Reasoning from connectives and relations between entities

Robert Mackiewicz 0 1 Philip N. Johnson-Laird 0 1 0 P. N. Johnson-Laird Department of Psychology, Princeton University , Princeton, NJ 08540, USA 1 Either Ann is taller than Beth or else Beth is taller than Cath, but not both 2 ) Department of Psychology, Warsaw School of Social Sciences and Humanities , Chodakowska 19/31, 03-815, Warsaw, Poland This article reports investigations of inferences that depend both on connectives between clauses, such as or else, and on relations between entities, such as in the same place as. Participants made more valid inferences from biconditionalsfor instance, Ann is taller than Beth if and only if Beth is taller than Caththan from exclusive disjunctions (Exp. 1). They made more valid transitive inferences from a biconditional when a categorical premise affirmed rather than denied one of its clauses, but they made more valid transitive inferences from an exclusive disjunction when a categorical premise denied rather than affirmed one of its clauses (Exp. 2). From exclusive disjunctions, such as Either Ann is not in the same place as Beth or else Beth is not in the same place as Cath, individuals tended to infer that all three individuals could be in different places, whereas in fact this was impossible (Exps. 3a and 3b). The theory of mental models predicts all of these results. - The logically correct answer is yes, and the validity of the inference depends both on the sentential connective or else, which interrelates the two clauses, and on the relations is taller than and is the tallest of, which interrelate entities. Previous studies have investigated how people understand connectives and make inferences from them, and how people understand and make inferences from relations. But no empirical studies have examined reasoning that hinges both on connectives and on relations, and this novel domain challenges theories of reasoning. The present article addresses this challenge and reports some new phenomena concerning such inferences. Relations in everyday discourse have various logical properties, of which perhaps the three most important are transitivity, symmetry, and reflexivity (Tarski, 1965, chap. V). A relation such as is in the same place as is transitive because if A is in the same place as B and B is in the same place as C, then A is in the same place as C. It is symmetric because if A is in the same place as B, then B is in the same place as A. And it is reflexive because A is in the same place as A. A relation such as is taller than is transitive, but it is asymmetric because if A is taller than B, then B is not taller than A, and it is also irreflexive because A is not taller than A. Relations do have other logical properties (see Tarski, 1965), but these properties are recondite and seldom, if ever, relevant to everyday discourse. Indeed, most relational terms in language have no important logical propertiesfor instance, A loves B is neither transitive nor intransitive, neither symmetric nor asymmetric, and neither reflexive nor irreflexive. Previous psychological studies have investigated how individuals make inferences from transitive relationsfor instance, A is bigger than B and B is bigger than C; therefore, A is bigger than C (e.g., Clark, 1969; Huttenlocher, 1968). Transitive inferences occur in everyday life, in intelligence tests, and in inferring economic preferences (Tversky & Kahneman, 1986). The difficulty of such inferences depends on the number of relations that must be integrated (Viskontas, Morrison, Holyoak, Hummel, & Knowlton, 2004), the distance between queried elements (Mynatt & Smith, 1977), and the elicitation of extraneous visual images (e.g., Knauff, Fangmeier, Ruff, & JohnsonLaird, 2003). Previous studies have also examined inferences based on two-dimensional spatial relations, temporal relations, and relations between relations, and have shown that the difficulty of inferences also depends on the number of possibilities in which the premises hold (e.g., Byrne & Johnson-Laird, 1989; Carreiras & Santamaria, 1997; Goodwin & Johnson-Laird, 2005, 2006; Schaeken, Johnson-Laird, & dYdewalle, 1996). Other studies have shown that some relations are pseudo-transitivefor instance, A is a blood relative of Bbecause individuals tend to make transitive inferences from them unless they are cued to counterexamples (Goodwin & Johnson-Laird, 2008). The connectives that occur in everyday discourse include if, and, and or. These connectives resemble those of sentential logic (see, e.g., Jeffrey, 1981), but the degree of resemblance is a matter of controversy, especially in the case of if (e.g., Evans & Over, 2004; Girotto & JohnsonLaird, 2004; Johnson-Laird & Byrne, 2002; Oaksford & Chater, 2007). Psychologists have carried out many studies of inferences from conditional assertions, if A then B, where the values of A and B are clauses, such as there is a vowel on a card (for reviews, see, e.g., Evans & Over, 2004; Johnson-Laird, 2006). These studies, as well as those of other connectives such as or else, concern inferences that hinge solely on the connectives, and not on relations between entities that are described in their individual clauses (e.g., Braine & OBrien, 1998; Johnson-Laird, Byrne, & Schaeken, 1992). Current accounts of reasoning include those based on formal rules of inference (e.g., Braine & OBrien, 1998; Rips, 1994), those based on suppositions (e.g., Evans & Over, 2004), those based on probabilistic considerations (e.g., Geiger & Oberauer, 2010; Oaksford & Chater, 2007), and those based on mental models (e.g., Johnson-Laird & Byrne, 1991). None of these accounts has hitherto been applied to inferences that depend on both relations and connectives. Formal rule theories, however, apply to such inferences, because these theories can postulate meaning postulates to capture the logical properties of relations. For example, the following meaning postulate captures the transitivity of is taller than: For any x, y, z, if x is taller than y, and y is taller than z, then x is taller than z. Proponents of such theories have also proposed meaning postulates for two-dimensional spatial reasoning (e.g., Hagert, 1984). But the case against meaning postulates has been presented elsewhere (e.g., Goodwin & JohnsonLaird, 2008). The suppositional theory of Evans and Over (2004) and the probabilistic theory of Oaksford and Chater (2007) have been applied primarily to if among the sentential connectives, which these theories analyze in terms of the conditional probability of the then clause, given that the if clause is true. Although both of these theories might be extended to other connectives and to reasoning that hinges on both relations and connectives, these extensions have yet to be made. Our aim is therefore to follow up the predictions of the theory of mental models rather than to try to pit one theory against the others, because of the uncertainty about their predictions. The mental model theory th (...truncated)