[Some somewhat random reflections]
Among the most intriguing findings in the literature on reasoning in recent years is that of content effects on the Wason selection task—the differential performance people display on the selection task, depending on whether the task is “descriptive” or “thematic”.[1] Wason and others found that in abstract descriptive versions of the task (e.g., “If a card has a vowel on one side then it has an even number on the other side”), only five to ten percent of subjects choose correctly. In more concrete, but still descriptive, versions, performance improves but not dramatically, with correct selection tending to remain at about 25 percent of subjects or fewer. By contrast, if conditionals are “thematic” in nature (e.g., deontic or stating a social contract), performance improves dramatically (Griggs & Cox, 1982; Cheng & Holyoak, 1985; Cosmides, 1989; Gigerenzer & Hug, 1992, Fiddick, Cosmides, & Tooby, 2000). For example, in the “drinking age problem” (Griggs and Cox, 1982), subjects are cued into the role of a bouncer in a Boston bar and told they’ll lose their job unless they enforce the deontic rule,
“If a person is drinking beer, then he must be over 20 years old.”
With “Drinking beer” as “P,” “Drinking ginger-ale” as “not-P,” “22 years old” as “Q,” and “19 years old” as “not-Q,” about 75 percent of subjects provide the correct P-and-(not-Q) response (Griggs and Cox, 1982 (73%); Cosmides, 1985)—a pattern of response generally found with deontic conditionals.
Content effects have been said to invalidate the mental logic approach to conditional reasoning (e.g., Beth and Piaget, 1966; Braine, 1978; Braine and O’Brien, 1991) in which “natural logic” is seen as an instantiation of standard logic (Manktelow and Over, 1987; Johnson-Laird and Byrne, 1991, p. 77). In its stead, two dominant approaches have been advanced to explain content effects:
(1) That content-specific schemata are invoked to represent and reason about thematic situations, enabling people to grasp the implications of thematic rules more effectively than those of descriptive rules.
(2) That domain-general reasoning mechanisms are used in both descriptive and thematic tasks, but factors that cause variations in performance on descriptive versions are, for various reasons, especially conducive to “correct” reasoning on thematic versions.
Variants of the content-specific approach include pragmatic reasoning schema theory (Cheng & Holyoak, 1985, 1989; Cheng, Holyoak, Nisbett, & Oliver, 1986; Politzer & Nguyen-Xuan, 1992) according to which we employ schemas, acquired by induction and stored in long-term memory, to reason about deontic rules; and social contract theory (Cosmides, 1985, 1989; Cosmides & Tooby, 1989; Gigerenzer & Hug, 1992; Fiddick, Cosmides, & Tooby, 2000; Beaman, 2002), which posits innate “Darwinian algorithms” specialized for reasoning about social contracts and detecting violators. The main evidence for content-specific views lies in the rapidity and effectiveness with which people reason about deontic and/or social contract rules versus the slower, more plodding performance displayed on descriptive rules. Social contract theorists have even taken subjects’ strong performance on social contract rules to suggest the existence of a dedicated “module”—a “special purpose computational mechanism” characterized by “high-speed” and “informational encapsulation” (Fodor 1983)—for detecting social contract violators. If a cheater detector as such has evolved, it would operate more or less automatically (at little or no cognitive cost) and independently of utility considerations.
Under the domain-general approach, on the other hand, conditional reasoning is a “central cognitive process” involving human synthetic abilities. Positing vaguer, more conscious reasoning mechanisms than domain-specific approaches, domain-general accounts typically center around “economic” considerations of cost and benefit—the cognitive cost of reasoning to a given conclusion versus the potential cognitive gain from appropriate data selection. While some theorists have emphasized the cost side in this “cost-benefit calculus” (e.g., the mental models approach of Johnson-Laird & Byrne, 1992), others have emphasized the benefit side (utility-maximization in Manktelow & Over, 1991 and Kirby 1994; and rational information-gain in Chater & Oaksford, 1995). The main evidence for a given domain-general account lies in the predictive power of a posited “metric” used to measure cost and benefit.
Another kind of account maintains that performance on the selection task is not guided by “reasoning” per se but by heuristics and biases involved in comprehension of the task. For example, Evans (1972) attributes a tendency of subjects to select P and Q cards (the most common selection on descriptive versions of the selection task) to a “matching bias,” i.e., a tendency to focus on the properties or propositions named in the conditional rule being considered. The strongest evidence for this account is that the tendency to select named properties or propositions is seen even when one or both the antecedent and consequent are negated. In the “relevance theory” approach of Sperber et al. (1995; see also Girotto, Cara, and Sperber, 2000; and Sperber and Girotto, 2002), performance is similarly seen as guided by “intuitions of relevance,” in this case by subjects’ interpretations of the communicative intentions of speakers. Since, as discussed below, these authors view performance on the selection task as wholly explained by relevance theory, effects on performance arising from the contents of the rules (or content effects) disappear. More generally, on such accounts “reasoning” (evaluating or inferring conclusions from premises) is seen as pre-empted by the comprehension process, so that the question of whether “reasoning” on the selection task is domain-general or domain-specific does not arise.
Sperber and Girotto (2002) raise a particularly troublesome issue about the selection task in their criticism of its use in experiments by Cosmides and collaborators, experiments designed to show the existence of an innate “cheater detector.” Sperber and Girotto argue that in such studies the Wason selection task is confused with a “trivially simple” categorization exercise in which, rather than reasoning deductively to search for counter-examples of a conditional rule, subjects deploy their knowledge of the concept of “cheater” to find potential cheaters. “Cheater” is a well-known compound concept that combines the properties “takes benefit” and “does not pay cost”; and not surprisingly, people readily pick out these properties (or behaviors that exemplify these properties) when asked to find potential “cheaters.” But such an exercise, Sperber and Girotto maintain, exhibits nothing more interesting than an ability to pick out the properties of any known compound concept, as seen, for example, in a similar “selection task” (op. cit.) in which subjects pick out the properties “moves through the air” and “does not have an engine” to identify a potential “glider.”
Sperber and Girotto’s argument may have broader implications than the authors observe. For it may apply to any version of the selection task. Taking Wason’s original abstract descriptive version of the task,
“If a card has a vowel on one side, then it has an even number on the other side.”
