In the selection task, individuals generally do not follow the deductive rules of standard logic. In the present research, a new abstract selection task was designed by extending the range of cards that the participants face (inspired by Manktelow, Sutherland and Over, 1995). It was used in two experiments to test predictions about the matching bias and the probabilistic approach of reasoning. By multiplying the number of cards, we showed a reduction of the pq response, indicating that the matching bias may partly be due to practical features of the task (experiment 1). Surprisingly, half of the participants unsystematically turned over specific cards in some categories. The post hoc justifications allowed us to distinguish several possible interpretations of the task, and differing strategies, (either deductive or inductive) in a bid to uncover contradictions. The result was replicated with a shorttime procedure. It also showed a progression of the logical p not-q response (experiment 2). We thus propose that a distinction be made between a pragmatic and a practical component of the matching bias. The features of the task also define a range of deductive and inductive strategies to solve the problem.
Several theories compete to explain the peculiarities of
human thinking and reasoning facing logical tasks. The
Wason’s selection task (1966, 1968) is considered to be a
paradigmatic deductive task aimed at studying these
cognitive processes and is still widely used today. In this
task, the participant is informed about a game where a card
always has one letter on one face and a number on the other.
A conditional rule is stated: “If there is a vowel on one side
of the card, then there is an odd number on the other side”.
Four cards are proposed: a vowel (antecedent p), a
consonant (not-p), an odd number (consequent q) and an
even number (not-q). The task is to select the cards which
determine whether the rule is true or false. Following
standard logic, there is only one correct response: turning
over the p and not-q cards. It has been consistently shown
that the selection task elicits correct responses only 5-10%
of the time, and that most of the subjects choose to turn over
only the p or the p and q cards, thus showing what appears
to be a confirmation bias
One of the most prevalent criticisms of the selection task is
about its lack of ecology. Several researches thus involved
experimental variations on the content of the cards or in the
instructions. It had been argued that responses are more
likely to conform to standard logic if a thematic content is
provided. However
To better ascertain the role of the array of cards in the
cognitive processes, a thematic LAST should be replaced by
an abstract one. As far as we know, no such study exists.
However some studies, related to decision-making and
expected utilities, suggested the importance of strategies
based on Bayesian rationality in order to reduce uncertainty.
In order to study the effects of this enlarged abstract
selection task and to have a fine-tuned interpretation of the
response patterns, we followed
We hypothesized that using an enlarged selection task
would change the types of response (H1). Following most
research in the domain, we wonder whether the frequency of
the p not-q responses may increase in comparison to
Participants Bachelor and master students of the faculty of Humanities and Social Sciences in Neuchâtel (Switzerland) participated voluntarily (N = 46; Mean age = 23;0; male/female = 17/29).
Material We used an enlarged selection task including 16 cards, with 4 cards per instance (p, not-p, q, not-q), and in each of the instances, one card was repeated (e.g. two “A” cards among the vowels). The arrangement of the cards was counterbalanced in four different orders. The instructions were provided as follow, translated into French: “Exercise (about 5 minutes): Here is a card game. All the cards have one letter on one face and a number on the other face. We place a few cards of this game before you. One rule states that if there is a vowel on one side of a card, then there is an odd number on the other side of the card. Among the cards in front of you, which card(s) need to be turned over in order to know whether the rule is true or false?” (see Appendix). The time period (5 minutes) is provided to ascertain that the participants take enough time to look at the cards. On the back of the page, the participant writes a justification (“Could you justify your response? Why did you choose this(these) card(s)?”), and then explains his/her method (“Could you describe how you reasoned?”). Once the participant begins to give their justification, they can no longer change their card choice. Finally, we asked the participant if they had ever seen a similar kind of task with cards. The results of those who recognized the task were withdrawn before analysis began.
Procedure The participants had free time to read and solve the task, and then to turn the page over and justify. Response analyses There is no statistical difference in the frequency of the logical p not-q response (ns) (H1 rejected). Nevertheless, there is a significant difference in the distribution of the three main classes of responses (p; pq; other), χ2(2, N = 46) = 18.203, p < .001. Follow-up analyses with residuals indicate that: more p response are provided (37% vs 21%, z = 7,3) as well as “other” responses (41% vs 26%, z = 7,0); pq responses diminished importantly (22% vs 53%, z = -14,4) (H2 corroborated) (see table 1, fig. 1). matching bias”, due to the presence of a number of cards. In case of four cards, the shape of the task invites people to actively pair the cards. In our task, this “practical matching bias” is removed. The pq responses are due to a residual pragmatic bias (the rule still focuses on p and q cases). Unexpected response diversity Surprisingly, a new class of response appeared in the 16-cards condition, involving greater response heterogeneity than in Wason’s study. Two classes of responses can now be distinguished, depending on whether the participants selected the cards “systematically” or “unsystematically”. For each instance, they were expected to select all or none of the cards in a category. Indeed, some people made purely “systematic” selections (i.e. only selecting 4 cards in one or several of the instances) (N = 24). But 22 participants (48%) made “unsystematic” selections (i.e. choosing only one, two or three cards among four).
