Broome and Me*

I want to understand the wide-scoping maneuver and why it fails (if indeed it does). In order to understand this I think I'll need (more) modal logic, but I might also need to understand more about wide-scoping itself.

I think I now understand the intuitive distinction: the "undetachable" O(p->q) is a "normative requirement" which tells you that if you believe p, you ought to believe q. It doesn't mean that p's truth obligates you to believe q.

Intuitively, this does seem to capture what is going on in the miners case: O(A->blA) means that, if we believe that they are in A, we ought to believe we should block A. The truth of their being in A isn't enough to rationally require us to believe we ought to block A, since we might not know where they are, in which case blocking A would be foolhardy.

Here is the paradox in the version which is NOT supposed to be helped by wide-scoping. Part of the key here is that deontically ideal worlds (rel. to a pt. of eval.) are a subset of the epistemically possible worlds (rel. to that pt. of eval.). So MUST(p->q) entails OUGHT(p->q). [Another way of putting this is OUGHT implies CAN. MUST(p) means CANNOT(~p), which entails NOT OUGHT(~p), which entails OUGHT(p).] [Note that this is only true in an unsatisfying sense, though--the unsatisfying sense in which Lincoln ought to have been assassinated, since it is now epistemically necessary that this was so.]

1. O(A -> BlA) Premise (in wide-scoped form)

2. O(B -> BlB) ''

3. M(A v B) Premise: either miners are in A or they are in B

4. M(BlA -> BlO) Blocking A entails blocking one shaft

5. M(BlB -> BlO) ''

6. O(A v B) Must implies ought

7. O(BlA -> BlO) ''

8. O(BlB -> BlO) ''

9. O(BlO) Dilemma, MP, MP [not sure how to write out all the steps that go on inside the scope of the "ought" operator in this kind of notation.]

While turning "ought implies can" into "must implies ought" is a bit fishy, surely the problem here still lies with the fact that step 6 generates only two cases for dilemma. M(A v B), from which (6) is descended, is intended to be taken trivalently: it doesn't decompose into M(A) v M(B). A disjunction may be necessary (or obligatory) without having a necessary (or obligatory) disjunct.

Surely this is the situation that putting "A v B" inside the scope of the O was supposed to prevent!

Could this help to explain why restricting MP is better than using wide-scoping?

***

John Broome, "Normative Requirements" Ratio 1999, XII, 4.

MacFarlane and Kolodny, "Ifs and Oughts," manuscript.