Mostly inspired by “some”:http://semanticsarchive.net/Archive/TI1OGVlY/iffiness.pdf “papers”:http://www-personal.umich.edu/~thony/note_on_if.pdf by Thony Gillies, I’ve been thinking a lot lately about the scope of modals inside conditionals. I started thinking about these again because of some issues that come up in a paper that John MacFarlane and Niko Kolodny are presenting at the “AOC”:http://aoc.web.arizona.edu/. But that paper isn’t online, so I won’t discuss (yet) the issues they raise, which are largely about deontic modals. Instead I want to go over a puzzle about disjunction that challenges a position I’m attracted to.

The basic setup of the puzzle comes from Thony. Imagine that I’ve lose a marble. The three places in my apartment that marbles go under are the sofa, the table and the desk. I’ve looked under the table, and the marble isn’t there. It seems I can truly say (1).

(1) if the marble isn’t under the desk, it must be under the sofa.

But it also seems I can truly say.

(2) It’s not the case that the marble must be under the sofa (since it might be under the desk).

And from these, by modus tollens, it would seem to follow that the marble is under the desk. But I’m clearly in no position to conclude that. What has gone wrong? There are three possible options that one could take here.

• Conclude that modus tollens is invalid.
• Conclude that the logical form of (1) is not as it appears, and really the modal ‘must’ takes wide scope over the conditional.
• Conclude that there is an equivocation between (1) and (2).

The first option seems desperate, particularly since we can use a similar argument to show that modus ponens is invalid. Obviously some people (e.g. McGee, Lycan) have rejected modus ponens, and for reasons not a million miles from what we’re considering here. But it seems like a move of last resort to me. By the end of this post we’ll come to something that might look like a new cost of the position.

The second option is, however, a very popular position. I used to believe it, but I’ve been convinced by Thony that it isn’t true. The problem concerns what happens with weak modals. So (3) seems to be false.

(3) If the marble isn’t under the desk, it might be under the table.

That’s false because I looked under the table, and I know it isn’t there. But let’s say that the modal ‘might’ takes wide scope, and the conditional is a material implication. So (3) is equivalent to (4).

(4) It might be the case that (the marble is under the desk or the table).

But (4) is true, and we wanted (3) to be false. Perhaps the problem isn’t the assumption that the modal in (3) takes wide scope, but that the conditional is a material implication. So let’s assume that the embedded conditional is some kind of epistemic conditional. (I.e. if p, q is roughly equivalent to must (not p or q).) Then we have a different problem. Imagine that I said (3) before I had looked under the table. Then it would be intuitively true. But (5) on this reading is arguably false.

(5) It might be the case that (if the marble isn’t under the desk it is under the table).

That’s arguably false because at that stage I know that _must (marble is under desk or table)_ is false. It’s starting to look like the problem is that ‘might’ takes wide scope, not the interpretation of the conditional. As I mentioned in “a previous post”:http://tar.weatherson.org/2007/11/08/gillies-on-wide-scopism/ these problems are even worse when we use deontic modals. So I think the wide scope solution is flawed.

That leaves us with the equivocation solution. Here’s the version of the equivocation story I prefer. (It’s basically a translation into static semantics of the dynamic semantics story Thony likes.) Epistemic modals have an attached plural variable. In standard settings the values of that variable are those propositions known, or perhaps available to be known, in the conversation. (There are some relativist bells and whistles I like here, but I’m ignoring those for this post.) Roughly, _Must (X) p_ is true iff some of the X collectively entail p, and _Might (X) p_ is true iff p is consistent with the X.

The complication comes in conditionals, particularly in the consequent of conditionals. In that case, I think the X is ‘partially bound’. All the propositions that are known are among the X, but so is the antecedent. So the LF of (1) is something like

(1a) If (marble not under desk), must (Y) (marble under sofa)

where the Y are all the propositions known plus the proposition that the marble isn’t under the desk. And the LF of (3) is something like

(3) If (marble not under desk), might (Y) (marble under table)

where Y is as above. This gives all the right results in these cases, and does so in something like a systematic way.

Now for the problem. Epistemic modals also behave oddly in the second disjunct of disjunctions. So both (6) and (7) sound OK to me.

(6) Either the marble is under the desk or it must be under the sofa.
(7) Either the marble is under the desk or else it must be under the sofa.

I think (7) is a little preferable to (6), a fact that will become a little important in what follows. Both (6) and (7) can be combined, it seems, with (2) to conclude (improperly) that the marble isn’t under the desk. Again we have three options.

• Conclude that disjunctive syllogism is invalid.
• Conclude that the modal in (6)/(7) takes wide scope.
• Conclude that there is an equivocation between (6)/(7) and (2).

Again, both of the first two options seem unhappy. Indeed, denying disjunctive syllogism for natural language ‘or’ seems to be an even more radical step than denying it for natural language ‘if’. (Various heterodox logicians might disagree here, but debating that would take us too far afield.) And the wide-scope move looks just disastrous, since it would make (8) acceptable.

(8) Either the marble is under the desk or it might be under the table.

So we need some kind of equivocation story. Following some (long) conversations with Ishani, I’m inclined to believe the following story.

Start with (7). Arguably the ‘else’ there means ‘if not’. In general, I think, it seems fine to analyse (9) as (10).

(9) p or else q.
(10) p or, if not p, q.

If ‘if’ means material implication, then (10) is just equivalent to _p or q_. But we might not read ‘if’ that way. In particular, we might say that modals in the consequents of conditionals have partially bound variables. So we might read (7) as (11), which we in turn analyse as (12).

(11) Either the marble is under the desk or, if not, it must be under the sofa.
(12) (Marble under desk) or (if (not under desk), must (Y) (under sofa)).

Again, Y consists of those propositions we know plus the antecedent, i.e. that marble isn’t under the desk. So we know the second disjunct is true, so the disjunction is known, so it is assertable. (And note that we don’t have the normal Gricean problems with redundant disjuncts because the first disjunct is a constituent of the second disjunct.)

So far so good. But what should we say about (6)? Well, I think we should say much the same thing. More precisely, I think we should say that there is an unpronounced ‘else’ in (6), and that ‘else’ means ‘if not’, and then the analysis is as for (7). The reason that (6) is a little less happy than (7) is that it only makes sense if we read in this unpronounced element, and the speaker should probably have pronounced this for us. The reason that we can read this element in is that in general _p or q_ and _p or else q_ are so close in meaning that we can freely substitute one for the other.

Note that this story is available to people who think that the problem with the argument from (1) and (2) is that modus tollens is invalid. Such people should think that the argument from (7) and (2) to the conclusion that the marble is not under the desk is not, strictly speaking, an instance of disjunctive syllogism. Rather, it requires the use both of modus tollens and disjunctive syllogism. And such people say that modus tollens is invalid. No wonder the argument fails!

But there is something a little odd about this position. If modus ponens is invalid, then it is possible for _p or else q_ to be true while _p or q_ is false. That’s a surprising, and I think a little unhappy, result. I much prefer the equivocation story.

Coming later: The same arguments run through with deontic modals rather than epistemic modals.