1. The 3rd London Reasoning Workshop 18-19/08/2007
Matching heuristic cannot explain matching bias in conditional reasoning. In honour of Jonathan Evans’ 60th Birthday
The 3rd London Reasoning Workshop　18-19/08/2007
Akira Nakagaki (Waseda University)

2. What is matching bias in propositional reasoning?
The phenomenon known as “matching bias” consists of a tendency to see cases as relevant in logical reasoning tasks when the lexical content of a case matches that of a propositional rule which applies to that case (Evans, 1998).
The matching effect has been first demonstrated in the truth table task (Evans, 1972) and then the Wason’s selection task (Evans & Lynch, 1966).
Matching bias is a robust phenomenon which has for many years presented a challenge to various theories of propositional reasoning (Evans, 1998).

3. Wason’s Selection Tasks (Wason, 1966)
Hypothetico-deductive reasoning
Very difficult task (usually around 10% correct)
Selection patterns: selection p, q (46%), selection p (33%), selection p,　q, ￢q (7%), selection p, ￢q (4%) （Johnson-Laird & Wason 1970）
Statement: If a card has E on the face, then it has 7 on the back. (p ⇒ q）
Task: Which card or cards do you have to turn over in order to decide whether the statement is true or false?
　Four Cards presented in AST　
（ｐ）
（￢ｐ）
（ｑ）
（￢ｑ） 7 5 E K

4. Wason’s Selection Tasks using negations paradigm (Evans & Lynch 1973)
Negation is introduced into an affirmative conditional p ⇒ q (If a card has E on the face, then it has 7 on the back).
StatementⅠ: p ⇒ q StatementⅡ: p ⇒ ￢q
　 StatementⅢ: ￢p ⇒ q 　 StatementⅣ: ￢p ⇒ ￢q
Task: Which card or cards do you have to turn over in order to decide whether the statement is true or false?
　Four cards presented in AST　
（ｐ）
（￢ｐ）
（ｑ）
（￢ｑ） E 7 K 5

5. Truth Table Tasks (Evans, 1972,1983, Nakagaki, 1998)
Tasks that examine truth conditions of a conditional.
Statement: If a card has A on the left, then it has 8 on the right. (p ⇒ q）
Task: Participants are asked to decide for each card whether it conforms or contradicts the statement.
Card presentation in TTT

6. What is a matching-heuristic in logical reasoning?
The heuristic-analytic theory proposed by Evans (1989) is based on the idea that people reason analytically about problem information that is pre-consciously selected as relevant at a prior heuristic stage.
In order to explain matching bias, Evans (1999) has proposed a matching-heuristic which works at the heuristic stage and directs attention to cases explicitly stated in the rules regardless of the presence of negations.
Matching bias account by Heuristic-analytic Theory 1998
Evans (1984, 1989) presented a general heuristic-analytic (H-A) theory of reasoning and decision making. The general idea was to explain biases in reasoning and judgemental tasks on the basis of selective attention to or representation of problem features. Heuristic processes are described as preconscious or preattentive and are essentially pragmatic in nature. The function of such processes is to represent selectively relevant aspects of the problem, and to retrieve and represent relevant information from memory. Relevance here is psychological and not necessarily logical as we shall see (the connection with the relevance theory of Sperber & Wilson, 1986, is discussed later). Analytic processes include the capability for deductive competence although the mechanism
was not specified at the time (subsequent links with mental model theory have been explored by Evans, 1993, 1995, Evans & Over, 1996a).
In the H-A theory, biases occur even when participants have the logical capability, through the analytic system, to make correct deductions. This is because logically relevant information may be selected out, or logically irrelevant information selected in, at the heuristic stage. The specific account of matching bias relies on two heuristics, the if- and “matching-heuristic”.
In the case of the Wason selection task it was argued that choices are made entirely heuristically: in effect, participants simply choose the cards that appear relevant. This is because the task only requires that participants indicate which cards they would choose, and does not elicit an explicit process of reasoning. The data in Fig. 2 are clearly consistent with such an account: the effects of the if heuristic are seen in the preference for TA and the avoidance of FA choices. The second—and additive—influence of the matching-heuristic is seen in the matching bias effect observed on all four cases. There appears to be no reason to propose any analytic reasoning on this task.
In the case of the truth table task, both heuristic and analytic stages are observed. Here the argument is that certain logical cases are filtered out as irrelevant and not considered by the analytic system. Cases with low relevance are ones that are not favoured by either heuristic, i.e. false antecedent cases (not favoured by if-heuristic) and mismatching cases (not favoured by matching heuristic). However, the cases that appear relevant are subject to analytic processing which determines whether they are classified as true or false. The if-heuristic is particularly influential on the verification task because it directs attention to the state of the world that might exist if the antecedent of the rule is true. Focusing on the true antecedent leads by a simple step of Modus Ponens to discovery of the TT case which is accepted by the analytic system. The falsification task is more difficult and is influenced by both heuristics. Thus people may consider the true-antecedent in line with the if-heuristic, but analytic reasoning is required to determine that the consequent must be false. Nevertheless, a number of people do succeed in discovering the correct TF case. It is also clear from the data that such discovery is made much more often when the case is favoured by the matching-heuristic. Choices of TF are maximal on the rule if p then not q (double match) and minimal on the rule if not p then q (double mismatch). On the latter, people more often propose FT (double match) as a falsifying case.
In summary, the selection task asks only which cards are relevant and is thus answered by use of the heuristic system only. The truth table task requires both a judgement of relevance and then—for relevant cases—further judgement of whether such cases are true or false. Hence the latter task requires analytic reasoning also.

