Psychological Review 1992, Vol. 99, No. 3, 418-439 Copyright 1992 by the American PsychologicalAssociation, Inc. 0033-295X/92/S3.00 Prepositional Reasoning by Model P. N. Johnson-Laird Princeton University Ruth M. J. Byrne Department of Computer Science University College, Dublin Dublin, Ireland Walter Schaeken University of Leuven Leuven, Belgium This article describes a newtheory of prepositional reasoning, that is,deductions depending on / or, and, and not. The theory proposes that reasoning isa semantic process based on mental models. It assumes that people are able to maintain models of only a limited number of alternative states of affairs, and they accordingly use models representing as much information as possible in an im- plicit way. They represent a disjunctiveproposition, such as "There isa circle or there is a triangle," by imagining initially2 alternative possibilities: one in whichthere isa circleand the other in which there is a triangle. This representation can, if necessary, be fleshed out to yield an explicit represen- tation of an exclusive or an inclusivedisjunction. The theory elucidates all the robust phenomenaof propositional reasoning. It also makes several novel predictions, which were corroborated by the results of 4 experiments. Propositional reasoning is ubiquitous in daily life. It consists of combining information from propositions containing such connectives as if, and, or, and not. For example, take the follow- ing premises: If there is fog, then the plane will be diverted. There is a fog. It is easy to draw the following conclusion: The plane will be diverted It is natural to suppose that the mind must contain a corre- sponding formal rule of inference: If A, then B. A .'. B It is also natural to suppose that the inference proceeds by matching the logical form of the premises to this rule of modus ponens. The rule can then be used to derive the conclusion. The dominant theoretical tradition is indeed that human Walter Schaeken is a research assistant funded by the Belgian Na- tional Fund for Scientific Research. We thankJonathan Evansfor his advice on the phenomena of propo- sitional inference and Malcolm Bauer for carrying out the multiple regression. We are also grateful to Martin Braine, who abides strictly by Marquisof Queensbury rules. Finally,we thank Mark Keane, Steve Palmer, Robert Sternberg, and anonymous referees fortheir criticisms of an earlier version of this article. Correspondence concerning this article should be addressed to P. N. Johnson-Laird, Department of Psychology, Princeton University, Princeton, New Jersey 08544. beings are equipped with formal rules of inference that enable them to make deductions. Versionsof such theories have been proposed by most of the those who haveworked on the psychol- ogy of deductive reasoning.The idea goes back to Boole in the nineteenth century, who wrote of his formalcalculus for propo- sitional reasoning, "The lawswehaveto examine are the lawsof one of the most important mental faculties. The mathematics we have to construct are the mathematics of the human intel- lect" (1847/1948, p. 7). In our era, the formal view has been advocated by Piaget and his colleagues: "Reasoning is nothing more than the propositional calculus itself" (Inhelder& Piaget, 1958, p. 305). Piaget's views about logic were idiosyncratic (e.g., see Braine & Rumain, 1983), but more recent proponents of formal rules have based their systems on orthodox logic. In particular, they have used the logical method of natural deduc- tion, which has separate rules of inference for each of the con- nectives, not, if, and, and or. Manytheorists have proposed such accounts of the psychology of propositional reasoning (e.g., Braine, 1978 Braine, Reiser, & Rumain, 1984 Johnson-Laird, 1975 Macnamara, 1986 Osherson, 1974-1976,1975 Pollock, 1989 Rips, 1983,1988 Sperber& Wilson, 1986). They all hold the viewaptly expressed by Rips (1983) in the following terms: . . . deductivereasoningconsists in the application of mental in- ference rules to the premises and conclusion of an argument. The sequence of applied rules forms a mental proof or derivation of the conclusion from the premises, where these implicit proofs are analogous to the explicit proofs of elementary logic (p. 40). Henceforth, we will refer to these accounts of reasoning as rule theories. Our view of deductive competence is that people are rational in principle, but they err in practice. Anyset ofdeductive prem- ises yields an infinite number of valid conclusions, but most of them are banal, such as an arbitrarynumberof conjunctions of 418

