Dynamic Probabilistic Entailment. Improving on Adams' Dynamic Entailment Relation

The inferences of contraposition (A ⇒ C ∴ ¬C ⇒ ¬A), the hypothetical syllogism (A ⇒ B, B ⇒ C ∴ A ⇒ C), and others are widely seen as unacceptable for counterfactual conditionals. Adams convincingly argued, however, that these inferences are unacceptable for indicative conditionals as well. He argued that an indicative conditional of form A ⇒ C has assertability conditions instead of truth conditions, and that their assertability ‘goes with’ the conditional probability p(C|A). To account for inferences, Adams developed the notion of probabilistic entailment as an extension of classical entailment. This combined approach (correctly) predicts that contraposition and the hypothetical syllogism are invalid inferences. Perhaps less well-known, however, is that the approach also predicts that the unconditional counterparts of these inferences, e.g., modus tollens (A ⇒ C, ¬C ∴ ¬A), and iterated modus ponens (A ⇒ B, B ⇒ C, A ∴ C) are predicted to be valid. We will argue both by example and by calling to the results from a behavioral experiment (N = 159) that these latter predictions are incorrect if the unconditional premises in these inferences are seen as new information. Then we will discuss Adams’ (1998) dynamic probabilistic entailment relation, and argue that it is problematic. Finally, it will be shown how his dynamic entailment relation can be improved such that the incongruence predicted by Adams’ original system concerning conditionals and their unconditional counterparts are overcome. Finally, it will be argued that the idea behind this new notion of entailment is of more general relevance.

New Paradigm Psychology of Conditional Reasoning and Its Philosophical Sources

There is a new Bayesian, or probabilistic, paradigm in the psychology of reasoning, with new psychological accounts of the indicative conditional of natural language and of conditional reasoning. Dorothy Edgington has had a major impact on this new paradigm, through her views on inference from uncertain premises, the relation between the probability of the indicative conditional, P(if p then q), and the conditional probability, P(q|p), and the use of the Ramsey test to evaluate conditionals. Accounts are given in this chapter of the psychological experiments in the new paradigm that confirm empirical hypotheses inspired by her work and other philosophical sources.

Conditional Reasoning Part I

Three types of conditional are distinguished: the material conditional, indicative conditional, and subjunctive/counterfactual conditional. The apparent difference in truth conditions of each is suggestive of different psychological procedures used in the evaluation of each. The psychology of the material conditional is then examined. Despite procedures in formal logic that are suggestive of sui generis imaginative states (e.g., “assuming” a proposition for conditional proof, or for reductio), we need not countenance the use of such states within the psychological procedures used to carry out the inferences. Further, work in psychology has long suggested that humans do not, as a rule, reason in accordance with normative standards appropriate to the material conditional. A popular alternative proposal in psychology is that conditional reasoning involves the use of mental models (Johnson-Laird & Byrne, 2002). The use of mental models is shown to be consistent with conditional reasoning involving only sequences of beliefs.

Conditionals and Abduction

The chapter gives a preliminary sketch of some cognitive differences between indicative conditionals and counterfactual conditionals relevant to the testing of hypotheses by experiment. They especially concern cases where the indicative conditional can be decided without new evidence while the counterfactual conditional cannot. They also show that the antecedent of a ‘counterfactual’ conditional need not be presupposed to be false. Differences connected with the past tense morphology of ‘would’ are explored. Cases are given where the morphology should be understood as expressing a ‘fake past’, modal rather than temporal.

The Logic of Conditional Belief

The logic of indicative conditionals remains the topic of deep and intractable philosophical disagreement. I show that two influential epistemic norms—the Lockean theory of belief and the Ramsey test for conditional belief—are jointly sufficient to ground a powerful new argument for a particular conception of the logic of indicative conditionals. Specifically, the argument demonstrates, contrary to the received historical narrative, that there is a real sense in which Stalnaker’s semantics for the indicative did succeed in capturing the logic of the Ramseyan indicative conditional.

Learning Conditional Evidence

Learning indicative conditionals and learning relative frequencies have one thing in common: they are examples of conditional evidence, that is, evidence that includes a suppositional element. Standard Bayesian theory does not describe how such evidence affects rational degrees of belief, and natural solutions run into major problems. We propose that conditional evidence is best modeled by a combination of two strategies: First, by generalizing Bayesian Conditionalization to minimizing an appropriate divergence between prior and posterior probability distribution. Second, by representing the relevant causal relations and the implied conditional independence relations in a Bayesian network that constrains both prior and posterior. We show that this approach solves several well-known puzzles about learning conditional evidence (e.g., the notorious Judy Benjamin problem) and that learning an indicative conditional can often be described adequately by conditionalizing on the associated material conditional.

What is wrong with false-link conditionals?

It is a common intuition that the antecedent of an indicative conditional should have something to do with its consequent, that they should be somehow connected. In fact, many conditionals sound unacceptable precisely because they seem to suggest a connection which is not there. Although the majority of semantic theories of conditionals treat this phenomenon as something pragmatic, for instance, something that is conversationally implicated, no one has offered a full-fledged pragmatic explanation of why missing-link, and, in particular, false-link conditionals strike us as odd. The aim of this paper is to explore the possibility that the link is an example of a conversational implicature. We discuss possible tests one can employ to identify conversational implicatures, and, ultimately, we show that the connection between a conditional’s antecedent and consequent fails them all.

Conditional Propositions and Conditional Assertions

A comparison of accounts of indicative conditional statements that treat them as a distinctive kind of conditional speech act—qualified assertion of the consequent qualified by the supposition of the antecedent—and accounts that treat them as unqualified assertions of propositions that are a function of antecedent and consequent. The aim is to reconcile the two accounts by making them precise in a common pragmatic framework, and showing that the former can be seen to be equivalent to a limiting case of the latter. It is argued that this way of representing the conditional assertion account helps to explain the relation between indicative and subjunctive conditionals.

Counterfactuals and Probability

This chapter continues the attempt, begun in chapter 10, to reconcile the thesis that conditionals have truth conditions with accounts such as Dorothy Edgington’s that aim to explain conditionals as expressing a distinctive kind of attitude represented by conditional probability. This time the focus is on subjunctive or counterfactual conditionals. It is argued that the propositional analysis helps to explain the cases, emphasized by Edgington, where counterfactual statements seem to be retrospective assessment of what was earlier said with an indicative conditional. It is also argued that the propositional analysis can allow for cases where counterfactuals have probability values but not truth-values, and more generally that it can help to explain the relationship between counterfactuals and objective chance.