The propositional nature of human associative learning is strongly supported by studies of trace eyeblink and fear conditioning, in which awareness of the contingency of a conditioned stimulus upon an unconditioned stimulus is a prerequisite for successful learning. Studies of animal lesion and human imaging suggest that the hippocampus is critical for establishing functional connections between awareness and trace conditioning.

The target article by Cohen Kadosh & Walsh (CK&W) raises questions as to the precise nature of the notion of abstractness that is intended. We note that there are various uses of the term, and also more generally in mathematics, and suggest that abstractness is not an all-or-nothing property as the authors suggest. An alternative possibility raised by the analysis of numerical representation into automatic and intentional codes is suggested.

Algorithmic-level specifications carry part of the explanatory burden in most psychological theories. It is, thus, inappropriate to limit a comparison and evaluation of theories to the computational level. A rational analysis considers people's goal-directed and environmentally adaptive rationality; it is not normative. Adaptive rationality is by definition non-absolute; hence, neither deductive logic nor Bayesian probability theory has absolute normative status.

Against Mitchell et al.'s assertions, we argue that (1) the concordance between learning and awareness does not support any particular learning theory, (2) their propositional approach is at odds with examples of learned behaviours that contradict beliefs about causation, and (3) the relative virtues of the two approaches in terms of parsimony is more ambiguous than Mitchell et al. suggest.

We comment on the similarities and differences of Mitchell et al.'s framework for understanding classical and operant conditioning and the theoretical framework put forth by Rumbaugh et al. (2007). We propose that all nonhuman and human learning may be based on amalgams created by co-occurring stimuli that share their response-eliciting properties and that these amalgams may be propositional in nature.

Oaksford & Chater (O&C) advocate Bayesian probability as a way to deal formally with the pervasive nonmonotonicity of common sense reasoning. We show that some forms of nonmonotonicity cannot be treated by Bayesian methods.

Cohen Kadosh & Walsh (CK&W) argue that recent findings challenge the hypothesis of abstract numerical representations. Here we show that because, like many other authors in the field, they rely on inaccurate definitions of abstract and non-abstract representations, CK&W fail to provide compelling evidence against the abstract view.

The consideration of deliberate versus automatic processing of numeric representations is important to math education, memory for numbers, and decision-making. In this commentary, we address the possible roles for numeric representations in such higher-level cognitive processes. Current evidence is consistent with important roles for both automatic and deliberative processing of the representations.

Oaksford & Chater (O&C) focus on patterns of typical adult reasoning from a probabilistic perspective. We discuss implications of extending the probabilistic approach to lifespan development, considering the role of working memory, strategy use, and expertise. Explaining variations in human reasoning poses a challenge to Bayesian rational analysis, as it requires integrating knowledge about cognitive processes.

The notion that human associative learning is a usually conscious, higher-order process is one of the tenets of organization theory, developed over the past century. Propositional/sequential encoding is one of the possible types of organizational structures, but learning may also involve other structures.

We argue that Cohen Kadosh & Walsh's (CK&W's) definitions of neural coding and of abstract representations are overly shallow, influenced by classical cognitive psychology views of modularity and seriality of information processing, and incompatible with the current knowledge on principles of neural coding. As they stand, the proposed dichotomies are not very useful heuristic tools to guide our research towards a better understanding of the neural computations underlying the processing of numerical quantity in the parietal cortex.

The probabilistic approach to human reasoning is exemplified by the information gain model for the Wason card selection task. Although the model is elegant and original, several key aspects of the model warrant further discussion, particularly those concerning the scope of the task and the choice process of individuals.

Proust's madeleine illustrates the automatic nature of associative learning. Although we agree with Mitchell et al. that no compelling scientific proof for this effect has yet been reported in humans, evolutionary constraints suggest that it should not be discarded: There is no reason by which natural selection should favor individuals who lose a fast and automatic survival tool.

Cohen Kadosh & Walsh (CK&W) argue that numerical representation is primarily non-abstract. However, in their target article they failed to consider recent behavioral priming experiments. These priming experiments provide evidence for an abstract numerical representation under automatic conditions.

Human cognition requires coping with a complex and uncertain world. This suggests that dealing with uncertainty may be the central challenge for human reasoning. In Bayesian Rationality we argue that probability theory, the calculus of uncertainty, is the right framework in which to understand everyday reasoning. We also argue that probability theory explains behavior, even on experimental tasks that have been designed to probe people's logical reasoning abilities. Most commentators agree on the centrality of uncertainty; some suggest that there is a residual role for logic in understanding reasoning; and others put forward alternative formalisms for uncertain reasoning, or raise specific technical, methodological, or empirical challenges. In responding to these points, we aim to clarify the scope and limits of probability and logic in cognitive science; explore the meaning of the “rational” explanation of cognition; and re-evaluate the empirical case for Bayesian rationality.

I argue that the dual-process account of human learning rejected by Mitchell et al. in the target article is informative and predictive with respect to human behaviour in a way that the authors' purely propositional account is not. Experiments that reveal different patterns of results under conditions that favour either associative or rule-based performance are the way forward.

We concur with Cohen Kadosh & Walsh (CK&W) that representation of numbers in the parietal cortex is format dependent. In addition, we suggest that all formats do not automatically, and equally, access analog magnitude representation in the intraparietal sulcus (IPS). Understanding how development, learning, and context lead to differential access of analog magnitude representation is a key question for future research.

We discuss the necessity of conscious thinking in the single-system propositional model of learning. Research from honeybees to humans suggests that associative learning can take place without the need for controlled reasoning or the development of beliefs of relationships between objects or events. We conclude that a single learning system is possible, but not if it depends on complex thinking.