Abstract
In “Desire as Belief” and “Desire as Belief II,” David Lewis (1988, 1996) considers the anti-Humean position that beliefs about the good require corresponding desires, which is his way of understanding the idea that beliefs about the good are capable of motivating behavior. He translates this anti-Humean claim into decision theoretic terms and demonstrates that it leads to absurdity and contradiction. As Ruth Weintraub (2007) has shown, Lewis’ argument goes awry at the outset. His decision theoretic formulation of anti-Humeanism is one that no sensible anti-Humean would endorse. My aim is to demonstrate that Lewis’ infelicitous rendering of anti-Humeanism really does undermine the force of his arguments. To accomplish this, I begin by developing a more adequate decision theoretic rendering of the anti-Humean position. After showing that my formulation of anti-Humeanism constitutes a plausible interpretation of the anti-Humean thesis, I go on to demonstrate that if we adopt this more accurate rendition of anti-Humeanism, the view is no longer susceptible to arguments like the ones Lewis has devised. I thereby provide a more robust response to Lewis’ arguments than has yet been offered, and in the process I develop a formulation of anti-Humeanism that creates the possibility for future decision theoretic arguments that, unlike Lewis’, speak directly to the plausibility of anti-Humeanism.
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Notes
Oddie (1994) contends that Lewis’s arguments fail to work against defenders of (3) who adopt a causal decision theory, and Byrne and Hajek (1997) and Oddie (2001) further develop this response to Lewis. Although I am persuaded by their analyses, I think focusing on whether Lewis’s arguments continue to work within a causal decision theory obscures the deeper problem with Lewis’s translation of anti-Humeanism. In any case, my aim is to show that a reasonable version of anti-Humeanism is immune to Lewis’s attack, and that anti-Humeans therefore need not take refuge in causal decision theory. Similarly, I take it that Hajek and Pettit (2004) make a convincing case that anti-Humeans who embrace what they call an indexicalist account of goodness or rightness can evade Lewis’s arguments, but my aim is to show that anti-Humeanism is independent of the debate over indexicalist metaethics.
For a different, and earlier, response to Lewis according to which Lewis has offered an infelicitous translation of the anti-Humean thesis into decision theory, see (Broome 1991). Byrne and Hajek (1997: 423–26) also suggest that an anti-Humean could evade Lewis’s arguments by rejecting Lewis’s formulation of anti-Humeanism. But neither Broome nor Byrne and Hajek nor Weintraub offers an alternative translation of the anti-Humean view, and as a result they do not test whether Lewis’s arguments work against a more plausible form of anti-Humeanism.
There is room for debate over whether desires should work this way, and so room for debate over whether we should interpret the value functions of decision theory as representing desires, but regardless of how desires behave it is absurd to think that beliefs about the good are (also) tied in this way to credences. See (Weintraub 2007) for a more detailed exposition of the absurdity of (3).
This may not always be the case. If, for instance, the change in C(A) is accompanied by a change in the agent’s expectation of how A would be realized, then it may generate a change in V(A) without producing a similar change in V(A). But in this sort of case, there is no reason to think that the change in her expectation of how A would be realized would not also produce a change in her belief about the goodness of A, as measured by \( \sum\nolimits_{y} {C\left( {g\left( A \right) = y} \right) \cdot y} \). In any event, the important point here is that if C(A) is the only thing that changes, then AV(A) will remain constant even though V(A) will change.
\( V\left( W \right) = C\left( {\left( {{{W \wedge M} \mathord{\left/ {\vphantom {{W \wedge M} W}} \right. \kern-\nulldelimiterspace} W}} \right) \cdot V\left( {W \wedge M} \right)} \right) + C\left( {\left( {{{W \wedge \neg M} \mathord{\left/ {\vphantom {{W \wedge \neg M} W}} \right. \kern-\nulldelimiterspace} W}} \right) \cdot V\left( {W \wedge \neg M} \right)} \right) \).
This calculates out as: \( \left( {{{0.25} \mathord{\left/ {\vphantom {{0.25} {0.75}}} \right. \kern-\nulldelimiterspace} {0.75}}} \right) \cdot 1000 + \left( {{{0.5} \mathord{\left/ {\vphantom {{0.5} {0.75}}} \right. \kern-\nulldelimiterspace} {0.75}}} \right) \cdot 500 = 667 \).
