Abstract
Does the desirability of a proposition depend on whether it is true? Not according to the Invariance assumption, held by several notable philosophers. The Invariance assumption plays an important role in David Lewis’ famous arguments against the so-called Desire-as-Belief thesis (DAB), an anti-Humean thesis according to which a rational agent desires a proposition exactly to the degree that she believes the proposition to be desirable. But the assumption is of interest independently of Lewis’ arguments, for instance since both Richard Jeffrey and James Joyce make the assumption (or, strictly speaking, accept a thesis that implies Invariance) in their influential books on decision theory. The main claim to be defended in this paper is that Invariance is incompatible with certain assumptions of decision theory. I show that the assumption fails on the most common interpretations of desirability and/or choice-worthiness found in decision theory. I moreover show that Invariance is inconsistent with Richard Jeffrey’s decision theory, on which Lewis’ arguments against DAB are based. Finally, I show that Invariance contradicts how we in general do and should think about conditional desirability.
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Notes
A means-end belief, such as a belief that \(\phi \)-ing will produce result \(X\), may on the Humean picture lead an agent to desire \(\phi \); but only if the agent already desires \(X\).
It is generally accepted that when an agent learns a proposition \(B\), her new credence in any proposition \(A\) should equal her old conditional credence in \(A\) under the evidential (or matter-of-factual) supposition that \(B\) (which in many cases can be quite different from contrary-to-factual suppositions; see footnote 4). Similarly, when an agent learns a proposition \(B\), the new desirability she assigns any proposition \(A\) should equal her old conditional desirability in \(A\) under the evidential supposition that \(B\). Invariance thus states that neither evidentially supposing nor learning \(A\) should affect \(A\)’s desirability.
I should point out that Lewis also considered and rejected a more plausible version of DAB, which allows for different degrees of goodness and states that a rational agent desires a proposition to the extent that she expects the proposition to be desirable (or good) [see Eq. 17 in Lewis (1988)]. Although this generalised version of DAB is arguably more plausible than the original version, the difference between the two is irrelevant to the topic of this paper. In particular, Lewis also uses the Invariance assumption to argue against the more general DAB thesis.
I should emphasise that I am assuming (like Jeffrey 1983, p. 90) that a desirability measure that is conditional on A represents the desires that the agent in question commits herself to having after she has come to fully believe A (rather than after A becomes true). I thank Wlodek Rabinowicz for encouraging me to make that clear. But even if a desirability measure that is conditional on A represents the desires the agent commits herself to having when A is true, Invariance would fail, for the reasons given in this paper, as long as her credences are not probabilistically independent of the truth.
Although \(A\) may strictly speaking count as ‘news’ for someone who hypothetically supposes \(A\), learning that \(A\) is true should not (given that the supposition is evidential) make any difference to the set of attitudes to which \(A\) has been hypothetically added. Hence, relative to that set, \(A\) does not count as news.
In keeping with the general interpretation of conditional value functions (see fn. 5), I am interpreting \(V_{A}\) as representing the causal efficacy that the agent in question expects various acts to have after she comes to fully believe that A has already been performed. Again, I thank Wlodek Rabinowicz for pressing me on this.
For instance, when supposing that Oswald did not kill Kennedy, most people (hypothetically) conclude that someone else did (since they strongly believe that Kennedy was murdered), but when supposing that Oswald had not killed Kennedy, most people (hypothetically) conclude that nobody killed Kennedy (since they don’t believe that there was a conspiracy to murder the president). See e.g. Adams (1975).
If you think that contrary-to-factually supposing \(A\) involves imagining yourself to be in a situation where \(A\) is true, rather than a situation where you believe \(A\) to be true, then Invariance still fails, provided that you think that the probability of you knowing \(A\) increases when \(A\) becomes true.
There might be special cases where you think that \(A\) is more desirable than \(B\) but you think that unlike \(B\), the value of \(A\) would be diminished if you make an effort to make it true. Hence, you might be willing to give up more to make \(B\) true than \(A\), even though you find the latter more desirable. I will set such cases aside. More importantly, when good or bad outcomes depend probabilistically, but not causally, on the alternatives an agent is faced with, value as measured by willingness to give up might differ from news value, as causal decision theorists have emphasised.
Some take this one step further, and take value, according to an agent, to be determined by her choice. I will leave such strict behaviourism aside in this paper.
I thank J. Robert G. Williams and Katie Steele for raising this issue.
