Sunday, July 31, 2005

The Mystery of the Disappearing Diamond

Jonathan Kvanvig has an interesting paper on Fitch's knowability paradox in which he argues that the paradox is a puzzle for everyone (not just those who believe that if a proposition is true then it is knowable). The puzzle is one of modal collapse; Fitch's argument shows that:
1. p --> possibly Kp
(if p then it is possible that p is known by some being at some time)
2. p --> Kp
(if p then p is known by some being at some time).
But how come the possibility operator can just disappear like that? Everyone has work to do explaining this, according to Kvanvig.

Thinking about the way the Fitch argument works suggests the following explanation:
Nothing of the form (p and not Kp) is knowable. But given 1, if anything of that form is true, then it is knowable. That's why, if 1 is true, nothing of the form (p and not Kp) is true. And that's why if 1 is true then 2 is true too.

Kvanvig must think there's something unsatisfying about this simple explanation. But what? Do we need further explanation of one or more of the claims it draws upon? I don't think so, but even if we do it could surely be given. More plausible, I think, is the thought that the simple explanation doesn't really explain the disappearance of the possibility operator; it just explains why 2 follows from 1.

If the worry is something like this then it is interesting. What counts as a good explanation of a fact does plausibly depend on (among other things) the way the fact is presented. The thought here would be that presenting the implication of 2 by 1 as a case of surprising modal collapse makes the simple explanation offered above inadequate (even if it is a good-enough explanation of the same fact under a different description).

But I am left with two questions:
A. What exactly is wrong with the simple explanation, considered as an explanation of the modal collapse? What virtue would a satisfying explanation have that this one lacks?
B. If we can answer question A, do we have any reason to think that there will be a good explanation of the modal collapse so described (i.e. an explanation that has the virtue we've identified as lacking in the simple explanation)? Or might it be that the best we can do is explain why 1 implies 2, leaving the fact that 2 takes the form of 1 without the 'possibly' a sort of coincidence?


Joe Salerno said...

Carrie leaves us with an interesting set of questions. I'll just attempt an answer to question A.

"A. What exactly is wrong with the simple explanation, considered as an explanation of the modal collapse? What virtue would a satisfying explanation have that this one lacks?"

The simple explanation reiterates why it is that 'all truths are knowable' entails 'all truths are known'. We might add that the converse entailment is trivial, if 'able' is read in accordance with a reflexive accessibility relation. Taken as an explanation, the proofs of these entailments are deficient in at least one respect. They do not explain the intuition that the believable principle, 'all truths are knowable', is semantically distinct from the unbelievable principle, 'all truths are known'. A satisfying explanation would tell us why it is that the apparent non-equivalence is so intuitive.

Carrie Jenkins said...

"A satisfying explanation would tell us why it is that the apparent non-equivalence is so intuitive."

Thanks for the comment, Joe. That's an interesting point. I think one thing that's behind the intuition you mention is that when we (or at any rate our innocent, pre-Fitch selves) hear 'all true propositions are knowable' we just don't think about true propositions of the form (p and not Kp). They don't exactly spring to mind. Yet according to the simple explanation, it is attention to these cases that reveals why 1 and 2 are equivalent (or rather, why the surprising direction of the equivalence holds). But if lack of attention to the relevant cases is all that's behind the intuition that 1 and 2 are not equivalent, then something like the simple explanation - supplemented, perhaps, with the claim that propositions of the form (p and not Kp) don't immediately spring to mind when we consider general claims about all true propositions - might do enough to explain the strength of that intuition.

Another point that I think might do some work here is that philosophers have tried to use 'all true propositions are knowable' to express a claim about the epistemic accessibility of reality, and that accessibility claim intuitively does not entail that all true propositions are known. But, as I argue here:
the epistemic accessibility claim is not well expressed by 'all true propositions are knowable'. So one explanation of the strength of the intuition among philosophers is that they mistakenly hear 'all truths are knowable' as expressing a claim which genuinely does not imply that all truths are known. said...

I just read your "AR and Epistemic Accessibility" and I wondered what you would reply to the following objection:
Your argument that the consequent of (5*) does not contradict (4) seems to rely on the implicit quantification(over beings and times) in Kp. If this is the only way out, we could still argue that "it is possible that I recognize tomorrow at 10am that [p] obtains" implies "tomorrow at 10am I will recognize that [p] obtains", since these claims do not contain any quantifiers. But isn't that equally unsatisfying?

Carrie Jenkins said...

