Condition (4) in the definition of a saturating counterpart relations (that is, there is no other mapping that satisfies the first 3 conditions but which results in W1 having lower harm when combined with HMV) seems to be a bit ad hoc and designed to get him out of various situations, like the absurd conclusion, without having independent appeal.
One way to motivate this is that it’s a generalization of symmetry. Counterparts are chosen so that their welfares match as closely as possible (after any personal identity-preservation constraints, which could be dropped), where the distance between two worlds is roughly measured in additive terms (rather than, say, the minimizing the maximum harm), which matches our additive aggregation for calculating harm.
If you took one world, and replaced all the identities with a disjoint set of identities of the same numbers, while preserving the distribution of welfare, adding condition (4) to the other conditions makes these worlds morally equivalent. If you switched the identities and changed exactly one welfare, then the mapping of identities would be one of the permissible mappings under condition (4). It picks out the intuitively correct mappings in these cases. Maybe condition (4) is unjustifiably strong for this, though.
Another way to look at it is that the harm in a given world is the minimum harm under all mappings satisfying conditions 1-3. Mappings which satisfy the minimum in some sense make the worlds most similar (under the other constraints and definition of harm).
Furthermore, if you were doing infinite ethics and didn’t have any other way to match identities between worlds (e.g. locations) or had people left over who weren’t (yet) matched (after meeting identity constraints), you could do something like this, too. Pairwise, you could look at mappings of individual identities between the two worlds, and choose the mappings that lead to the minimum (infimum) absolute aggregate of differences in welfares, where the differences are taken between the mapped counterparts. So, this is choosing the mappings which make the two worlds look as similar as possible in terms of welfare distributions. The infimum might not actually be attained, but we’re more interested in the number than the mappings, anyway. If, within some distance of the infimum (possibly 0, so an attained minimum), all the mappings lead to the same sign for the aggregate (assuming the aggregate isn’t 0), then we could say one world is better than the other.
One way to motivate this is that it’s a generalization of symmetry. Counterparts are chosen so that their welfares match as closely as possible (after any personal identity-preservation constraints, which could be dropped), where the distance between two worlds is roughly measured in additive terms (rather than, say, the minimizing the maximum harm), which matches our additive aggregation for calculating harm.
If you took one world, and replaced all the identities with a disjoint set of identities of the same numbers, while preserving the distribution of welfare, adding condition (4) to the other conditions makes these worlds morally equivalent. If you switched the identities and changed exactly one welfare, then the mapping of identities would be one of the permissible mappings under condition (4). It picks out the intuitively correct mappings in these cases. Maybe condition (4) is unjustifiably strong for this, though.
Another way to look at it is that the harm in a given world is the minimum harm under all mappings satisfying conditions 1-3. Mappings which satisfy the minimum in some sense make the worlds most similar (under the other constraints and definition of harm).
Furthermore, if you were doing infinite ethics and didn’t have any other way to match identities between worlds (e.g. locations) or had people left over who weren’t (yet) matched (after meeting identity constraints), you could do something like this, too. Pairwise, you could look at mappings of individual identities between the two worlds, and choose the mappings that lead to the minimum (infimum) absolute aggregate of differences in welfares, where the differences are taken between the mapped counterparts. So, this is choosing the mappings which make the two worlds look as similar as possible in terms of welfare distributions. The infimum might not actually be attained, but we’re more interested in the number than the mappings, anyway. If, within some distance of the infimum (possibly 0, so an attained minimum), all the mappings lead to the same sign for the aggregate (assuming the aggregate isn’t 0), then we could say one world is better than the other.