Illusions in quantified reasoning: How to make the impossible seem possible, and vice versa

All Bare C. 0 1 0 This research was supported in part by ARPA (CAETI) Contracts N6600l-94-C-6045 and N6600l-95-C-8605 and by the Xian-Lin Ji Foundation of Peking University, China . We thank the members of our laboratory , who provided much helpful advice: Patricia Barres, Victo ria Bell, Zachary Estes, Yevgeniya Goldvarg, and Mary Newsome. We are grateful to Monica Bucciarelli, who investigated the construction of external models in quantified reasoning. We also thank Michael Oaks ford, David O'Brien, Tom Ward, and three anonymous reviewers for their be addressed to Y.Yang or P.N. Johnson-Laird, Department of Psychol ogy , Green Hall, Princeton University , Princeton, NJ 08544 ( 1 YINGRUI YANG and P. N. JOHNSON-LAIRD Princeton University , Princeton, New Jersey The mental model theory postulates that reasoners build models of the situations described in premises, and that these models normally represent only what is true. The theory has an unexpected consequence. It predicts the existence of illusions in inferences. Certain inferences should have compelling but erroneous conclusions. Two experiments corroborated the occurrence of such illusions in inferences about what is possible from disjunctions of quantified assertions, such as, "at least some of the plastic beads are not red." Experiment 1 showed that participants erroneously inferred that impossible situations were possible, and that possible situations were impossible, but that they performed well with control problems based on the same premises. Experiment 2 corroborated these findings in inferences from assertions based on dyadic relations, such as, "all the boys played with the girls." Logical reasoning is a fundamental human ability, but what underlies it is a matter ofcontroversy. Theorists argue that it depends on rules that have a specific content (Cheng & Holyoak, 1985), on memories for previous cases of reasoning (Kolodner, 1993), on innate modules attuned to specific contents (Cosmides, 1989; Cummins, 1996), and on connectionist networks of associations (Shastri & Ajjanagadde, 1990). Each of these possibilities may playa part, but none can explain the whole story, because human reasoners can make deductions about matters of which they have no prior knowledge. For example, if one is told All the beads are plastic. All the plastic things are red. - All A are B. .. All A are C. This rule would enable one to draw the deduction above, and the system of rules proposed by Rips would allow many deductions to be made, although no current psycho logical theory based on formal rules is "complete" in the sense that none allows for all possible valid deductions to be drawn. On the other hand, some theorists have argued that the untrained mind is not equipped with formal rules of in ference, but rather relies on the general principle that a valid deduction is one in which the conclusion must be true given that the premises are true (Johnson-Laird, 1983; Johnson-Laird & Byrne, 1991; Polk & Newell, 1995). This principle is put into practice by constructing mental models of the premises, formulating a conclusion on the basis of such models-if none is provided by a helpful interlocutor-and checking its validity by examining whether or not it holds in the models of the premises. A fundamental principle ofthis model theory, the "prin ciple of truth," is that reasoners normally represent only what is true in order to minimize the load on working memory. This principle is subtle because it is applicable at two levels. First, mental models represent only true possibilities. Second, within those true possibilities, they represent the literal propositions (affirmative or negative) in a premise only when they are true. If a proposition is false, its negation is true. But, what reasoners fail to rep resent is falsity-that is, false affirmative propositions in the premises and false negative propositions in the premises. We can best illustrate the way the principle works with an example. Consider an exclusive disjunction about a hand of cards that contains two literals, one negative and one affirmative: At the first level of the principle of truth, reasoners con struct two alternative models corresponding to the two true possibilities: where each model is shown on a separate line, and "~,, denotes negation. Each model corresponds to a true pos sibility, given the disjunction-that is, each model cor responds to a true row in a truth table. At the second level of the principle, each model represents only those literal propositions in the disjunctive premise that are true within the possibility. Hence, the first model represents the truth of the negative disjunct in the premise (there is not a king in the hand), but it does not represent the fal sity of the affirmative disjunct (there is an ace in the hand). Similarly, the second model represents the truth of the affirmative disjunct in the premise, but it does not represent the falsity of the negative disjunct. Thus, men tal models normally represent the literals in the premises when they are true in the true possibilities, but not when they are false. Reasoners make mental "footnotes" to keep track of what is false. The footnotes make it possible to flesh out the models explicitly. And only fully explicit models of what is possible represent both the true and the false lit erals in each model. Thus, the fully explicit models of the disjunction above are: Fully explicit models can be used to reason in a completely valid way akin to the use of truth tables. The principle of truth concerns the normal interpreta tion of assertions, particularly assertions containing log ical terms, such as sentential connectives and quanti fiers. Clearly, individuals can keep track of what would be false in the case of simple categorical assertions, such as: It is raining. At what point reasoners cease to bear falsity in mind is an empirical question, but previous studies suggest that this point is normally reached when a set of premises calls for more than one model-that is, when the premises are compatible with several possibilities (see, e.g., Johnson-Laird & Savary, 1996). In such a case, mental footnotes about falsity are likely to be soon forgotten, and so the predictions presented here depend only on mental models, not on fully explicit models. One of the authors (J.-L.), while developing a computer implementation of the mental model theory, discovered an odd output to a particular inference. He first thought there was a bug in the program. When he worked out the correct answer by hand, however, he discovered that the program was correct and that he had succumbed to an il lusion. Indeed, the principle of truth has an unexpected consequence. It predicts the existence of inferences that yield illusory conclusions-that is, conclusions that nearly everyone draws, that seem compelling, but that are invalid. After the discovery ofpotential illusions, Rips (1994) tried his PSYCOP progr (...truncated)