7. How does a matching-heuristic explain matching bias?
In the WST, participant tend to select only p or p & q cards regardless of the presence of negations in the rule, because the matching-heuristic directs attention to the p and q cards that are explicitly stated in the rules. Responses reflect only heuristic judgments of relevance and analytic reasoning processes do not come in to play.
The TTT requires both a judgement of relevance and, for relevant cases, further judgement of whether such cases are true or false. The TTT requires analytic reasoning also.
The revised heuristic-analytic theory of reasoning 2006 p.388
A matter of some debate has been whether selection task responses can be explained entirely by heuristic processes. Evans (1989) gave an account of selections that seemed to require operation only of two heuristics: an if heuristic and a not, or matching, heuristic (Evans, 1998). The if heuristic would cause people to attend to the A and ignore the D, since if powerfully induces a process of hypothetical thinking on the supposition that the antecedent holds (Evans & Over, 2004). The matching heuristic causes items named explicitly in the statement to be perceived as relevant. This would also favor A over D but also 3 over 7. When negated components are introduced into the conditional statement (as was first shown by Evans & Lynch, 1973), people tend to switch logical choices to maintain matching responses, a phenomenon known as matching bias (Evans, 1998). For example, if the rule is phrased as “If there is an A one side of the card then there is not a 3 on the other side of the card,” most participants will choose the A and 3 cards, which are now the logically correct choices. The choice of consequent cards seems almost entirely determined by matching, whereas antecedent card choices continue to reflect also a preference for true antecedent over false antecedent. Hence, an additive combination of if and matching heuristics seems adequate to account for typical data sets.
In spite of appearances, I believe that for most participants,the analytic system is actively engaged on this task. First, there are good a priori reasons why it should be. The task is of an abstract nature, which may encourage activation of analytic reasoning as a factor additional to those shown in Figure 3. The instructions refer to truth and falsity and, by implication, necessity (choose only those cards . . .). The population tested -generally,undergraduate students—are also bright enough for more than the10% or so who solve the abstract problem to be engaging in analytic reasoning. Certainly, the same population can provide much higher normative solution rates on other deductive reasoning tasks studied in the literature.
There are actually several forms of evidence suggesting that analytic reasoning is engaged on the abstract selection task. 1 Verbal protocols 2 Evidence comes from card inspection times.
If analytic reasoning is involved in selection task choices, why do heuristics seem to account so strongly for the selections made? The answer lies in the satisficing principle and shows again how cognitive biases can occur in the analytic, as well as the heuristic, system. Most people (except those of very high cognitive ability) treat verification and falsification as though they were symmetrical. Participants will happily justify a choice of a matching card combination on the grounds that it will prove the rule true or prove it false, as was originally shown by Wason and Evans (1975). Since the standard instruction refers to discovering whether the statement is true or false, the analytic system satisfices (accepts the heuristically cued choice) whenever it can find a verification or falsification justification. In practice, this means that heuristically cued choices on the selection task will nearly always be accepted.
This discussion of the abstract selection task illustrates all the main features of the extended heuristic-analytic theory. Potential choices are typically restricted to those that are heuristically cued. The analytic system is engaged to scrutinize the choices but, due to a built-in cognitive bias in most people, will satisfice by accepting cases thatcould be true without confirming their logical necessity, hence explaining the common choice of the 3 card. Those very high in intelligence, however, can successfully inhibitthe heuristically generated choices and pay attention to the 7 card.

