The Probability of Conditionals : The Psychological

The two main psychological theories of the ordinary conditional were designed to account for inferences made from assumptions, but few premises in everyday life can be simply assumed true. Useful premises usually have a probability that is less than certainty. But what is the probability of the ordinary conditional and how is it determined? We argue that people use a two stage Ramsey test that we specify to make probability judgements about indicative conditionals in natural language, and we describe experiments that support this conclusion. Our account can explain why most people give the conditional probability as the probability of the conditional, but also why some give the conjunctive probability. We discuss how our psychological work is related to the analysis of ordinary conditionals in philosophical logic. 3 Many psychological experiments have been run on ordinary indicative conditionals in natural language. In the relatively older literature (reviewed by Evans, Newstead, & Byrne, 1993), a typical experiment would contain an ordinary conditional as a major premise, with its antecedent or consequent, or a negation of its antecedent or consequent, as the minor premise. The participants would be asked to assume these premises, and then whether they endorsed some conclusion that might, or might not, validly follow. For example, participants were given conditionals of the form if p then q as the major premise and p as the minor premise, and what was recorded was the frequency of endorsement of the conclusion q in the valid inference Modus Ponens (MP). Other participants were given the same major premise but not-q as the minor premise, and a record was made of the frequency of endorsement of the conclusion not-p in the valid inference Modus Tollens (MT). The general finding was that MP is endorsed with a significantly higher frequency than MT. 1. Psychological theories of the conditional Two major psychology theories of the ordinary indicative conditional in natural language were developed to try to explain the results of experiments on these conditionals. Braine and O'Brien (1991) proposed the existence of a kind of mental natural deduction system, with introduction and elimination inference rules for the ordinary conditional and other sentential connectives (see also Rips & Marcus, 1977, and Over, 2003a). Their proposal was that these inference rules supplied all the semantics that was necessary for negation, 4 conjunction, disjunction, and the conditional. For example, the very meaning of their conditional was supposedly based …