PROPOSITIONAL REASONING 419 a premise with itself. Logically untutored individuals never draw such conclusions. In general, they eschew conclusions that contain less semantic information than premises. Hence, sup- pose they are asked what follows logically from the following premise: Anne is at the party and Alan is at the game. They do not spontaneously draw the conclusion: Anne is at the party. Similarly, they seek conclusions that are more parsimonious than the premises. Hence, suppose they are asked what follows from the following premises: Betty is here. Brian is at work. They do not spontaneously draw the conclusion: Betty is here and Brian is at work. They also do not bother to repeat in their conclusions what is asserted categorically by a premise (cf. Grice, 1975). They try instead to draw conclusions that make explicit some informa- tion only implicit in the premises. In short, to deduce is to maintain semantic information, to simplify, and to reach a new conclusion. Where there is no valid conclusion that meets these three constraints, logically untrained individuals declare that nothing follows from the premises (Johnson-Laird, 1983).Any theory of how people reason should accordingly reflect these constraints, though they be may emergent properties of other principles. In thisarticle, weaim to present a newexplanation of preposi- tional reasoning. Our hypothesis is that the underlyingdeduc- tive machinerydepends not on syntactic processes that use for- mal rules but on semantic procedures that manipulate mental models. This theory is in part inspired by the model-theoretic approach to logic. Semantic procedures construct models of the premises, formulate parsimonious conclusions from them, and test their validityby ensuring that no alternative models of the premises refute them. This approach is akin to the analysis of problem solving as a heuristic search through a problem space in which each state corresponds to a mental model (Ne- well, 1990 Newell & Simon, 1972 Simon, 1990). Various sorts of mental models have been postulated as un- derlying deduction (e.g., see Erickson, 1974 Guyote & Stern- berg, 1981 Levesque, 1986 Newell, 1981 Polk & Newell, 1988). Our view is that models have a structure that corre- sponds directly to the structure of situations. Each individual in a situation isrepresented by a corresponding mental token, and the properties of individualsand the relations among them are likewise modeled in an isomorphic way (see Johnson-Laird, 1983, p. 419-447). This theory has been applied to spatial rea- soning (Byrne & Johnson-Laird, 1989), to reasoning withsingle quantifiers (see Johnson-Laird & Bara, 1984 Johnson-Laird & Byrne, 1989), and to reasoning with multiplequantifiers (John- son-Laird, Byrne, & Tabossi, 1989). It has not, however, been applied to propositional reasoning. Hence, as many critics have pointed out (Braineet al., 1984 Evans, 1987 Rips, 1986,1990), the model theory has been radically incomplete. In the present article, which supersedes the suggestions in Johnson-Laird and Byrne (1991), we remedy this deficiency. Wepresent a compre- hensive model theory of propositional reasoning. Our plan is to consider those connectives that can be accom- modated within rule theories, namely, not, and, or, and if. We begin by outlining the distinction in logic between rules and models. Wedescribe some representative rule theories in psy- chology and the contrasting model theory, including its com- puter implementation. This theory motivates a reanalysis of the major experimental studies, and we show how it explains their principal phenomena. The theory leads to novel predictions, and we report some experiments designed to test them. Finally, we consider the chief differences between rules and models. Rules and Models in Logic Logicians distinguish between reasoning based on formal rules of inference (proof-theoretic methods) and reasoning based on models (model-theoretic methods). A proof-theoretic method uses formal rules of inference to derive conclusions from premises in a syntactic way. Here, for example, is a formal rule of inference for inclusivedisjunction: A or B, or both. Not-A. Therefore, B, where A and B can be any propositions. The rule can be used to make the following deduction: Lisa is in Cambridge or Ben is in Dublin, or both. Lisa is not in Cambridge. Therefore, Ben is in Dublin. The disjunctive rule is part of most psychological theories based on formal rules (e.g., Braine, 1978 Johnson-Laird, 1975 Rips, 1983). A formal calculus can be given a semantic interpretation in terms of models, and the standard model-theoretic method for the propositional calculus is based on truth tables. The mean- ing of each connective is specified by a truth table. An inclusive disjunction of two propositions, A or B or both, istrue provided that at least one of the two propositions is true, and is false only if they are both false. This truth-functional definition can be stated in a truth table, where T denotes true, and F denotes false, and each row states a separate possibility: A T T F F B T F T F A or B, or both T T T F The connectives of the propositional calculus can all be defined by truth tables. Strictly speaking, it is a mistake to assign truth values to sentences in natural language:The same sentence can be used to assert many different propositions for example, the sentence "I felt ill yesterday" asserts different propositions de- pending on who asserts it and when it is asserted. Hence, it is

