Psychological Bulletin © 2012 American Psychological Association 2012, Vol. 138, No. 3, 427–457 0033-2909/12/$12.00 DOI: 10.1037/a0026841 Theories of the Syllogism: A Meta-Analysis Sangeet Khemlani and P. N. Johnson-Laird Princeton University Syllogisms are arguments about the properties of entities. They consist of 2 premises and a conclusion, which can each be in 1 of 4 “moods”: All A are B, Some A are B, No A are B, and Some A are not B. Their logical analysis began with Aristotle, and their psychological investigation began over 100 years ago. This article outlines the logic of inferences about syllogisms, which includes the evaluation of the consistency of sets of assertions. It also describes the main phenomena of reasoning about properties. There are 12 extant theories of such inferences, and the article outlines each of them and describes their strengths and weaknesses. The theories are of 3 main sorts: heuristic theories that capture principles that could underlie intuitive responses, theories of deliberative reasoning based on formal rules of inference akin to those of logic, and theories of deliberative reasoning based on set-theoretic diagrams or models. The article presents a meta-analysis of these extant theories of syllogisms using data from 6 studies. None of the 12 theories provides an adequate account, and so the article concludes with a guide—based on its qualitative and quantitative analyses—of how best to make progress toward a satisfactory theory. Keywords: logic, mental models, reasoning, rules of inference, syllogisms Consider this inference: the monks were most adept at formulating their arguments in syllogisms. Even though modern logicians relegate syllogisms to In some cases when I go out, I am not in company. one small part of logic, psychological studies of reasoning with Every time I am very happy I am in company. Therefore, in some cases when I go out, I am not very happy. (1) determiners, such as some and all, have almost all concerned syllogistic reasoning. It is an example of a syllogism, though it is not in the usual formal The ability to reason is at the core of human mentality, and dress used in textbooks of logic. It is also valid, where “a valid many contexts in daily life call for inferences, including decisions inference is one whose conclusion is true in every case in which all about goals and actions; the evaluation of conjectures and hypoth- its premises are true” (Jeffrey, 1981, p. 1). Among Western think- eses; the pursuit of arguments and negotiations; the assessment of ers, Aristotle was the first to analyze syllogisms, and they were evidence and data; and above all science, technology, and culture. central to logic until the second half of the 19th century. Scholastic Reasoning based on quantifiers enters into all these pursuits, as the logicians thought that almost all arguments purporting to be logical following sorts of assertion illustrate: could be expressed in syllogisms, and this tradition continued into the 20th century. The BBC once broadcast a monastic debate, and Any experiment containing a confound is open to misinterpretation. No current word processor spontaneously corrects a user’s grammar. Every chord containing three adjacent semitones is highly dissonant. (2) This article was published Online First January 30, 2012. If such premises are combined with others, they validly imply Sangeet Khemlani, Navy Center for Applied Research in Artificial various sorts of conclusion. In daily life, individuals reason in a Intelligence, Naval Research Laboratory, Washington, D.C.; P. N. Johnson-Laird, Department of Psychology, Princeton University. variety of contexts, and often so rapidly that they are unaware of This research was supported by a National Science Foundation Graduate having made an inference. Consider the discussion in the following Research Fellowship to Sangeet Khemlani and by National Science Foun- example: dation Grant SES 0844851 to P. N. Johnson-Laird to study deductive and Speaker 1: If you drop this cup it’ll break. probabilistic reasoning. We are grateful to the following individuals for Speaker 2: It looks pretty solid to me. taking the time to correct our accounts of their theories, where necessary, Speaker 1: Yes, but it’s made from porcelain. (3) and for other helpful comments: Nick Chater and Mike Oaksford (proba- bility heuristics), Marilyn Ford (verbal substitution), Bart Geurts (the The first speaker has made a tacit inference from the beliefs that monotonicity theory), Ken Gilhooly (matching), Thad Polk (the verbal porcelain is fragile and that fragile things break if they are reasoning model), Russ Revlin (illicit conversion), Lance Rips (the dropped. The context of a potential inference affects the likelihood PSYCOP model), and Keith Stenning (the source-founding model). We with which it will be made. For instance, when individuals reason thank Hua Gao, Catrinel Haught, Niklas Kunze, Max Lotstein, Gorka about a matter that elicits a moderate emotion, they tend to Navarrete, Janani Prabhakar, and Marco Ragni for their help and criticisms. Correspondence