Dual-goal facilitation in Wason's 2–4–6 task: What mediates successful rule discovery?

The standard 2–4–6 task requires discovery of a single rule and produces success rates of about 20%, whereas the dual-goal (DG) version requests discovery of two complementary rules and elevates success to over 60%. The experiment examined two explanations of DG superiority: Evans' (1989) positivity-bias account, and Wharton, Cheng, and Wickens' (1993) goal-complementarity theory. Two DG conditions were employed that varied the linguistic labelling of rules (either positively labelled Dax vs. Med, or mixed-valence “fits” vs. “does not fit”). Solution-success results supported the goal-complementarity theory since facilitation arose in both DG conditions relative to single-goal tasks, irrespective of the linguistic labelling of hypotheses. DG instructions also altered quantitative and qualitative aspects of hypothesis-testing behaviour, and analyses revealed the novel result that the production of at least a single descending triple mediates between DG instructions and task success. We propose that the identification of an appropriate contrast class that delimits the scope of complementary rules may be facilitated through the generation of a descending instance. Overall, our findings can best be accommodated by Oaksford and Chater's (1994) iterative counterfactual model of hypotheses testing, which can readily subsume key elements of the goal-complementarity theory.     

Dual-goal facilitation in Wason's 2-4-6 task: what mediates successful rule discovery?

The standard 2-4-6 task requires discovery of a single rule and produces success rates of about 20%, whereas the dual-goal (DG) version requests discovery of two complementary rules and elevates success to over 60%. The experiment examined two explanations of DG superiority: Evans' (1989) positivity-bias account, and Wharton, Cheng, and Wickens' (1993) goal-complementarity theory. Two DG conditions were employed that varied the linguistic labelling of rules (either positively labelled Dax vs. Med, or mixed-valence “fits” vs. “does not fit”). Solution-success results supported the goal-complementarity theory since facilitation arose in both DG conditions relative to single-goal tasks, irrespective of the linguistic labelling of hypotheses. DG instructions also altered quantitative and qualitative aspects of hypothesis-testing behaviour, and analyses revealed the novel result that the production of at least a single descending triple mediates between DG instructions and task success. We propose that the identification of an appropriate contrast class that delimits the scope of complementary rules may be facilitated through the generation of a descending instance. Overall, our findings can best be accommodated by Oaksford and Chater's (1994) iterative counterfactual model of hypotheses testing, which can readily subsume key elements of the goal-complementarity theory.     

Contrast class cues and performance facilitation in a hypothesis-testing task: evidence for an iterative counterfactual model

Hypothesis-testing performance on Wason’s (Quarterly Journal of Experimental Psychology 12:129–140, 1960) 2–4–6 task is typically poor, with only around 20% of participants announcing the to-be-discovered “ascending numbers” rule on their first attempt. Enhanced solution rates can, however, readily be observed with dual-goal (DG) task variants requiring the discovery of two complementary rules, one labeled “DAX” (the standard “ascending numbers” rule) and the other labeled “MED” (“any other number triples”). Two DG experiments are reported in which we manipulated the usefulness of a presented MED exemplar, where usefulness denotes cues that can establish a helpful “contrast class” that can stand in opposition to the presented 2–4–6 DAX exemplar. The usefulness of MED exemplars had a striking facilitatory effect on DAX rule discovery, which supports the importance of contrast-class information in hypothesis testing. A third experiment ruled out the possibility that the useful MED triple seeded the correct rule from the outset and obviated any need for hypothesis testing. We propose that an extension of Oaksford and Chater’s (European Journal of Cognitive Psychology 6:149–169, 1994) iterative counterfactual model can neatly capture the mechanisms by which DG facilitation arises.     

