Academic journal article Memory & Cognition

# Reasoning with Conditionals: Does Every Counterexample Count? It's Frequency That Counts

Academic journal article Memory & Cognition

# Reasoning with Conditionals: Does Every Counterexample Count? It's Frequency That Counts

## Article excerpt

A series of experiments investigated what determines people's degree of belief in conditionals and their readiness to draw inferences from them. Information on the frequency of exceptions to conditional rules was contrasted with information about the number of different disabling conditions causing these exceptions. Experiments 1 and 2, using conditionals with arbitrary contents, revealed a strong effect of frequency information and no effect of disabling information. Experiment 3 established that, in the absence of frequency information, the disabling condition information used in Experiments 1 and 2 affected belief in the conditionals and inference acceptance, as has been found in many previous studies (Byrne, 1989; DeNeys, Schaeken, & d'Ydewalle, 2003b). Experiment 4 extended the results of Experiments 1 and 2 to everyday conditionals. The results show that belief in a conditional, and the confidence in inferences subsequently drawn from it, both depend on the subjective conditional probability of the consequent given the antecedent. This probability is estimated from the relative frequency of exceptions regardless of what causes them.

"Exceptions confirm the rule," says a German proverb. The view expressed by it summarizes people's everyday experience that nothing in this world is certain and that belief in a general rule should not be completely abandoned in the face of exceptions that show up every now and then. In research on reasoning with conditional statements such as ifp then q, a probabilistic perspective has been advanced over the last 10 years (see, e.g., Anderson, 1995; Evans & Over, 2004; Oaksford & Chater, 1994, 2001). According to this view, people hold a degree of belief in a conditional statement that depends on the subjective conditional probability of the consequent given the antecedent. Some researchers pursuing the probabilistic approach assume that people evaluate their belief in a conditional rule according to a procedure called the Ramsey test (Evans & Over, 2004, ch. 2). When evaluating the belief in a rule such as "if you open the fridge, then a light inside goes on," people thus suppose p in a mental simulation (in which the fridge door is opened) and then relate the number of pq cases to the number of p-q cases ("-" denotes the negation of q) within this suppositional framework (Evans, Over, & Handley, 2005; Over & Evans, 2003). This comes down to comparing rule-confirming cases (in which the fridge was opened and the light went on) with exceptions (in which the fridge was opened and the light did not go on). The higher this ratio is-that is, the fewer exceptions to confirming cases there are-the higher people's belief in the conditional rule will be.

Extensive testing of this idea was first provided by Evans, Handley, and Over (2003) and Oberauer and Wilhelm (2003). Both research groups independently designed a task that has come to be known as the probabilistic truth table task. In this task, participants are provided with a conditional statement and explicit frequency information about the four cases that comprise the conditional's truth table-the conjunctions of pq, p-q, -pq, and -p-q. Participants are men asked to evaluate their belief in the conditional statement considering the frequency information given. In several studies (Evans et al., 2003; Oberauer, Geiger, Fischer, & Weidenfeld, 2007; Oberauer & Wilhelm, 2003), the conditional probability P(q|p) had the largest influence on people's belief judgments.

If conditionals allow for exceptions, inferences from conditionals must have a degree of uncertainty. Research on reasoning with conditional statements as premises corroborates this assumption. A large body of research has investigated the role of counterexamples on inferences drawn from conditional statements (see, e.g., Byrne, 1989; Cummins, 1995; De Neys, Schaeken, & d'Ydewalle, 2003b; Markovits & Potvin, 2001). Byrne and colleagues (Byrne, 1989; Byrne, Espino, & Santamaria, 1999) were among the first researchers to systematically investigate effects of counterexamples on inference tasks such as modus ponens (MP) and modus tollens (MT). …

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