4.3 Deductive Reasoning

4.3.1 Disjunction tasks

The THOG task

The most famous disjunction problem, the THOG task, was created by Wason and is shown in Figure 4.1 (Wason & Brooks, 1979). Readers unfamiliar with the THOG problem should read it now before moving on.

The answer to the THOG problem is that the white square and black circle cannot be THOGs while the white circle must be a THOG, but only 35% of

In front of you are four designs:

Black Square, White Square, Black Circle, White Circle

You are to assume that I have written down one of the colours (black or white) and one of the shapes (square or circle). Now read the following rule carefully. If, and only if, any of the designs includes either the colour I have written down, or the shape I have written down, but not both, then it is called a THOG. I will tell you that the Black Square is a THOG.

Each of the designs can now be classified into one of the following categories: A. Definitely is a THOG

B. Insufficient information to decide C. Definitely is not a THOG

I have brought a deck of cards. It contains only these four types of card: Black Square, White Square, Black Circle, White Circle

I deal one for myself from the deck, and I won't show it to you. Now I'll deal you each a card, and I will pay for a dinner for each person who has a card including either the colour of my card, or the shape of my card, but not both.

(The four cards above are given to Rob, Tim, Paul and John, respectively.) Without showing you my card, I can tell you that I owe Rob a dinner. Which card do you think I could have? And do you think that I have to pay for a dinner for someone else? If so, for whom?

Figure 4.2: The Pub problem, a contextualised version of the THOG task.

participants in Wason and Brooks’s (1979) study gave this response. The prob- lem states that the black square is a THOG, which means that the experimenter must be thinking of a white square or a black circle (a THOG shares one char- acteristic with the design the experimenter is thinking of). If the experimenter is thinking of a white square then the black circle (shares neither characteristic) and white square (shares both) can be ruled out as THOGs. If the experimenter is thinking of the black circle then the black circle (shares both) and white square (shares neither) can be ruled out as THOGs. Under both alternatives, a white circle shares one characteristic, and is therefore a THOG.

Girotto and Legrenzi (1989) created the pub problem, a reformulation of the THOG problem using realistic content. The problem is about a character called Charles who plays a game with four friends in a pub. The problem stated by Charles appears in Figure 4.2.

This problem is analogous to the abstract problem (in this case the answer is that John is also owed a dinner), yet 89% of people answered correctly accord- ing to Girotto and Legrenzi (1989). Furthermore, when Girotto and Legrenzi (1993) simply gave the name SARS to the hypothesised shape in the abstract version of the task, so that a THOG has one feature in common with a SARS, they observed 70% correct performance. The explanation given for the difficulty of the original THOG problem is called confusion theory and argues that people

p q p or q

T T F

T F T

F T T

F F F

Table 4.1: Truth Table for the exclusive disjunction ‘p or q’ where T = true and F = false.

simply treat the exemplar THOG as if it was the design chosen by the experi- menter. They then look for other designs that have one feature in common with the exemplar (Newstead, Girotto & Legrenzi, 1995). It is suggested that when people have to keep several hypotheses in mind at once, as with the exclusive disjunction in the THOG problem, they experience a cognitive overload and re- sort to more intuitive strategies. In this case, the intuitive strategy is to match the values of the exemplar with the test cases.

Truth Table tasks

Disjunctive reasoning can also be measured with a Truth Table task. A Truth Table is used in logic to demonstrate how the truth or falsity of each variable determines the validity or invalidity of a proposition about those variables. For example, Table 4.1 presents a Truth Table for the exclusive disjunction rule ‘p or q’. The fact that the disjunction is exclusive means that either p or q must be true, but not both. 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 disjunctive rule true (valid) or false (invalid). In an inclusive disjunction, either p or q must be true, but both can be true as well. The Truth Table for an inclusive disjunction is shown in Table 4.2.

Truth Table tasks given to participants to measure their conceptions of

p q p or q

T T T

T F T

F T T

F F F

Table 4.2: Truth Table for the inclusive disjunction ‘p or q’ where T = true and F = false.

I order wine I order water I’ll order wine or water

T T T/F?

T F T/F?

F T T/F?

F F T/F?

Table 4.3: Truth Table for the disjunction rule ‘I’ll order wine or water’, where T = true and F = false.

the conditional include the truth and falsity of the variables but leave the rule column blank for the participant to complete, i.e. the participant decides whether each combination of variables makes the rule true or false. This can be given in thematic as well as abstract form, as demonstrated in Table 4.3 for the disjunction ‘I’ll order wine or water’. By asking participants to complete the final column we can infer how logical people are in their assessment of dis- junctions and whether they prefer an exclusive or inclusive interpretation of the disjunction.

Evans (1993) reviewed a set of studies that used abstract disjunctive Truth Table tasks. He found that the not-p not-q case was always rated false, as it should be under both exclusive and inclusive readings, but the p not-q and not-p q cases were rated true about 80% of the time, despite both being true under both readings. This suggests that people do not reason entirely logically with disjunctions. As for a preference for exclusive or inclusive readings, the findings were inconsistent. In some studies there was a clear preference for an exclusive reading (where the p q case is rated false), in some there was a clear preference for an inclusive reading (where the p q case is rated true) and in others there was no clear preference (Evans, 1993).

Disjunctive Inference task

In a Disjunctive Inference task, participants are given a disjunctive rule along with a premise about that rule, followed by a conclusion derived from the rule and premise. The participant then assesses whether the conclusion is valid or invalid. For example:

Rule: Either A or B Premise: not B Conclusion: A

p q Material Conditional Biconditional

t t t t

t f f f

f t t f

f f t t

Table 4.4: Truth Table for ‘if p then q’ where t = true and f = false.

inferences are ‘Either p or q; not p; q’ and ‘Either p or q; not q; p’. Both of these inferences are valid under both the exclusive and inclusive readings. The affirmation inferences are ‘Either p or q; p; not q’ and ‘Either p or q; q; not p’. Under an exclusive reading, both affirmation inferences are valid. Under an inclusive reading, the conclusions may or may not be true, so the inferences are invalid (not necessarily true).

As with the Truth Table task, the Disjunctive Inference task can be given with thematic as well as abstract content. For example, ‘My sister keeps trop- ical fish, which are either angels or neons; they’re not angels; therefore they’re neons’. As with the Truth Table task, this type of task can be used to assess the extent to which participants conform to normative logic and whether they prefer an exclusive or inclusive reading.

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