How to eliminate illusions in quantified reasoning

All Bare C. 0 0 This research was supported by the Xian-Lin Ji Foundation ofPeking University. We thank Jonathan Baron , Mike Oaksford, Steven Sloman, and an anonymous referee for their helpful criticisms of an earlier version of the paper. We also thank the members ofour laboratory for much ad vice: Patricia Barres , Monica Bucciarelli, Victoria Bell, Zachary Estes, Y Yang, Department of Psychology , Green Hall, Princeton University , Princeton, NJ 08544 ( The mental model theory postulates that reasoners build models of situations described in premises. These models normally make explicit only what is true according to the premises. The theory has an unexpected consequence. It predicts the existence of illusions in inferences: Certain inferences should have compelling but erroneous conclusions. Previous studies have corroborated the existence of such illusions. The present study reports the first effective antidote to them. For example, most people incorrectly answer "yes" to the following problem: Only one of the following statements is true . . . fAt least some of the plastic beads are not red.!None of the plastic beads are red.! Is it possible that none of the red beads are plastic? In two experiments, we progressively eliminated this fallacy and others by using instructions designed to overcome the bias toward truth. The difference between the illusory and the control problems disappeared when the participants were instructed to work out both the case in which the first premise was true and the second premise was false and the case in which the second premise was true and the first premise was false. - . . All these sonatas have harmonic relations that can be handled by a context-sensitive grammar. Even if one knows nothing about the relevant set of sonatas or about context-sensitive grammars, one can still grasp the validity of the inference. The conclusion resulting from the inference must be true given that the premises are true. The ability to make valid inferences about ab stract matters is presumably a precursor to the acquisi tion of logic and mathematics. Yet it is controversial. Some theorists argue that the making of inferences de pends on formal rules that are akin to those of a logical calculus and that reasoners construct a chain of inferen tial steps akin to those of a proof (e.g., Braine, 1998; Rips, 1994). According to such theories, reasoning is a syntac tic process: The logical form ofthe premises is recovered, and then formal rules are applied to the premises in order to derive a proof in a sequence of syntactic steps (Yang, Braine, & O'Brien, 1998). For example, the following rule of inference can be used to make the preceding in ference about the sonatas: All A are B. .. All A are C. An alternative theory postulates that reasoning is a se mantic process. According to this theory, reasoners con struct mental models of the situations described by the premises, and they test the validity of conclusions by checking whether they are consistent with all the models of the premises (e.g., Johnson-Laird & Byrne, 1991). Thus, the inference about the sonatas can be made from a single model of the premises. Reasoners assume a small but arbitrary number of tokens to stand for the relevant set of sonatas. They tag each token designating a sonata in order to indicate that the piece is tonal and then that it has harmonic relations that can be handled by a context sensitive grammar. This model supports the conclusion that all the sonatas have harmonic relations that can be handled by a context-sensitive grammar. Reasoners can search for alternative models of the premises that refute this conclusion, but there is no such model and so the conclusion is valid. The controversy between these two theories is long standing. Each theory can account for certain empirical phenomena, and there are few crucial results that corrob orate one theory and refute the other. But, one seemingly innocuous assumption of the model theory has led to a discovery that, as we shall see, may be able to resolve the controversy. In order to minimize the load on working memory, rea soners represent as little information as possible. Accord ingly, a fundamental assumption ofthe mental model the ory is: the principle of truth, which states that the mental models of a set of assertions represent only the true possibilities according to the assertions, and each model of a true possibility represents the literal propositions in the premises (affirmative or negative) only when they are true within the possibility. A literal is a proposition that contains no sentential connectives and that is either af firmative or negative. Thus, the conjunction: There is a circle, but there is not a triangle contains two literals (there is a circle, there is not a trian gle). The principle of truth is subtle, because it applies at two levels. At one level, mental models do not represent those possibilities that are false according to the prem ises. In the case of the preceding conjunction, for exam ple, reasoners construct a single mental model ofthe only true possibility: where "~" denotes negation. Mental models do not rep resent the three cases in which the conjunction is false (the presence of a circle and a triangle, the absence of a circle and the presence of a triangle, and the absence of both a circle and a triangle). We explain below how peo ple try to represent the false possibilities when a task calls for them. As the previous example shows, mental models do represent negative assertions provided that they are true. Negative assertions are a well-known cause of difficulty in reasoning (see, e.g., Evans, Newstead, & Byrne, 1993), but this difficulty does not override the principle oftruth; people can represent possibilities cor responding to true negative assertions. The principle of truth applies at a second level, which concerns the representation of the literals in an assertion. Thus, the exclusive disjunction There is a circle or else there is not a triangle has two mental models, one for each of its true possibil ities, which we represent on separate lines: As these models illustrate, a literal in an assertion is rep resented in a possibility only if it is true in that possibil ity. Hence, the first of these models represents explicitly that there is a circle, but it does not represent that the lit eral, there is not a triangle, is false in this possibility. Like wise, the second model represents explicitly that there is not a triangle, but it does not represent that the literal, there is a circle, is false in this possibility. According to this theory, reasoners try to remember what is false, but these "mental footnotes" soon tend to be forgotten, es pecially when the assertions contain several connectives. Mental models are sensitive to the sentential connec tive; they represent only the possibilities that are true de pending on the connective. And within these possibilities, they (...truncated)