Hypothesis-testing strategies: Why two goals are better than one

An intriguing finding in the hypothesis-testing literature concerns a large increase in the proportion of subjects who discover a rule when they are asked to determine two rules rather than that rule alone. This finding is based on Wason's (1960) “2 4 6” task, in which subjects try to discover a rule (ascending numbers) by generating and testing number triples. They are initially given an example (“2, 4, 6”) of the rule that leads to overly specific hypotheses. With single-goal (SG) instructions, subjects try to discover the correct rule and are told whether each triple proposed fits the rule. With dual-goal (DG) instructions, correct and incorrect categories are labelled, respectively, as DAX and MED. Subjects try to discover both rules and are told whether each proposed triple is DAX or MED. Two explanations of why DG subjects do better at rule discovery than SG subjects are tested: the quantity of information and the testing of complementary rules using the prevalent positive-test strategy. Results support the latter explanation: DG subjects outperform SG subjects only if they know the rules are complementary, and that SG subjects' performance does not improve when required to test more triples before announcing their first rule. A third explanation, the positivity of the linguistic label of the feedback, is ruled out. Understanding the superiority of DG instructions might suggest a general method for enhancing rule discovery.     

Inducing a Rule in Wason's 2-4-6 Task: A Test of the Information-Quantity and Goal-Complementarity Hypotheses

In the standard 2-4-6 induction task, subjects are instructed to discover the rule generating sequences of three numbers by inventing number triples for which they receive immediate feedback. The rule is “ascending numbers”. Performance is greatly aided with Dual Goal (DG) instructions that ask subjects to discover two rules, one that generates “Dax” triples (equivalent to “yes” instances with Single Goal [SG] instructions) and another that generates “Med” triples (equivalent to “no” instances). The present study eliminates two explanations for this effect suggested by Wharton, Cheng, and Wickens (1993). Experiment 1 tested their Information-Quantity hypothesis that the effect results simply from the DG subjects testing more triples prior to proposing a rule. Our DG subjects were more likely to solve the problem and produced more “negative” triples than SG subjects when both groups generated exactly 15 triples. Two further groups received feedback only after generating all 15 triples, and again DG subjects were more likely to solve the problem and to generate more “negative” triples. Experiment 2 tested Wharton et al.'s Goal-Complementarity hypothesis that success under DG instructions hinges on preserving the complementary representation of the two rules. We compared SG instructions with three types of DG instructions that suggested different types of triples (Dax, Med, both Dax and Med, neither Dax nor Med). DG instructions were more effective in promoting successful rule discovery regardless of differences in rule complementarity. Our analysis of the heterogeneity of the examplars generated with DG instructions in both experiments suggest that success on the 2-4-6 task is as much a consequence of the breadth of hypotheses that subjects entertain as it is a consequence of the testing strategy.     

Sensible reasoning in two tasks: Rule discovery and hypothesis evaluation

The hypothesis testing skills of undergraduates were measured in two tasks: the 2-4-6 rule discovery task in which students generate and assess hypotheses, and a hypothesis evaluation task, which requires only the assessment of hypotheses. The results of Experiments 1 and 2 show that the students consistently employed a disconfirmation strategy when assessing hypotheses, but employed a counterfactual inference strategy when they also were required to generate the hypotheses. The results of Experiment 3 suggest that the selection of the hypothesis testing strategy reflected a balance between the logical requirements of the task and the desirability of possible outcomes. Taken together, the findings support a more consistent picture of human rationality across tasks, and suggest alternatives to accounts of confirmation bias.     

Traces of reasoning with pragmatic schemas

Holyoak and Cheng's (1995) account of Wason's (1966) selection task was evaluated by testing participants' recognition memory for the rule. To accommodate the finding that participants' selections are systematically influenced by manipulating their perspective on the rule to be tested, Holyoak and Cheng put forward a development of Pragmatic Reasoning Schema (PRS) theory, according to which the rule being tested is mapped onto different schematic rules depending on the perspective taken. The instantiated schema then becomes the basis for reasoning, and the different schemas encourage the selection of different cases. We hypothesised that if participants interpret the rule by instantiating a particular schema, then they should falsely recognise a sentence corresponding to the instantiated schema a short time after performing the task. An experimental test provided limited support for this hypothesis.     

