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.

Counterfactuals and Abduction

We argue that counterfactuals and indicative conditionals are not so different. Certain notorious differences previously observed between pairs of indicative and counterfactual sentences are actually due to the presence of an anti-abductive modal auxiliary (would) in the consequent of the counterfactual. But such auxiliaries, of which will is another example, span the counterfactual-indicative divide.

Restriction without Quantification: Embedding and Probability for Indicative Conditionals

Many modern theories of indicative conditionals treat them as restricted epistemic necessity modals. This view, however, faces two problems. First, indicative conditionals do not behave like necessity modals in embedded contexts, e.g., under ‘might’ and ‘probably’: in these contexts, conditionals do not contribute a universal quantification over epistemic possibilities. Second, when we assess the probability of a conditional, we do not assess how likely it is that the consequent is epistemically necessary given the antecedent. I propose a semantics which solves both problems, while still accounting for the data that motivated the necessity modal view. The account is based on the idea that the semantics of conditionals involves only a restriction of the relevant epistemic state, and no quantification over epistemic possibilities. The relevant quantification is contributed by an attitude parameter in the semantics, which is shifted by epistemic modals. If the conditional is asserted, the designated attitude is acceptance, which contributes a universal quantification, producing the effect of a restricted necessity modal.

Applications in Epistemology

This chapter applies ranking theory to two problems in epistemology and the philosophy of science: conceptual belief change, including logical learning, as well as learning indicative conditionals. The chapter concludes with a defense of rigidity.

Indicative conditionals: probabilities and relevance

We propose a new account of indicative conditionals, giving acceptability and logical closure conditions for them. We start from Adams’ Thesis: the claim that the acceptability of a simple indicative equals the corresponding conditional probability. The Thesis is widely endorsed, but arguably false and refuted by empirical research. To fix it, we submit, we need a relevance constraint: we accept a simple conditional$$\varphi \rightarrow \psi$$φ→ψto the extent that (i) the conditional probability$$\mathrm{p}(\psi |\varphi )$$p(ψ|φ)is high, provided that (ii)$$\varphi$$φis relevant for$$\psi$$ψ. How (i) should work is well-understood. It is (ii) that holds the key to improve our understanding of conditionals. Our account has (i) a probabilistic component, using Popper functions; (ii) a relevance component, given via an algebraic structure of topics or subject matters. We present a probabilistic logic for simple indicatives, and argue that its (in)validities are both theoretically desirable and in line with empirical results on how people reason with conditionals.

De Finettian Logics of Indicative Conditionals Part II: Proof Theory and Algebraic Semantics

In Part I of this paper, we identified and compared various schemes for trivalent truth conditions for indicative conditionals, most notably the proposals by de Finetti (1936) and Reichenbach (1935, 1944) on the one hand, and by Cooper (Inquiry, 11, 295–320, 1968) and Cantwell (Notre Dame Journal of Formal Logic, 49, 245–260, 2008) on the other. Here we provide the proof theory for the resulting logics and , using tableau calculi and sequent calculi, and proving soundness and completeness results. Then we turn to the algebraic semantics, where both logics have substantive limitations: allows for algebraic completeness, but not for the construction of a canonical model, while fails the construction of a Lindenbaum-Tarski algebra. With these results in mind, we draw up the balance and sketch future research projects.

Chasing Hook: Quantified Indicative Conditionals

The chapter looks at indicative conditionals embedded under quantifiers, with a special emphasis on ‘one-case’ conditionals as in No query was answered if it came from a doubtful address. It agrees with earlier assessments that a complete conditional (with antecedent and consequent) is embedded under a quantifier in those constructions, but then proceeds to create a dilemma by showing that we can’t always find the right interpretation for that conditional. Contrary to earlier assessments, Stalnaker’s conditional won’t always do. The chapter concludes that the embedded conditional in the sentence above is a material implication, but the if-clause also plays a pragmatic role in restricting the domain of the embedding quantifier. That an appeal to pragmatics should be necessary at all goes with Edgington’s verdict that ‘we do not have a satisfactory general account of sentences with conditional constituents’.

Counterfactuals and Probability

Dorothy Edgington has been a resolute defender of an NTV account of conditionals, according to which a conditional does not express a proposition that makes a categorical claim about the world, but instead make a qualified claim, or express a conditional belief, qualified by or conditional on the proposition expressed by the antecedent. Unlike some philosophers who defend an NTV view for indicative conditionals, but not for subjunctive or counterfactual conditionals, Edgington argues for the more radical thesis that both kinds of conditionals should be given a non-propositional analysis. This chapter considers Edgington’s NTV account of subjunctive conditionals, the role of objective probability in the account, and its relation to the possible-worlds propositional analysis of subjunctive conditionals.