4.3 Deductive Reasoning

4.3.2 Conditional tasks

Conditional reasoning is the process of drawing conclusions from rules based on ‘if’. As Manktelow (1999) argued, the word ‘if’ has probably sparked more interest from psychologists, philosophers and logicians than any other word in the English Language.

Conditional rules come in many forms, including ‘p only if q’, ‘q if p’, ‘p if and only if q’ and the most commonly used form, ‘if p then q’. According to formal propositional logic, each of these statements should be treated as the material conditional except for ‘p if and only if q’, which is biconditional. Under a material conditional reading, the rule is only proved false when p is true and q is false. Under a biconditional reading, p implies q but q also implies p, so the rule is false when p is false and q is true but also when p is true and q is false. The material conditional and biconditional readings are represented in Figure 4.3 as Euler diagrams and in Figure 4.4 as Truth Tables.

p q p

q

Figure 4.3: Euler diagrams to represent ‘if p then q’ under the a) material con- ditional interpretation, where p implies q, and b) biconditional interpretation, where p implies q and q implies p.

Wason Selection Task

The most famous task in the psychology of reasoning is the Wason Selection Task (WST), which was designed to measure conditional reasoning. It was developed in the 1960s and has spawned a great deal of research since, so much so, that one journal stopped publishing any research that used it (Manktelow, 1999). Figure 4.4 displays the task. Participants are shown four cards that each have a letter on one side and a number on the other side. Two cards are letter side up, say A and D, and two are number side up, say 3 and 7. Participants are asked to chose those cards, but only those cards, that they would need to turn over in order to tell whether a rule such as ‘if there is an A on one side of the card then there is a 3 on the other side’ is true or false. In this case, the A and 7 cards would need to be turned over. If the A card had a not-3 number on the other side, or if the 7 had an A on the other side, then the rule would be falsified. However, hundreds of participants across many studies have failed to make this choice. Most chose the A card and sometimes the 3 card too (Wason & Johnson-Laird, 1972; Evans, 1993).

Matching bias (Evans, 1998) is commonly observed on the Wason Selection Task (Wason, 1968). The typical response, A or A and 3, still tends to be given even when the rule is changed from ‘if A then 3’ to ‘if A then not 3’, despite the ‘A, 3’ response then becoming logically correct. This result has been interpreted as indicating that participants simply match the cards to the rule rather than using any systematic reasoning strategy. Wason and Evans (1975) demonstrated that participants showed no awareness of their bias in verbal reports, suggesting that it is an unconscious attentional bias.

The WST has been investigated with thematic content as well as abstract content and in many cases, this has been found to improve performance. Wason

Each of the cards below has a letter on one side and a number on the other side. Select all those cards, but only those cards, which would have to be turned over in order to discover whether the rule is true.

Rule: If there is an A on one side of the card, there is a 3 on the other side.

A

D

3

7

Figure 4.4: A Wason Selection Task example

and Shapiro (1971) gave participants the rule “every time I travel to Manchester I travel by train”, with the cards ‘Manchester’, ‘Leeds’, ‘train’ and ‘car’. While success on the abstract version of the task tends to be lower than 10%, in the thematic case 10 out of 16 participants selected the correct cards: Manchester and car. The facilitative effect of thematic content doesn’t always hold true though, it appears only to help when the rule is deontic rather than descriptive, i.e. conveying a rule, permission or obligation, such as ‘if a person is drinking alcohol then they must be over 18 years of age’ or ‘if a person is on the train then they must have a valid ticket’. It has been argued that this is because familiar rules elicit evolutionarily developed schema, for the purposes of such things as cheater detection (Cosmides, 1989; Cosmides & Tooby, 1992).

Despite its popularity the effectiveness of the WST as a measure of reasoning has been challenged. Sperber et al. (1995) have suggested that the task doesn’t necessarily measure conditional reasoning at all. Instead, performance is highly influenced by relevance-guided mechanisms that pre-empt any reasoning mech- anisms. When faced with a reasoning problem, or any other text, we first need to comprehend the information given and this will include inferring the writer’s intended meaning. In the case of inference tasks participants need to infer or evaluate conclusions derived from premises, so although their interpretation of the premises may be influenced by relevance principles, it is explicitly clear that they must go further and engage in reasoning processes as well. In the case of the selection task, participants are not asked to reason from premises to con- clusions but are instead asked to judge the relevance of each of the cards to the rule. In this case, the judgments of relevance that come from the comprehension

p q if p then q

T T T

T F F

F T T

F F T

Table 4.5: Truth Table for ‘if p then q’ where T = true and F = false.

process provide an intuitive answer to the problem and there is no explicit need to engage in any further reasoning. This may be the source of the pervasive matching bias.

This interpretation of the selection task was supported by six studies across two papers (Sperber et al., 1995; Sperber & Girotto, 2002). Sperber and his colleagues have shown that success rates in the task can be dramatically ma- nipulated by altering the relevance factors of the content. Success in descriptive versions of the task can be increased to over 50%, more in line with the success rates usually found with deontic versions (Sperber et al., 1995). Furthermore, success in deontic versions can be reduced to below 20%, similar to the rates usually found with descriptive and abstract versions (Girotto, Kemmelmeir, Sperber & van der Henst, 2001).