Evoking the permission schema: The impact of explicit negation and a violation-checking context

Cheng and Holyoak (1985) proposed that realistic reasoning in deontic contexts is based on pragmatic schemas such as those for assessing compliance with or violation of permission and obligation rules, and that the evocation of these schemas can facilitate performance in Wason's (1966) selection task. The inferential rules in such schemas are intermediate in generality between the content-independent rules proposed by logicians and specific cases stored in memory. In one test of their theory, Cheng and Holyoak demonstrated that facilitation could be obtained even for an abstract permission rule that is devoid of concrete thematic content. Jackson and Griggs (1990) argued on the basis of several experiments that such facilitation is not due to evocation of a permission schema, but, rather, results from a combination of presentation factors: the presence of explicit negatives in the statement of cases and the presence of a violation-checking context. Their conclusion calls into question both the generality of content effects in reasoning and the explanation of these effects. We note that Jackson and Griggs did not test whether the same combination of presentation factors would produce facilitation for an arbitrary rule that does not involve deontic concepts, as their proposal would predict. The present study tested this prediction. Moreover, we extended Jackson and Griggs' comparisons between performance with an abstract permission rule versus an arbitrary rule, introducing clarifications in the statement of each. No facilitation was observed for an arbitrary rule even when explicit negatives and a violation-checking context were used, whereas strong facilitation was found for the abstract permission rule under the same conditions. Performance on the arbitrary rule was not improved even when the instructions indicated that the rule was conditional rather than biconditional. In contrast, a small but reliable degree of facilitation was obtained for the abstract permission rule, with violation-checking content even in the absence of explicit negatives. The theory of pragmatic reasoning schemas can account for both the present findings and those reported by Jackson and Griggs.     

Is understanding regret dependent on developments in counterfactual thinking?

Children's understanding of counterfactual emotions such as regret and relief develops relatively late compared to their ability to imagine counterfactual worlds. We tested whether a late development in counterfactual thinking: understanding counterfactuals as possibilities, underpinned children's understanding of regret. Thirty 5‐ and 6‐year‐olds completed tasks assessing counterfactual thinking and understanding regret. Performance on the counterfactual task was better than that on the regret task. We suggest that thinking about counterfactuals as possibilities is a necessary but not sufficient cognitive development in children's understanding of regret. We discuss how other developments in counterfactual thinking may underpin children's emotional understanding.     

Attention as a cueing function during kindergarten children's dimensional change task performance

The purpose of this experiment is to test whether shift flexibility in kindergarten children is a joint function of rule‐usage and inhibition of attention. Sixty‐six children were given either a distraction or facilitation condition in a computerized version of the dimensional change card sort task. In the distraction condition, the background of the post‐shift matching stimulus was inconsistent with the relevant matching dimension. In the facilitation condition, the background of the post‐shift matching stimulus was consistent with the relevant matching dimension. Results revealed that children made few errors in the standard version of the shift task, thereby supporting Cognitive complexity and control theory's contention that 5‐year‐old children shift easily across dimensions due to their use of higher‐order setting rules to solve contradictions. The proportion of errors increased however in the distraction condition suggesting that attention to the background interfered with children's ability to shift between dimensions. Therefore, these data provide evidence that refocusing attention to dimensions along with rule‐use processes affect shift flexibility and argue for the inclusion of both factors into theoretical accounts of shift performance. Copyright © 2009 John Wiley & Sons, Ltd.     

Unique behavioral strategies in visuomotor learning: Hope for the non-learner

The existence of individual differences in motor learning capability is well known but the behaviors or strategies that contribute to this variability have been vastly understudied. What performance characteristics distinguish an expert level performer from individuals who experience little to no success, those labeled non-learners? We designed a rule-based visuomotor task which requires identification (discovery) and then exploitation of specific explicit and implicit task components that requires a specific movement pattern, the task rule, for goal achievement. When participants first attempt the task, they are informed about the goal, but are naïve to the task rule. Therefore, the purpose of this experiment is to determine how acquisition of both implicit and explicit task components, the inherent elements of the task rule, reveals differing strategies associated with performance and task success. We test the hypothesis that an examination of performance will reveal sub-groups with varying levels of success. Further, for each subgroup, we expect to find a unique relationship between visual Time-in-Target feedback (a measure of success) and subsequent updating of each task component. Out of 32 non-disabled adults, we identified three distinct sub-groups: (Low Performer/Non-Learner (LP, N = 9), Moderate Performer (MP, N = 12) and High Performer (HP, N = 11)). A quantitative analysis of behavioral patterns reveals three findings: First, the LP sub-group demonstrated significantly lower task success which was associated with difficulty identifying the explicit component of the task. Second, the HP sub-group acquired the two task components in parallel over practice. Third, when both explicit and implicit component performance is plotted across sub-groups, a task component continuum emerges that seamlessly progresses from low to moderate to high performer groups. An exploratory analysis reveals that self-reported level of prior lifetime accumulation of video game and physical activity experience is a significant predictor of individual task performance (R² = 0.50). In summary, what appears to be a key distinction between varying levels of human rule-based motor learning is the process by which feedback is used to update performance of inherent elements of the task rule. Evidence of a performance continuum and limited prior experience suggests that Low Performer/Non-Learners are generally inexperienced with these kinds of tasks, although the role of genetics and other innate learning capabilities in visuomotor learning is still largely unknown. These findings provoke new research directions toward probing the differential performance strategies associated with expertise and the development of interventions aimed to convert non-learners into learners.     

