Theory Comparison for Generalized Quantifiers

Theory Comparison for Generalized Quantifiers Marco Ragni a (marco.ragni@cognition.uni-freiburg.de) Henrik Singmann b (henrik.singmann@psychologie.uni-freiburg.de) Eva-Maria Steinlein a (eva.steinlein@jupiter.uni-freiburg.de) a Center for Cognitive Science, b Department of Psychology University of Freiburg Abstract Premises and conclusions in classical syllogistic reasoning are formed using one of four quantifiers (All, Some, Some not, None). In everyday communication and reasoning, however, statements such as “most” and “few” are formed as well. So far only Chater and Oaksford’s (1999) Probability Heuristics Model (PHM) makes predictions for these so-called general- ized quantifiers. In this article we (i) extend existing and de- velop new theories, (ii) develop multinomial processing tree (MPT) models for these theories, and (iii) conduct an ex- periment to test the models. The models are evaluated with G 2 , Akaike’s (AIC) and Bayesian Information Criteria (BIC), and Fisher’s Information Approximation (FIA). Mental model- based accounts and PHM provide an equal account to the data. Keywords: Syllogistic Reasoning; Generalized Quantifiers; MPTs; Model Selection Introduction Syllogistic reasoning derives a conclusion from two quanti- fied statements. Each statement is formed using one of the four quantifiers: all (A), some (I), some not (O), or none (E). Consider, a simple syllogism, e.g., Some X are not Y . Few Y are Z. From these premises a reasoner could conclude “Few X are Z” although again (logically) nothing follows. An additional advantage of using generalized quantifiers is that they can provide a new benchmark for theories of classical syllogis- tic reasoning. The remainder of this article is structured as follows: We first review the only existing theory for the generalized quan- tifiers most and few – the PHM (Chater & Oaksford, 1999). In a second step we extend successful approaches for classical syllogistic reasoning. Specifically, we extend the Matching Hypothesis and develop two additional mental model-based accounts – one based on minimal models and the other on preferred mental models. Finally, all theories are formalized via multinomial processing tree (MPT) models and their em- pirical performance is assessed on data from a new experi- ment. Theories for Generalized Quantifiers Established Theory: Probability Heuristics Model All cognitive scientists are intelligent. Some intelligent people are rich. Most individuals erroneously infer from these statements that “Some cognitive scientists are rich” (Khemlani & Johnson- Laird, 2012). The only logically valid response is, how- ever, that nothing follows. A meta-analysis by Khemlani and Johnson-Laird (2012) investigating twelve theories shows that there is still no psychological theory that provides a com- prehensive account for human syllogistic reasoning. The as- sessed theories can be categorized into one of the following three classes: heuristic theories, logical theories, and model- based theories. The findings indicate that heuristic-based theories (such as the Matching Hypothesis by Wetherick & Gilhooly, 1990) and the Probabilistic Heuristics Model (PHM by Chater & Oaksford, 1999) and model-based theories (such as the Theory of Mental Models, see Johnson-Laird, 2006) performed better overall than theories of mental logic. It has been criticized that classical quantifiers are too lim- ited with respect to everyday contexts. The quantifiers A and E are too strict, not allowing any exceptions, whereas I and O are considered too weak requiring only a single individ- ual (Pfeifer, 2006). In contrast, everyday human reasoning is “based [. ] on beliefs, in which there are varying degrees of confidence” (Evans, 2002, p. 980). In accordance with that, we consider the generalized quantifiers most (M) and few (F), e.g., Chater and Oaksford (1999) proposed a Probability Heuris- tics Model (PHM) that accounts for the four classical and two generalized quantifiers M and F. It predicts that conclusions are chosen by their likelihood. Chater and Oaksford (1999) argue that quantifiers can be ordered according to their re- spective informativeness: A > M > F > I > E >> O with A being the most informative quantifier as no excep- tions are allowed. Simple heuristics like the min-heuristic and p-entailments describe the conclusion drawn by individ- ual reasoners. The min-heuristic postulates that the quanti- fier of the conclusion is the same as the one in the least in- formative premise. Hence, for Example (2) above, the min- heuristic predicts reasoners conclude that “Some X are not Z” as the first premise (containing O) is less informative than the second one (containing F). The conclusion drawn by this heuristic is called min-conclusion. If more than one conclu- sion can be derived from a given problem, the following con- clusions are called p-entailments (i.e., they are probabilisti- cally entailed) of the min-conclusion. For Example (2) the min-conclusion “Some X are not Z” entails that “Some X are Z. ” The predictions were evaluated in two experiments (Chater & Oaksford, 1999). Participants received two premises and