Hi Kim,

Thanks for an interesting comment. Is your point that if we assume:
(a). p --> possibly I recognise [p] at 10am tomorrow,
we can infer (by Fitch-like reasoning):
(b). p --> I will recognise [p] at 10am tomorrow?

If so, I'm not sure I'm seeing how it's an objection to what I say in the paper. There are certainly lots of things in the *vicinity* of WVER* which would generate Fitch-like worries if we accepted them. My point is just that we don't need to accept anything that generates Fitch-like worries in order to accept the anti-realist claim that reality is epistemically accessible, since that claim is properly expressed by WVER* (which, according to me at least, is not Fitchable). I take it that most of the people who believe that reality is epistemically accessible will actually reject (a), so won't be specially worried by the fact that it is Fitchable.

Perhaps it would help to stress that my AR&EA paper is not supposed to address the questions, recently raised by Kvanvig, that I was discussing in this post. The Fitchability of (a), and indeed that of the original (p --> possibly Kp), surely do raise interesting questions for everyone of the kind Kvanvig identifies. But I wasn't trying to address them in that paper - all I was trying to argue was that Fitch-style worries present no *special* problems for the anti-realist.

Does that help? said...

> Is your point that if we >assume:
> (a). p --> possibly I recognise
> [p] at 10am tomorrow,
> we can infer (by Fitch-like
> reasoning):
> (b). p --> I will recognise [p] > at 10am tomorrow?
Yes, that's my point, I just realized I didn't express it as clearly as I should have.
As I understood you, you want to say that "p-><>Kp" isn't the right formalisation of the anti-realist claim that "reality is epistemically accessible". Then you go on trying to say where exactly the mistake lies.
Surely, the same mistake would be made in formalising "p is epistemically accessible by me at 10am" as "p-><>Ktmp"(where "Ktmp" means m knows p at t). If the same mistake is made, we should be able to apply the same solution to it. Otherwise we couldn't really say we solved the problem. And that's why I think it's an objection to your account, because even on your account (a) is fitchable.
Thanks for your time, I hope this is more understandable.

Carrie Jenkins said...

Hi Kim,

I'm still not sure I'm seeing a problem; I'm happy to allow that (a) is fitchable. In my paper I'm not proposing a solution that could be applied to every kind of Fitch-like problem, just one that gets mind-dependence anti-realists out of a mess.

I haven't really thought about what it would be for reality (or some chunk of it) to be epistemically accessible *by me at 10am tomorrow*, but I guess the best way to formalize that idea is as: p --> possibly I recognise [p] at 10am tomorrow. The fact that this is Fitchable is interesting, in the same way that the fact that the original p --> possibly Kp is Fitchable is interesting. But neither of these facts seems to be a particular problem for the mind-dependence anti-realist I was trying to defend, since this anti-realist wouldn't defend either claim.

Maybe this'll help explain where I'm coming from. The way I see it, *one* of the things that's wrong with accepting that 'all truths are epistemically accessible by me at 10am tomorrow' and then formalizing this as 'p --> possibly I know p at 10am tomorrow' is that really what you mean is 'p --> possibly I recognise [p] at 10am tomorrow'. But another thing that's wrong with it is that it's just plain false, even when properly formulated. Fitch-like reasoning is one thing that can be used to show that it's false, as you note, but since we already thought it was false anyway, this isn't specially worrying, though it is still interesting (for the same kind of reason as it's interesting that p --> Kp follows from p --> possibly Kp). I think you're right that an analogue of my proposal can't rescue the claim that reality is epistemically accessible to me at 10am tomorrow from Fitch-like problems. But I don't want to rescue it - I'm only interested in rescuing anti-realism. said...

Hi Carrie,

okay, this is probably my last post on this topic, I think I'm getting on your nerves. ;-)
I agree that (a) is plain false and I agree that we all knew that before and I agree that the antirealist would not claim (a). I also know that you only want to rescue AR.
My worry was that a satisfactory defense of AR should not only rescue the knowability principle from triviality, but offer a general explaination of what goes wrong in fitch-like cases, including the case of (a). I think this is the main point of disagreement between us and I don't know how to argue for my view.
Thx for the illuminating discussion.

Carrie Jenkins said...

"I think I'm getting on your nerves. ;-)"

Not at all! It's been really interesting. I don't actually think anything 'goes wrong' in Fitch-like cases, but I think you and Kvanvig are right that a (totally general) explanation of why it *feels* like something's gone wrong would be a good thing to be able to provide. But I don't think that providing one is part of defending AR, since (in my view) AR amounts to a claim that *isn't* Fitchable. So addressing the issues surrounding Fitchable things seems tangential to a defence of AR.