The study of neuronal specialisation in different cognitive and perceptual domains is important for our understanding of the human brain, its typical and atypical development, and the evolutionary precursors of cognition. Central to this understanding is the issue of numerical representation, and the question of whether numbers are represented in an abstract fashion. Here we discuss and challenge the claim that numerical representation is abstract. We discuss the principles of cortical organisation with special reference to number and also discuss methodological and theoretical limitations that apply to numerical cognition and also to the field of cognitive neuroscience in general. We argue that numerical representation is primarily non-abstract and is supported by different neuronal populations residing in the parietal cortex.

The recognition that contentful universals are rare and often “banal” does not undermine the fact that most non-universal but recurring patterns of language are amenable to explanation. These patterns are sensical or motivated solutions to interacting and often conflicting factors. As implied by the Evans & Levinson's (E&L's) article, linguistics would be well served to move beyond the essentialist bias that seeks universal, innate, unchanging categories with rigid boundaries.

Defending first-person introspective access to own mental states, we argue against Carruthers' claim of mindreading being prior to meta-cognition and for a fundamental difference between how we understand our own and others' mental states. We conclude that a model based on one mechanism but involving two different kinds of access for self and other is sufficient and more consistent with the evidence.

Reasoning with conditionals is often thought to be non-monotonic, but there is no incompatibility with classical logic, and no need to formalise inference itself as probabilistic. When the addition of a new premise leads to abandonment of a previously compelling conclusion reached by modus ponens, for example, this is generally because it is hard to think of a model in which the conditional and the new premise are true.

McKay & Dennett (M&D) suggest that some positive illusions are adaptive. But there is a bidirectional link between memory and positive illusions: Biased autobiographical memories filter incoming information, and self-enhancing information is preferentially attended and used to update memory. Extending M&D's approach, I ask if certain false memories might be adaptive, defending a broad view of the psychosocial functions of remembering.

Oaksford & Chater (O&C) argue that a rational analysis is required to explain why a functional process model is successful, and that, when a rational analysis is intractable, the prospects for understanding cognition from a functional perspective are gloomy. We discuss how functional explanations can be arrived at without seeking the optimal response function demanded by a rational analysis, and argue that explaining function does not require optimality.

McKay & Dennett (M&D) look for adaptive misbeliefs that result from the normal, though fallible, functioning of human cognition. Their account can be substantially improved by the addition of two elements: (1) significance of a belief's testability for its functionality, and (2) an account of reason appropriate to understanding systemic misbelief. Together, these points show why religion probably is an adaptive misbelief.

This response defends the view that human metacognition results from us turning our mindreading capacities upon ourselves, and that our access to our own propositional attitudes is through interpretation rather than introspection. Relevant evidence is considered, including that deriving from studies of childhood development and other animal species. Also discussed are data suggesting dissociations between metacognitive and mindreading capacities, especially in autism and schizophrenia.

Evans & Levinson (E&L) claim Kiowa number as a prime example of the semantically unexpected, threatening both Universal Grammar and Linguistic Universals. This commentary, besides correcting factual errors, shows that the primitives required for Kiowa also explain two unrelated semantically unexpected patterns and derive two robust Linguistic Universals. Consequently, such apparent exceptionality argues strongly for Universal Grammar and against E&L.

The type of processing of numerical dimensions varies greatly and is governed by context. Considering this flexibility in tandem with a fuzzy demarcation line between automatic and intentional processes, it is suggested that testing the effect of notation should not be confined to automatic processing, in particular to passive viewing. Recent behavioral data satisfying the authors' stipulations reveal a considerable, though perhaps not exclusive, core of common abstract processing.

The past 50 years have seen an accumulation of evidence suggesting that associative learning depends on high-level cognitive processes that give rise to propositional knowledge. Yet, many learning theorists maintain a belief in a learning mechanism in which links between mental representations are formed automatically. We characterize and highlight the differences between the propositional and link approaches, and review the relevant empirical evidence. We conclude that learning is the consequence of propositional reasoning processes that cooperate with the unconscious processes involved in memory retrieval and perception. We argue that this new conceptual framework allows many of the important recent advances in associative learning research to be retained, but recast in a model that provides a firmer foundation for both immediate application and future research.

Belief in evolved belief systems stems from using a population-genetic model of evolution that misconstrues the developmental relationship between genes and behaviour, confuses notions of “adapted” and “adaptive,” and ignores the fundamental role of language in the development of human beliefs. We suggest that theories about the evolution of belief would be better grounded in a developmental model of evolution.

Oaksford & Chater (O&C) aim to provide teleological explanations of behavior by giving an appropriate normative standard: Bayesian inference. We argue that there is no uncontroversial independent justification for the normativity of Bayesian inference, and that O&C fail to satisfy a necessary condition for teleological explanations: demonstration that the normative prescription played a causal role in the behavior's existence.

Conditional universals have always interested linguists more than unrestricted universals, which are often impossible to demonstrate empirically because categories cannot be defined in a cross-linguistically meaningful way. But deep dependencies have not been confirmed by more recent empirical research, and those universals with solid empirical support mostly relate to scalar patterns that can plausibly be related to processing cost.

The theory put forward by Cohen Kadosh & Walsh (CK&W) proposing that semantic representations of numerical magnitude in the parietal cortex are format-specific, does not specify how these representations might be constructed over the course of learning and development. The developmental predictions of the non-abstract theory are discussed and the need for a developmental perspective on the abstract versus non-abstract question highlighted.

Like number representation, basic arithmetic seems to be a natural candidate for abstract instantiation in the brain. To investigate this, researchers have examined effects of numeral format on elementary arithmetic (e.g., 4+5 vs. four+five). Different numeral formats often recruit distinct processes for arithmetic, reinforcing the conclusion that number processing is not necessarily abstracted away from numeral format.

Nothing in McKay & Dennett's (M&D's) target article deals with the issue of how the adaptivity, or some other aspect, of beliefs might become a biological adaptation; which is to say, how the functions discussed might be coded in such a way in the brain that their development was also coded in gametes or sex transmission cells.

Oaksford & Chater (O&C) rely on a data fitting approach to show that a Bayesian model captures the core reasoning data better than its logicist rivals. The problem is that O&C's modeling has focused exclusively on response output data. I argue that this exclusive focus is biasing their conclusions. Recent studies that focused on the processes that resulted in the response selection are more positive for the role of logic.

We discuss findings on evaluative conditioning (EC) that are problematic for the “conscious reasoning/propositional knowledge” account of learning, namely, dissociations between conscious beliefs and acquired (dis)liking. We next argue that, both for EC and for Pavlovian learning in general, conditioned responding cannot rationally be inferred from propositional knowledge type “CS refers to/signals US,” and that, therefore, performance cannot be explained.

Universal Grammar promotes the study of an idealization of language behavior and language learning. In examining the diversity of actual behavioral strategies used to achieve linguistic goals, Evans & Levinson (E&L) move towards studying language as a behavior. This approach can benefit from studying communicative and cognitive capacities more broadly – across species. We exhort like-minded linguists to cast off the remaining intellectual shackles of linguistic speciesism.