8. Evidences against the matching-heuristic theory
Evans’ own experiment by which he tried to demonstrate the matching heuristic (Evans, Experiment 1, 1995).
Results of disjunctive reasoning tasks (Evans & Johnson-Laird 1969; Evans, & Newstead, 1980; Roberts, 2002).
Results of Wason’s selection task with two rules(Feeney & Handly, 2000).
Results of non-standard Wason’s selection task(Nakagaki, 2000).
Developmental study of Wason’s selection task (Nakagaki, 1992).

9. EvidenceⅠ： The matching-heuristic theory was refuted by his own experiment
Evans conducted an experiment by which he tried to demonstrate the matching heuristic (Evans, Experiment 1, 1995).
In WST, participants were asked to decide to which of the four cards the rule appears to be relevant.
Contrary to his expectation, participants did not show any matching bias in his critical experiment.
Nevertheless, Evans interpreted this result not as a refutation of his theory but as a change of what is relevant to the task.

10. EvidenceⅡ： Results of disjunctive reasoning tasks
There is no matching bias in disjunctive selection tasks (Wason & Johnson-Laird 1969; Nakagaki, 1990; Roberts, 2002).
There is no matching bias in disjunctive truth table tasks (Evans, & Newstead, 1980, Experiment 1).
They described themselves their experiment as “providing a powerful test of the matching bias hypothesis (matching-heuristic)”.
But they found no evidence of the matching bias.

11. EvidenceⅢ： Suppression of q card selections in WST
What will happen if participants are presented with the WST with two rules, p⇒q and r⇒q.
Since the matching heuristic is preconscious and non-logical, it is predicted that this WST would increase participants’ selections of the q card due to its enhanced relevance.
In spite of this prediction, the introduction of the additional rule in WST suppressed q card selections substantially (Feeney & Handly, 2000).

12. EvidenceⅣ： Anti-matching bias in non-standard WST.
What will happen if participants are asked to select cards which verify or falsify the rule in WST. This task is called “non-standard WST”.
Since the matching heuristic is preconscious and non-logical, it is predicted that this instruction would not change participants’ card selection.
In spite of this prediction, they showed anti-matching bias in non-standard WST using negation paradigm (Nakagaki, 2000).

13. EvidenceⅤ： Anti-matching bias in prescriptive WST
A developmental study of prescriptive WST revealed different patterns of card selection in junior high school students (Nakagaki, 1992).
Since the matching heuristic is a very simple one, it is predicted that younger generation would be more prone to matching bias than adults.
In spite of this prediction, they tended to select not-p and not-q cards even in WST with p⇒q，and showed anti-matching bias in prescriptive WST using negation paradigm.

14. Conclusion: Does the matching-heuristic really explain matching bias?
The matching-heuristic directs attention to cases explicitly stated in the rule. However, no explanation is found why people judge these cases as relevant to the task.
If the matching heuristic includes relevance judgment as well, it is a re-description of the phenomenon. In other words, it is a tautological explanation of matching bias.
Matching bias is not a non-logical response (Evans1998 p.48) at all, but it is the best possible logical solution for the participant’s system of propositional operations (Nakagaki, 2005).
The CP theory postulates that the case pq is always pregnant in four forms p⇒q, p⇒￢q, ￢p⇒q and ￢p⇒￢q, no matter whether and where negation is introduced in p⇒q. When participants reason about or interpret a conditional (¬)p⇒(¬q), they are always concerned about the co-occurrence between two events p and q. This is not only because the events p and q have positive expressions in the conditional (Wason, 1965), but more importantly because the core psychological meaning of the conditional p⇒q is an assertion “two events p and q must co-occur”. Note that this pregnance is unavoidable for the participants whose logicality is quasi-conditional, because the case pq is the only verifying one in the world they are thinking about.
What is more important is that this pregnance does not change even if negation is introduced into p⇒q. However, the core meaning of p⇒￢q and ￢p ⇒q is now negation of the co-occurrence between p and q, that is, the assertion “two events p and q must not co-occur”. In any event, the pregnance (ce qui s’impose à l’ésprit) is always a status of the case pq. Therefore participants are liable to fall into the preconception that two events p and q must co-occur when the case pq becomes pregnant as a verifying case or the preconception that they must not co-occur when it becomes pregnant as a falsifying case. The CP theory explains the M bias by the CP effect caused by the introduction of negation into conditionals.

15. Thank you for your attention.
End of my presentation
Presenter: Akira Nakagaki (Waseda University)