One might consider an old fashioned doomsday type argument - nuclear technology and a pessimistic view of human nature with a constant small probability of nuclear war for each generation of politicians.]]>

I've been thinking a little about this kind of way of arguing for additivity (thanks for the pointer!)

Here's one thing I've come up against. Suppose you've got your separable component propositions, and they're "relational", in the sense your treatment of Kant is. That is: we've got a "slot" for desert-and-happiness, with cells specifying a degree of happiness/unhappiness, and also whether the person is deserving or not.

One of the key technical assumptions you need is "restricted solvability". And roughly, that tells you that if you've got two complete propositions differing only in which component proposition they have in this slot, with different values, and then you've got some other complete proposition x whose value lies between them, then you can find a substitution for the component in the first pair that matches the value of x.

It's a sort-of continuity assumption.

Now, in something like the Kant case, continuity is going to come, intuitively, from varying degree of happiness. So suppose the pair we start with are both deserving people with different degrees of happiness. Then you'd hope to be able to match any intermediate value by finding an intermediate level of happiness for a deserving person. Seems okay.

But suppose you have a deserving person, and an undeserving person, each with different levels of happiness/unhappiness. Then it seems more substantive ethical assumption to assume that you can find some substitution for that component of overall value that will match any other realized intermediate value (picture a situation where the happiness/unhappiness of the deserving is always more important than the happiness/unhappiness of the undeserving, so the realizable values form two disconnected "islands").

So I think this assumption bears thinking about, if you're going to make the very general argument here (I was thinking of this in connection to arguments for the additivity of overall accuracy of a credence function, and that's also relational in this way---the accuracy of a given credence turns on whether the proposition which is its content is true or false. So I think that hidden in the solvability axiom is going to be a particular assumption about the way that accuracy-given-truth and accuracy-given-falsity relate).

Now, I wonder whether one can overcome this just by embedding the whole structure in a bigger one (filling in the gaps, as it were), deriving an additive representation for that, and then cutting back to the original. After all, solvability is a richness assumption, and embeddability often helps out for that in other cases.

The other technical assumptions look more innocent to me, fwiw---it seems like separability (/independence) and this restricted solvability are the two where the real action is.

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Assume Alt(exh(AvB > C)) = { exh(A > C), exh(B > C) } = { (A > C) & ~(B > C), (B > C) & ~(A > C) }.

Then exh(exh(AvB > C)) = (AvB > C) & ~[(A > C) & ~(B > C)] & ~[(B > C) & ~(A > C)].

~[(A > C) & ~(B > C)] & ~[(B > C) & ~(A > C)] is equivalent to (A > C) <-> (B > C).

Assuming that AvB > C entails (A > C) v (B > C), it follows that exh(exh(AvB > C)) entails (but is not equivalent to) A > C and B > C.

In your examples, one or both of A and B in the antecedent is complex, which might introduce some more alternatives, but I'd expect the above entailment to remain.

As far as I can tell, this also gets (9)/(10) right.

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Still assuming your calculation is correct, I have a query. Take scenario 1 in the paper (Both children are on the right and the seesaw is balanced) and evaluate the following:

(3a') If Blue or both of them were on the left, then the seesaw would be unbalanced.

In that scenario I feel like (3a') is false, because if Blue was on the left, the seesaw would be balanced! But if exh(exh(Av(A&B)>C)) is equivalent to A&B>C, then we should expect (3a') to be true, since if both children were on the left, it would indeed be unbalanced.

Perhaps this is also relevant. In the paper, exhaustifying Av(A&B) to obtain (A&-B)v(A&B) was forcing a dilemma on the friends of the exhaustification. They get the (3a)-(4a)-(4b) trio right, but then they could not get the (9a)-(10a)-(10b) trio right. I cannot tell whether double exhaustification is helping with that or not. Is it?

Also thank you so much for the discussion!]]>

Could you tell more how you thought double exhaustification would help?]]>