Similarly, \( V\left( {\neg W} \right) = C\left( {\left( {\neg {{W \wedge M} \mathord{\left/ {\vphantom {{W \wedge M} {\neg W}}} \right. \kern-\nulldelimiterspace} {\neg W}}} \right) \cdot V\left( {\neg W \wedge M} \right)} \right) + \) \( C\left( {\left( {{{\neg W \wedge \neg M} \mathord{\left/ {\vphantom {{\neg W \wedge \neg M} {\neg W}}} \right. \kern-\nulldelimiterspace} {\neg W}}} \right) \cdot V\left( {\neg W \wedge \neg M} \right)} \right) \).
This calculates out as: \( \left( {{{0.25} \mathord{\left/ {\vphantom {{0.25} {0.25}}} \right. \kern-\nulldelimiterspace} {0.25}}} \right) \cdot \left( { - 1000} \right) + \left( {{0 \mathord{\left/ {\vphantom {0 {0.25}}} \right. \kern-\nulldelimiterspace} {0.25}}} \right) \cdot \left( { - 500} \right) = - 1000 \).
See (Joyce 1999) for a discussion of imaging.
In evaluating AV(W), I shall also suppose that \( \left( {\left( {\neg W \wedge \neg M} \right) \bullet W} \right) \) is \( \left( {W \wedge \neg M} \right) \), although this judgment of proximity is irrelevant given that the credence for \( \left( {\neg W \wedge \neg M} \right) \) is 0. I include it in the evaluation of AV(W) only for completeness:
$$ \begin{aligned} AV\left( W \right) & = C\left( {W \wedge M} \right) \cdot \left( {V\left( {W \wedge M} \right) - V\left( {\neg W \wedge M} \right)} \right) + C\left( {\neg W \wedge M} \right) \cdot \left( {V\left( {W \wedge M} \right) - V\left( {\neg W \wedge M} \right)} \right) + \\ \quad C\left( {W \wedge \neg M} \right) \cdot \left( {V\left( {W \wedge \neg M} \right) - V\left( {\neg W \wedge \neg M} \right)} \right) + C\left( {\neg W \wedge \neg M} \right) \cdot \left( {V\left( {W \wedge \neg M} \right) - V\left( {\neg W \wedge \neg M} \right)} \right). \\ \end{aligned} $$This calculates out as: \( 0.25 \cdot \left( {1000 - \left( { - 1000} \right)} \right) + 0.25 \cdot \left( {1000 - \left( { - 1000} \right)} \right) + 0.5 \cdot \left( {500 - \left( { - 500} \right)} \right) + 0 \cdot \left( {500 - \left( { - 500} \right)} \right) = 1500. \)
I will only go into as much detail in recreating Lewis’s arguments as necessary to show that they do not apply to my preferred decision theoretic rendering of the anti-Humean position.
Intuitively, the difference between AV(W) and AV(W/W) stems from the fact that the assumption that Tim has won changes his relative subjective probabilities for having bet on the Yankees as opposed to the Mariners, resulting in a change in how good it is to win, by his lights. Formally:
$$ \begin{aligned} AV\left( {{W \mathord{\left/ {\vphantom {W W}} \right. \kern-\nulldelimiterspace} W}} \right) & = C\left( {{{\left( {W \wedge M} \right)} \mathord{\left/ {\vphantom {{\left( {W \wedge M} \right)} W}} \right. \kern-\nulldelimiterspace} W}} \right) \cdot \left( {V\left( {W \wedge M} \right) - V\left( {\neg W \wedge M} \right)} \right) + C\left( {{{\left( {\neg W \wedge M} \right)} \mathord{\left/ {\vphantom {{\left( {\neg W \wedge M} \right)} W}} \right. \kern-\nulldelimiterspace} W}} \right) \cdot \left( {V\left( {W \wedge M} \right) - V\left( {\neg W \wedge M} \right)} \right) + \\ \quad C\left( {{{\left( {W \wedge \neg M} \right)} \mathord{\left/ {\vphantom {{\left( {W \wedge \neg M} \right)} W}} \right. \kern-\nulldelimiterspace} W}} \right) \cdot \left( {V\left( {W \wedge \neg M} \right) - V\left( {\neg W \wedge \neg M} \right)} \right) + C\left( {{{\left( {\neg W \wedge \neg M} \right)} \mathord{\left/ {\vphantom {{\left( {\neg W \wedge \neg M} \right)} W}} \right. \kern-\nulldelimiterspace} W}} \right) \cdot \left( {V\left( {W \wedge \neg M} \right) - V\left( {\neg W \wedge \neg M} \right)} \right). \\ \end{aligned} $$This calculates out as:
$$ \left( {{{0.25} \mathord{\left/ {\vphantom {{0.25} {0.75}}} \right. \kern-\nulldelimiterspace} {0.75}}} \right) \cdot \left( {1000 - \left( { - 1000} \right)} \right) + 0 \cdot \left( {1000 - \left( { - 1000} \right)} \right) + \left( {{{0.5} \mathord{\left/ {\vphantom {{0.5} {0.75}}} \right. \kern-\nulldelimiterspace} {0.75}}} \right) \cdot \left( {500 - \left( { - 500} \right)} \right) + 0 \cdot \left( {500 - \left( { - 500} \right)} \right) = 1333. $$This is an essential element in the proof of Lewis’s Initial Lemma on (Lewis 1996: 310).