1 – Unsystematic selections are common in any card combination. In particular, 4 of the 17 participants gave an unsystematic p response, 2 participants of 9 gave an unsystematic pq response, and 1 over 3 gave an unsystematic p not-q response.
2 – In this “unsystematic” category, some “partial” selections are made (few cards selected in specific instances) (N = 18). Some people also made “purely partial” selections by choosing few cards in each of the four instances (N = 4). No-one gave a mix of partial and systematic selection (few cards in some instance, all cards selected in other instances) (N = 0).
3 – All instances (p, not-p, q, not-q) are confronted to both selection styles, systematic or not. The relative importance of unsystematic over systematic responses is particularly increased for not-p and not-q cards: participants generally selected these cards unsystematically (see fig. 2). 45 40 35 30 25 20 15 10 5 0
Less participants chose the pq response. This suggests
that the matching bias is less effective, and the objective
shape of the task partially accounts for this bias. We
propose to distinguish a “pragmatic matching bias”, due to
the stated rule, as claimed by
els Vow
Consonants
Even
Odd Kinds of partial response Several unsystematic strategies can be distinguished: (i) choosing only one card in each desired instance (ex : “A”, “K”, “3”); (ii) choosing only different cards in the desired instances (ex : “A”, “I”, “U”, “K”, “D”, “P”); (iii) choosing only repeated cards in the desired instances (ex: two “A” and two “2” cards). Finally, 4 participants (over 22) mixed up these strategies. Post hoc justifications Written justifications are a first means to understanding the reasoning behind these strategies. The participants giving an unsystematic pq response (N = 3) justified their strategy in reference to the repeated cards. For instance: “We need to look at all the different cards (the other A or 2 are not chosen) to confirm the rule that some cards can follow the rule and others not”.
Among the unsystematic p responses (N = 4), one participant selected only two “A” cards and explained that he first considered the rule as true, then chose one “A” card among the vowels, and finally another “A” to check again. Another participant only chose three different vowels and did not explain the exclusion of the repeated “A” card. Another chose three different cards explaining that she expected specific even numbers related to the letters, seemingly looking for a meaning to this abstract task; she thus tried to discover a different underlying rule rather than directly dismissing the one stated in the instructions.
One participant chose only one of each repeated “A” and “3” cards, giving an unsystematic p not-q response (N = 1): “there is no need to check all the cards with a vowel for instance, as we know that every time there is one vowel, then, on the back there is an even number”. This ambiguous explanation suggests that he tried to know whether all the cards are following this rule or following another one. Remaining questions In this experiment, people tended to use undocumented strategies, apparently illogical but reasonable. Rather than just checking the validity of the stated rule, they may use strategies to discover whether there is another rule governing the cards that possibly contradicts the stated rule, or any other specific meaningful rule instead. It is perhaps possible that the time allocated in the first study and the complexity of the task could explain the surprising response patterns.
For a better comparison, we balanced two groups, some participants having the classical 4-card task and others having the 16-card task. We hypothesized a difference in the response distributions (H1). The 16-cards task would imply more p not-q response (H2). Following the first study, we hypothesized a diminished pq frequency in the 16-cards (H3). Finally, if unsystematic choices are due to a lack of time to observe the cards and give a response, shortening the time constraint would imply more “partial” responses (H4).
Participants Ninety-eight french speaking students of a swiss Higher Teacher Training Institute participated voluntarily (N = 98; Mean age = 28;6 ; male/female = 34/64): 50 participants were randomly involved in the 4cards condition, and 48 participated in the 16-cards condition.
Material The same material was used as in the first study. Procedure The participants had one minute only to individually read and solve the problem. The experimental conditions occurred at the same time in eight separated rooms. Both experimental conditions occurred in each room.
Response analyses Grouping the responses in three main classes (p, pq, other), no significant difference was found in the distributions of the conditions, χ2 (2, N = 98) = 5,032, p = .081 (H1 not corroborated, see table 2, fig. 3)1.
PQ
P-Q
P-PQ-Q
Interestingly, there are more p not-q selections in the 16cards (N = 7) than in the 4-cards group (N = 1), p = .026 (one-sided Fisher’s exact test, Cramer’s V = 0.230) (see fig. 3). With more subjects, a progression of the p not-q expected response appeared (H2 corroborated). The observed strategies for this p not-q response are similar to those reported in Study 1 for the partial responses: (i) five participants unsystematically circled a few not-q cards, 1The distribution of the 4-cards condition does not differ with Wason’s (1968) distribution, ns. Identically, the responses of the 16-cards conditions do not differ from study 1, ns. suggesting that they did not apply a purely deductive reasoning or that they did so with uncertainty; two of them circled a single p and a single not-q card; (ii) one participant chose two different p cards, and three different not-q cards, thus avoiding selecting the repeated cards; similarly, another selected one of the repeated p, and two different not-q cards (one of the repeated cards); (iii) one participant selected the repeated p and not-q cards.