Graphical representation fosters discovery in the 2-4-6 task

In the 2-4-6 rule discovery task, reasoners seek to discover a rule that governs the arrangement of three numbers (or triple). The to-be-discovered rule is “ascending numbers”. Upon being given the triple 2-4-6 as an initial example, however, reasoners tend to formulate algebraically specific hypotheses. Traditionally, this task is conducted primarily from an internal representation of the triples and candidate hypotheses. More recently, substantial representational effects have been demonstrated wherein an external representation of the dimensions of the problem space facilitated successful rule discovery. In the two experiments reported here, an interactive graphical representation was created by concurrently plotting each triple produced by the participants. In Experiment 1, participants who performed the task with this external representation were more likely to discover the rule than were a group of control participants. Experiment 2 replicated the effect but also assessed participants' hypotheses for each triple generated. Results indicated that a graphical representation of the triples fostered the development of hypotheses that were less constrained by the implied algebraic specificity of the initial triple.     

Further investigation on the mnemonic effect of gestures: Their meaning matters

Three experiments are reported that examine the nature of knowledge underlying performance in the invariant learning task. Previous research (Bright & Burton, 1994; McGeorge & Burton, 1990) has supported an account of performance based on the implicit abstraction and application of a rule pertaining to the invariant feature. In contrast, we found effects in both the digit and clock invariant tasks that are difficult to explain solely in terms of subjects acquiring the experimenters' rule. In all three experiments, manipulation of test item properties that are independent of the invariant feature led to a detriment in performance that is not predicted by an account based on the experimenters' rules. Furthermore, the use of an on-line measure of awareness (confidence ratings) provided some evidence that performance is mediated by low confidence explicit knowledge.     

False belief understanding: The influence of person,grammatical mood,counterfactual reasoning and working memory

The present study had two major goals. The first goal was to assess the relative difficulty among different versions of the unexpected contents task by systematically varying the dimensions of grammatical mood (indicative vs. subjunctive) and person (self vs. other), and to examine the correlational pattern between these different versions of the unexpected contents task and the unexpected locations task. The second goal was to examine the specificity of the relation between false belief understanding and counterfactual reasoning after controlling for age and working memory ability. One hundred, 3‐ to 6‐year‐old, children were administered two measures of false belief understanding (two versions of the unexpected contents task and two versions of the unexpected locations task), two measures of counterfactual reasoning and a working memory measure. Results showed that performance on the unexpected contents task did not significantly differ across conditions. However, only the conditions of the unexpected contents task that concerned another person's false belief correlated significantly with the unexpected locations task. Moreover, counterfactual reasoning was found to explain a significant amount of variance in the unexpected locations task, even after controlling for age and working memory performance. Findings are discussed in the context of different accounts of the development of theory of mind, and in the context of different interpretations of the relation between false belief understanding and counterfactual reasoning.     

