How Diagrams Can Improve Reasoning

We report an experimental study on the effects of diagrams on deductive reasoning with double disjunctions,/i?/' example: Raphael is in Tacoma or Julia is in Atlanta, or both. Julia is in Atlanta or Paul is in Philadelphia, or both. What follows? We confirmed that subjects fmd it difficult to deduce a valid conclusion, such as Julia is in Atlanta, or both Raphael is in Tacoma and Paul is in Philadelphia. In a preliminary study, the format of the premises was either verbal or diagrammatic, and the diagrams used icons to distinguish between inclusive and exclusive disjunctions. The diagrams had no effect on performance, ln the main experiment, the diagrams made the alternative possibilities more explicit. The subjects responded faster (about 35 s) and drew many more valid conclusions (nearly 30%) from the diagrams than from the verbal premises. These results corroborate the theory of mental models and have implications for the role of diagrams in reasoning. How can diagrams help people to reason? The question has a long philosophical history, but a shoti psychological one. Philosophers, for example, have worried about how the use of a single diagram in a geometric proof might mislead geometers (Beth, 1971). In a pioneering psychological article, Larkin and Simon (1987) distinguished the role of diagrams in three separate sorts of processes; search, recognition, and inference (see also Tabachneck & Simon, 1992). Larkin and Simon said diagrams can make it easier to fmd relevant information: One can scan from one element to another element nearby much more rapidly than one might be able to find the equivalent information in a list of numbers or verbal assertions. Diagrams can make it easier to identify instances of a concept: An iconic representation can be recognized faster than a verbal description. Their symmetries can reduce the number of cases that need to be examined. However, about inference, Larkin and Simon (1987) wrote: In view of the dramatic effects that alternative representations may produce on search and recognition processes, it may seem surprising that the differential effects on inference appear less strong. Inference is largely independent of representation if the information content of the two sets of inference rules [one operating on diagrams and the other operating on verbal statements] is equivalent—i.e. the two sets are isomorphs as they are in our examples, (p. 71) Address correspondence to Malcolm I. Bauer, Department of Psychology, Princeton University, Green Hall, Princeton, NJ 08544. Barwise and Etchemendy (1992) have argued that the truth behind the adage that a picture is worth a thousand words is that diagrams and pictures are good at presenting a wealth of specific conjunctive information, "it is much harder to use them," they said, "to present indefinite information, negative information, or disjunctive information" (p, 80). Such information is often better conveyed by sentences, and so their pedagogical program, Hyperproof. makes use of both diagrams and sentences. It appears that these researchers are skeptical about how diagrams can aid in inference, especially reasoning depending on disjunctions or negations. The present article provides a theoretical basis for why diagrams can help with such reasoning, and describes two experiments that give empirical support to our claims. The results have implications for the role of imagery in reasoning, and we comment briefiy on this point as well. A deduction is valid if its conclusion must be true given that its premises are true. Formal logic provides methods of testing validity, and nearly all psychologicai theories of reasoning have postulated the existence of formal rules of inference in the mind (Braine, 1978; Inhelder & Piaget, 1958; Rips. 1983; Smith, Langston. & Nisbett, 1992). An inference is difficult, according to such theories, if it calls for a long chain of steps in the derivation of its conclusion, or if it calls for a rule of inference that is difficult to access or to apply. Diagrams are unlikely to affect performance, however, for a reason similar to the one adduced by Larkin and Simon (1987): Once the logical form of the problem has been extracted from a diagram, the same chain of deductions based on the same rules of inference should unfold. The theory of mental models tells a very different story (Johnson-Laird, 1983; Johnson-Laird & Byrne, 1991). Granted the definition of a valid deduction (see above), any method of testing validity is in effect a method of ensuring that the conclusion holds in all the possible states of affairs characterized by the premises. Insteadof relying on formal rules of inference, the model theory postulates a more direct process mirroring the examination of possibilities: Individuals reason by (a) constructing a model, or models, based on the information in the premises and background knowledge, (b) formulating a conclusion that is true of the tnodel and subject to other constraints, such as parsimony, and (c) searching for alternative models in which the conclusion does not hold. If there is no such alternative model, then the conclusion is valid. In general, deductions depending on multiple models should be difficult, and erroneous conclusions to them should be consistent with the truth of the premises, because the subjects consider some, but not all, of the possible tnodels. These predictions, which cannot be made by any existing theories based on formal rules, have been corroborated in studies of all the main sorts of deduction (see Johnson-Laird & Byrne, 1991). Deductive reasoning often depends on taking into account altemative possibilities, and a major source of errors is the 372 Copyright © 1993 American Psychological Society VOL. 4, NO. 6, NOVEMBER 1993 PSYCHOLOGICAL SCIENCE Malcolm I. Bauer and P.N. Johnson-Laird difficulty of holding in mind several models simultaneously. A premise such as "All the women in the room are psychologists'" is consistent with the existence, or not, of psychologists in the room who are not women, and with the existence, or not, of individuals in the room who are neither women nor psychologists. Many deductions, however, can be drawn without explicitly representing such possibilities. For example: All the women in the room are psychologists. All the psychologists in the room are Russians. Therefore, all the women in the room are Russians. A similar phenomenon occurs with reasoning based on conditionals (see Johnson-Laird, Byrne. & Schaeken. 1992). So, in what circumstances are reasoners forced to consider alternative possibilities? One way in which to elicit such representations is, according to the model theory, to use a disjunction, that is, a premise of the form "A or B," where A denotes one proposition and B denotes another proposition. Much evidence exists to show that disjunctions are harder to think about than conjunctions. Osherson (1974-1976), for example, in characterizing children's and adolescent's deductive competence, observed that disjunctions are harder than conjunctions. Many studies of concept attainment bear out this claim (e.g., Bruner, Goodnow, & Austin, 1956; Neisser & Weene, 1962). Likewise, studies of deduction have also corroborated it (Braine, Reiser, & Rumain, 1984). Theories based on formal rules of inference cannot explain the phenomenon, but merely assume post hoc that rules for conjunction are easier to accessortoapply than those for disjunction (Braine et al.. 1984; Rips, 1983). However, the difference follows directly from the model theory because a conjunction calls for only one explicit model, whereas a disjunction calls for at least two explicit models. If the model theory is right, then there should be a breakdown in deductive performance as the number of models increases beyond the capacity of working memory. One way in which to increase models is to use so-called double disjunctions, such as Raphael is in Tacoma or Julia is in Atlanta, or both. Julia is in Atlanta or Paul is in Philadelphia, or both. What follows? The premises support five altemative models: