More on benefiting infinitely many people

Once again let’s suppose that there are infinitely people on a line infinite in both directions, one meter apart, on positions numbered in meters. Suppose all the people are on par. Fix some benefit (e.g., saving a life or giving a cookie). Let L_n be the action of giving the benefit to all the people to the left of position n. Let R_n be the action of giving the benefit to all the people to the right of position n.

Write A ≤ B to mean that action B is at least as good as action A, and write A < B to mean that A ≤ B but not B ≤ A. If neither A ≤ B nor B ≤ A, then we say that A and B are noncomparable.

Consider these three conditions:

• Transitivity: If A ≤ B and B ≤ C, then A ≤ C for any actions A, B and C from among the {L_k} and the {R_k}.

• Strict monotonicity: L_n < L_n + 1 and R_n > R_n + 1 for all n.

• Weak translation invariance: If L_n ≤ R_m, then L_n + k ≤ R_m + k and if L_n ≥ R_m, then L_n + k ≥ R_m + k, for any n, m and k.

Theorem: If we have transitivity, strict monotonicity and weak translation invariance, then exactly one of the following three statements is true:

1. For all m and n, L_m and R_n are incomparable

2. For all m and n, L_m < R_n

3. For all m and n, L_m > R_n.

In other words, if any of the left-benefit actions is comparable with any of the right-benefit actions, there is an overwhelming moral skew whereby either all the left-benefit actions beat all the right-benefit actions or all the right-benefit actions beat all the left-benefit actions.

Proposition 1 in this paper is a special case of the above theorem, but the proof of the theorem proceeds in basically the same way. For a reductio, assume that (i) is false. Then either L_m ≥ R_n or L_m ≤ R_n for some m and n. First suppose that L_m ≥ R_n. Then the second and third paragraphs of the proof of Proposition 1 show that (iii) holds. Now suppose that L_m ≤ R_n. Let L_k^* = R_−k and R_k^* = L_−k. Say that A≤^*B iff A^* ≤ B^*. Then transitivity, strict monotonicity and weak translation invariance hold for ≤^*. Moreover, we have L_m ≤ R_n, so R_−m≤^*L_−n. Applying the previous case with  − m and  − n in place of n and m respectively we conclude that we always have L_j>^*R_k and hence that we always have L_j < R_k, i.e., (ii).

I suppose the most reasonable conclusion is that there is complete incomparability between the left- and right-benefit actions. But this seems implausible, too.

Again, I think the big conclusion is that human ethics has limits of applicability.

I hasten to add this. One might reasonably think—Ian suggested this in a recent comment—that decisions about benefiting or harming infinitely many people (at once) do not come up for humans. Well, that’s a little quick. To vary the Pascal’s Mugger situation, suppose a strange guy comes up to you on the street, and tells you that there are infinitely many people in a line drowning in a parallel universe, and asks you if you want him to save all the ones to the left of position 123 or all the ones to the right of position  − 11, because he can magically do either one, and nothing else, and he needs help in his moral dilemma. You are, of course, very dubious of what he is saying. Your credence that he is telling the truth is very, very small. But as any good Bayesian will tell you, it shouldn’t be zero. And now the decision you need to make is a real one.