against monotonic deontic orderings

February 2, 2009 in Ethics, Language, Philosophy, Semantics

Following up on the earlier post, I think we can pinpoint the source of the version of Kolodny and MacFarlane’s “paradox” that afflicts state of the art accounts of the conditional a little bit better.

Consider a deontic ordering on worlds ≤. Let w be ideal in a set of worlds I iff for all v in I: if v ≤ w, then w ≤ v.

Let a deontic ordering ≤ be monotonic[*] just in case for any two sets of worlds I and I‘ such that I ⊆ I‘ and any w: if w in I and w is ideal in I’, then w is ideal in I. Monotonicity is a presupposition of traditional definitions of deontic orderings (e.g., Kratzer’s). It is, I think, fairly intuitive: if a world is ideal in a set, then reducing its competition (shrinking the set) should not diminish its rank.

Monotonicity causes problems in cases like the following (from Kolodny and MacFarlane’s “Ifs and Oughts“).

Ten miners are trapped either in shaft A or in shaft B, but we don’t know which. Flood waters threaten to flood the shafts. We have enough sandbags to block one shaft, but not both. If we block one shaft, all the water will go into the other shaft, killing any miners inside it. If we block neither shaft, both shafts will fill halfway with water, and just one miner, the lowest in the shaft, will be killed (“Ifs and Oughts”, p. 1).

It seems like all the following should be consistent (indeed, most people are probably inclined to endorse all three, given a description of the case).

• (if in A)(ought block A) (if they’re in A, we ought to block A)
• (if in B)(ought block B) (if they’re in B, we ought to block B)
• not-ought(block A or block B) (it’s not the case that: we should: block A or block B)

Assuming monotonicity and a Kratzer semantics for conditionals with modalized consequents, however, they are inconsistent.

Kratzer semantics for conditionals with modalized consequents:

• c is a context.
• C is a c-supplied function from worlds to sets of worlds. C is the modal base.
• C+p is a function mapping a world i into the set of p-worlds in C(i).
• d is a c-supplied deontic selection function, mapping from sets of worlds into their subsets. d selects the ideal worlds from a set of worlds relative to a c-supplied deontic ordering.
• (if p)(ought q) is true at <c,i> iff d(C+p(i)) ⊆ {j: j satisfies q}

Suppose p and p’ partition C(i), and suppose each of the following is true at <c,i>.

• (if p)(ought q)
• (if p‘)(ought q‘)
• not-ought(q or q‘)

Since (if p)(ought q) is true at <c,i>, the ideal worlds in C+p(i) are all q worlds.

Since (if p’)(ought q’) is true at <c,i>, the ideal worlds in C+p’(i) are all q’ worlds.

Since not-ought(q or q‘) is true at <c,i>, there are ideal worlds in C(i) where both q and q’ are false.

Assuming monotonicity, since p and p’ partition C(i), there are either ideal worlds in C+p(i) where both q and q’ are false or either ideal worlds in C+p’(i) where both q and q’ are false. It follows that either (if p)(ought q) is false at <c,i> or (if p’)(ought q‘) is false at <c,i>.

I believe this result generalizes to context-shifty semantic accounts of the conditional like Thony Gillies’. In other words, the problem seems fully general. The lesson is that the culprit isn’t the semantics for the conditional. Rather, it’s assuming that the deontic ordering on worlds is monotonic.

[*] Note on terminology. I call the ordering monotonic because the fact that standardly defined deontic orderings are monotonic follows from the monotonicity of the (partial) identity function from I‘ to I with respect to ≤.