Mitchell et al.'s claim, that their propositional theory is a single-process theory, is illusory because they relegate some learning to a secondary memory process. This renders the single-process theory untestable. The propositional account is not a process theory of learning, but rather, a heuristic that has led to interesting research.

In documenting the dizzying diversity of human languages, Evans & Levinson (E&L) highlight the lack of universals. This suggests the need for complex learning. Yet, just as there is no universal structure, there may be no universal learning mechanism responsible for language. Language is a behavior assembled by many processes, an assembly guided by the language being learned.

It is a foundational prediction of evolutionary theory that human beliefs accurately approximate reality only insofar as accurate beliefs enhance fitness. Otherwise, adaptive misbeliefs will prevail. Unlike McKay & Dennett (M&D), we think that adaptive belief systems rely heavily upon misbeliefs. However, the case for positive illusions as an example of adaptive misbelief is weak.

The dual-code proposal of number representation put forward by Cohen Kadosh & Walsh (CK&W) accounts for only a fraction of the many modes of numerical abstraction. Contrary to their proposal, robust data from human infants and nonhuman animals indicate that abstract numerical representations are psychologically primitive. Additionally, much of the behavioral and neural data cited to support CK&W's proposal is, in fact, neutral on the issue of numerical abstraction.

I agree with Oaksford & Chater (O&C) that human beings resemble Bayesian reasoners much more closely than ones engaging standard logic. However, I have many problems with their “rational analysis” framework, which appears to be rooted in normative rather than ecological rationality. The authors also overstate everyday rationality and neglect to account for much relevant psychological work on reasoning.

Conflation of our unique human endowment for language with innate, so-called universal, grammar has banished language from its biological home. The facts reviewed by Evans & Levinson (E&L) fit the biology of cultural transmission. My commentary highlights our dedicated learning capacity for vocal production learning as the form of our language endowment compatible with those facts.

I argue that adaptive, self-directed misbeliefs are likely more prevalent and important than McKay & Dennett (M&D) claim. Humans often falsely interpret their own behavior in terms of culturally afforded categories. Despite their falsity, such self-interpretations are often adaptive because of our disposition to behave consistently with them. This makes us easier to interpret by similarly enculturated interactants.

In this commentary, we assess the propositional approach to associative learning not only in terms of veridicality and falsifiability, but also in heuristic value. We remark that it has furthered our knowledge and understanding of human, as well as animal, associative learning. At the same time, we maintain that models developed from the association formation tradition continue to bear great heuristic value as well.

Bayesian Rationality (Oaksford & Chater 2007) illustrates the strengths of Bayesian models of cognition: the systematicity of rational explanations, transparent assumptions about human learners, and combining structured symbolic representation with statistics. However, the book also highlights some of the challenges this approach faces: of providing psychological mechanisms, explaining the origins of the knowledge that guides human learning, and accounting for how people make genuinely new discoveries.

In this commentary, I support and augment Cohen Kadosh & Walsh's (CK&W's) argument that numerical representations are not abstract. I briefly review data that support the non-abstract nature of the representation of numbers between zero and one, and I discuss how a failure to test alternative hypotheses has led researchers to erroneously conclude that numerals automatically activate their semantic meaning.

Cohen Kadosh & Walsh (CK&W) neglect the solid empirical evidence for a convergence of notation-specific representations onto a shared representation of numerical magnitude. Subliminal priming reveals cross-notation and cross-modality effects, contrary to CK&W's prediction that automatic activation is modality and notation-specific. Notation effects may, however, emerge in the precision, speed, automaticity, and means by which the central magnitude representation is accessed.

One of the most striking features of “Bayesian rationality” is the detail with which behavior on logical reasoning tasks can now be predicted and explained. This detail is surprising, given the state of the field 10 to 15 years ago, and it has been brought about by a theoretical program that largely ignores consideration of cognitive processes, that is, any kind of internal behavior that generates overt responding. It seems that an increase in explanatory power can be achieved by restricting a psychological theory.

The commentaries raise a host of challenging issues and reflect a broad range of views. Some commentators doubt that there is any convincing evidence for adaptive misbelief, and remain (in our view, unduly) wedded to our “default presumption” that misbelief is maladaptive. Others think that the evidence for adaptive misbelief is so obvious, and so widespread, that the label “default presumption” is disingenuous. We try to chart a careful course between these opposing perspectives.

Understanding the universal aspects of human language structure requires comparison at multiple levels of analysis. While Evans & Levinson (E&L) focus mostly on substantive variation in language, equally revealing insights can come from studying formal universals. I first discuss how Artificial Grammar Experiments can test universal preferences for certain types of abstract phonological generalizations over others. I then discuss moraic onsets in the language Arrernte, and how its apparent substantive variation ultimately rests on a formal universal regarding syllable-weight sensitivity.

We propose that learning has a top-down component, but not in the propositional terms described by Mitchell et al. Specifically, we propose that a host of learning processes, including associative learning, serve to imbue the representation of the conditioned stimulus (CS) with affective meaning.

The main conclusion to draw from Mitchell et al's article is that it is difficult to disentangle cognitive and learning processes in contingency and causal experiments. More compelling evidence for human associate learning comes from research where, because of the type of events involved, participants are unable or unlikely to think about the relationships between the events.

Bayesian rationality is an important contribution to syllogistic inference, but it has limitations. The claim that confidence in a conclusion is a function of informativeness of the max-premise is anomalous because this is the least probable premise. A more plausible account is that confidence is inversely related to complexity. Bayesian rationality should be supplemented with principles based on cognitive complexity.

In our opinion, the discontinuity between extant human and nonhuman minds is much broader and deeper than most researchers admit. We are happy to report that Evans & Levinson's (E&L's) target article strongly corroborates our unpopular hypothesis, and that the comparative evidence, in turn, bolsters E&L's provocative argument. Both a Universal Grammar and the “mental continuity” between human and nonhuman minds turn out to be modern myths.

Cohen Kadosh & Walsh (CK&W) present convincing evidence indicating the existence of notation-specific numerical representations in parietal cortex. We suggest that the same conclusions can be drawn for a particular type of numerical representation: the representation of time. Notation-dependent representations need not be limited to number but may also be extended to other magnitude-related contents processed in parietal cortex (Walsh 2003).