Barbey & Sloman (B&S) have written a brilliant paper, but they are not explicit about the logic of the elementary set operations they appeal to. As Boolean algebra shows, this logic is the same as elementary propositional logic. The set operations of taking the complement, intersection, and union, which are necessary for natural sampling, are the same Boolean operations (up to isomorphism) as negation, conjunction, and disjunction in propositional logic. In addition to making this general point, I would give more prominence to the logical rule that the authors relegate to the target article's Note 3: the excluded middle. This rule states that all propositions of the form “p or not-p” are logically true. Its central place in natural sampling cannot be explained by the Swiss-army-knife model of the mind. This point reinforces B&S's criticisms of that model.

In the “mind as Swiss army knife” model, the mind consists of many domain-specific modules for reasoning and decision making. There is, for example, supposed to be a domain-specific module for inferences about cheaters in social arrangements (Cosmides Reference Cosmides1989), as well as a separate module for natural sampling. The human mind is not supposed to have a content-independent logical ability to reason about cheaters, natural sampling, or other matters in general. Some supporters of this model even deny that logic is a normative standard for judging rationality, calling it useless “baggage” (Todd & Gigerenzer Reference Todd, Gigenenzer, Gigerenzer and Todd1999, p. 365).

Logic is not, however, useless “baggage” for natural sampling problems, which absolutely depend on the logic of the set operations, including the rule of the excluded middle. Problems that appeal to this rule can be hard for people, but can become easier when the relevance of the rule is made explicit. B&S also refer to a difficult logical problem, called THOG, which becomes easier when its logical form is made transparent using logical trees that reveal nested sets (Griggs & Newstead Reference Griggs and Newstead1982). Consider the following problem (from Levesque Reference Levesque1986) that is simpler to describe than the THOG set up, but which illustrates the same points:

We could call this the “Ann problem.” I have modified it to be about cheating, but it could have any content. It is a hard problem for most people. They respond with “Cannot tell,” though the correct answer is “Yes “(Toplak & Stanovich Reference Toplak and Stanovich2002). The Ann problem is difficult because the relevance of the excluded middle rule is not transparent. It can, however, be made easier by adding the rule explicitly: that Ann is either a cheater or not a cheater. Logicians would say that the Ann problem requires non-constructive inference. The non-constructive step is the application of the excluded middle rule. Thanks to the rule, we can know, from “above,” a priori and logically that Ann is either a cheater or not a cheater, although we never learn which she is in the reasoning – that is the non-constructive aspect. We cannot reduce this reasoning to constructive processing from “below” by a domain-specific module (Over & Evans, forthcoming). Such non-constructive inference is the purest example of rule-based thought in a dual-process model, either in the type that B&S endorse or in other types (Evans Reference Evans2007).

The Ann problem could also be made easier by putting it into a logical tree form, which is so often used to make probability problems easier in natural sampling (Over 2007). The tree would begin with two branches, with one for the possibility of Ann as a cheater and the other for the possibility of Ann as a non-cheater. The tree would reveal that a cheater is looking at a non-cheater in either case. This form of the Ann problem would be closely analogous to a natural sampling problem. Kleiter (Reference Kleiter, Fischer and Laming1994) used logical trees in his seminal work on natural sampling, and others have followed him in this, but not always with the realization that the trees are purely logical constructions (Kleiter realises this, but Zhu & Gigerenzer Reference Zhu and Gigerenzer2006 do not). Such a tree necessarily depends on the rule of the excluded middle.

It is challenging to think of a context in which knowing whether a cheater is looking at a non-cheater could tell us something important about cheating in the real world. With more space, we could discuss possibilities. In any case, it is clear that actual natural sampling could help us judge how far we could trust someone not to be cheater, depending on the number of times he has, or has not, cooperated with us in the past. B&S specify the conditions under which natural sampling can actually be useful and not merely an abstract exercise. Even then, natural sampling presupposes that we can apply the rule of the excluded middle to construct exhaustive subsets. True, we may find it impossible in a practical sense to apply the rule to a vague term, like “is depressed,” rather than a precise one, like “has cancer.” But in that case, actual natural sampling cannot be fully carried out. We could not actually complete the task of constructing exhaustive subsets of the depressed and not depressed. Since “is depressed” is vague, it has borderline cases: people we cannot classify as either being depressed or, alternatively, not being depressed. We cannot do without the excluded middle and other logical rules in natural sampling and in other general reasoning about cheating, probability, and many other matters. We cannot adopt a Swiss army model of the mind and call logic useless “baggage,” while relying on its rules for natural sampling and constructing logical trees (as in Zhu & Gigerenzer Reference Zhu and Gigerenzer2006). The alternative is a dual-process theory that gives logic its proper place in our thought, which is limited but still necessary (Evans & Over Reference Evans and Over1996).