—the task may, in some sense, be said to involve utilizing one’s “concept” of “card with vowel on one side and odd (or ‘non-even’) number on the other side” to find such a card. Of course, one must first arrive at such a concept in order to use it. And to arrive at the concept, one must at least recognize the nature of the task—that it involves finding counter-examples in order to falsify the proposed conditional rule. Thus if people performed the selection task by this route, their performance may fairly be called “reasoning.” The conclusion in Wason’s original work of course is that with descriptive conditionals people generally do not reason in this way.
But what about thematic versions of the task? To find a violator of a conditional rule stating an obligation or entitlement, as in deontic rules, one may consult the concept of “violator,” just as to find a cheater on a social contract, one may consult the concept of “cheater.” And conversely, in the drinking age problem, people would plausibly be just as good at finding “lawful beer drinkers” as unlawful ones—or for that matter, “people to whom the drinking laws do not apply,” provided they understand the concept. Or if asked to determine who is endangering oneself in a given situation, is it not inherent to the concept of self-endangerment that one is both “taking a risk” and “not taking the proper precaution,” however these properties are instantiated? Thus, Sperber and Girotto’s argument appears to associate thematic versions of the selection task, where subjects look for violators of a given rule, with “categorization,” and descriptive versions, where subjects seek to determine whether a rule is true or false, with “reasoning.” The view that thematic versions of the selection task involve categorization rather than reasoning appears to be an aspect of Sperber and Girotto’s overall argument, which is that performance on the selection task is determined by how the subject understands the communicative intentions of a speaker. If the subject is in effect instructed to find a “violator,” a “cheater,” a “glider,” or a “lawful beer drinker,” one will find them by consulting one’s concepts for these things.
In the account presented here, as with relevance theory, performance on the selection task is seen as guided by perceived communicative intentions. But here it is argued that communicative intentions are grasped not solely through “intuitions of relevance,” as Sperber et al. (1995) claim, but through a variety of means both semantic and syntactic. One such communicative intention, easily grasped, is that of universal-quantification of a conditional statement, an intention effected in two ways: through empirical guaranty (e.g., one has seen all instances of x, and can therefore confirm that “all x are F”); and by stating what I shall call a “formal rule,” a structural relation, inhering in the social or natural environment, in which universal quantification is implicit (e.g., that some property F is a structural feature of in all entities x). As shown below, conditional rules that are understood as universally-quantified in either of these ways evoke strong performance on the selection task. The distinction between “reasoning” and “categorization,” sharply drawn by Sperber and Girotto, is not maintained here, since, as discussed below, there is no fundamental syntactic difference between descriptive and thematic “formal rules,” and hence no basis for distinguishing the processes involved in “logical” performance on the two versions of the task.
The implication of this account is that “content effects” are real but related to content only in the sense that quantification of conditional rules must be unambiguously universal to evoke strong performance on the selection task. In thematic cases, a rule is assumed to exist; thus the rule must apply universally over a defined range of entities. In descriptive cases, it is not clear that a rule exists—it is the subject’s task to determine whether a rule exists—and so universal application cannot be assumed except in exceptional cases, as discussed below. It is this simple difference—unambiguous universal quantification of thematic rules, unambiguous universal quantification only in some cases of descriptive rules—that explains content effects on the selection task.
Formal and Epistemic Rules
“Correct” performance on descriptive versions of the selection task can easily be elicited, as seen, for example, in Sperber et al. (1995), discussed below, in which a reliable “recipe” for facilitating correct performance is advanced. A grasp of conditional implication is therefore not alien to human reasoning. However, two factors appear to undermine this ability.
First, material implication does not appear to hold in cases where the consequent is true regardless of the antecedent. For example, one might say, “If you’re hungry, then there are $20 dollars in the drawer”—a plausible statement but seemingly not an example of material implication. In particular, contraposition (the inference schema whereby the premises if P then Q and not-Q lead to not-P) is invalid in this case, since one may still be hungry even if there are not $20 in the drawer (not-Q). Given such loose usage of conditional phraseology, people may not be pre-disposed to immediately grasp universal material implication when that is in fact the meaning of a statement. Braine and O’Brien (1991, p. 187) suggest that since contraposition is not valid for all P, Q, people may not derive it as an inference schema, a failure that may account for the widespread failure to make modus tollens deductions as well as for the failure to select not-Q in the selection task. However, the fact that people easily perform these deductions under certain conditions (described below) suggests that there may be another explanation (as discussed below). In any case, it can be argued that the true semantics of such statements do not in fact entail that the consequent is true regardless of the antecedent. For example, “If you’re hungry, then there are $20 dollars in the drawer” may plausibly be interpreted deontically as, “If you’re hungry, then you may have $20 dollars that are in the drawer,” to which contraposition applies. Braine and O’Brien give the example, “If you want some water, there is a fountain in the hall,” a statement that appears unlikely to admit of contraposition. However, the sentence (in most contexts) is not a conditional at all but a simple statement to the effect that “There is a fountain in the hall,” with the if-clause present only to justify providing the information. Moreover, it is possible to construct a context in which the statement has a clear conditional meaning. For example, we might imagine a magical world where objects are willed into and out of existence by mental acts, and thus be able to say, “If you want some water, then a fountain is [will appear] in the hall”—a statement to which contraposition applies.
Second, even in cases where an “if…then” structure is taken to imply universal conditional implication—that all entities that have the feature P also have the feature Q—logically there are two possible interpretations of such a statement: (1) that property Q just happens to occur whenever P occurs, perhaps coincidentally or as a result of other factors that in future may not be present; (2) that the properties P and Q are structurally related. The former case, which may correspond to statistical regularity, allows for the possibility that there is no mechanism behind the conditional relation. The latter case, on the other hand, in which a structural relation between P and Q is recognized, defines the conditions in which people are most likely to select P-and-(not-Q) in the selection task.
But when is a structural relation between P and Q recognized? Structural relations are characteristic of what I shall call “formal rules,” where “formal” implies that a conditional statement has an auxiliary that binds the antecedent and consequent formally. Examples of formal rules are deontic rules as in the drinking age problem in which the conditional statement can be written (following parsing similar to Fodor, 2000):
“It is required that, if a person drinks beer, he is over 20 years old.”
