Abstract
This paper explores the different ways in which conditionals can be carriers of good and bad news. I suggest a general measure of the desirability of conditionals, and use it to explore the different ways in which conditionals can have news value. I conclude by arguing that the desirability of a counterfactual conditional cannot be reduced to the desirability of factual propositions.
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Notes
There is generally not a clean-cut distinction between indicative conditionals and counterfactuals, but the former are usually expressed in indicative mood, while the latter are often expressed in subjunctive mood and can have antecedents that the speaker knows to be false.
The same conditional can have positive desirability w.r.t. some propositions but negative desirability w.r.t. others. In this case, for instance, the conditional has negative factual desirability w.r.t. A but positive counterfactual desirability w.r.t. \(\lnot \text {A}\). As I formalise this in Sect. 4.1: Having conditioned the desirability function on \(\text {A}\), the value of the conditional is negative. But the value of the conditional is positive when we have conditioned on \(\lnot \text {A}\).
For the present purposes, it does not matter whether the conditionals that I discuss are actually true or not; all that matters is whether the agents in question believe them to be true. The same holds below when I discuss the probabilistic information that a conditional carries and the counterfactual desirability of a conditional.
The regret Alice feels in this situation, when she knows that she could have chosen differently, could be seen as what causes the undesirability of her situation (over and above the normal burdens of unemployment). But on another reading, the regret she feels reflects the undesirability of her situation. The idea is that it especially bad to be unemployed if one previously could have chosen a guaranteed long-term job, and Alice’s feeling of regret reflects this. However, if we assume that Alice’s feeling of regret is what causes the counterfactual undesirability of the conditional in question (w.r.t. a situation where she finds herself unemployed) then the question arises as to whether feelings of regret can be rational (as a referee reminds us). For if not, then there is no need to account for them in normative decision theory. I will not attempt to argue here that such feelings can be rational. Instead, I will just register my agreement with the (Humean) approach to rationality according to which feelings are neither rational nor irrational. As Loomes and Sugden (1982) put it:
[P]sychological experiences of regret and rejoice cannot properly be described in terms of the concept of rationality: a choice may be rational or irrational, but an experience is just an experience ... [I]f an individual does experience such feelings, we cannot see how he can be deemed irrational for consistently taking those feelings into account. (820)
Weak Centring is logically equivalent to Modus Ponens, and is relatively uncontroversial. Strong Centring is much more controversial: Does the fact that the sky is blue and grass is green really entail that if the sky were blue then the grass would be green? Counterfactuals where the antecedent is not related to the consequent in the right way—that is, the former does not make the latter more likely to be true—certainly do sound odd. However, as David Lewis reminds us, we should not conflate oddness with falsity. While it might be a rule of pragmatics that one should not assert a counterfactual unless the consequent is related to the antecedent in the right way, Strong Centring could still, he points out, be a truth of semantics (Lewis (1986), p. 28).
Note that \(\text {A}\wedge \lnot \text {B}\) is not an element of the partition of \(\text {A}\rightarrow \text {B}\) since I am assuming Weak Centring: \(\text {A}\rightarrow \text {B}\) entails that \(\text {A}\wedge \lnot \text {B}\) is false (so the two are mutually inconsistent). To take an example, the assumption is that if I drop the plate and it doesn’t break then it is false that if I drop the plate (or were to drop the plate) it will (or would) break. We can, however, extend the measure to one where Weak Centring is not assumed, by adding this element (or, strictly speaking, by adding \(\text {A}\wedge \lnot \text {B}\wedge (\text {A}\rightarrow \text {B})\)) to the partition. Moreover, we can extend the measure to one where Strong Centring is not assumed, by replacing the first element (i.e., \(\text {A}\wedge \text {B}\)) of the partition by \(\text {A}\wedge \text {B}\wedge (\text {A}\rightarrow \text {B})\).
It is worth emphasising that while I build on papers that use the multidimensional possible world semantics to extend Jeffrey’s theory, the plausibility of Eqs. 1 and 2 does not depend on that semantics. In fact, since these equations are, given CEM and Centring, simply instances of Jeffrey’s desirability formula, any extension of Jeffrey’s theory to conditionals should entail these equations, assuming CEM and Centring, but the more complicated formulas (see last fn. and Eq. 3) if the extension does not assume CEM and Centring.
As a referee points out, the CEM will have odd-sounding implications when the antecedent is not related to the consequent in the right way. Hence, I have to appeal to Lewis’ defense of Strong Centring, cited in fn. 8, to explain how the CEM can be true despite having odd-sounding implications.
Consider for instance the interaction between our confidence in the counterfactual ‘if we were to toss the coin it would come up heads’ and ‘if we were to toss the coin it would come up tails’.
Contrary to received wisdom, even David Lewis, the principal enemy of CEM, actually took what his theory entails about the Bizet-Verdi example to be a counterintuitive implication of his view, rather than an argument against CEM, as a close reading of his discussion of the example reveals (Lewis (1986), pp. 80–82).
