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Might be worth noting that standard deontic logic does not validate (i) even if we assume that the actual world is among the deontically perfect worlds.
(i) [O(p --> q) --> .p --> Oq]
Since our world is obviously sub-optimal, we should not expect a detachment principle like (i) to hold. And (iii) is not valid unless there is a unique moral standard for each world. That is effectively to hold that there is a single correct moral principle for each world. SDL does validate (iii).
What you seem to be after in your representation of the deontic conditional is the dyadic operator in (i').
(i') O(Q / P), (read) in all of the morally best worlds at which P is true, Q is true. And it might be the case that in all of the best worlds in which you murder someone, you do so gently. But then we seem to get an analogue of Forrester's Paradox. Since Q entails P and since (ii') is true,

(ii') |-(Q ->P) -> [O(Q / P)-> O(P/P)]

So in the best worlds at which Smith murders Jones gently, Smith ought to murder Jones gently. Yikes!


Here's the analogue I mentioned.

1. O(P/P) [it is obligatory that Smith murder Jones given that he has done so]

But intuitively (for whatever that's worth),

2. ~O(P/P) [it is not obligatory that Smith murder Jones, given that he has done so].

Kenny Easwaran

It seems to me that the questionable premise is the second one. It certainly seems false that just because p-->q happens to be true that O(p)-->O(q) would be true. It would be a more plausible condition if N(p-->q) were true, where N is a necessity operator, possibly even logical necessity. This seems to be the case in the deduction given, because necessarily, if Smith murders Jones gently then Smith murders Jones. But even here, it seems a bit odd to me to think that obligation can "see through" the logical connection here. After all, even if it is obligatory that one save the lives of the five people tied to the trolley track, and the only way to save their lives is to push the fat man in the way of the trolley, it doesn't seem obviously obligatory that one push the ft man in the way of the trolley.


Of course that inference rule is either (i) if
|-p->q then |-Op->Oq or (ii) N(p-->q)->(Op->Oq). Otherwise it is clearly invalid. Judging by the example (in which we have an entailment from mudering gently to murdering) I think he has in mind (ii).

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