by an increase in capacity (such as working memory), this leaves unanswered the question of where the specialist machinery comes from in the first place. This position is in stark contrast to ARP, which does not posit any special machinery for analogical completion and argues that such abilities simply fall out of existing general-purpose semantic memory systems.
R7. Wider issues
A number of commentators raise miscellaneous higher-level issues concerning metaphysical assumptions and the methodological approach embodied within ARP. Our responses to these meta-issues are collected here.
R7.1. Metaphysical assumptions
Linhares is concerned with some of the metaphysical assumptions underlying ARP, namely, the representational module assumption and the associated assumption of metaphysical realism. The central issue in both cases is the nature of mental representation. Thus, Linhares suggests that “mental objects” are dynamically built, consisting of a concept with associated sets of connotations, whereas “[a]nalogies are mechanisms by which a mental object acquires connotations from different mental objects.” In contrast, ARP (or at least its connectionist implementation) assumes that mental representations can be adequately characterised as sets of features. Consider first the abstract claims of ARP. Although Linhares expresses no opinion about relational priming per se, he agrees with our view of relations as transformations between mental representations, and with the corollary that the application of such transformations during analogy effectively results in the acquisition of source features or connotations by a target mental object. What, though, are connotations? The word itself is slightly ethereal and mysterious. Linhares, however, provides some clues: they are (a) “rules for carving waves,” (b) “relations between mental objects,” and (c) “properties of particular objects.” The first of these appears to be a restatement of the representational module assumption that Linhares denies, while the third is entirely equivalent to the featural representations that we have assumed. The difference then between Linhares' position and ARP reduces to the claim that mental objects consist, in addition to featural representations, of an atomic concept that may stand in some relation to other atomic concepts. Therefore, we see Linhares' position as an elaboration of our position. That elaboration may well turn out to be justified when more complex cognitive abilities are considered, but the simulations reported in our target article demonstrate that it is not necessary for the target phenomena.
It is also important to be clear about the extent to which our representational assumptions shape the ARP model's behavior. The ARP model would produce equivalent results regardless of the precise featural representations chosen, provided that appropriate abstract regularities over the domain hold (e.g., that the act of cutting affects some but not all features of an object that may be cut, but preserves all features of an object that cannot be cut). Thus, the ARP model requires that mental objects have a core (featural) representation that is invariant within an individual, but not, as a strong form of metaphysical realism would suggest, a representation that is invariant across individuals. Such core representations are presumably acquired throughout early development as the child becomes attuned to regularities within its environment, or as Linhares might argue, acquires “rules for carving waves.” We have no concerns in accepting this, given its widespread and apparently successful use throughout cognitive modelling (e.g., Elman et al. Reference Elman, Bates, Johnson, Karmiloff-Smith, Parisi and Plunkett1996; Mareschal & Thomas Reference Mareschal and Thomas2007; Rogers & McClelland 2004; Rumelhart & McClelland 1986; Shultz Reference Shultz2003).
R7.2. Computational-level theory and computational complexity of ARP
Two meta-theoretical issues concerning ARP are raised by Wareham, Rooij, & Müller (Wareham et al.). First, they are concerned with our failure to specify a computational-level theory of our target domain (in the sense of Marr Reference Marr1982), and second, they are concerned with our failure to discuss the computational complexity of ARP. In our view the computational-level theory – the input-output function – is given by the empirical effects underlying our seven observations. These observations relate the inputs that the developing child is exposed to (both in terms of increasing world knowledge and the specific analogy problems provided to children in the various developmental studies cited) and the outputs that the child produces (when presented with an analogy for completion). We therefore do not see this issue as a major concern, although it is perhaps behind several other commentaries that, as discussed above, object to our equating analogy with performance on the a:b::c:d task employed originally by Piaget et al. (Reference Piaget, Montangero, Billeter and Piaget1977) and later by Goswami and Brown (Reference Goswami and Brown1989; Reference Goswami and Brown1990).
The computational complexity of ARP, on the other hand, is a more difficult question and Wareham et al. are right to raise it. We agree that there is an important role for computational complexity results to play in cognitive theorizing, particularly when considering issues of scaling. As Wareham et al. point out, Structure Mapping Theory (SMT) is computationally intractable, and this is, or should be, of concern to advocates of SMT. With regard to the computational complexity of ARP, it is again relevant to stress that our connectionist model, with its specific architecture, training procedure, and algorithm, is presented as an implementation of ARP that demonstrates the theory's plausibility, but alternative approaches to the implementation of the theoretical claims of ARP (i.e., of relational priming and relations as transformations between representations) are surely possible. The question is therefore not one of the computational complexity of contrastive Hebbian learning, but whether there exists a computationally tractable implementation of ARP. This is an open question, but the inherently parallel nature of priming gives us reason to be cautiously optimistic about the scaling behavior of the theory.
R7.3. Relation to neurophysiology
An alternative meta-theoretical position is taken by Speed, who is concerned with the neurophysiological processes underlying analogical abilities – Marr's (1982) implementation level – and our failure to provide a theory at this level. Specifically, Speed argues that analogy is an emergent property not of more basic psychological processes such as relational priming, but of more basic physiological processes. Unfortunately, she does not suggest what these physiological processes might be. Although we accept that psychological processes must eventually be grounded in physiological ones (see Mareschal et al. Reference Mareschal, Johnson, Sirois, Spratling, Thomas and Westermann2007), we find this reductionist perspective to be of limited use. To accept it would lead one to discard the entire field of cognitive psychology in favor of neurophysiology. While neurophysiology has told us much about the functioning of the brain, it has told us relatively little of the functioning of the mind.
We could of course have speculated more about the neural mechanisms underlying ARP (and the first of our models is at least broadly compatible with what is known of neural processing mechanisms; see O'Reilly Reference O'Reilly1996), but any comments would have been entirely speculative. Worse, in our view, is that they would conflate levels of explanation. We firmly believe that the neural level is important, but our model is concerned with the cognitive level. One may wish for well-grounded models of how “the brain, as a physical system, [can] take raw sensory information and perform higher-order cognition using that information (e.g., solving Raven's Progressive Matrices),” as Speed writes, but the field is not yet in a position to develop such theories. In any case, to do so would in our view be greatly facilitated by first developing a cognitive-level theory of how people solve Raven's Progressive Matrices, and this is the level at which ARP is conceived. In a similar vein before considerations of neuropsychological problems (e.g., autism or depression) can be mapped onto connectionist models (see the commentary by Swain & Swain) we must have a good understanding of the relations between the components of the model (e.g., hidden units, connections) and the brain (see also, Mareschal & Thomas Reference Mareschal and Thomas2007).
R8. Conclusion
Where, then, does this leave us? Young children, infants as young as 12 months old even, show signs of analogical ability (as do some chimpanzees). ARP accounts for such abilities, and for many of the improvements in analogizing seen in the first decade of the child's life. We accept, though, that we need to get to the other end – to adult-level competence. As argued throughout, our view is of analogy as an umbrella for a variety of concepts and processes – a view also explicitly endorsed by numerous commentators (Borges et al., Sloutsky, Wasserman) and denied by none. We accept that adult-level analogical competence involves processes beyond relational priming. We suspect, though (given what we know of the evolution and development of our cognitive abilities), that it does not involve an “analogy module,” as rival models seem to claim, but instead that it is firmly grounded in recycling old parts.
ACKNOWLEDGMENTS
We are grateful to European Commission grant NEST-029088 (ANALOGY) and the Economic and Social Research Council, United Kingdom.
