Conditional Probability and the Cognitive Science of Conditional Reasoning

This paper addresses the apparent mismatch between the normative and descriptive literatures in the cognitive science of conditional reasoning. Descriptive psychological theories still regard material implication as the normative theory of the conditional. However, over the last 20 years in the philosophy of language and logic the idea that material implication can account for everyday indicative conditionals has been subject to severe criticism. The majority view is now apparently in favour of a subjective conditional probability interpretation. A comparative model fitting exercise is presented that shows that a conditional probability model can explain as much of the data on abstract indicative conditional reasoning tasks as psychological theories that supplement material implication with various rationally unjustified processing assumptions. Consequently, when people are asked to solve laboratory reasoning tasks, they can be seen as simply generalising their everyday probabilistic reasoning strategies to this novel context. Given the centrality of the conditional, if p then q, to any account of human reasoning one might have expected some convergence in their treatment across the range of disciplines involved in the cognitive sciences. The explanatory scheme in cognitive science allows an important role both for normative disciplines, such as logic and the philosophy of language, and for the descriptive disciplines such as the psychology of reasoning. For a cognitive scientist, cognition is computation (Oaksford, 1997; Pylyshyn, 1984). Computational explanation is multi-levelled (e.g., Anderson, 1990; Marr, 1982; Pylyshyn, 1984). The computational (Marr, 1982) or rational (Anderson, 1990) level, specifies a normatively justified and descriptively adequate model of some cognitive phenomenon (Oaksford and Chater, 1996, 1998). This is implemented at the algorithmic or performance level. Normally there are aspects of the data that are not explained at the computational level that must be addressed by the algorithmic level. For example, the computational level will not have much to say about speed of processing, which provides an important source of constraint on cognitive theories. Given such an explanatory scheme one might have expected to find cognitive psychologists constructing algorithmic level theories using the current best guess as to the appropriate normative theory of the conditional. Although a reasonable expectation it has not turned out that way. Address for correspondence: Mike Oaksford, School of Psychology, Cardiff University, PO Box 901, Cardiff, CF10 3YG, UK. Email: [email protected] Mind & Language, Vol. 18 No. 4 September 2003, pp. 359–379. #Blackwell Publishing Ltd. 2003, 9600GarsingtonRoad, OxfordOX4 2DQ,UKand 350Main Street,Malden,MA02148,USA. In the psychology of reasoning, the two main theoretical approaches, mental logic (e.g., Rips, 1994; Braine and O’Brien, 1998) and mental models (e.g., Johnson-Laird and Byrne, 1991), have adopted material implication as their computational level theory of the conditional. That is, the conditional is treated as truth-functional. Experimental work has concentrated on the indicative or straight (Bennett, 1995) conditional, e.g., if Oswald didn’t shoot Kennedy, then someone else did, or, if you turn the key the car starts. In the philosophy of language and logic, ‘. . . the majority view [is] that straight conditionals are a matter of subjective conditional probabilities’ (Bennett, 1995, p. 332). One consequence of this view is that people, ‘do not use ‘‘if’’ to express propositions, evaluable in terms of truth’ (Edgington, 1995, p. 280). And if conditionals do not have truth conditions then they would not appear to be truth functional, as many psychological theories assume. Note though this view has its detractors (Lewis, 1976). Thus, rather than a happy coincidence of opinion, the psychology of reasoning and normative theories of the conditional seem to be passing each other by. This may be because in the psychology of reasoning the protective belt (Lakatos, 1970) of assumptions defending the truth functional view have not yet been penetrated. These different domains deal with different data. The philosopher of logic and language deals with intuitively compelling examples, whereas the psychologist deals with experimental results. One way of protecting the truth functional view of the conditional against intuitively compelling counterexamples is Grice’s proposal that pragmatic conversational factors may intervene. For example, according to material implication, a sufficient condition for believing that ‘if the moon is made of cheese, then Mars is green’, is that you believe the antecedent to be false, i.e., you believe that the moon is not made of cheese. But this seems to be insufficient grounds for such a belief. However, perhaps examples like this don’t count because although this claim has the truth conditions of the material conditional, its assertability conditions are different. It has been suggested that this conditional would only be asserted if someone believed that the conditional probability of Mars being green given that the moon is made of cheese is high (Jackson, 1987). (See Edgington (1995), for further arguments against positions like this.) A way of protecting the truth functional approach has also emerged in the psychology of reasoning. As we have argued, computational explanation in cognitive science is multi-levelled. Given an appropriate computational level theory, there is a choice of cognitive representations and processes in which to implement that theory at the algorithmic level. So, for example, in psychology experiments people are much less likely to endorse inferences corresponding to modus tollens (MT, if p then q, :q, therefore :p) than modus ponens (MP, if p then q, p, therefore q). Mental logicians (e.g., Rips, 1994) explain this finding by proposing that people do not possess an inference schema for MT, but must generate these inferences by the more complex reductio ad absurdum argument form. Mental modellers (Johnson-Laird and Byrne, 1991) on the other hand, who adopt a more semantic 360 M. Oaksford and N. Chater # Blackwell Publishing Ltd. 2003 approach, argue that people do not represent all the conditions that make a conditional true. So they initially represent the conditional, if p then q, in working memory as,