There is no scarcity of literature on analogies. They have been a recurrent topic in contemporary philosophy of science, in particular after the publication of Mary Hesse’s Models and Analogies in Science (1966), and they have attracted a great deal of attention in cognitive psychology over the past 30 years. Drawing on insights from this work and grounding his analysis on a wide range of examples from science and mathematics (the index lists 47 different ones), Bartha offers a novel perspective on analogical reasoning. Before going into a more detailed presentation of the contents of the book, I would like to mention several important respects in which By Parallel Reasoning stands out from the existing literature on analogies. First, it takes as its starting point an extremely rich and diverse set of examples of analogies that are taken from such different domains as anthropology, law, mathematics, philosophy, and physics. Second, it focuses on analogical arguments and investigates what it takes to make the conclusions of such arguments plausible (where plausibility is cashed out both in modal terms, as prima facie plausibility, as well as in probabilistic terms, as degrees of plausibility). Third, it proposes a normative theory for evaluating analogical arguments, which takes into consideration crucial differences between various kinds of analogical arguments. Fourth, Bartha’s systematization of various types of mathematical and scientific arguments is an important methodological achievement and constitutes a model “at a level of detail intermediate between an elementary ‘commonsense’ description of analogical arguments and a meticulous case study” (236). Finally, Bartha presents philosophical justifications for the use of analogies in science and mathematics. These points, which are presented and discussed with great erudition, make Bartha’s book a milestone in the study of analogical reasoning.

Bartha’s analysis of analogical arguments is based on Hesse’s influential model of analogies, which consists of “vertical” relations within the source and target domains and “horizontal” relations between the two. However, whereas Hesse restricts the vertical relations to causal relations, Bartha loosens this requirement and introduces the notion of prior associations in the source, which can be logical, causal, explanatory, or correlative. Hesse’s horizontal relations between analogs are replaced by Bartha with a general principle of potential for generalization. Thus, prior association and potential for generalization are the two dimensions used for classifying analogical arguments in Bartha’s articulation model; the criteria for evaluating these arguments depend on their analysis along these dimensions. In other words, Bartha’s model does not evaluate analogical arguments according to a single form or standard. Rather, the evaluation varies depending on the kind of argument under consideration. This is a remarkable departure from almost all previous models of analogies and one of the main strengths of Bartha’s account. Moreover, he does not slide down the slippery slope of relativism but identifies general argument patterns that induce particular requirements for their evaluation. In sum, the articulation model can be applied to radically different analogical arguments and it demands different standards for their assessment.

The nine chapters of By Parallel Reasoning fall into four main parts. In the first part (Chapters 1–3), Bartha motivates his theory by critically reviewing existing ideas about analogies and analogical reasoning. These include some commonsense considerations for evaluating analogical arguments that can be traced back to Aristotle, to Hesse’s theory, and to views that take analogical arguments to be deductions, single-case inductions, and sampling arguments. For each of these accounts, Bartha shows that it is too narrow to cover analogical arguments in general. He does this by skillfully presenting counter arguments, often drawn from the realm of mathematics. He also argues that a characterization of analogies that is based only on the positive, negative, and neutral analogies between source and target domains does not have enough resources to distinguish between good and bad analogies. To overcome this limitation, Bartha introduces the notion of “prior associations.” In addition to philosophical theories of analogy, he also discusses computational theories, both symbolic and connectionist (see Gentner et al. 2001, for an overview). These are mainly employed for a restricted aspect of analogical reasoning, namely for establishing a mapping between two domains. The process of making an analogical inference is commonly split up into four phases: retrieval, mapping, transfer, and learning (viii). In contrast, Bartha’s analysis begins with actual analogical arguments, in which “correspondences between the source and target are given” (99). As he points out, computational theories are highly sensitive to the way the domains are represented by the user (61), a fact to which little attention is usually paid. Bartha’s main objection to structuralist approaches is that they aim at establishing the most systematic mapping, but that this does not necessarily lead to the most plausible argument. Despite the fact that a more systematic mapping does not necessarily imply a stronger analogical argument, Bartha later ranks multiple analogies according to the amount of overlap between the source and target (278), thus invoking the criterion he earlier criticized. Bartha identifies interesting and fruitful ideas in the computational approaches (88)—he discusses various versions of structure-mapping theories, the constraint-satisfaction model, the multiconstraint theory, case-based reasoning, and others—but ultimately does not consider them conducive to his goal; while computational models are psychologically motivated and aim at maximizing overall similarity, Bartha aims at a normative theory that generalizes logical or explanatory relationships.