concerning this article should be addressed to Sangeet outperform individuals who are not in an emotional state, includ- Khemlani, Intelligent Systems Section, Code 5515, Navy Center for Ap- ing their own performance on matters that do not elicit an emotion plied Research in Artificial Intelligence, Naval Research Laboratory, 4555 (Johnson-Laird, Mancini, & Gangemi, 2006). In psychological Overlook Avenue SW, Washington, DC 20375. E-mail: skhemlani@ experiments, participants are happy to make inferences about gmail.com hypothetical individuals. One exception occurs with people from a 427 428 KHEMLANI AND JOHNSON-LAIRD nonliterate culture, who are biased to make inferences from per- were unable to account for syllogistic reasoning, then they would sonal knowledge, but who will reason hypothetically provided that have little hope of making sense of reasoning in general. This no one can have the relevant knowledge, such as inferences about argument seems pertinent today. a remote planet (Dias, Roazzi, & Harris, 2005). There are now 12 main sorts of theory of syllogistic reasoning Some psychologists demand that studies of reasoning should be and reasoning based on monadic premises, that is, premises in ecologically valid: “Those who study first-order logic or variants which assertions assign only properties to individuals. One reason thereof, such as mental rules and mental models, ignore the eco- for the plethora of theories is that theorists have different goals. logical and social structure of environments” (Hertwig, Ortmann, Early theorists, for instance, presupposed some sort of deductive & Gigerenzer, 1997, pp. 105–106). In fact, abstract mathematics mechanism and sought only to explain causes of error (e.g., Chap- and logic are often outside the ecological and social context of man & Chapman, 1959; Woodworth & Sells, 1935). Current environments. Much the same applies to Sudoku puzzles, which theorists aim to characterize heuristics that yield conclusions and are popular in both East Asia and the Western world. Their disregard logical validity (e.g., Chater & Oaksford, 1999). Other solution depends on pure deduction—arithmetical calculations theories provide an account of logical reasoning with less regard play no part—and on understanding the quantified principle that for the intuitive responses that naive individuals make (e.g., every row, column, and member of each of the nine 3 ϫ 3 groups Johnson-Laird & Steedman, 1978; Rips, 1994). Likewise, some of cells contains each of nine digits. The puzzles derive from accounts of logical reasoning focus on the evaluation of monadic Euler’s study of Latin squares, and their domain is open-ended conclusions within the scope of the first-order predicate calculus— (i.e., the puzzles can be larger than 9 ϫ 9 squares) and computa- the logic in which quantifiers, such as “some artists,” range only tionally intractable (see Lee, Goodwin, & Johnson-Laird, 2008). over individuals (e.g., Rips, 1994). But other theories consider a The puzzles have no linguistic context beyond the statement of the broader variety of monadic inferences, such as the following puzzle itself, and they have no ecological validity either for us or example: for our protohominid ancestors. They show, however, that logi- cally untrained individuals can make deductions from triply quan- More than half the people in the room are English. tified assertions, and their popularity establishes that some people More than half the people in the room are American. І At least someone in the room is both English and American. (4) not only exercise logical ability in abstract problems, but also enjoy doing so. Without this ability, as Piaget realized, it is Quantifiers, such as “more than half the people,” cannot be rep- impossible to account for the development of logic and mathemat- resented in first-order logic (Barwise & Cooper, 1981), because ics (see, e.g., Beth & Piaget, 1966). they call for quantification over sets rather than individuals. Another context in which logically untrained individuals are Hence, theories that invoke only standard first-order logic cannot called on to reason explicitly is when taking a test, such as the hope to explain these inferences, or any that hinge on related Scholastic Assessment Test or the Graduate Record Examination. determiners, such as “more than a third,” “more than a quarter,” And, of course, some people have to reason in the psychological “more than a fifth,” etc. laboratory as participants in studies of reasoning. In such cases, the The existence of 12 theories of any scientific domain is a small instructions often call for them to draw conclusions that follow of disaster. If psychologists cannot converge on a single theory of necessity from the premises (see, e.g., Table 5). Most people have monadic reasoning, then the last 30 years of research have failed little difficulty in understanding the task, and even children of 9 or at best made so little progress that skeptics may think that years of age are able to make appropriate inferences (Johnson- cognitive science itself is not feasible.