With a formal rule it is clear, from the statement itself or from its context, under what authority the antecedent and consequent are combined. This authority or principle of synthesis can be explicitly stated in an “outcome clause” that, though typically omitted, is implicit in the rule. In the drinking age rule, it may be added in brackets as in:
“It is required that, if a person drinks beer, he is over 20 years old [to conform with the drinking laws].”
That is, drinking beer and being over 20 years old are bound to each other in the speaker/hearer’s mind by law.
More abstractly, one could write,
“It is required, for R, that, in all instances of P, Q.”
—where the term following the connective (R) is a blank that signifies recognition of some (perhaps hidden and not necessarily well understood) structural relation that entails the universal conditional implication of Q whenever P. Such structural relations can be social or physical laws, rules of etiquette, social contracts, configurations of the physical environment such that safety is threatened (or recognized as threatened) under certain circumstances, etc.
What I shall call an “epistemic” rule, by contrast, expresses no formal relation but merely a perceived regularity in which there is no salient relation between antecedent and consequent. For example, the statement, “If a bird is a swan, it is white,” will not generally be understood to include an auxiliary that binds the antecedent and consequent formally, as in, “It is required that, if a bird is a swan, it is white.” The only authority that can bind antecedent and consequent formally in conditionals expressing facts about nature (e.g., “requiring” that all swans are white) is a scientific law. And although descriptive rules can be understood as scientific laws (as discussed below), generally they are not. Thus, “If a bird is a swan, it is white,” is not plausibly explicated by:
“It is required that, if a bird is a swan, it is white [to conform with scientific law].”
(Or alternatively, “It is a scientific law that, if a bird is a swan, it is white.”)
Formal and epistemic rules can be distinguished in terms of what I shall call their “implicit syntax.” Two aspects of the implicit syntax of formal rules distinguish them from epistemic rules. One is that, as Fodor (2000) argues, epistemic rules are about entities with the property P also having the property Q, in other words, a simple predicate-argument statement such as “Swans are white.” Formal rules, on the other hand, are about Q being required. The latter interpretation is hopefully captured by the above parsing of formal rules where the P clause, enclosed in commas, expresses not the sentence topic but (adjectivally) the range of entities over which the requirement that Q (where “requirement that Q” is the sentence topic) applies. The other aspect of syntax that distinguishes formal from epistemic rules is the presence of an implicit outcome clause in the former but not the latter. It is the presence of the implicit outcome clause that determines that the sentence topic is a requirement and not some entity with the property P.
More crucially, however, formal and epistemic rules differ in terms of quantification. While application of a formal rule is universal within the range of entities x having the property P, epistemic rules apply not necessarily to all such entities but only to at least some (and possibly all) of them. These different quantifications have enormous implications for what testing strategy one may rationally employ in the selection task. In epistemic cases, finding an instance of P that is not an instance of Q will not disconfirm the rule, since the rule under this interpretation only states that at least some instances of P are also instances of Q. On the other hand, the more instances of P that are found to be instances of Q, the stronger is one’s evidence that the rule is true. Thus, with epistemic rules a confirmatory “positive test strategy” is potentially more informative than a falsificationist one. And indeed, as noted above, the most common response on selection tasks involving epistemic rules is the confirmatory P and Q response.
When is universal quantification of an epistemic conditional unambiguous?
With descriptive conditional rules, as noted, there are two ways of effecting universal-quantification: through empirical guarantee and through statement of a formal rule (which also provides a guarantee though not an empirical one). The key difference between the two is the agent of the guarantee. For an empirical guarantee, the agent is the speaker him or herself—someone who has observed all instances of x and can confirm that all x are F. For a formal rule, it is a law or norm external to the speaker. How is a guarantee of either sort communicated?
One way—perhaps the most common—is through denial of an existentially-quantified statement. If in response to a statement of the form
x(Fx), one asserts: Denied(
x(Fx))
[There
does not exist an entity x such that x is
F.] —one in effect asserts:
x(~Fx)
[For
all x, x is not F.]
In contrast to a typical universally-quantified epistemic statement (“All swans are white”), whose universal-quantification will tend not to be taken literally, universal-quantification here, by virtue of the semantics of the situation, is part of the meaning of the statement.
As a practical illustration, consider the following colloquy:
Ms. Jones: But your mother’s relatives have red hair.
Mr. Benson: Yes, all members of my family have red hair. If one is a Benson, then one has red hair.
The conditional statement is a standard epistemic conditional and should not evoke widespread “logical” performance (i.e., selection of P-and-(not-Q): “Bensons” and “non-red-headed”) on the selection task. Although the statement, taken literally, asserts that all Bensons are red-headed, for reasons discussed above it does not have the force of a guarantee. Thus there is room for skepticism, for example, that the speaker really has in mind all members of his family while making his comments. His statement may therefore be construed as probabilistic, and so a rational response on the selection task might be P-and-Q (“Bensons” and “red-heads”), as indeed is the most frequent response on epistemic versions of the selection task.
However, consider the following modified version of the colloquy[2]:
Ms. Jones: But your mother’s relatives have brown hair.
Mr. Benson: No, all members of my family have red hair. If one is a Benson, then one has red hair.
Here there is a guarantee of a universal association, one that arises not from an “outcome clause” as in formal rules, but from our intuition that the speaker has in mind all members of the Benson family. Why would we have this intuition? Fundamentally, because Mr. Benson’s statement is a denial of an existentially-quantified statement and so guarantees a negative. And semantically, a guarantee of a negative is universally-quantified. In terms of a chain of semantic entailment, where the response type entails the locution which entails the quantification, we have:
Denial of existentially-quantified statement (response type) à guarantee of negative (locution) à universal quantification (quantification)
In comprehending Mr. Benson’s communication as a guarantee that no Bensons have brown hair or other than red hair, his audience must understand him as considering all possible counter-examples (all Bensons who might have brown hair or are other than red-headed). Even if his audience doubts that Mr. Benson has in fact considered all possible counter-examples, it is part of the meaning of his statement that he has.