By considering philosophical and computational theories of analogy in his book, Bartha overcomes traditional disciplinary boundaries. This allows him to formulate the most comprehensive and sophisticated account of analogical arguments to date. Nevertheless, he does not claim to provide an exhaustive theory (151). His own theory, the articulation model, is introduced and discussed in the second and longest part, which takes up almost half of the book (Chapters 4–6). Analogical arguments are classified by the direction (predictive, explanatory, functional, and correlative) and the mode (deductive and inductive) of their prior associations. With the aid of illustrative examples, Bartha carefully discusses generalities of the resulting types of arguments, both in informal and formal terms. As an example of the latter, he formulates the canonical form of a predictive/probabilistic analogy as “Q because φ+ and Π, despite φ,” for a hypothetical analogy Q, sets of relevant contributing and counteracting factors φ+ and φ, and factors Π that are absent from the source (114). Depending on the nature of the prior associations, for Bartha, “the proper way to evaluate an analogical argument is to assess its potential for generalization” (195), and for each of the types of arguments mentioned, he discusses specific criteria for their assessment. The advantage of Bartha’s approach, namely of covering the widest range of analogical arguments possible, without pressing them into a single mold (57), turns out to be somewhat of a disadvantage with regard to the presentation of his model, which could easily come across as an overly complex taxonomy (e.g., within mathematical analogies, he distinguishes further between those based on algebraic, geometric, and asymptotic similarities and within scientific models between those based on feature matching, formal, and parametric similarities). To counter the complexity of his classification, Bartha’s presentation is very methodical and he always uses examples of analogical arguments to illustrate his points. A short introduction and conclusion at the beginning and end of each section also help the reader to stay on top of the material.

Two different justifications for analogical arguments in science are provided in the third part of the book (Chapters 7–8). Bartha’s “top down” justification is based on a pragmatic analysis of the costs and benefits of adopting a working hypothesis. Using an original meta-analogy between the use of analogies in legal and scientific reasoning, he argues that analogical arguments are well suited to attain certain aims, such as coherence, simplicity, and unification, that are generally valued in scientific inquiry (in particular during “normal science,” in Kuhn’s sense). In the second, “bottom up,” justification of analogical arguments, Bartha adduces symmetry arguments in support of analogical reasoning and also shows how to incorporate his theory into a Bayesian framework for updating beliefs.

By Parallel Reasoning concludes with a broader outlook on the role of analogies (Chapter 9). Here, Bartha explores the use of analogical arguments during phases of “revolutionary science” (again in Kuhn’s sense). These often involve analogies between remote branches of science or between branches of science and non-scientific domains, which are accommodated within the general framework by loosening the requirements on prior associations. Bartha ends his book with an exploration of the relation between analogies and thought experiments, an argument that some form of normative theory of analogical arguments should be an essential component of computational and psychological theories of analogy, and some thoughts on what to do with defective analogical arguments. As small additions to Bartha’s considerations, I would like to raise two points here, which admittedly pertain to issues that he only touches upon. First, it is a general problem for analyses of mathematical analogies to accommodate two approaches that differ in what they take as primitive for a characterization of analogies. On the one hand, there is the approach that focuses on relating the objects and relations of the source with those of the target domain, like the approach based on structure-preserving mappings, of which isomorphisms are a particular kind. On the other hand, there is the approach that takes regularities or laws in the two domains as basic. Bartha’s idea of allowing for different kinds of analogies is very promising in this regard. Indeed, his discussion of mathematical analogies that are based on algebraic similarities (161–165) fits well with the second of these approaches. However, influenced by the first approach, he also maintains that it “seems a reasonable constraint on any analogy relation” that an admissible analogy relation be a function between the objects and relations of the source to those of the target (161). It is arguable whether this constraint can be applied to analogies between groups of different cardinalities (Schlimm 2008) or to analogies between 2- and 3-dimensional geometry. My second point concerns what do to when analogical arguments fail. Bartha suggests to “either find a novel representation of the prior association or modify the conclusion of the argument, so that the problematic features cease to be critical” (331). There is, however, at least one other way of repairing a failed analogy, namely by modifying the representation of the target domain. An example of this way of proceeding is Stone’s work on the connection between Boolean algebras and topology. The initial analogy based on superficial similarities failed, but Stone was able to repair it by mapping the operation of ring addition not to addition in Boolean algebras, as he had initially proposed, but to the operation of symmetric difference (see Schlimm 2009, pp. 226–230). To conclude, if one expects a single simple account of analogical reasoning from this book, one will be disappointed. Moreover, its broad scope and deep and nuanced discussions do not make this book a light read. However, the reader who takes the time to work though it will be rewarded. Bartha’s analyses offer a wealth of penetrating insights into analogical arguments, and his articulation model is the most comprehensive attempt at a normative theory of analogical reasoning to date. It is a truly impressive achievement, opening up a promising new perspective for the future study of analogical arguments in both philosophy and cognitive science.