Intuitively, assuming that Tim has bet on the Mariners makes determining the absolute value of winning easy: 1000 if it is true that he wins minus (−1000) if it is not true. Formally:
$$ \begin{aligned} AV\left( {{W \mathord{\left/ {\vphantom {W M}} \right. \kern-\nulldelimiterspace} M}} \right) & = C\left( {{{\left( {W \wedge M} \right)} \mathord{\left/ {\vphantom {{\left( {W \wedge M} \right)} M}} \right. \kern-\nulldelimiterspace} M}} \right) \cdot \left( {V\left( {W \wedge M} \right) - V\left( {\neg W \wedge M} \right)} \right) + C\left( {{{\left( {\neg W \wedge M} \right)} \mathord{\left/ {\vphantom {{\left( {\neg W \wedge M} \right)} M}} \right. \kern-\nulldelimiterspace} M}} \right) \cdot \left( {V\left( {W \wedge M} \right) - V\left( {\neg W \wedge M} \right)} \right) + \\ \quad C\left( {{{\left( {W \wedge \neg M} \right)} \mathord{\left/ {\vphantom {{\left( {W \wedge \neg M} \right)} M}} \right. \kern-\nulldelimiterspace} M}} \right) \cdot \left( {V\left( {W \wedge \neg M} \right) - V\left( {\neg W \wedge \neg M} \right)} \right) + C\left( {{{\left( {\neg W \wedge \neg M} \right)} \mathord{\left/ {\vphantom {{\left( {\neg W \wedge \neg M} \right)} M}} \right. \kern-\nulldelimiterspace} M}} \right) \cdot \left( {V\left( {W \wedge \neg M} \right) - V\left( {\neg W \wedge \neg M} \right)} \right). \\ \end{aligned} $$This calculates out as:
$$ \left( {{{0.25} \mathord{\left/ {\vphantom {{0.25} {0.5}}} \right. \kern-\nulldelimiterspace} {0.5}}} \right) \cdot \left( {1000 - \left( { - 1000} \right)} \right) + \left( {{{0.25} \mathord{\left/ {\vphantom {{0.25} {0.5}}} \right. \kern-\nulldelimiterspace} {0.5}}} \right) \cdot \left( {1000 - \left( { - 1000} \right)} \right) + 0 \cdot \left( {500 - \left( { - 500} \right)} \right) + 0 \cdot \left( {500 - \left( { - 500} \right)} \right) = 2000. $$Intuitively, assuming Tim has won leaves two possibilities. If he bet on the Mariners, then it was good to have bet on them (and won) because it was a bigger payoff. But if he bet on the Yankees, then it would actually have been worse, in terms of expected utility, to have bet on the Mariners, as long as we suppose that there was an even chance of having lost had he done so. In formally evaluating AV(M/W), I shall suppose that \( \left( {\left( {\neg W \wedge M} \right) \bullet \neg M} \right) \) is \( \left( {W \wedge \neg M} \right) \), and that \( \left( {W \wedge M} \right) \) and \( \left( {\neg W \wedge M} \right) \) are equally close to \( \left( {W \wedge \neg M} \right) \). These suppositions may be controversial, but they are irrelevant because conditionalizing on M reduces the credences for \( \left( {W \wedge \neg M} \right) \) and \( \left( {\neg W \wedge \neg M} \right) \) to 0. As before, I include these proximity judgments only for completeness:
$$ \begin{aligned} AV\left( {{M \mathord{\left/ {\vphantom {M W}} \right. \kern-\nulldelimiterspace} W}} \right) & = C\left( {{{\left( {W \wedge M} \right)} \mathord{\left/ {\vphantom {{\left( {W \wedge M} \right)} W}} \right. \kern-\nulldelimiterspace} W}} \right) \cdot \left( {V\left( {W \wedge M} \right) - V\left( {W \wedge \neg M} \right)} \right) + C\left( {{{\left( {\neg W \wedge M} \right)} \mathord{\left/ {\vphantom {{\left( {\neg W \wedge M} \right)} W}} \right. \kern-\nulldelimiterspace} W}} \right) \cdot \left( {V\left( {\neg W \wedge M} \right) - V\left( {W \wedge \neg M} \right)} \right) + \\ \quad C\left( {{{\left( {W \wedge \neg M} \right)} \mathord{\left/ {\vphantom {{\left( {W \wedge \neg M} \right)} W}} \right. \kern-\nulldelimiterspace} W}} \right) \cdot \left( {0.5 \cdot \left( {V\left( {W \wedge M} \right) - V\left( {W \wedge \neg M} \right)} \right) + 0.5 \cdot \left( {V\left( {\neg W \wedge M} \right) - V\left( {W \wedge \neg M} \right)} \right)} \right) + \\ \quad C\left( {{{\left( {\neg W \wedge \neg M} \right)} \mathord{\left/ {\vphantom {{\left( {\neg W \wedge \neg M} \right)} W}} \right. \kern-\nulldelimiterspace} W}} \right) \cdot \left( {0.5 \cdot \left( {V\left( {W \wedge M} \right) - V\left( {\neg W \wedge \neg M} \right)} \right) + 0.5 \cdot \left( {V\left( {\neg W \wedge M} \right) - V\left( {\neg W \wedge \neg M} \right)} \right)} \right). \\ \end{aligned} $$This calculates out as:
$$ \begin{gathered} \left( {{{0.25} \mathord{\left/ {\vphantom {{0.25} {0.75}}} \right. \kern-\nulldelimiterspace} {0.75}}} \right) \cdot \left( {1000 - 500} \right) + 0 \cdot \left( { - 1000 - 500} \right) + \left( {{{0.5} \mathord{\left/ {\vphantom {{0.5} {0.75}}} \right. \kern-\nulldelimiterspace} {0.75}}} \right) \cdot \left( {0.5 \cdot \left( {1000 - 500} \right) + 0.5 \cdot \left( { - 1000 - 500} \right)} \right) + \hfill \\ 0 \cdot \left( {0.5 \cdot \left( {1000 - \left( { - 500} \right)} \right) + 0.5 \cdot \left( { - 1000 - \left( { - 500} \right)} \right)} \right) = - 167. \hfill \\ \end{gathered} $$Intuitively, if we now assume that Tim would have won regardless of whom he had bet on (rather than simply assuming that he won his actual bet), then the calculation is easy: betting on the Mariners and winning is worth 1000, not betting on them and winning is worth 500, so the absolute value of betting on them is 500. In the formal evaluation, for simplicity, this time I will leave out the parts of the equation that are multiplied by 0:
$$ AV\left( {{M \mathord{\left/ {\vphantom {M W}} \right. \kern-\nulldelimiterspace} W}} \right) = C\left( {{{\left( {W \wedge M} \right)} \mathord{\left/ {\vphantom {{\left( {W \wedge M} \right)} W}} \right. \kern-\nulldelimiterspace} W}} \right) \cdot \left( {V\left( {W \wedge M} \right) - V\left( {W \wedge \neg M} \right)} \right) + C\left( {{{\left( {W \wedge \neg M} \right)} \mathord{\left/ {\vphantom {{\left( {W \wedge \neg M} \right)} W}} \right. \kern-\nulldelimiterspace} W}} \right) \cdot \left( {V\left( {W \wedge M} \right) - V\left( {W \wedge \neg M} \right)} \right). $$This calculates out as: \( \left( {{{0.25} \mathord{\left/ {\vphantom {{0.25} {0.75}}} \right. \kern-\nulldelimiterspace} {0.75}}} \right) \cdot \left( {1000 - 500} \right) + \left( {{{0.5} \mathord{\left/ {\vphantom {{0.5} {0.75}}} \right. \kern-\nulldelimiterspace} {0.75}}} \right) \cdot \left( {1000 - 500} \right) = 500. \)
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Acknowledgements
Thanks to James Joyce and two anonymous referees for their helpful comments and criticisms.
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Daskal, S. Absolute value as belief. Philos Stud 148, 221–229 (2010). https://doi.org/10.1007/s11098-008-9323-0
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DOI: https://doi.org/10.1007/s11098-008-9323-0