When to trust the data: Further investigations of system error in a scientific reasoning task

When evaluating experimental evidence, how do people deal with the possibility that some of the feedback is erroneous? The potential for error means that evidence evaluation must include decisions about when to “trust the data.” In this paper we present two studies that focus on subjects’ responses to erroneous feedback in a hypothesis testing situation—a variant of Wason’s (1960) 2–4–6 rule discovery task in which some feedback was subject tosystem error: “hits” were reported as “misses” and vice versa. Our results show that, in contrast to previous research, people are equally adept at identifying false negatives and false positives; further, successful subjects were less likely to use a positive test strategy (Klayman & Ha, 1987) than were unsuccessful subjects. Finally, although others have found that generating possible hypotheses prior to experimentation increases success and task efficiency, such a manipulation did little to mitigate the effects of system error.     

Causes and Counterparts

It follows from David Lewis's counterpart-theoretic analysis of modality and his counterfactual theory of causation that causal claims are relativized to a set of counterpart relations. Call this Shlewis's view. I show how Shlewis's view can provide attractively unified solutions to similar modal and causal puzzles. I then argue that Shlewis's view is better motivated, by his own lights, than the view Lewis actually held, and also better motivated than a similar approach which relativizes causal claims to sets of ‘contrast events’.     

Dual processes,evolution and rationality

In three experiments we tested hypotheses derived from the goal specificity literature using a real-world physics task. In the balance-scale paradigm participants predict the state of the apparatus based on a configuration of weights at various distances from the fulcrum. Non-specific goals (NSG) have been shown to encourage hypothesis testing, which facilitates rule discovery, whereas specific goals (SG) do not. We showed that this goal specificity effect depends on task difficulty. The NSG strategy led to rule induction among some participants. Among non-discoverers, SG participants were faster and more accurate on difficult problems than NSG participants. The use of misleading exemplars (scale configurations that obscured the rule governing outcomes) led to fixation on inappropriate hypotheses for NSG but not SG participants. When more diagnostic learning exemplars were used, NSG non-discoverers still performed worse than SG participants on difficult problems. SG participants also outperformed NSG participants on a post-test of difficult problems. These findings qualify the generality of goal specificity effects.     

When seeing is knowing: the role of visual cues in the dissociation between children's rule knowledge and rule use

The Dimensional Change Card Sort (DCCS) task requires children to switch from sorting cards based on shape or color to sorting based on the other dimension. Typically, 3-year-olds perseverate, whereas 4-year-olds flexibly sort by different dimensions. Zelazo and colleagues (1996, Cognitive Development, 11, 37-63) asked children questions about the postswitch rules and found an apparent dissociation between rule knowledge and rule use, namely that 3-year-olds demonstrate accurate knowledge of the postswitch rules despite sorting cards incorrectly. Here, we show that children's success with these questions is grounded in their use of available visual cues; children who fail sorting use the target cards to correctly answer questions, and when the cards are unavailable they guess. This suggests that there might not be a dissociation between children's rule knowledge and rule use in the DCCS.     

What Kind of Normativity is the Normativity of Grammar?

The overall goal of this article is to show that aesthetics plays a major role in a debate at the very center of philosophy. Drawing on the work of David Bell, the article spells out how Kant and Wittgenstein use reflective judgment, epitomized by a judgment of beauty, as a key in their respective solutions to the rule‐following problem they share. The more specific goal is to offer a Kantian account of semantic normativity as understood by Wittgenstein. The article argues that Wittgenstein's reason for describing language as a collection of language games is to allow for a perspective that shows those games as internally purposive without any extralinguistic purpose. This perspective also allows for that union of the general rule and its particular application in practice that the original paradox of rule‐following is wanting.     

Inhibition of action rules

Inhibition is often proposed as an important executive-control process, but its existence is difficult to establish empirically. Recently, Mayr and Keele (2000) reported that it takes longer to switch to a recently disengaged task set (i.e.,n−2 task-set repetitions) than to a less recently disengaged task set. Thissetalternation cost may indicate inhibition of the disengaged task set. The primary goal of the present study was to test the inhibition account of set-alternation costs against an important alternative account based on automatic episodic retrieval. The episodic retrieval account predicts reaction time facilitation instead of costs for the special case ofcomplete n−2 repetitions of both task sets and all stimulus— response aspects. A new task-switching paradigm, in which action rules varied on a trial-by-trial basis while the relevant stimulus dimension remained invariant, allowed the implementation of a high proportion of completen−2 repetitions. Consistent with the inhibition view, set-alternation costs were obtained even for these constellations.