What we don't need to save ordinary conditionals from

It’s been a while since I posted anything. Mostly this is because life has been pretty hectic lately. In September we spent a few weeks in Gdańsk, but now we’re moving every few weeks between various places in the UK, visiting different universities and trying to get some research done meanwhile. This semester I’m mostly based in Bristol as a British Academy Visiting Fellow, working with Oystein Linnebo on the dynamic approach to abstraction principles, and doing some directed reading (on groundedness with Hannes Leitgeb and on axiomatic theories of truth with Leon Horsten). These days, however, we’re hanging out in Scotland, currently visiting Arche Research Centre in St. Andrews, taking off for a few other places tomorrow.

I gave a talk here on Tuesday about nominalistic approaches to neologicism, and decided to stick around for the Arche/CSMN graduate conference. Today, I managed to catch a talk by Ernest Lepore followed by an interesting talk about counterfactuals by Daniel Berntson (with a commentary by Guðmundur Andri Hjálmarsson). Daniel’s talk was titled Saving Ordinary Counterfactuals and was devoted to the problems that quantum indeterminacy (or related phenomena) are supposed to raise for our intuitions about ordinary counterfactuals. The whole thing was very clear and quite interesting. I do have one minor worry, though - I don’t think the problem that Daniel is trying to address exists... Counterfactually: if there were such a problem, Daniel’s approach would be a neat way to approach it. But let’s start from the beginning...

The problem

Intuitively we accept the counterfactual:

(1) If I were to throw a champagne glass off the top of Empire State Building, it would break.

Supposedly, quantum mechanics informs us also that:

(2) There is some chance that a glass thrown off the top of the Empire State Building will quantum tunnel to the moon without breaking.

If indeterminism is true and (2) expresses objective probability, (2) seems to entail (3):

(3) If I were to throw a champagne glass of the top of the Empire State Building, it might safely quantum tunnel to the moon.

This entails:

(4) If I were to throw a champagne glass off the Empire State Building, it might not break.

Now, Daniel suggests that "(4) puts pressure on us to give up (1)" and that there is an "inescapable clash" in the infelicitous assertion (5):

(5) If I were to throw a champagne glass off the Empire State building, it would break; and furthermore, it might not break.

The strategy

In order to save the truth of (1) within the Lewis-Stalnaker approach, Daniel suggests replacing:

A>B iff all of the closest A-worlds are B-worlds.

with

A>B iff the vast majority of the closest A-worlds are B-worlds.

The underlying idea now would be that (1) is made true because most of the closest worlds where the glass is dropped are worlds where it is broken, whereas (4) emphasizes that not all closest possible worlds are worlds where the glass is broken.

Say we put aside the issue of how we are to count possible worlds and they ratios if there are infinitely many of those. There still are some problems that come along with this solution. Most prominently, agglomeration (A>B, A>C hence A>B&C) and transitivity (A>B, A^B>C hence A>C) fail. To fix these issues, Daniel introduces the notion of being almost true, and says that certain claims, even though they aren’t strictly speaking true on this semantics, are still almost true, like when we have a counterfactual which doesn’t preserve probability ratio, but whose consequent is only slightly less probable than the antecedent. There are some bells and whistles to play around with here, but this should be enough for the set-up.

The worry

First, observe that a necessary (but not obviously sufficient) condition for thinking that (5) is a problem is the acceptance of (1) and (4). Strictly speaking, so far Daniel has shown how to preserve the truth of (1), but didn’t say explicitly how to make sense of (4).

In fact, Daniel introduces might-conditionals by saying:

A >m> B iff ~(A>~B)

That is, a might conditional A >m> B is supposed to come out true iff it is not the case that ~B is true in the vast majority of the closest A-worlds.

Alas, this reading of might-conditionals doesn’t support the truth of (4), because given that all the relevant worlds where the glass is broken are worlds where it is not the case that it tunnels safely to the moon, (4) still comes out false if it is to be read as a might-conditional.

Second, it’s not clear where the clash really is. I would certainly be worried if I had intuitive reasons to believe:

(5’) If I were to throw a champagne glass off the Empire State building, it would break; and furthermore, it wouldn’t break.

But so far, I don’t. (5) certainly doesn’t entail (5’). Given that (4) shouldn’t be constructed as a might-counterfactual if its truth is to be preserved, what job exactly is "might" doing in (4) and (5)?

Well, I’m inclined to say that that are at least two ways to accept (1), (4) and (5) even if we play along with non-probabilistic Lewis-Stalnaker semantics.

Story 1. (1) says that in every closest possible world, where I drop the glass, it’s broken, whereas (4) says that there are still some accessible but less similar worlds, where even though the glass is dropped, it’s not broken.

Story 2 (1) says that in every closest possible world, where I drop the glass, it’s broken, and (4) says that it is possible that the glass is dropped and not broken.

Personally, I prefer story 2, because it assigns less content to "might", and because (5) as it is, when read along the reading suggested in story 2, entails that the world where the glass is broken is not the closest one anyway.

I don’t even have the intuitions that (5) displays any sort of clash to start with. Imagine I say:

If I were to join you for conference drinks tonight, I would be hungover the next day. Well, in fact, it’s highly unlikely that I would decide to drink only water, in which case I might feel good the next day even if I go. So if I were to join you for conference drinks tonight, I might feel good the next day, but I think I wouldn’t.

This doesn't seem to contain any contradiction whatsoever.

Interestingly, we did a poll and around half of the audience thought that (5) was problematic, and around half that it wasn’t.

Comments

Tuomas said…
Just to add to your poll, I'm in agreement with you about (5) not being particularly problematic. When we say things like (1) I think that we don't generally mean that the consequent is absolutely necessary.
Rafal Urbaniak said…
Actually, I just chatted about these things with Hannes and he suggests there is a 2005 paper which argues that the way I indicated leads to certain troubles. I have to dig the paper up and take a look at it, stay tuned.:)