As in study 1, there was a reduction of pq selections, p = .027 (one-sided Fisher’s exact test, Cramer’s V = 0.216). This pattern is comparable to study 1, with a decrease from 50 to 25% (H3 corroborated). This result apparently confirms a matching bias reduction and the time allocation does not particularly change this tendency. Further experiments may reveal small differences.
Description of unsystematic strategies Among the 16-card group, one can find systematic selections (N = 22), and unsystematic selections (N = 26). The proportions are not significantly differing from study 1, ns (H4 not corroborated). The proportions of unsystematic selections (54%) are very similar to Study 1 (48%). This makes it difficult to account for these unsystematic strategies by time pressure. As a consequence, it has to be supposed that the enlarged task reveals differentiated problem interpretations. In the partial responses, one can distinguish either purely unsystematic selections (selecting few cards in certain instances) (N = 21) or a mix of unsystematic and systematic selections (few cards selected in some instance, and all cards in other) (N = 5). Further investigations are currently under process.
This study set out to show that the performances to the Wason selection task are partly due to its objective features. In line with a half century of literature, very few participants gave the expected p not–q response (less than 15%), either systematically (selecting all p and not–q cards) or unsystematically (selecting only a few p and/or not–q cards). Extending the objective part of the selection task does not radically imply more logical responses, although a small effect did arise. The 16-card selection task involved a little progression of this expected response, and this may reinforce the idea of a non-ecological 4-card selection task. When presented with more items, a few people tended to choose p and not-q cards, often unsystematically, potentially using inductive reasoning. Nevertheless, analysing the other responses is informative on the role of the task’s shape.
Generally explained as a matching bias effect or as
biconditional understanding, we observe a reduction of pq
responses when there were more numerous items at the
participants' disposal. In our opinion, this means that the
problem was understood differently, not only due to the
items mentioned in the conditional statement, but also
because of the objective context the statement refers to. This
allows a supposed two-sided matching bias. Indeed the
situation for the subject is shaped by attentional cues, which
here is the rule, but equally by the contextual structure. We
therefore propose to distinguish the two dimensions, namely
a “pragmatic component” and a “practical component” of
the matching bias. This practical component, which is not
located in the individual, also explains why a “matching
bias” is often considered as robust in spite of instructional
variations
We noticed an increased heterogeneity of the responses,
larger than in the traditional selection task. Among them,
some surprising responses occurred, which do not enter a
priori in any category of logical argument: about 50% of the
participants decided to turn over specific cards in some
instances. These "partial" responses are diversely justified.
Rather than being a random procedure due to external
constraints and heuristic processes, these responses appear
regularly, with or without time pressure, as a consequence
of several unsystematic but reasonable strategies. Two main
reasons support the use of these unsystematic strategies.
Firstly, partial selections can be intended to explore the
problem with a view to “seeing later”, i.e. find out
something else about the context and the cards, about the
hidden meanings of the rule, about the experimenter’s
intentions… Secondly, these selections could be the result
of a specific interpretation of the problem and of resulting
deductive reasoning. Some participants supposed that all the
cards of the game are governed by a rule which is
potentially different from the rule stated by the
experimenter: the participant has thus to know whether all
the cards follow the stated rule or if they follow a different
hypothetical rule, which may contradict the stated rule. In
this interpretation, it would not be relevant to turn over all
the cards of a category, as they are supposed to follow a
common rule. Turning over a single vowel or a few vowels
could be sufficient enough to determine the existence of an
unknown rule of the game. We thus observed unsystematic
strategies driven either by inductive or by
hypotheticodeductive rationalities. We did not find a way to explain our
results using a probabilistic approach, though we agree with
It is possible that these deductive as well as inductive rationalities and these strategies were also used with the classical selection task. Indeed, they would be confounded, due to the limited range of possible responses. In Wason’s selection task, only one card is proposed for each logical instance and is considered as representative (for a logician). Hence, when a participant selects a card, we cannot know if he wants to select a category rather than a single case. It is supposed that the analysis confounded differing rational interpretations and reasoning strategies on the basis of their common final response. Doing so, we lack a fine understanding on how cognition proceeds. In short, processes are hidden by one-off responses.
Following
We are especially grateful to Daniel Dukes (Center of Cognitive Science at University of Neuchâtel)whose generous suggestions and support have invaluably contributed to the article. Exercice (environ 5 minutes): On a un jeu de cartes. Toutes les cartes comportent une lettre d’un côté et un chiffre de l’autre. On pose quatre cartes. Une règle indique : « s’il y a une voyelle sur un côté d’une carte, alors il y a un chiffre pair sur l’autre côté de la carte ». Parmi les cartes posées, quelle(s) carte(s) fautil absolument retourner pour savoir si la règle est vraie ou fausse) (Entourez-la/les).