An example of a moral quandary from inconsistent obligations
I would think that the point of obligation is a calculus of fulfilment of duty. That is: if O(p), and if q is a necessary condition for p, then O(q). That is, O(p), p⇒q ⊨ O(q). This is neither factual nor deontic detachment, of course, but in effect an evaluation of contractual dependencies.
I will borrow my example of the nature of social dimension of propositions to produce a paradox having nothing to do with factual detachment, but having the same character of the interplay between actions and obligations. If p = "you lie" and q = "you hurt other people's feelings",
then we would normally say that O(¬p) and O(¬q). However, there are some circumstances where telling people the truth would hurt other people's feelings: where ¬p⇒q; and in particular to avoid hurting people's feelings you must lie, ¬q⇒p.
- From O(¬p) and ¬p⇒q we infer O(q);
- From O(¬q) and ¬q⇒p we infer O(p).
Thus we obtain the absurdities O(p) & O(¬p), and O(q) & O(¬q).
This conundrum arises from simple reasoning regarding the nature of obligation: the absurdities seem to arise unavoidably due to the interplay between truth-values of propositions and obligations to fulfill them — of the same sort that factual detachment gives rise to, but by a different means.
On the interplay between is and ought in deontic logic
In the comments above, I have of course tried to suggest that simply being interested simultaneously in the truth of propositions and the obligatoriness of propositions, as deontic logic is, introduces an element of multidimensionality. The conundrum of my example above does not require one to entertain multidimensional values for propositions, but it does strongly suggest that any usable logic of obligations which also concerns itself with the actions actually taken must take into account the interplay between truth-values p and obligation-values O(p). That is: the contradiction arises not because O(¬p) and O(¬q) are inherently contradictory, but because introducing the purely logical ¬p⇒q introduces a contradiction of obligations, in precisely the same way that one observes in the Gentle Murder and Chisholm "paradoxes". That is: in my example above, we should infer
O(¬p), O(¬q), ¬O(p) ⊨ ¬(¬p⇒q)
or possibly instead either
O(¬p), ¬p⇒q ⊨ ¬O(¬q)
O(¬q), ¬p⇒q ⊨ ¬O(¬p).
If we adopt the corresponding purely syntactical developments as rules of inference, the absurdity of the situation would then be manifest by a logical contradiction of the form O(¬p) & ¬O(¬p), or something similar.
One might think to formulate a second-order obligation — ∀p:O(¬[O(p) & O(¬p)]), which states that there should be no catch-22s — no inconsistent obligations. We then observe the absurdities as failings of the system itself to be "morally-satisfiable" (i.e. for all of the obligations to actually be fulfillable). Formulating such a meta-obligation would not prevent the inconsistent obligations from actually arising, however; it would just make the failure slightly starker, because it would represent a transgression of the form A & O(¬A), where in this case A = O(p) & O(¬p).
It seems quite clear to me that in these cases, the absurdities O(p) & O(¬p) arise due to the fact that some non-obligational premise — "you will kill someone" in the Gentle Murder Paradox, "you will not go to the party" in Chisholm's paradox, or "telling the truth will hurt someone's feelings" in my example — is incompatible with the imposed obligations. One might say that the conditional in my own example is somehow substantially different, but I don't see how this helps. Indeed, if we recognise that the material implication ¬p⇒q is equivalent to p v q, we may immediately see that
(p & O(¬p)) v (q & O(¬q))
follows by dilemma. Thus the social transgressions p & O(¬p) which are blatant in Chisholm's and the Gentle Murder paradoxes are all but immediately manifest in my example as well: but it only uses the rule of contractual dependency. This suggests that the moral dimension of propositions, which is introduced on a syntactical level by deontic logic, is accurately represented by factual detachment.
I find it is simply more efficient to recognise that if one wishes to have operators which project propositions to truth-values regarding their social acceptability, it essentially follows that one has constructed a logic in which absurdities of a moral character might arise from statements of fact which transgress the boundaries of social acceptability. Indeed, it is difficult for me to imagine how to avoid this possibility if one wishes to reason simultaneously with propositions p and their obligatoriness O(p) in any way where one could have any bearing on the other.