Leech et al. present a computational (connectionist) approach to explain analogy-making from a developmental perspective. At the outset we would like to emphasize that this is a very compelling and advanced approach: Tackling the problem from a developmental point of view enables the authors to highlight completely new aspects of analogy-making. This approach reflects the infantile learning process, from comparisons based mainly on superficial similarity to a controlled and advanced strategy of analogical comparison based on structural systematicity. So far, analogy models have always been inspired by adult analogical reasoning. Emphasizing the learning process, it makes sense to approach this problem from an infantile developmental perspective: Leech et al.'s model is based on a neural network implementing a Hebbian learning algorithm which is able to enhance bit by bit the strategy for analogy-making and explicitly model the development from superficial similarity to structural similarity. More precisely, the network can learn causal relations using a transformation of an object (e.g., “apple”) by a causal agent (e.g., “knife”) to achieve a representation of a transformed object (e.g., “cut apple”). The network extends this ability step by step to different domains and cross-mapping analogies; it can model the relational shift from surface similarity to relational similarity; and finally, it is trained on analogies involving multiple objects and multiple relations. There exists no other comparable analogy model modeling strategic learning – current analogy models can only model analogical learning by analogical transfer.
Another interesting capability is the creative potential of the analogy model: A trained network can creatively construct completely new objects when a relation is applied to a new (target) object. However, this capability must also be seen critically: Any relation can be applied to any (suitable or unsuitable) object and always leads to some result, which might be completely meaningless and absurd.
Inspired by research on infantile development, the authors investigate mainly analogies used in previously conducted analogy experiments with children. These are typically proportional analogies, that is, a-is-to-b-as-c-is-to-what analogies. All of these analogies are based only on a single, common relation, which is the same in source and target. We argue that such analogies are oversimplified – the task in these examples is applying the same relation to a new target object rather than making an analogous transfer. The target object is in fact very similar to the source object with respect to the applicability and the outcome of the relation. The “analogical” mapping required to solve the analogy is very small. We do not deny that such oversimplified analogies are necessary to investigate the initial analogical abilities of very young children; however, an analogy model (if it is not limited to modeling the analogy-making capability of 1-to-5-year-old infants, who anyway have only a very limited ability of analogy-making) must foremost have the capability to solve analogies with a minimum analogous structure. Therefore, the authors extend their investigation to so-called adult analogies. But in fact, their “adult analogy” differs from the “child analogy” only in that it comprises not just one single identical relation, but a set of single identical relations. The crucial characteristic of an analogy is, however, the common structure, that is, the analogous relations which are part of a common higher-order structural system. The analogy research community agrees on the fact that “systematicity” is the decisive point that is crucial for an analogy. Interesting analogies usually contain mappings that are only partially consistent. The sets of common relations considered by the authors are only connected via having the same source or target domain (see Fig. 16 in sect. 4.2 of the target article) and therefore have a flat and not a higher-order hierarchic structure. Principally it would be possible to extend the proposed approach to cover higher-order structures as well, by allowing the input of the neural network to represent composed structures – for example, as in the Structure-Mapping Engine (SME), where not only the object sun, but also composed structures such as the mass of the sun mass(sun) and the greater than relation between the masses of sun and planet mass(sun)>mass(planet) can each be represented by a single node. But still, this extended approach would only allow for composing relations with objects, not for decomposing them again.
We also see a problem in the expressive power, which arises in almost all analogy models: Except for logic-based analogy models (such as heuristic-driven theory projection [HDTP]; Gust et al. Reference Gust, Kühnberger and Schmid2006), no analogy model is able to express complex rules (often requiring quantified variables and logic operations). Many analogies in the physical and mathematical domains require this expressivity to be correctly modeled (for examples, see Gust et al. Reference Gust, Krumnack, Kühnberger, Schwering, Geibel and Jain2007).
A further aspect that should be mentioned is a restriction imposed on the involved relations: all simple analogies discussed in the target article map only on the same relations. If semantically similar relations (e.g., “Iraq occupied Kuwait,” but “Germany annexed Austria”) are represented closely in the neural network, we can assume that semantically similar relations can also be mapped on each other (which is an advantage of the neural network approach to symbolic approaches). However neural network approaches cannot map two analogous, but semantically different, relations on each other. In the case of the Rutherford analogy, the gravitational force in the solar system and the Coulomb force in the atom are analogous, but they cannot be mapped on each other. At a more general level, both relations could be described as two attracting forces. Many analogy models avoid this mapping problem by representing relations already at the required level of abstraction. However, the generalization of gravitation and Coulomb force to attracting force is part – actually it is the result!! – of the analogical reasoning and therefore must not be skipped. Like most other analogy models, the approach discussed here does not support this level of abstraction.
Most of our criticism does not refer specifically to the analogy model by Leech et al., but rather is directed at general shortcomings of most (subsymbolic) approaches. However, subsymbolic approaches have certain advantages in handling vagueness and uncertainty and in modeling learning. To bring together the strengths and overcome the shortcomings we see a lot of potential in connecting both approaches via neuro-symbolic integration.