Due to this controversy, the WST was ruled out as a measure of reasoning ability for this thesis. Had mathematics students been found to change in WST performance alongside their mathematics study, it would be unclear whether the change had been in conditional reasoning ability or in interpretation processes.

Truth Tables

Truth Table tasks were discussed in Section 4.3.1 as a measure of disjunctive reasoning, but they can also be used for measuring conditional reasoning. Table 4.5 presents a Truth Table for the conditional rule ‘if p then q’. Again, each line represents a different combination of truth and falsity of the values p and q, and the final column denotes whether that combination makes the conditional rule true or false. Again, Truth Table tasks can be given to participants to complete in thematic as well as abstract form, as demonstrated in Table 4.6 for the conditional ‘If it rains then I take an umbrella’.

By asking participants to decide which lines of the table they consider to be valid, it is possible to determine which reading of the conditional most closely matches their interpretation. As discussed in Section 2.2, there are at least four ways in which people may interpret a conditional statement: material con-

If the letter is H then the number is 5 The letter is H

Conclusion: The number is 5 Yes

No

Figure 4.5: Example Modus Ponens item from the Conditional Inference task.

ditional (that endorsed by formal propositional logic), biconditional, defect- ive conditional and conjunctive conditional, and these differ in terms of Truth Tables (see Table 2.2).

Conditional Inference Task

In a Conditional Inference task, participants are given a conditional rule along with a premise about that rule, followed by a conclusion derived from the rule and premise. The participant then deduces whether the conclusion is valid or invalid. Alternatively, participants may generate a conclusion that they consider to be valid, but this is far less common in the literature and so the evaluation version is focused on here.

Figure 4.5 shows a typical conditional inference task item with a valid con- clusion. This is an example of a Modus Ponens inference, one of four inference types used in the task. The four inferences, modus ponens (MP), denial of the antecedent (DA), affirmation of the consequent (AC) and modus tollens (MT) are shown in Table 4.7, along with the four rule forms created by rotating the presence of negations, and whether the inferences are considered valid according to the four interpretations.

An abstract (using only letters and numbers) 32-item version of the condi- tional inference task was used by Inglis and Simpson (2008, 2009a) to compare mathematics and non-mathematics students’ reasoning behaviour. The task in-

it rains I take an umbrella if it rains then I take an umbrella

T T

T F

F T

F F

Table 4.6: Truth Table for the conditional rule ‘if it rains then I will take an umbrella’, where T = true and F = false.

If the letter is S then the number is 6 The number is not 6

Conclusion: The letter is not S Yes

No

a) Modus tollens b) Denial of the antecedent

If the letter is M then the number is 4 The letter is not M

Conclusion: The number is not 4 Yes

No

Figure 4.6: Example items from the Conditional Inference task showing a) a Modus Tollens inference and b) a Denial of the Antecedent inference.

cluded the four inference types, each presented four times with the four different rule forms shown in Table 4.7, which were created by varying the position of negatives. This created 16 items, with explicit negations. In a further 16 items the problems were identical in structure except that the negations were implicit (e.g. ‘not 3’ might be represented as ‘6’). Figure 4.6 shows some example items from the task.

Inglis and Simpson (2009a) found that mathematics students outperformed non-mathematics students (based on the material conditional being the norm- ative reading, see Section 2.3), even when the groups were matched for general intelligence. The mathematics students did not improve in task performance over the course of a year, but the initial difference left open two possibilities: the mathematics students may have improved in conditional reasoning during

MP DA AC MT

Conditional Pr Con Pr Con Pr Con Pr Con

if p then q p q ¬p ¬q q p ¬q ¬p

if p then ¬q p ¬q ¬p q ¬q p q ¬p

if ¬p then q ¬p q p ¬q q ¬p ¬q p

if ¬p then ¬q ¬p ¬q p q ¬q ¬p q p

Minor Premise Type Affirmative Denial Affirmative Denial Material Validity Valid Invalid Invalid Valid Defective Validity Valid Invalid Invalid Invalid Biconditional Validity Valid Valid Valid Valid Conjunctive Validity Valid Invalid Valid Invalid Table 4.7: The four inferences and conditional statement types with and without negated premises (Pr) and conclusions (Con). The symbol ¬ should read ‘not’. At the bottom, the validity of each inference under each interpretation is given.

pre-university but post-compulsory study of mathematics, in line with the TFD, or it could be the case that people with more normative reasoning styles are disproportionately filtered into studying post-compulsory mathematics.

In an interview study conducted by Inglis (2012), eight stakeholders in the mathematics community (e.g. members of the education committees of the Insti- tute of Mathematics and its Applications and the London Mathematical Society) were asked to look at the Conditional Inference task and rate their agreement with the statement “This task captures some of the skills that studying ad- vanced mathematics develops”. Of those eight participants, six strongly agreed with the statement (five on a five-point Likert scale) and two agreed (four on a five-point Likert scale). One participant even went as far as to say that “If studying A-level maths doesn’t make you better at that, there is something wrong with the syllabus”.

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