Thursday, August 23, 2012

Paradoxes of comparison

There are three sets, A, B and C, each consisting of the same number of people, whose lives are endangered by the same sort of danger, and whose future prospects as far as you know are on par. There are also three hungry kids, x, y and z, who will survive if you don't give them breakfast, but who would benefit from your giving them breakfast once (you have no opportunity to do anything more for them). Suppose you have a choice between two actions:

  1. Save all the people in A and feed nobody.
  2. Save all the people in B and feed x.
Now, it sure seems like 2 is the better action than 1. One might even formulate a general principle:
  • (*) If an action saves the same number of lives and feeds more hungry children, and all else is on par, then it's better.
But actually this is false. For suppose that B is a proper subset of A, and there are 100 people in A who are not in B. Since we said that A and B have the same number of people, this can only be the case when A and B are infinite sets. In this scenario, if one goes for 2, there will be 100 people whom one won't be saving. So we should modify (*), perhaps to:
  • (**) If an action saves the same number of lives and feeds more hungry children, and the sets of lives saved and children fed are disjoint between the two actions, and all else is on par, then the action is better.

But paradox ensues when we specify that A and B have no people in common, but C is a subset of A missing 100 people, and then add the option:

  1. Save all the people in C and feed y and z.
For by one application of (**), 2 is better than 1, and by another application of (**), 3 is better than 2. But 3 is not better than 1, because the 100 people in A who aren't in C die if you go for 3, and that's not balanced out by the two children fed.

(There is a literature on infinite utilities, and I am not claiming any originality for this case.)

One could take this as yet another argument against the transitivity of "better than". But that doesn't get us out of paradox, since denying that transitivity is itself paradoxical. Moreover, there is already something paradoxical in having to deny (*)—that principle sure seemed plausible.

We could conclude that one can't have infinite sets of people, and make this be one of the family of arguments against actual infinites. Maybe.

But I want to do something else here. I think this, like a number of other paradoxes (which need not all involve infinity; I have a hunch that White's puzzle, as per Joyce's reply discussed in the link, is in this family), is due to us having two ways of comparing. We have an uncontroversial and unproblematic inclusion or domination comparison. It is uncontroversial that all other things being equal, if you can save all the people in A or all the people in B, and the people in A are a proper subset of those in B, then you should save the people in B. It is uncontroversial that if p entails q, then q is at least as likely as p. And so on.

But we also insist on comparing apples to oranges, comparing where there is no inclusion or domination relation. Typically, five oranges are more valuable than one apple, and five apples are more valuable than one orange. To make such comparisons we often assign numbers—say, cardinalities, utilities, prices or probabilities—to the things we are comparing, but we can also just make ordinal comparisons without assigning numbers (I didn't assign any utilities when I gave the ethical story).

I think a lot of paradoxes have the consequence that comparisons without domination are fishy. They need not satisfy transitivity. They might suffer from some arbitrariness. In the ethical sphere, this can be manifested in incommensurability of options. In probability theory, this surfaces in difficulties surrounding infinite sample spaces or nonmeasurable sets (as in White's puzzle, since nonmeasurable sets and non-exact probabilities are of a piece, I think).

Yet we need comparisons-without-domination.

So what should we do? In the ethical sphere, perhaps what we need is basically what Aquinas says about the order of charity. Aquinas thinks that when choosing between an equal benefit to one's parent or to a stranger, one should bestow the benefit on one's parent. But what if the benefit to the stranger is greater? If only slightly greater, we should still benefit our parent. But if much greater, we should benefit the stranger. But where is the line drawn? Aquinas refuses to answer. There is no rule, it seems. Rather, this is just somehting for the wise and virtuous agent to know. And maybe there is an analogue to this answer in the case of the non-ethical paradoxes.

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