The Additive Assumption Underlying Contrast Cases

A common way to argue for a philosophical claim is to set up a pair of contrasting cases. All factors are held equal across both cases except the factor in question. Tweaking a factor has the effect of isolating the contribution the factor makes to the overall value of the scenario. Jason Stanley (2005) used this method by varying the stakes (low/high) between cases. In a recent paper, Mark Schroeder borrowed and adapted Stanley’s cases to argue that stakes are not the only relevant factor when it comes to the subject-sensitivity of ‘knows’. For Schroeder the relative costs of making errors matter as well. Practical rationality theorists would be well-advised to heed the work of Shelly Kagan (1988). Kagan identified the additive assumption underlying contrast cases. In this post, I explore this assumption in an effort to help the future avoid repeating the past.

One of the problems of using contrast case methodology to make generalizations (i.e. rational belief is subject-sensitive because it varies with the stakes or the relative costs of errors) is that it often commits the additive fallacy. Generalizing from contrast cases it is assumed that a certain factor (e.g. the subject-sensitivity of rational belief) will make a similar difference in other cases. Using this assumption the goal is to show that the favored generalization is able to yield intuitive predictions across many contexts and cases.

A strategy to undercut contrast case arguments is to take away the contrast case proponent’s ability to generalize his conclusion or transport his conclusion into more complex cases. The proponent’s methodology involves isolating the contribution of certain factors. Once the weight of a factor is isolated it is inferred that this factor will have a similar weight in general or in different contexts. In Schroeder’s paper this found detailed expression in his model of the cost of type-2 error. He then used this model to show it correctly predicted what was going on in further cases. Contrast case opponents argue against such methodology by arguing for context-dependence. Kagan did this by showing the additive assumption is false. As Kagan (1988) remarked about the contribution of a factor in a given case:

if the contribution is dependent not only on the single feature, but on other factors as well, then despite the preservation of the feature considered in isolation, a changed context could easily result in a significantly altered contribution (p. 24).

Underlying the additive fallacy is the assumption that the governing function is additive. The governing function maps the value of the factors onto the overall status of the act. If the function is additive it is assumed one can sum the separate contributions of the individual factors (i.e. costs of errors) to determine the overall status of the act. Schroeder’s model presupposes the governing function is additive and costs make independent contributions that can be summed to determine the overall value of the act.

When Schroeder presented his paper at Stanford Krista Lawlor raised the question of how Schroeder’s model would handle a more complex scenario. Krista’s example involved two couples in the same car in a forced choice situation where the choice of either couple would prevent the other couple from reaching their goal. I took Krista’s example to imply that the governing function cannot be strictly additive. That is, the right assessment of the scenario is not accomplished by simply adding the costs of error for each individual couple. What might be required is a multiplier governing function. This occurs when factors do not make contributions to be summed; rather, clusters of factors must be multiplied to generate the correct assessment of the case. Kagan (1988) used the suffering of Trixie and Fritz as an example of the need for a multiplier governing function.

The problem is that the additive assumption requires this same model even when Trixie is morally to blame for having performed the act which created the situation. Perhaps, for example, Trixie had attempted to harm Fritz, but her scheme had gone awry, harming them both, and her more significantly. The additive model can point to her guilt as a reason for not helping Trixie, but it is forced to maintain that the magnitude of her suffering provides just as strong a reason to aid her as if she were innocent. Yet many would want to reject exactly this latter claim: the suffering of the guilty simply does not count as much as the suffering of the innocent (p. 20).

In Krista’s example a second factor comes into play: either couple’s choice will be responsible for plight of the other couple. That is, Hannah and Sarah will be partially responsible for Bill and Tom missing their mortgage payment. Viewed in an additive manner Hannah and Sarah’s choice makes a positive independent contribution to the overall status of the act for Hannah and Sarah and a negative contribution to the status of the act for Bill and Tom. Simply adding up the costs and benefits may not yield the correct assessment of the case because we need to account for how the choice of Hannah and Sarah will be responsible for plight of Bill and Tom. We might want to say the benefits for Hannah and Sarah do not count as much because they are achieved at the expense of Bill and Tom. Another complication is a situation of malice where Hannah and Sarah make their choice in an effort to make Bill and Tom suffer. Such cases would be better handled by multiplying factors instead of simply adding them up. Krista’s intuition of the weirdness of Schroeder’s view applied to more complex cases may, implicitly, be grounded on the idea that his model, which presupposes the additive assumption, cannot adequately capture what is going on when factors multiply in important ways.

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