Illusions in modal reasoning

YEVGENIYA GOLDVARG 0 1 P. N. JOHNSON-LAIRD 0 1 0 We thank the members of our laboratory , who have provided much helpful advice: Patricia Barres, Victoria Bell, Zachary Estes, Mary Newsome, Fabien Savary, Lisa Torreano, Isabelle Vadeboncoeur, and Yingrui Yang. We thank Geoffrey Keene for advice on modal logic, Lance Rips for testing his PSYCOP program with an illusory inference , Bonnie Meyer for carrying out the "think aloud" study, and Walter Schaeken and Robert Mackiewicz for a suggestion about formal rule theories. We also thank many other colleagues, too numerous to name, for their ideas and alternative hypotheses. We are grateful to Tom Ward, Simon Handley, and an anonymous reviewer, for their helpful criti ment of Psychology , Green Hall, Princeton University , Princeton, NJ 08544 ( 1 Princeton University , Princeton, New Jersey According to the mental model theory, models represent what is true, but not what is false. One unexpected consequence is that certain inferences should have compelling, but invalid, conclusions. Three experiments corroborated the occurrence of such illusions in reasoning about possibilities. When problems had the heading "Only one of the premises is true," the participants considered the truth of each premise in turn, but neglected the fact that when one premise is true, the others are false. When two-premise problems had the heading "One of the premises is true and one is false," the participants still neglected the falsity of one of the premises. As predicted, however, the illusions were reduced when reasoners were told to check their conclusions against the constraint that only one of the premises was true. We discuss alternative explanations for illusory inferences and their implications for current theories of reasoning. - where each row denotes a separate model, "memory" de notes a model of the flaw being in memory, "software" denotes a model of the flaw being in the software, and the third model combines these two models into one. If naive individuals are asked to list the possibilities given the disjunction above, they tend to list precisely the pre ceding models (Johnson-Laird & Savary, 1996). Each model represents only what is true in a particular possi bility. Hence, for the first possibility, the model represents that it is true that the flaw is in memory. It is also false in this case that the flaw is in the software, but, according to the principle of truth, models do not usually represent false information explicitly. Similarly, for the second possibil ity, the model represents that the flaw is in the software, but it does not represent explicitly that it is false that the flaw is in memory. The theory postulates that reasoners make "mental footnotes" to keep track of the information about what is false, but that these footnotes are soon likely to be forgotten. In contrast to mental models,jully explicit models of the disjunction represent the false components in each pos sibility, using negations that are true: where "~,, denotes negation. These three fully explicit models match the three rows that are true in a truth table of the assertion. It is important to bear in mind that the principle of truth concerns the failure to represent what is false, not a failure to represent negation. If a negative asMental Models Fully Explicit Models Note-Each line represents a model of a separate possibility, "~,, de notes negation, and " ..." denotes a model with no explicit content. "Iff" denotes "if and only if," and the only difference between its men tal models and those for "if" are in the mental footnotes that are used to construct fully explicit models. where P denotes the set of models corresponding to the proposition p, Q denotes the set of models correspond ing to the proposition q, and R denotes the set of models corresponding to the proposition r. Hence, when indi viduals think about the truth of the proposition p, they should not bring to mind the falsity ofthe other two propo sitions, q and r, especially if the propositions p, q, and r are themselves ofany degree of complexity. In most cases, reasoners can still reach the correct conclusion even if they fail to represent what is false, but in some cases, as we shall see, the theory predicts that they will draw an in valid conclusion. Suppose that p, q, and r, are themselves disjunctive propositions, say, about a particular hand of cards: There is a king in the hand or there is an ace, or both. (p) sertion in a proposition is true, then it will be repre sented, but cases where it is false will not be represented. There is a queen in the hand or there is an ace, or both. (q) Table I presents the mental models and the fully explicit There is a jack in the hand or there is a ten, or both. (r) models for each of the main sentential connectives, as generated by a computer program implementing the the We cannot express the disjunction of disjunctions with ory at different levels of expertise. Fully explicit models a premise ofthe form: "There is a king in the hand or there can be constructed from mental models provided that one is an ace, or both, or else there is a queen in the hand or has access to the footnotes indicating what is exhaustively there is an ace, or both, or else, there is a jack in the hand represented in the mental models (Johnson-Laird & or there is a ten, or both." That would be too confusing. Byrne, 1991). But, there is a simpler alternative way to express the dis The model theory applies to modal reasoning-that is, junction of disjunctions, using the following rubric: reasoning about what is possible and what is necessary Only one of the following premises is true about a partic (Bell & Johnson- Laird, 1998). Other theories of reasoning ular hand of cards: might also be able to explain the phenomenon of modal reasoning. Thus, it might be feasible to complete Osher There is a king in the hand or there is an ace, or both. son's (1976) formal rules for modal reasoning in such a There is a queen in the hand or there is an ace, or both. way that the lengths of the derivations of the various sorts There is a jack in the hand or there is a ten, or both. of conclusion predict their relative difficulty. Alternatively, current theories of necessary deductions (e.g., Braine & This formulation is equivalent to the following: p or else O'Brien, 1991, 1998; Rips, 1994)might be extendedto deal q or else r, where these variables have as values the ap with modal reasoning by adding formal rules for drawing propriate disjunctions above. The two ways ofexpressing conclusions about possibilities. Hence, a key question is the exclusive disjunction have the same truth conditions, whether the mental model theory predicts some unique the same mental models, and the same fully explicit phenomenon that would be difficult, ifnot impossible, for models. Given the preceding premises and asked the fol the other current theories of reasoning to predict. lowing question: The an (...truncated)