On the other hand, in the first colloquy Mr. Benson’s response is an affirmation of an existentially-quantified statement, and as an affirmation it can be either a guarantee of a positive (as literally Mr. Benson’s conditional statement is) or an existentially-quantified statement of support. Thus we have:
Affirmation of existentially-quantified statement à guarantee of positive or existentially-quantified statement of support à universal or existential quantification
Given this ambiguity, Mr. Benson’s audience has leeway in how it interprets his statement. In particular, if one is prone to taking the universal-quantification of epistemic conditionals with a grain of salt, one may interpret Mr. Benson’s statement as existentially-quantified.
The evocation of “logical” performance on the selection task by this colloquy follows a “recipe” put forward in Sperber et al. (1995) for facilitating “logical” performance in cases of descriptive rules. Although, as shown below, all examples discussed by Sperber et al. are formal rules rather than empirical guarantees, the situational semantics is the same in either case: if a conditional statement is a denial of existentially-quantified statement, it must be construed as unambiguously universally-quantified.
Relevance Theory
To be “relevant,” new information must have “cognitive effects,” i.e., it must add new beliefs or cause old beliefs to be abandoned. Sperber et al. list three ways in which a conditional statement may have relevance, each associated with different card selections in the selection task. For example, (1) If P then Q may imply P→Q (i.e., of any given card having the feature P, it has the feature Q), in which case only the P card may be checked. Or (2) If P then Q may imply that there are instances of P and Q, so that subjects may select P and Q to “singly verify or jointly falsify this existential assumption.” Finally (3), per the definitions of “if” and “then,” If P then Q may imply that all entities having the property P also have the property Q, a statement interchangeable with the implication that no entities not having the property Q have the property P. Only under interpretations of the latter type should people perform logically on the selection task. But what would make subjects interpret a descriptive conditional statement in this manner? According to Sperber et al., the rule must be interpreted as a denial of the occurrence of P-and-(not-Q) cases. “Relevance” then consists in the fact that the rule contradicts a positive existential statement or, equivalently, implies a negative universal statement, since both entail that a prior belief, namely, that cases of P-and-(not-Q) exist, is untrue.
For example, suppose a science teacher asserts:
“There exist some birds that are not feathered.” (1)
The teacher’s assistant in a lab session then contradicts the teacher by saying:
“If an animal is a bird, then it is feathered.” (2)
If subjects are presented with cards that say “Albatross,” “Moose,” “Feathered,” and “Non-feathered,” and are asked to check whether the stated conditional (2) is true or false, relevance theory predicts they will make the logical P-and-(not-Q) selection (“Albatross” and “Non-feathered”) because the conditional statement is relevant by virtue of its denial that there are cases of P-and-(not-Q). Subjects thus look for cases of P-and-(not-Q) (non-feathered birds), since finding such cases will alter prior beliefs (have cognitive effects).
However, there is reason to suspect that interpreting a descriptive conditional as a denial of P-and-(not-Q) cases generates in the subjects’ mind a formal rule with implicit syntax such as “It is required that (or is a law of nature that), if an animal is a bird, then it is feathered.” If so, the teaching assistant in effect says, “[No, the professor was wrong because it’s a law of nature that] if an animal is a bird, it is feathered.” Such a statement is equivalent not merely to “there are no cases of P-and-(not-Q)” but to “it is impossible that there are cases of P-and-(not-Q).” Since only a law or formal rule can guarantee a negative in this manner, it seems plausible that subjects would have such a rule in mind in interpreting the teaching assistant’s statement. Indeed, scientific laws are commonly expressed as the denial of the possibility of some phenomenon, e.g., “There are no cases of a falling body not falling at 32 feet per second per second (barring friction).”
Sperber et al. conduct an experiment in which subjects are presented with the following context story:
“Imagine that you are a journalist and that you are preparing a piece on the small principality of Bagurstan. The reigning Prince studied Economics at Oxford and has imposed a radical form of liberalism upon his country. In Bagurstan, people retire at 65, students get a salary, but they have no social security, no right to work, no unemployment benefits, no minimum salary, no maternity leave. Yet, the Prince claims that in Bagurstan there are no serious economic or social problems. Economic mechanisms alone allow everyone to find a satisfactory solution.”
Appended to the story are the following sentences, each given to one of two subject groups (n = 20 in each):
[relevance condition] “For instance, the Prince asserts, in my country, if a person is of working age, then this person has a job.”
[irrelevance condition] “Of course, the Prince adds, in my country, if a person is older than 65, then this person is without a job.”
Presented with cards stating, “AGE: 32,” “AGE: 79,” “JOB: YES,” and “JOB: NO,” 70 percent of subjects selected P-and-(not-Q) under the relevance condition and 25 percent selected it under the irrelevance condition. Sperber et al. attribute this result primarily to their manipulation of effect (and to a lesser extent effort) in the two conditions. The context story describes a country under a “radical” free market economic regime that lacks elements of a social safety net such as social security and a right to work. One may then wonder whether the country suffers from social ills such as unemployment, a situation that in the relevance condition corresponds to cases of P-and-(not-Q). The Prince, however, denies the occurrence of P-and-(not-Q) by asserting that “economic mechanisms” ensure full employment, per free market theory. The “relevance” of the Prince’s statement lies in its denial of preconceptions one may have reasonably had, given the context story. Subjects therefore, according to Sperber et al., look for counter-examples of the form the Prince claims are impossible, namely, cases of P-and-(not-Q). By contrast, in the irrelevance condition, the Prince’s statement cannot be interpreted as a denial of some prior existentially-quantified hypothesis since one would have expected that people over 65 do not work (indeed it was stated that they do not).
However, it is arguable that in the relevance condition the Prince’s statement has the force of a formal rule. That is, the Prince asserts not just that there is no unemployment, but that it is (theoretically) impossible that there should be unemployment. Why would this be? In part because of aspects of the context story—references to studying “Economics at Oxford” where one presumably learns the theory being imposed, “radical form of liberalism,” “economic mechanisms,” etc. But also, the implication of formal rule arises from the structure of any context story that evokes “relevance” in the way outlined in Sperber et al. A conditional statement that is a denial of some prior assumption, even if tentatively held, typically asserts not merely that there are no cases of P-and-(not-Q) but that it is impossible that there should be such cases. The reason for this, as noted above, is that a conditional statement that denies an existentially-quantified proposition is a guarantee of a negative, and thus may be said to take on the structure of a scientific theory. This is the insight behind Popperian falsificationism—that scientific theories do not affirm positive propositions but guarantee negative ones. Thus the Prince’s statement might be explicated as:
“It is required that, if a person is of working age, this person has a job [in conformity with economic theory].”
Or perhaps more aptly,
“It is an economic law that, if a person is of working age, this person has a job.”
If this is so, then in this context looking for factual counter-examples and looking for violations of a rule are one and the same.
To separate the influence of relevance from that of formal rules, it is necessary to create a context in which the Prince’s statement is not an obvious denial of P-and-(not-Q) cases but is explicitly a statement of a formal rule. For example, we might have:
“Imagine that you are a journalist and that you are preparing a piece on the small principality of Bagurstan. The reigning Prince studied Economics at Oxford and has imposed a radical form of liberalism upon his country. In Bagurstan, people retire at 65, students get a salary, but they have no social security, no right to work, no unemployment benefits, no minimum salary, no maternity leave. Yet, the Prince claims that in Bagurstan there are no serious economic or social problems. Economic mechanisms alone allow everyone to find a satisfactory solution. Indeed, as you wander around the principality, things look quite rosy. Interviewing residents at random, you find no one of working age who is out of work. You see signs of wealth everywhere. All the residents appear to drive luxury automobiles, live in large houses, and eat well. You conclude that Bagurstan is a place where the theory of free markets and reality coincide. Later, expounding on his economic philosophy, the Prince remarks that wherever free market liberalism is imposed, the ‘automatic mechanism’ of the ‘invisible hand’ ensures full employment. The Prince says: ‘if a person is of working age, then this person has a job’.”
If the formal rule account of the selection task is correct, subjects should perform about as well with this context story as with the one provided by Sperber et al., even though this one doesn’t follow their “recipe” for evoking correct performance.
The Account so Far
(1) For a conditional rule to evoke widespread “logical” performance on the selection task, it must be unambiguously universally-quantified.
(2) To be seen as unambiguously universally-quantified, a conditional rule must guarantee a negative, whether the basis for this guarantee is an abstract rule (as in formal rules) or specific knowledge of a speaker (as in epistemic rules).
(3) For an epistemic rule to guarantee a negative (and thus be understood as universally-quantified), it must be a denial of an existentially-quantified statement.
(4) For a descriptive formal rule to guarantee a negative (and thus be understood as universally-quantified), it must be a denial of an existentially-quantified statement.
(5) Absent an empirical guarantee or formal rule interpretation of an epistemic rule, such rules are (shrewdly) interpreted as existentially-quantified.
An implication of this account is that content effects on the selection task arise mainly from ambiguity regarding universal quantification, and therefore should disappear if conditional statements are unambiguously universally quantified. This account contrasts with content-specific approaches in that it posits no special ability people have of grasping the implications of deontic and/or social contract rules versus those of descriptive or epistemic rules. And it contrasts with domain-general accounts in that it denies the fundamental explanatory power of the various cost-benefit “metrics” that have been proposed to account for performance on the selection task. Performance instead is seen as determined by nothing more glamorous than correct sentence comprehension, which is sometimes affected by syntax (or “implicit syntax”), as in formal rules, and sometimes by semantics, as in epistemic rules. The larger implication is that human beings are not inept reasoners, as some early commentators on Wason’s results suggested, but “pragmatic virtuosos” (to borrow a term from Girotto et al., 2000), with language comprehension abilities finely tuned to syntax and subtle semantics arising from context.
In the next section, I consider some arguments that have been advanced in favor of domain-general accounts and consider how data provided in support of these may be better explained by this account.
Utility
For example, Manktelow and Over (1991) argue that to grasp the implications of a deontic rule, subjects must represent the utilities of parties affected by the rule. This requires adopting the perspective of either the agent, who enforces the rule, or the actor, who conforms (or does not conform) to the rule. In a rejoinder to a response of Johnson-Laird and Byrne (1992) that mental models of deontic conditionals require no representation of utilities, Manktelow and Over (1992) provide an experiment involving a “weak preference task” and a “strong preference task,” corresponding to scenarios of weak and strong utilities, respectively. In the “weak preference task,” subjects were told that people filled in “record cards” according to the rule:
(WPT) “If your number is 75, then you may write G on your card.”
Presented with cards showing 75 (P), 55 (not-P), G (Q), and N (not-Q), the subjects’ task was to turn over all and only those cards needed to find whether mistakes were made. In the “strong preference task,” subjects were presented with cards that show test results of students where cards with G mean that the students are eligible for a prize, with the students themselves filling in their cards according to the rule:
(SPT) “If your score is 75, then you may write G on your card.”
The task in this case was to find whether any students cheated.
Since these are permission rules, the correct response is (not-P)-and-Q, a response given by one out of 13 subjects (8 percent) in the weak preference case and ten out of 14 (71 percent) in the strong preference case. In both cases, the subject’s task is to enforce a rule. The difference is that the students filling out the cards under the strong preference rule have something at stake (utility) in their actions, while those filling out cards under the weak preference rule realize neither benefits nor costs from their actions. It thus appears that the greater utility involved in the strong preference task enables subjects to better grasp the implications of the rule. But why would this be? One possibility is that in taking the perspective of a party that rations benefits (prizes), one gains utility from not having one’s rule violated—thus in detecting cheaters. In contrast, utility gained from finding people who make inconsequential errors, as in the weak preference rule, is minimal. Thus a possible interpretation of the experimental results is that the deontic selection task is a utility maximization exercise (a view taken in Kirby, 1994; and Oaksford and Chater, 1995). The greater the amount of utility involved, the more likely that subjects will be cued to the nature of the rule, motivating a proper grasp of the rule’s implications.
However, another possible interpretation is that the amount of utility involved cues subjects to the nature of the rule in the absence of unambiguous linguistic information about the nature of the rule. In the present case, the rules differ in the relative strengths of two connotations of the word “may.” “May” could mean “may (but need not),” a plausible interpretation in the weak preference case, with the parenthetical “but need not” undermining any sense that the rule involves permission or entitlement. Or it could mean “are entitled to,” the force of which is much stronger in the strong preference case, undermining the parenthetical “but need not” since not writing G when there is a prize (or utility) for doing so is unlikely to be recognized as an option. With the weak preference rule, interpreting “may” as “may (but need not)” leaves ambiguous whether there even is a rule, since nothing would seem to bind an agent to perform a task to achieve any particular outcome. Thus it would be implausible to explicate the weak preference rule as:
(WPT’) “It is permitted that, if your number is 75, you write G on your card [in order to conform with the rules].”
The statement under this interpretation may even be taken as an epistemic rule where subjects conceive that their task is to determine whether people follow a rule as opposed to whether people err in trying to follow a rule. By contrast, there is no implausibility in explicating the strong preference rule as:
(SPT’) “It is permitted that, if your score is 75, you write G on your card [to signify your eligibility for a prize].”
What is important to note is that although recognition of utility may influence one’s interpretation of “may,” the proximate determinant of performance on the selection task (once one’s interpretation of “may” is determined) is the interpretation itself. To recognize “may” as implying a permission, one must find cues that entitlement is involved. Amount of utility (the high utility of winning a prize versus the low utility of merely conforming to a rule) may well provide such a cue, but beyond encouraging a formal rule interpretation of a conditional, it may have no effect on the selection task.
If utility is just a cue that encourages a formal rule interpretation of a conditional, then it should be possible to facilitate correct performance on the selection task without altering the amount of utility but simply by disambiguating the statement to clarify that it states a formal rule. This can be done by adding the word “only” before “if” in either the weak or the strong preference rule. For the strong preference rule we get the semantically equivalent:
(SPT’’) “Only if your score is 75, then you may write G on your card.”
For the weak preference rule we get the unambiguous:
(WPT’’) “Only if your number is 75, then you may write G on your card.”
Adding “only” to the weak preference rule (a tactic that has been shown to cause the difference in difficulty between modus ponens and modus tollens to disappear (Evans, 1977b; Roberge, 1978)) provides unambiguous universal quantification. It clarifies that only in those instances where one’s number is 75 may G be written on one’s card, viz. that in all instances of G, one’s number must be 75. Thus it unambiguously entails that there is a stricture or rule expressed in the statement, but adds no utility other than the satisfaction one might get from following rules (now that a rule is recognized). Note that adding “only” does not change the connotation of the word “may,” which in the weak preference case still connotes “may (but need not)” since entitlement is still not implied. However, clarity that the rule is universally-quantified may accomplish the same thing that the sense of entitlement or utility does in the strong preference case.
Another experiment that may suggest that quantitatively represented utility is key to performance on the deontic selection task is in Kirby (1994). In Experiment 4 of that paper, subjects are presented with the drinking age rule (Griggs & Cox, 1982) described above, but with both the probability and utility of finding violators varied by condition. Probability is varied, in decreasing order, by three conditions that specify not-Q as “19-years-of-age,” “12-years-of-age,” and “4-years-of age,” respectively. Utility is varied, in order of decreasing utility, by the three conditions DON’T MISS (where one may be fired for missing a guilty person), CHECK (where one may get a bonus for finding violators), and DON’T CHECK (where one may be fired for checking an innocent person), respectively. As expected, it is found that higher probability and greater utility both correlate with correct card selection. Ceteris paribus, not-Q selections are most common for not-Q = “19-years-of-age,” next most common for not-Q = “12-years-of-age,” and least common for not-Q = “four-years-of-age.” Similarly, not-Q selections are most common under DON’T MISS, next most common under CHECK, and least common under DON’T CHECK. So finding violators of social rules appears, at least in part, to require proper motivation, where motivation is measured by subjective expected utility (probability times amount of utility).
Yet while checking for violators of social rules may be more or less well motivated depending on context, the more fundamental issue is not how one performs the task but competence to perform the task in the first place. It makes little sense to check whether four-year-olds are drinking beer, but obviously this does not mean people fail to grasp that four-year-olds are not allowed to drink beer. Similarly, though in the DON’T CHECK condition one may be fired for checking innocent people, this does not imply a failure to grasp the implications of the rule. Indeed the DON’T CHECK directions would be unintelligible if subjects did not grasp the difference between “innocent” and “guilty” specified by the rule.
To say that grasping the logic of conditional statements is an operation of a language comprehension module and so is insensitive to one’s utilities would be too simple, since clearly recognition of utilities can aid in interpretation of conditional statements. However, the fact that utility is not necessary for comprehension of deontic conditionals implies that utility operates only as a cue, playing a disambiguating role when language itself is ambiguous. All of this suggests that representation of utility may be a tool in a larger toolbox dedicated to language comprehension. Manktelow and Over’s strong and weak preference rules provide an illustration of how representation of utility may fill this role.
The key difference between the two rules lies in the perceived motivation to write “G” on the part of those filling out the cards. In the weak preference case, there is no motivation to write “G” since there is nothing to be gained by doing so. In the strong preference case, writing “G” is motivated by the fact that one thereby indicates entitlement to a “prize.” The difference in subjects’ performance on the two rules appears to arise from subjects’ recognition that the strong preference rule is a prohibition against writing “G” (if one’s score is other than 75) while the weak preference rule inspires no such recognition. Why would subjects have these differing perceptions? One possibility is that these differences arise from subjects’ inferences about the intentions behind the rules. If there is nothing to motivate people to write “G” (as in the weak preference case), there is no purpose in prohibiting it. The rule would then not be motivated as a rule. But if “students” filling in the cards can benefit illicitly by writing “G”, then the rule’s purpose is to prohibit an action that people are likely to commit. The suggestion here is that in comprehending these rules, subjects use “theory of mind”—inferences about the intentions of a “speaker” in instituting them.
