We doubt two claims made by Oaksford & Chater (O&C), one linguistic and one about inference, as they relate to conditionals (Bayesian Rationality, Reference Oaksford and ChaterOaksford & Chater 2007, henceforth BR as in target article). We do not think that the case has been made that sentences of the form “If A then B” have a semantics diverging from the material implication of propositional logics. We focus here on the related claim that human inference is probabilistic.

Classical Propositional Logic (CPL) is often claimed to be inadequate for explaining our spontaneous propositional inferences. The claim is based on the observation that, whereas CPL is monotonic, human inferences seem to be non-monotonic. For example, it is argued that the validity of some inferences, such as the one in (1a), may be cancelled by addition of another premise, such as the premise R in (1b). The claim is that now that John has broken his left leg, John will no longer run, even if the weather is fine. This is claimed to show the inadequacy of CPL as a tool to explain our propositional inferences, because in CPL, addition of another premise does not influence the provability of the sequent, as is shown in the inference from top to bottom in (1c).

(1a) (Premise 1) P→Q: If the weather is fine, John runs for a mile.

(Premise 2) P: The weather is fine.

(Conclusion) 3. Q: John runs for a mile.

The valid inference: Whenever Premise 1 and Premise 2 are true, 3 is also true.

(1b) R: John breaks his left leg.

(1c) P, P→Q ⊢ Q;>P, P→Q, R ⊢ Q Weakening

However, it is not clear that the addition of the premise R in (1b) to the set of premises in (1a) actually cancels out the validity of the initial sequent, P, P→Q ⊢ Q, in our propositional inferences. The point becomes clearer if we add the new proposition R to the set of premises, as in P, P→Q, R ⊢ Q. With the specification of the premise propositions as in (1a) to (1b), we do not find this revised sequent to be valid, but this is only because we no longer find the proposition “If the weather is fine, John runs for a mile” (= P→Q) to be true when we reevaluate the inference. In contrast, if we force ourselves to assume that all the three premise propositions P, P→Q, R are true in a model of interpretation, then in that model, we have to conclude Q. Some might find it difficult to think of such an interpretation model, because common-sense knowledge tells us that a person normally does not run with a broken leg, but we may sometimes make a claim such as, “If the weather is fine, John (always) runs for a mile. It does not matter if he gets injured. He always runs for a mile.” Thus, it is not impossible to force ourselves to think of models in which the proposition, “If the weather is fine, John runs for a mile” is true despite the fact that John has broken his leg. In such models, whenever P is true, Q is also true: that is, the inference goes through.

As is clear from the foregoing discussion, typical interpretation data that allegedly show that monotonic logic cannot capture our propositional inferences include a change of models in which we evaluate the propositional sequents. The valid inference in (1) can be re-stated as “In each model in which P and P→Q are both true, Q is also true.” The cancellation of the truth of Q arises because in the new model in which we “re-evaluate” the sequent, the premise P→Q is no longer true (or we find it more difficult to think of a model in which both P→Q and R are true). Because the initial valid inference in (1a) concludes R as a true proposition only on condition that P→Q and P are both true, this revision does not really involve the cancellation of the validity of the initial inference.

Given that specification of models in which logical formulas/sequents are evaluated is not part of either the syntax or the semantics of propositional logic languages, it is not clear that alleged non-monotonicity of reasoning, which arises because of the revision of models, requires formalising logical inference as non-monotonic or probabilistic. No doubt people hold beliefs with varying degrees of strength, and a result of reasoning is that these degrees of strength are changed. But one can agree with O&C on this point without thinking that the formal inference system itself is probabilistic.

Space here does not permit discussion of how some propositions are accepted and others rejected, but we are sympathetic to O&C's claim that heuristics that are sensitive to information gain must be involved, with the caveat that it cannot simply be information that is sought, but information that is important to the reasoner at a reasonable processing cost. This recalls discussion of relevance in Gricean pragmatics and Sperber and Wilson's relevance theory.

As Oakford & Chater note, for one's belief in the conditional in (2) it matters whether one discovers, for example, an unstarted car or is told that a car did not start.

A pragmatic explanation in terms of the tendency of speakers to produce utterances relevant to their audience is natural. Effects of the order in which information is presented (see BR, pp. 157ff) also require such an explanation, we believe.

This raises a methodological point. To understand human reasoning, both classical logic and O&C's probabilistic account of conditionals and of inference must be supplemented by accounts of processing, and of the pragmatics of utterance interpretation. Thus, it is not obvious that the probabilistic account is more parsimonious.