420 R JOHNSON-LAIRD, R. BYRNE, AND W SCHAEKEN propositions, not sentences, that have truth values (Strawson, 1950). Deductions can be made using truth tables rather than for- mal rules. Each premise is used to eliminate otherwise possible combinations of the atomic propositions that occur as constitu- ents of the premises. Thus, the deduction above concerns four possibilities of two atomic propositions: Lisa is in Cambridge T T F F Ben is in Dublin T F ft T F The first premise, Lisa is in Cambridge or Ben is in Dublin, or both, eliminatesthe fourth possibility in the table, which is not compatible with the truth of this premise. The second premise, Lisa is not in Cambridge, eliminates the first 2 possibilities. When you have eliminated the impossible, then whatever re- mains must be the case. What remains is, of course, the third possibility, in which it is true that Ben is in Dublin. This conclu- sion therefore follows validly from the premises, and the deduc- tion is made solely by using the meanings of the premises to eliminate possibilities. This method needs procedures for con- structing truth tables and for eliminating possibilities from them, but it does not need any formal rule of inference, such as the rule for disjunction that we described earlier. The fact that semantic procedures do not depend on formal rules of inference can be hard to grasp. Skeptics often ask, "Where do the truth tables come from���surely one needs to know the formal rules to construct the truth tables?" The an- swer is that truth tables are merelya systematic wayofspelling out a knowledge of the meanings of connectives. To know the meaning of an inclusive disjunction,A or B, is to know that the assertion is true if at least one of the propositions is true and that it is false only if both of the propositions are false. The skeptic, however, persists: "Surely this knowledge must derive from formal rules of inference?" In fact, the formal rules for propositional connectives are consistent with more than one possible semantics (e.g., see Kneale & Kneale, 1962, p. 678). Hence, although it is sometimes suggested that the meaning of a term derivesfrom, implicitly reflects,or is nothing more than the rules of inference for it, this idea is unworkable (see Osher- son, 1974-1976, Vol. 3, p. 253 Johnson-Laird, 1983, p. 41 Prior, 1960). On the contrary, the rulesof inference must reflect the meaningsof the connectives. The meaning of an assertion relates it to the world, and the meaning of a connective makes a contribution to these truth conditions. A rule of inference en- ables a reasoner to pass from a set of premisesto a conclusion in a purely formal way,but this step is constrained by the truth conditions of the assertions. A major part of modern logicconcerns the relations between proof-theoretic methods that rely on formal rules and model- theoretic methods that rely on the meanings of expressions. Logicians have proved that any propositional inference that is valid according to the truth-table method can be derived using the formal rules of the propositional calculus. The calculus is therefore said to be complete. Logicians have also proved that any propositional inference that can be derived using the calcu- lus can also be validated using truth tables. The calculus is therefore said to be sound(e.g., see Jeffrey, 1981). What must be emphasized, particularly in the context of psychological the- ories, is that the two approaches are distinct: One is syntactic and based on formal rules of inference, and the other isseman- tic and based on the meanings of connectives. To deny this point is to deny the significance of these proofsofcompleteness and soundness. Rule Theories of Propositional Reasoning Natural deduction has been advocated as the most plausible account of mental logic by many theorists (e.g., Braine, 1978 Braine et al., 1984 Johnson-Laird, 1975 Macnamara, 1986 Osherson, 1974-1976, 1975 Pollock, 1989 Rips, 1983,1988 Sperber & Wilson, 1986), and at least one simulation program uses it to construct both forward and backward chainsof infer- ence (Rips, 1983). All of these theories posit an initial process of recovering the logical form of the premises. Indeed, what they have in common outweighs their differences, but here we outline three of them to enable readers to make up their own minds. Johnson-Laird (1975) proposed a theory of propositional reasoning partly based on natural deduction. It distinguishes between primary and auxiliaryrulesof inference. The primary rules include the rule for disjunction presented earlier and the rule for modus ponens: If A, then B. Therefore, B. The following is the rule introducing disjunctiveconclusions: A Therefore, A or B, or both. This leads to deductions that throw semantic information away that is, the conclusion rulesout fewer statesof affairs than does the premise. Valid inferences that reduce information in this way,as we noted above, are not spontaneously drawn by logically untutored reasoners and strike them as odd or absurd (e.g., see Matalon, 1962). Yet, without this rule, it would be difficult to make the following inference: If it is frosty or it is foggy, then the game will not be played. It is frosty. Therefore, the game will not be played. Johnson-Laird therefore proposed that the rule (and others like it) is an auxiliary one that can be used only to prepare theway for a primary rule, such as modus ponens. Braine and his colleagues described a series of formal the- ories based on natural deduction (see Braine, 1978 Braine & Rumain, 1983). Their rules differ in format from Johnson- Laird's (1975) in two ways. First, and and or can connect any number of propositions, and so,forexample, therule introduc- ing the conjunction of premises has the following form in their theory: