A mental model theory of set membership

A mental model theory of set membership Sangeet Khemlani 1 , Max Lotstein 2 , and Phil Johnson-Laird 3,4 sangeet.khemlani@nrl.navy.mil, mlotstein@gmail.com, phil@princeton.edu US Naval Research Laboratory, Washington, DC 20375 USA University of Freiburg, Freiburg, Germany Princeton University, Princeton NJ 08540 USA New York University, New York, NY 10003, USA Abstract In contrast, this inference is valid: Assertions of set membership, such as Amy is an artist, should not be confused with those of set inclusion, such as All artists are bohemians. Membership is not a transitive relation, whereas inclusion is. Cognitive scientists have neglected the topic, and so we developed a theory of inferences yielding conclusions about membership, e.g., Amy is a bohemian, and about non-membership, Abbie is not an artist. The theory is implemented in a computer program, mReasoner, and it is based on mental models. The theory predicts that inferences that depend on a search for alternative models should be more difficult than those that do not. An experiment corroborated this prediction. The program contains a parameter, σ, which determines the probability of searching for alternative models. A search showed that its optimal value of .58 yielded a simulation that matched the participant’s accuracy in making inferences. We discuss the results as a step towards a unified theory of reasoning about sets. 5. Viv is a judge. Judges are lawyers. Therefore, Viv is a lawyer. Keywords: quantifiers, reasoning, sets, syllogisms. Introduction Quantifiers raise problems for linguists in their syntax and semantics (e.g., Peters & Westerstahl, 2006; Steedman, 2012). And they raise problems for cognitive scientists in their mental representation and roles in inference (e.g., Johnson-Laird, 2006; Oaksford & Chater, 2007; Rips, 1994). Our goal is to elucidate quantifiers by considering a neglected topic: assertions of set membership. Consider, for example, these three assertions: 1. Viv is a judge. 2. Judges are lawyers. 3. Judges are appointed in different ways. Assertion (1) states that an individual is a member of a set, and assertion (2) states that one set is included in another set. Assertion (3) is about a set, but it states, not that it is included in another set, but that it is a member of another set, i.e., the set of judges is a member of the set of those who are appointed in different ways. The difference matters because membership is not a transitive relation. Hence, the following inference is not valid: 4. Viv is a judge. Judges are appointed in different ways. Therefore, Viv is appointed in different ways. In formal logic, the second premise is equivalent to: anyone who is a judge is a lawyer, i.e., inclusion is defined in terms of membership. The distinction between inferences (4) and (5) is therefore subtle, and psychological theories need to recognize the difference between them. A powerful way in logic to represent the meaning of quantifiers, such as “all judges”, is as sets of sets. This method was popularized by Montague (see the accounts in, e.g., Johnson-Laird, 1983; Partee, 1975; Peters & Westerstahl, 2006). But, an alternative representation treats quantified assertions as stating relations between sets (Boole, 1854, Ch. XV). More recently, psychologists have had the same idea (e.g., Ceraso & Provitera, 1971; Geurts, 2003; Johnson-Laird, 1970; Politzer, van der Henst, Luche, & Noveck, 2006). On this account, the assertion all judges are lawyers means that the set of judges is included in the set of lawyers. Likewise, the assertion no judges are inmates means that the intersection of the set of judges and the set of inmates is empty. The advantage of this treatment is that it readily extends to quantifiers that cannot be captured in standard logical accounts (such as Rips, 1994). The assertion most judges are men, contains a quantifier “most judges” that cannot be defined using the quantifiers of first-order predicate calculus (Barwise & Cooper, 1991). Its relational meaning is simple: the cardinality of the intersection of the set of judges and the set of males is greater than the cardinality of the set of judges that are not males (see Cohen & Nagel, 1934). So, how do naive individuals reason about set membership? The aim of the present paper is to answer this question. Our answer is based on the theory of mental models. We accordingly begin with an outline of the theory from which we derive one principal prediction about such inferences. We report an experiment that corroborates this prediction. We use a computer program, mReasoner, to simulate performance, and show that the simulation provides a satisfactory fit with the experimental results. Finally, we draw some general conclusions about the psychology of set membership.