Theory of mind—sometimes called mind-reading, social cognition, pragmatic understanding, folk psychology or intuitive psychology—refers to the human ability to place oneself in the minds of others in order to discern their goals and interpret their behavior. Such an ability, it has been argued (Pinker, 2002), underlies much of cultural transmission by enabling humans to acquire behaviors whose adoption requires not just imitation but an understanding of the objectives of such behaviors—whether this be household chores, fixing a mechanical device, playing a sport or game, designing a system of some kind, etc. It has also been argued that theory of mind is central to word learning in that it enables us to grasp what people intend to refer to in using words or phrases (Bloom, 2000). For example, in one experiment an experimenter told children, “I am going to plonk Big Bird,” then performed two actions. After one action the experimenter said “There” and after the other the experimenter said “Whoops.” When later asked to “plonk Big Bird,” two-year olds tended to imitate the “intentional” action (indicated by “There”), not the “accidental” one (indicated by “Whoops”) (Tomasello & Barton, 1994; recounted in Bloom, 2000, 65).
With respect to language comprehension generally, theory of mind would enable people to understand utterances or communications in terms of their intended meanings, not just those that follow from grammar. (In many instances, of course, people speak ungrammatically in which case nothing follows from grammar.) Indeed it seems plausible that this is an important part of what we do in conversation—place ourselves in the minds of speakers to grasp their intended meanings and place ourselves in the minds of hearers to monitor what they are most likely to understand from our own speech. The rules of language or syntax provide rich means for precise communication of highly complex, abstract, and subtle information. The bottom line, however, is simply communication of information, however this can be achieved, even if not strictly in accordance with rules. And if language is ambiguous, inferences must be made.
When there is ambiguity in language, how might one place oneself in “another’s mind” and so draw correct inferences about a speaker’s intentions? With respect to conditional rules, two kinds of situations appear to involve theory of mind. One is the case just described where potential gain on the part of “students” skirting a rule cues subjects to the intention behind the rule—to forbid certain behavior. The latter locution requires universal quantification over the range of “actors” affected by the rule. The second type of instance that involves theory of mind is illustrated in the colloquy of Mr. Benson above, and occurs when a conditional rule, by virtue of being a denial of an existentially-quantified statement, guarantees a negative. Guaranteeing a negative, like the intention to forbid, is a locution that requires universal quantification. The difference between these two cases is that while Manktelow and Over’s strong preference rule is a formal rule, Mr. Benson’s statement is an epistemic rule where universal quantification arises from the personal guarantee of Mr. Benson that he has checked all potential instances of non-red-headedness among the Bensons and can confirm that no Bensons have other than red hair.
The evocation of “logical” reasoning by stating a conditional as a denial of an existentially-quantified statement follows a “recipe” for facilitating such reasoning on the selection task as set forth in Sperber et al. (1995) and following from Sperber and Wilson (1986). Similarly to theory of mind, relevance theory concerns intuitions about communicative intentions. In contrast to theory of mind, however, it provides a fully general account of the selection task (as discussed below) while theory of mind provides only a framework for understanding some aspects of performance. In particular, theory of mind seems not to be invoked when syntax or semantics unambiguously implies that a conditional is universally-quantified. An illustration is provided above with the improvement in performance on the selection task arising from placement of “only” before “if” in Manktelow and Over’s weak preference rule. What subjects presumably understand under this condition is not that there is any good reason for prohibiting people from writing “G” on their cards (a theory of mind construct) but that there is some, perhaps arbitrary, regulation prohibiting it. That is, subjects recognize that there is some semantic element additional to the antecedent and consequent, some outcome (compliance or lack thereof) that relates the antecedent and consequent systematically.
It is an argument of this paper that, when stated as a formal rule (like WPT’), a conditional statement is akin to a “logical form” representation of the rule, and that the implications of such a rule are computed more or less automatically, regardless of content. Content on this account primarily affects, via theory of mind, the plausibility of a formal rule interpretation of a conditional. But once a formal rule interpretation of a conditional rule is made, people clearly can grasp the implications of conditional rules precisely, as seen in their performance on the selection task on deontic and/or social contract conditionals and on descriptive conditionals in certain contexts. However, if a conditional rule is not conceived as a formal rule, the default interpretation appears to be that the speaker intends not universal- but existential-quantification and thus is not really stating a conditional rule at all but observing an empirical regularity. The latter is another example of theory of mind considerations.
Mental Models
According to the mental models theory, people perform deductions in three stages:
(1) Comprehension of premises: construction of mental models of states of affairs described by the premises;
(2) Conclusion based on model(s) of premises: formulating a parsimonious description, not explicitly stated in the premises, of models constructed;
(3) Checking conclusion against premises: revisiting one’s premises to try to construct alternative models that are inconsistent with the conclusion based on the initial models.
Errors occur, according to the theory, when people fail to consider all possible models of the premises and therefore fail to find counter-examples to the conclusions from their initial models. To use one of Johnson-Laird and Byrne’s (1991) examples:
(MM1) “If there is a circle, then there is a triangle.”
It is postulated that one’s first model will be a straightforward representation of the premises:
[O] ∆
….
where brackets indicate that the antecedent is exhaustively represented, so that upon further fleshing out of models, no additional models will also contain a circle. Thus the reasoner understands at this point that the statement is universally-quantified—that in all instances of a circle, there is a triangle. This explains one aspect of the selection task—that subjects almost invariably select P. It is selected to try to falsify one’s initial model.
Further fleshing out of models should make apparent that if the antecedent is not satisfied, the consequent may still be satisfied, i.e., there may still be a triangle. Thus two models may now be represented as:
[O] [∆]
~O [∆]
….
And a final fleshing out gives us all three possible models of the given rule:
[O] [∆]
[~O] [∆]
[~O] [~∆]
Since only one model contains the exhaustively represented “~∆”, one should again recognize universal-quantification—that in all instances where there is not a triangle, there is not a circle. So in the selection task, if one has fleshed out one’s models to this point, one should recognize that by turning over the “~∆” card, one would falsify the rule if one finds a circle. And so one should perform logically, selecting P-and-(not-Q) (“O” and “~∆”).
However, constructing such models is cognitively taxing (running up against limits of processing capacity and/or working memory), and if one does not fully flesh out these models, one will not recognize the second potential way of falsifying the rule. One will then commit the characteristic error on the selection task of failing to select not-Q.
