In Bayesian Rationality (Oaksford & Chater Reference Oaksford and Chater2007, henceforth BR), Oaksford & Chater (O&C) propose an important probabilistic theory of conditional reasoning. They identify the probability of a conditional, P(if p then q), with the conditional probability, P(q|p). Following the normative literature, they call this identity the Equation, and we will also take it for granted. There is substantial direct evidence for the Equation as a description of people's probability judgements (Over et al. Reference Over, Hadjichristidis, Evans, Handley and Sloman2007). O&C take an impressive indirect approach: They show how much they can explain in the psychology of conditional reasoning using the Equation. However, their main account occupies the halfway Bayesian house of assuming that contingent minor premises are certain.

The simplest application of O&C's main account is to Modus Ponens (MP): inferring q from the major premise if p then q and the minor premise p. They assume that the probability of the minor premise is 1. They thus hold that the probability of q after MP, P(q), is P(q|p), with P(p)=1. Their assumption is not fully Bayesian. For a full Bayesian, committed to strict coherence, a contingent proposition should not have a probability of 1. That would imply that it could never be disconfirmed. O&C also assume that the minor premise is certain in their treatment of other conditional inferences. To illustrate the limitations of this assumption, consider Modus Tollens (MT). This is inferring, for example, “Global warming will not continue” (not-p) from “If global warming continues then London will be flooded” (if p then q) as the major premise and “London will not be flooded” (not-q) as the minor premise. (Over et al. Reference Over, Hadjichristidis, Evans, Handley and Sloman2007, has evidence that conditionals like this satisfy the Equation.)

Looking in detail at O&C's description of MT in temporal reasoning, we begin (using, we hope, clearer symbols) with the prior conditional probability, Pold(q|p), and the prior marginal probabilities, Pold(p) and Pold(q). They derive a formula for MT, by elementary steps in probability theory, for inferring Pold(not-p|not-q) from those prior probabilities. They suppose that we later “learn” the minor premise, not-q, and so acquire a new belief state. They take “learn” in the extremely strong sense of meaning that Pnew(not-q)=1. This allows them to identify the probability of the conclusion of MT, Pnew(not-p), with Pold(not-p|not-q). They note that we may lower our belief in the major premise of MT when we learn not-q (Stevenson & Over Reference Stevenson and Over1995): learning the minor premise sometimes “alters” Pold(q|p) by lowering it. We use, they hold, this “modified” lower probability, along with Pold(p) and Pold(q), in their derived formula to recalculate Pold(not-p|not-q), and that is then taken as a revised Pnew(not-p). However, when the probability of the minor premise, not-q, becomes 1, the probability of q becomes 0. The probability of the major premise, if p then q, should consequently collapse to 0 (because the conditional probability is then 0), and not merely be “modified” to varying relatively low values. Of course, there is no technical problem in putting Pold(q|p)=0 into O&C's formula to recalculate Pold(not-p|not-q). The problem is that making Pold(q|p)=0 in every case is a serious limitation. (There is a technical problem if the probability of p becomes 0, since that makes the conditional probability undefined. We set this possibility aside, although it is a problem for O&C's account of denying the antecedent.)

Stevenson and Over (Reference Stevenson and Over2001) found that people had more confidence in some minor premises than in others, depending on the level of expertise of the person who asserted the premise. Their results suggest that, even if people heard an expert on the Thames Barrier assert “London will not be flooded,” they would not assign it a probability of 1, and might not even judge it to have a very high probability. Of course, people sometimes claim that they are “certain” or “sure” of a contingent proposition, but we should interpret them as saying that their confidence in it is relatively high, and not that it can never be disconfirmed. They might, in turn, interpret an experimenter who tells them to assume that a minor premise is “certain” as merely indicating that its probability is high. But what confidence do people have in the conclusion of a conditional inference as they become more (or less) confident of the minor premise? Rightly starting to move on from their halfway house, O&C suggest that a way to answer this question is to adopt Jeffrey's rule of conditionalization (Evans & Over Reference Evans and Over1996a; Jeffrey Reference Jeffrey1983).

The relevant form of Jeffrey's rule for MP is:

The normative status of this rule is debated (Howson & Urbach Reference Howson and Urbach1993), but to what extent do people generally conform to it? We and our collaborators investigated this question using categorical inferences (Evans et al. Reference Evans, Stevenson, Over, Handley and Sloman2004; Hadjichristidis et al., in preparation). An example is inferring “Falcons have the neurotransmitter Dihedron” (q) from “Robins have the neurotransmitter Dihedron” (p). One way to look at these inferences is to see them as having an implicit conditional (if p then q) as a major premise. We used a plausible manipulation of Pnew(p), for example, “Scientists are 80% certain that Robins have the neurotransmitter Dihedron.” We analysed our data with a regression that aimed to predict judgements of Pnew(q) using as predictors Pold(q|p)Pnew(p) and Pold(q|not-p)Pnew(not-p). Only Pold(q|p)Pnew(p) had a consistent influence on participants' judgements about Pnew(q).

These results imply that O&C are too limited, in their main account, in making Pnew(q) depend only on Pold(q|p). But the results also imply that they are not limited enough in later suggesting, more generally, that Pnew(q) will be determined by the full use of Jeffrey's rule. The evidence so far is that Pnew(q) will primarily depend on Pold(q|p)Pnew(p) and so, on just the probabilities of the major and minor premises. However, these results are only for one kind of implicit MP inference. The descriptive adequacy of Jeffrey's rule should be much more extensively investigated.