Thursday, November 01, 2012

Knowledge and Modals in Consequents of Conditionals

Modals interact in a characteristic way with conditionals. Suppose it’s next Wednesday morning, and I haven’t checked the news in a while. Consider:
  1. Obama probably won the election.
  2. If Romney won Ohio, Obama probably won the election.
Assuming that the last time I looked at the polls, they looked roughly as they do now, (1) is true in my mouth Wednesday morning, and (2) is false. When I say (1), I say something like, ‘most of the epistemically nearest worlds are worlds in which Obama won’. When I say (2), I restrict the worlds I’m looking at: paying attention only to those worlds in which Romney won Ohio, most of the epistemically nearest of them are Obama-winning worlds. I knew going in that the winner of Ohio is likely to win overall, whichever candidate that is. (But I know it’ll probably be Obama.) So (3) is true in my mouth Wednesday morning:
  1. If Romney won Ohio, Romney probably won the election.
Let’s suppose that as a matter of fact, Romney did win Ohio, contrary to my evidence. Still, since I haven’t gotten the bad news yet, my evidence still favors Obama’s having won the election. So when I say (1), it’s true. So is (3). If we look naively, this will appear puzzling. It looks like a counterexample to modus ponens, for the following are all true (not assertable by me Wednesday morning, but true):
  • Romney won Ohio.
  • If Romney won Ohio, Romney probably won the election.
  • Obama probably won the election.
Call the inference from X and a sentence of the form "if X, Y" to Y, naive modus ponens. Naive modus ponens leads us wrong in this case.

The solution to this puzzle, of course, is that modals and conditionals interact in a subtler way than is recorded in the surface grammar of (3). The ‘probably’ modal takes wide scope; “if p, probably q” says that, restricting attention to the p worlds, q is probable. Relatedly, I can’t perform naive modus tollens on my probability conditional: Obama probably won; if Romney won, then Obama probably didn’t win; therefore, Romney didn’t win.

The same goes for ‘might’ and ‘must’. Suppose I have seen election results for every state except Ohio, and I know for certain that the winner of Ohio won the election. Then I may truly say:
  1. If Romney won Ohio, Romney must have won the election.
  2. If Obama won Ohio, Obama must have won the election.
It doesn’t follow from the fact that Romney did win Ohio that I’d express a truth if I said “Romney must have won the election”. Indeed, it’s false—for all I know, Obama might have won. Sentence (4) says that, of all the Romney-winning-Ohio worlds, he wins the election in them.

This is all very different from the way that conditionals interact with non-modal claims. Suppose I truly say to myself:
  1. If the carpenter was here today, the picture is on the wall.
Suppose also that the carpenter was there then (the place and time where I said (6)). This entails that the picture was on the wall. Or if the picture is not on the wall, the truth of (6) entails that the carpenter wasn’t there. In other words, with non-modal consequents, you can perform naive modus ponens and modus tollens on conditionals.

Knowledge patterns with the modals. Suppose you’re trying to decide whether to trust someone. I might truly say:
  1. If he’s lying, you know he’ll just deny everything later.
This can be true even though (a) he is lying, and (b) you don’t know that he’ll deny everything later. For all you know, he’s honest, and will confirm everything. Indeed, you know that you don’t know he’ll deny everything later. But you can’t reason from this known fact and (7) to the conclusion that he isn’t lying. So naive modus ponens and modus tollens are mistakes here, just as in the cases of the obvious modals like might, must, and probably.

I think this is pretty decent evidence in favor of views like mine that treat ‘knows’ as either something a lot like a modal or a literal instance of a modal. I say, broadly with David Lewis, that ‘knows p’ is an evidential quantifier: it says of a given set of worlds that one’s evidence eliminates all the not-p worlds. When it appears in the consequent of a conditional, it’s very natural to restrict the set with the antecedent. So “If X, S knows p” says, first restrict your attention only to the X worlds; S’s evidence eliminates the not-p worlds that remain.


  1. Sure, but the widescoping approach is not universally good. One can't wide-scope these:

    1) If P, Necessarily P
    2) If P, it ought to be that P
    3) If my wife is cheating on me, I know that she is

    As these all go from false to true. Of course, that does not distinguish 'know' from other modals.

  2. True; there are subtleties at work. My point is that knowledge and the modals should be in the same camp.