This is not to say that to evaluate the desirability of a counterfactual we always need to assume the falsity of its antecedent. As will become apparent, a counterfactual can have both factual and counterfactual desirability, and it is only when trying to isolate the latter that we should suppose the falsity of the antecedent.
I should emphasise that the role the conditioning is playing here is to isolate the effect in question; but it is not the case that the conditional acquires factual/counterfactual desirability when we condition on A/\(\lnot \text {A}\) (I thank a referee for encouraging me to clarify this point). Later I will argue that the same conditional, \(\text {A}\rightarrow \text {B}\), can simultaneously have (factual) desirability w.r.t. A and (counterfactual) desirability w.r.t. \(\lnot \text {A}\). And we of course cannot simultaneously condition on A and its negation. Nevertheless, while the same conditional has the potential for both types of desirability, we can isolate each type by conditioning on the right proposition.
For comparison, suppose that being charming is both desirable when one is single and when one is in a relationship. One cannot (without contradiction) simultaneously condition on the proposition that one is single and on the proposition that one is in a relationship (assuming that these are inconsistent propositions). But one can isolate the ‘single-desirability’ that being charming has (and measure this effect) by conditioning on the first proposition and the ‘relationship-desirability’ that being charming has by conditioning on the latter.
We get this from 2 since \(P_\text {A}(\text {A}\mid \text {A}\rightarrow \text {B})=1\), \(P_\text {A}(\lnot \text {A}\mid \text {A}\rightarrow \text {B})=0\), and, by Weak Centring, \(P_\text {A}(\text {B}\mid \text {A}\rightarrow \text {B})=1=P_\text {A}(\lnot \text { B}\mid \text {A}\rightarrow \lnot \text {B})\).
By this definition: \(Des_{\text {A}}(\text {A}\wedge \text {B})-Des_{\text {A}} (\text {A}\wedge \lnot \text {B})\dot{=}Des(\text {A}\wedge \text {B}\wedge \text {A})-Des(\text {A})-[Des(\text {A} \wedge \lnot \text {B}\wedge \text {A})-Des(\text {A})] =Des(\text {A}\wedge \text {B})-Des(\text {A}\wedge \lnot \text {B})\).
The above equalities in conjunction with Eq. 2 directly give us:
$$\begin{aligned}&Des(\text {A}\rightarrow \text {B})-Des(\text {A}\rightarrow \lnot \text {B})\\&\quad =Des(\text {A}) \cdot P(\text {A}\mid \text {A}\rightarrow \text {B})-Des(\text {A})\cdot P(\text {A}\mid \text {A}\rightarrow \lnot \text {B})\\&\qquad +\,Des(\lnot \text {A})\cdot P(\lnot \text {A}\wedge \text {B}\mid \text {A}\rightarrow \text {B})-Des(\lnot \text {A})\cdot P(\lnot \text {A}\wedge \text {B}\mid \text {A}\rightarrow \lnot \text {B})\\&\qquad +\,Des(\lnot \text {A})\cdot P(\lnot \text {A}\wedge \lnot \text {B}\mid \text {A}\rightarrow \text {B})-Des(\lnot \text {A})\cdot P(\lnot \text {A}\wedge \lnot \text {B}\mid \text {A}\rightarrow \lnot \text {B}) \end{aligned}$$From this we get Eq. 7, since in general \(P(\alpha \wedge \beta )+P(\alpha \wedge \lnot \beta )=P(\alpha )\) .
When we update on \(\lnot \text {A}\wedge \lnot \text {B}\), \(P(\lnot \text {A}\wedge \lnot \text {B}\mid ... )\) becomes 1 (and all the other probabilities in 2 become 0.) Now by the definition of conditional desirability, \(Des_{\lnot \text {A}\wedge \lnot \text {B}}(\text {A}\rightarrow \text {B})-Des_{\lnot \text {A}\wedge \lnot \text {B}}(\text {A}\rightarrow \lnot \text {B})=Des(\lnot \text {A}\wedge \lnot \text {B}\wedge (\text {A}\rightarrow \text {B}))-Des(\lnot \text {A}\wedge \lnot \text {B})-[Des(\lnot \text {A}\wedge \lnot \text {B}\wedge (\text {A}\rightarrow \lnot \text {B}))-Des(\lnot \text {A}\wedge \lnot \text {B})]=Des(\lnot \text {A}\wedge \lnot \text {B}\wedge (\text {A}\rightarrow \text {B}))-Des(\lnot \text {A}\wedge \lnot \text {B}\wedge (\text {A}\rightarrow \lnot \text {B}))\).
Here I am again assuming the Conditional Excluded Middle, as a referee reminds me.
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Acknowledgments
I have benefited from discussing this paper with Richard Bradley, Branded Fitelson and Katie Steele. I would also like to thank the audience of the Sixth Workshop on Decisions, Games, & Logic, where an earlier version of this paper was presented, for their questions and comments. My work on this paper was partly supported by a grant from AXA Research Fund (14-AXA-PDOC-222).
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Stefánsson, H.O. Desirability of conditionals. Synthese 193, 1967–1981 (2016). https://doi.org/10.1007/s11229-015-0823-0
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DOI: https://doi.org/10.1007/s11229-015-0823-0