Abstract
Adams’ thesis is generally agreed to be linguistically compelling for simple conditionals with factual antecedent and consequent. We propose a derivation of Adams’ thesis from the Lewis-Kratzer analysis of if-clauses as domain restrictors, applied to probability operators. We argue that Lewis’s triviality result may be seen as a result of inexpressibility of the kind familiar in generalized quantifier theory. Some implications of the Lewis-Kratzer analysis are presented concerning the assignment of probabilities to compounds of conditionals.
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Notes
Consider a domain with four students a, b, c and d. Suppose that in case all of them work hard, only a, b, c will get an A. Suppose that only a and b actually work hard, though, and both get an A. In this case (b) seems true, but (a) false, since all of those who work hard get an A.
See in particular the Appendix to Schlenker (2006) on the parallel between “most” and “probably”, and Schlenker (2004) on “if” and “the”. The tight link between when-clauses and if-clauses features also prominently in Lycan’s (2001) account of conditionals, based on his work with M. Geis. See Lycan (2001) chapter 1, for syntactic evidence that “if...then...” constructions pattern as relative-clauses constructions. In particular, Lycan draws essentially the same generalization Kratzer makes about English: “English incorporates no binary sentential connective expressed by ‘if’ ”(2001: p. 91). Unlike Kratzer, however, Lycan appears skeptical about the validity of Adams’ thesis and tends to dismiss the significance of the Lewis triviality results. He writes: “it would be very surprising if the semantics of natural language conditionals reflected the probability calculus in any simple way” (2001: p. 90). We agree, but our point is precisely that Lewis’s triviality results essentially confirm the view that “if” does not behave as a binary sentential connective.
Our clarification of this point owes a lot to a discussion with S. Kaufmann, who pointed out that conditionalization is only one among several ways of articulating domain restriction for probability operators. One informal guess about the situation may be put as follows. It seems that domain restriction implies unambiguously that p A (A) = 1. As stressed notably by R. Jeffrey, \(p_A(C) = p(C | A)\; {\rm iff (i)} p_A(A) = 1\) (certainty) and (ii) \(p_A(C | A) = p(C | A)\) (rigidity). It is unclear whether domain restriction implies rigidity. In case it does not, the Lewis-Kratzer analysis of if-clauses implies Adams’ thesis under the assumption of rigidity, and the derivation of Adams' thesis holds only in a specific domain—where rigidity holds. We leave the exploration of this point and of its implications for future work.
We are indebted to D. Rothschild for this nice way of putting it.
Interestingly, Kaufmann (2004) has put forward the idea that a rule such as (29), which he calls local conditional probability, might be a rational rule of belief update, and for that matter that it could provide a counterexample to Adams’ thesis. We agree with Kaufmann that it may be common practice to reason by cases according to (29), however we believe it is a reasoning fallacy, close to the base rate fallacy, which should not bear on the validity of Adams’ thesis. In our view, a theory of reasoning according to local probability is certainly relevant for the psychology of conditional probability, but we are skeptical that it should be part and parcel of the semantics of conditional probability. In that, we share the skepticism expressed by Douven (2008) in his criticism of the normative character of Kaufmann’s rule. To borrow a distinction nicely made by S. Kaufmann in conversation, we agree that an account of local probability is certainly relevant for a performance theory of conditionals and probability, but we doubt that it should be part of the corresponding competence theory.
In Adams’ logic, the conditional is treated as a binary sentential connective, but this connective always takes highest priority over Boolean formulae, and cannot embed other conditional formulae. Because of that, a formula (A ⇒ B) in Adams’ logic may be read as equivalent to a formula of the form [Probably](A, B) in the kind of languages with explicit probability operators that we introduced, and a Boolean formula ϕ as a formula of the form [Probably](ϕ). In Adams’ logic, ψ is a consequence of ϕ if for every probability distribution \(p, p(\phi)\leq p(\psi)\) . We could imagine to define an analogous notion of consequence between formulae prefixed by [Probably], namely to impose that [Probably](ψ) is a consequence of [Probably](ϕ) iff every model (W, p, V) that makes any formula [α](ϕ) true makes [α](ψ) true. However we are not interested in logical consequence in this paper. Our focus is primarily on the logical form of conditional sentences.
We are indebted to B. Spector for this example and for the judgment. Spector came up with the example in response to an objection made by D. Rothschild (2009) to a particular prediction of Kratzer’s account, namely the prediction that ‘there are n/m chances that p’ and ‘there are n/m chances that it is true that p’ need not be equivalent. Spector’s original judgment concerned the same pair in which ‘it is true that’ is used instead of ‘necessarily’. See D. Rothschild (forthcoming) for a more recent discussion of Spector’s example.
The so-called Independence Principle implies for any two-sided conjunction (McGee 1989: 500) that \(p((Even \Rightarrow >3) \wedge (Odd \Rightarrow <3)) = {\frac{1}{p(Even \vee Odd )}} \times [p(Even \wedge >3 \wedge Odd \wedge <3) + p (\neg Even \wedge Odd \wedge <3) \times p(Even \Rightarrow >3) + p(Even \wedge >3 \wedge \neg Odd ) \times p( Odd \Rightarrow <3)]\), which can be simplified as:
$$ p((Even \Rightarrow >3) \wedge (Odd \Rightarrow <3)) = p ( Odd \wedge <3) \times p(Even \Rightarrow >3) + p(Even \wedge >3) \times p( Odd \Rightarrow <3), $$therefore:
$$ p((Even \Rightarrow >3) \wedge (Odd \Rightarrow <3)) = 1/6 \times 2/3 + 2/6 \times 1/3 = 2/9. $$This can be derived from McDermott’s three-valued semantics where a conditional is neither true nor false if the antecedent is false and a conjunction is true if one of the conjuncts is true and the other neither true nor false.
A connection may be established between the present analysis of nested conditionals and the account outlined by Gibbard (1981), in which Gibbard considers that nested conditionals can be assigned probabilities if they are equivalent to a sentence that expresses a proposition. In this, our account converges with specific intuitions of the ‘no truth value’ view of conditionals, but as repeated throughout the paper, we do not endorse such a view, since we see the Kratzerian analysis as a better way of articulating some of the good intuitions contained in it. See also von Fintel (2006).
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Acknowledgments
Special thanks to D. Rothschild, B. Spector, D. Bonnay, R. Bradley, H. Leitgeb for helpful discussion, and to P. Schlenker, G. Politzer and S. Kaufmann for their detailed comments and valuable criticisms. We also thank audiences in Amsterdam, Bristol, Paris, Geneva and Göttingen, and particularly O. Roy, S. Duca, P. Keller and M. Schwager for their invitations, as well as participants at ESSLLI 2008 in Hamburg, where we taught a course on conditionals. We gratefully acknowledge the ANR-DFG project “Hypothetical Reasoning” for support.
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Égré, P., Cozic, M. If-Clauses and Probability Operators. Topoi 30, 17–29 (2011). https://doi.org/10.1007/s11245-010-9087-y
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DOI: https://doi.org/10.1007/s11245-010-9087-y