An important aspect of this argument is that people construct models in the order specified in the above example. In particular, they construct the second model, in which the antecedent is false, before constructing the third model, in which both the antecedent and the consequent are false. Why would people construct models in this order? The posited ordering may reflect considerations of relevance as discussed in Evans (Evans and Beck, 1981 ) who argues that the word “if” directs attention to the proposition that follows it. If so, then one’s immediate search for an alternative to the original model would lead to a model where the antecedent is false, the purpose being to consider whether in such a case the consequent can still be true. It is only after one has considered both possible truth values for the antecedent that one then turns to the possibility of an alternative truth value for the consequent and so recognizes that if the consequent is false, so also is the antecedent.
To show how logical performance on the selection task can be facilitated, Johnson-Laird and Byrne (1991) consider the alternative proposition
(MM2) “There is a circle only if there is triangle.”
This is equivalent to the modification of Manktelow and Over’s weak preference rule above (WPT”), with the truth conditions of the original statement preserved but the semantic content somewhat altered. Johnson-Laird and Byrne argue that such an assertion leads people directly to the two models:
[O] ∆
~O [~∆]
….
Given these initial models, one immediately has both critical insights needed for the selection task: that in all instances in which there is a circle, there is also a triangle, and that in all instances in which there is not a triangle, there is not a circle. Of course, constructing two models is more effortful than constructing one, and so subjects should grasp the first point more readily than the second, as indeed the data bear out. But in addition, constructing three models, as is required to perform the selection task on (MM1), is more effortful still, and so the account accurately predicts that correct performance will be significantly more common on (MM2) than on (MM1).
The question is why, after constructing the first model of (MM2), subjects now construct as their second model one that requires three steps to arrive at when they construct models of (MM1). Part of the answer may lie in Evans’ “relevance” analysis—that the placement of “if” in (MM2) focuses greater attention on the consequent than it does in (MM1) where “if” occurs in the antecedent. But Johnson-Laird and Byrne also point to the quantificational nature of “only” as bringing to mind the only other possible model, besides the first one, that universally-quantifies the rule.
Both explanations jibe with the account presented here. As noted above, modifying a conditional rule by adding “only” before “if” unambiguously implies universal quantification. And since there is no other possible interpretation, people may construct models of (MM2) in the manner postulated by Johnson-Laird and Byrne. But in addition, Evans’ “relevance” approach may relate to perceived predicate-argument structure. For example, in drawing attention to the proposition following the word “if,” an indicative statement like “If there is a circle, then there is a triangle” may be interpreted as a declarative sentence in the manner of “Circles are always found with triangles.” In the latter, “circles” is the argument of which the property “always found with triangles” is predicated. The conditional rule thus exploits a basic property of syntax: focusing on or putting into the background “different parts of a proposition, so as to tie the speech act into its context of previously conveyed information and patterns of knowledge of the listener” (Pinker and Bloom, 1992, p. 460). The word “if” may indeed focus greater attention on the proposition that follows it, but perhaps it does so by marking it as the argument of a larger proposition.
But a different interpretation is possible. Following the account presented here, the unambiguous universal quantification of “only if” may cause the statement to be interpreted as stating a requirement that there are triangles (when there are circles). The sentence topic then would not be a circle but the requirement that there is a triangle (whenever there is a circle). This makes the statement a formal rule that can be explicated as:
It is by way of the modification of conditionals brought about by adding “only if” that the mental models approach accounts for performance on deontic and/or social contract rules. For example, in the drinking age problem, the rule, “If one drinks beer, then one must be at least 20 years old,” is seen to be interpreted as:
“One drinks beer only if one is over 20 years old.”
~Drinks beer [~Over 20]
….
At the heart of the mental models approach is the idea that “mental models have the same structure as human conceptions of the situations they represent (Johnson-Laird, 1983)” (Johnson-Laird & Byrne, 1991), so that situations or states of affairs are conceived not in terms of structural relationships that underlie them but as mental layouts of what is possible for various premises. In the account here, by contrast, rules are an inherent part of the representation of formal rules (though not of epistemic rules). Thus, in the drinking age problem, rather than compare instances to possible states of affairs given by mental models, people carry in their minds the idea of a rule that has consequences that vary depending on whether the rule is violated or obeyed. It is the presence of represented rules and outcomes from violation or obedience that distinguishes formal from epistemic rules.
Another point of contrast is that the mental models approach assumes that after constructing models, people seek to falsify hypotheses, including those of epistemic conditional rules. But this will not be the case if, as argued above, people use a positive test strategy for such rules (as some, e.g., Klayman and Ha, 1987, have argued is generally the case in a probabilistic environment). Epistemic statements conjoin properties or propositions whose association is not necessarily tied to an underlying structure or rule. If an epistemic statement is not unambiguously universally-quantified, then it is probabilistic, and testing it is a matter of gathering evidence in the form of positive instances.
Formal rules, on the other hand, are structural relationships. And as a logical matter, to represent two discrete properties P and Q as structurally related, some principle of combination (a hypothesis, established principle, rule, theory, or ideology) is needed that entails such a connection. Otherwise, representing such properties as combined would be random and pragmatically dysfunctional, not hallmarks of human synthetic capabilities. Thus it seems likely that humans have rule-modeling abilities that allow them to combine variables of particular categories or ranges by virtue of some model that specifies criteria for licit combination. Conditional statements can activate such rule modeling abilities, but only provided they contain (implicitly or explicitly) some criterion by which antecedent and consequent can be lawfully combined.
[1]In the task (developed by Peter Wason (Wason, 1966)), the subject is presented with a conditional rule of the form If P then Q and shown four cards that separately represent instances of P, not-P, Q, and not-Q. On the reverse sides of the P and not-P cards is information about whether Q or not-Q also occurs, and on the reverse sides of the Q and not-Q cards is information about whether P or not-P also occurs. The subject is asked to select all and only those cards that must be turned over to check whether the rule is true (or is being obeyed). Since a conditional rule is violated when P occurs but Q does not, the subject should try to falsify the statement by turning over the P card (to see if not-Q is on the back) and the not-Q card (to see if P is on the back).
[2] The example in the second colloquy follows a “recipe” laid out by Sperber et al. for reliably facilitating correct performance on the selection task. The idea, as discussed below, is to set up a context such that a conditional rule is a denial of a previous belief. The “relevance” of the conditional then lies in this denial prompting subjects to look for counter-examples. The recipe works, but my interpretation of why it works is different from Sperber et al.’s.
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What Americans Believe
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