In the latter case, \(A\) would also be true (unless \(A\) is inconsistent with \(C\)).
This means that stated in von Neumann and Morgenstern’s framework, the Invariance assumption is, technically speaking, meaningless. The same is not true when the assumption is stated in Jeffrey’s framework: given his framework, the assumption is meaningful but false, as I have argued.
I thank an anonymous referee for Synthese for pointing out the relevance of Weintraub’s and Daskal’s papers to my argument.
I thank J. Robert G. Williams for raising this objection.
Such propositions need not be atomic. (Indeed, to make this response compatible with Jeffrey’s framework, we must assume that the maximally specific propositions in question are not atomic.) A proposition \(A\) that is maximally specific in all respects that is relevant to its value might be further divided into two propositions that differ in the outcome of coin toss \(C\), provided that the outcome of \(C\) is irrelevant to \(A\)’s value. Hence, \(A\) may be maximally specific, in the required sense, but not atomic.
Rather than assuming that \(A\) includes everything that is important for its value, we might assume that the agent whose desires \(V\) represents knows everything that is important for determining \(A\)’s value. In that case we might think of \(V\) as measuring ‘informed value’. (I thank an anonymous referee for raising this issue.) But although it might be true for an informed value function \(V\) that for any proposition \(B\) such that \(B\not =A\), \(V_{B}(A)=V(A)\), the same does not, for all the same reasons that have been given above, hold when \(B=A\).
In other words if \(Br\) represents bread and \(Bu\) butter, then for you: \(Des_{Br}(Bu)=Des(Bu\wedge Br)=Des(\$x)\).
So \(Des(Br)=Des(\$y)\).
That is, since more money is (let us assume) better than less money, \(Des(\$x+\$z)>Des(\$x)=Des(Br\wedge Bu)\).
Are there no \(A\) and \(B\) such that for some rational agents, \(Des(A)=Des(B)\) but nevertheless \(Des_{A}(B)\not =Des_{B}(A)\)? Possibly, but it is unclear that such examples should count as counterexamples in the present context. Suppose I find attaining a state of nirvana equally d esirable as receiving a billion pounds. Then presumably attaining a state of nirvana, given that I have a billion pounds, is also desirable. However, once I have attained a state of nirvana, worldly possessions are of no value to me. Hence, given that I have attained a state of nirvana, receiving a billon pounds is no more desirable than learning the truth of a tautology. So we have a counterexample to the assumption that whenever \(Des(A)=Des(B)\), \(Des_{A}(B)=Des_{B}(A)\). The problem with this purported counterexample is that it is a mischaracterisation of conditional desirability. This might be best explained by taking an example from conditional probability. I think that conditional on being drunk, I drive very badly. However, when I am drunk, I think I drive very well. The former evaluation is what is reflected by the conditional probability I now assign to the proposition that I drive well conditional on being drunk; the latter is, for these purposes, irrelevant. Similarly, how I would evaluate a billion pounds in a state of nirvana is not relevant for evaluating the formula for conditional desirability. Rather, what we need to consider is how I now evaluate a billion pounds given that I have attained a state of nirvana. But when we do that, it is unclear that a counterexample like this can be made to the formula.
There do not seem to be any \(A\) and \(B\), such that for a rational agent, \(Des(A)\not =Des(B)\) but nevertheless \(Des_{A}(B)=Des_{B}(A)\).
James M. Joyce has in personal communication pointed out that one way of making sense of Jeffrey’s formula for conditional desirability, is to think of it as measuring total welfare, rather than incremental changes in welfare. I agree that on that interpretation of conditional desirability, Jeffrey’s formula is very natural. However, I take it that the fact that Jeffrey’s formula strikes one as having counterintuitive implications (as discussed in this section) is evidence that we tend to think of the desirability of A given B in terms of the incremental changes in welfare that A brings when B is believed true, rather than in terms of the total welfare when A and B are true.
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Acknowledgments
I have benefited from discussing the content of this paper with Richard Bradley, Johannes Himmelreich, Alan Hájek, James M. Joyce, Wlodek Rabinowicz, Katie Steele and J. Robert G. Williams.
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Stefánsson, H.O. Desires, beliefs and conditional desirability. Synthese 191, 4019–4035 (2014). https://doi.org/10.1007/s11229-014-0512-4
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DOI: https://doi.org/10.1007/s11229-014-0512-4