Some arguments are probably valid: Syllogistic reasoning as communication

Some arguments are probably valid: Syllogistic reasoning as communication Michael Tessler, Noah D. Goodman {mtessler, ngoodman}@stanford.edu Department of Psychology, Stanford University Abstract Syllogistic reasoning lies at the intriguing intersection of natu- ral and formal reasoning, of language and logic. Syllogisms comprise a formal system of reasoning yet use natural lan- guage quantifiers, and invite natural language conclusions. How can we make sense of the interplay between logic and language? We develop a computational-level theory that con- siders reasoning over concrete situations, constructed proba- bilistically by sampling. The base model can be enriched to consider the pragmatics of natural language arguments. The model predictions are compared with behavioral data from a recent meta-analysis. The flexibility of the model is then ex- plored in a data set of syllogisms using the generalized quan- tifiers most and few. We conclude by relating our model to two extant theories of syllogistic reasoning – Mental Models and Probability Heuristics. Keywords: Reasoning; language; QUD; Bayesian model Figure 1: How will the reasoner interpret the experimenter’s argument? Consider for a moment that your friend tells you: “Every- one in my office has the flu and, you know, some people with this flu are out for weeks. ” Do you respond with “Everyone in your office has the flu. ” Do you respond with “Pardon me, there is no inference I can draw from what you just said. ” Or do you respond “I hope your officemates are not out for weeks and I hope you don’t get sick either. ” The first response – while true – does not go beyond the premises; the second response attempts to go beyond the premises by strict classical logic, and fails; the final response goes beyond the premises, to offer a conclusion which is probably useful and probably true. This cartoon illustrates a critical dimension along which cognitive theories of reason- ing differ: whether the core and ideal of reasoning is deduc- tive validity or probabilistic support. A separate dimension concerns the extent to which principles of natural language— pragmatics and semantics—are necessary for understanding reasoning tasks. In this paper, we explore the idea that the formalism of probabilistic pragmatics can provide insight into how people reason with syllogisms. The form of the argument above resembles a syllogism: a two-sentence argument used to relate two properties (or terms: A, C) via a middle term (B); the relations used in syl- logisms are quantifiers. Fit into a formal syllogistic form, this argument would read: All officemates are out with the flu Some out with the flu are out for weeks Therefore, some officemates are out for weeks The full space of syllogistic arguments is derived by shuffling the ordering of the terms in a sentence (“All A are B” vs. “All B are A”) and changing the quantifier (all, some, none, not all 1 ). Most syllogisms have no valid conclusion, i.e. there 1 For distinctiveness, we will refer to the quantifiers as such. Note is no relation between A & C which is true in every situa- tion in which the premises are true. This is the case with the argument above. Often in these cases, however, people are perfectly comfortable drawing some conclusion. A recent meta-analysis of syllogistic reasoning showed that over the population, the proper production of no valid conclusion re- sponses for invalid arguments ranged from 76% to 12%. For valid arguments, the accuracy of producing valid conclusions ranged from 90% to 1% (Khemlani & Johnson-Laird, 2012): people do not seem to find drawing deductively valid conclu- sions particularly natural. Perhaps because of this divergence between human behav- ior and deductive logic, syllogistic reasoning has been a topic of interest in cognitive psychology for over a hundred years (St¨orring, 1908), and before that in philosophy, dating back to Aristotle. Syllogisms are undoubtedly logical; indeed, the syllogistic form was the only formal system of logic for mil- lennia. At the same time, the arguments use natural language quantifiers and invite natural language conclusions; precisely pinning down the meaning and use of quantifiers has been an ongoing area of inquiry since Aristotle (e.g. Horn, 1989). Many theories of syllogistic reasoning take deduction as a given and try to explain reasoning errors as a matter of noise during cognition. Errors, then, may arise from im- proper use of deductive rules or biased construction of log- ical models. Many other kinds of reasoning, however, can be well-explained as probabilistic inference under uncertainty. Probability theory provides a natural description of a world in which you don’t know exactly how many people are in the that in sentence-form, the last two quantifiers are typically presented as “No A are B” and “Some A are not B”, respectively.