September 06, 2006

The Logical Necessity of Some Ignorance

Modest modal epistemic reasoning reveals the equivalence of the following two principles:

(1) Any truth can be known.

(2) All truths are known.

Jon Kvanvig, in his latest book The Knowability Paradox (2006), poses a challenge to anyone who accepts the validity of the reasoning---viz., explain the loss of the apparent logical distinction between these two principles. Here's one way to go.

It is logically necessary that each of the above propositions is false. Logically necessary propositions often appear to express different thoughts, especially if one involves a concept not involved in the other.

Why think that the above propositions are necessarily false? An appendix to Nicholas Rescher's Epistemic Logic (2005) inspires an answer. It is a logical fact that there are more truths than knowables. Knowledge requires thought, and we could at most think a countable number of propositions. However, the true propositions themselves are uncountable. A diagonalization argument is required to make this stick, but it shouldn't be difficult to construct one for, say, a class of truths about the rational numbers. So if (i) there are more truths about the rational numbers than things that can be known about the rational numbers and (ii) the proof of this rests on no non-logical facts, then it is logically necessary that there is an unknown truth. (2) is logically false. And by the same reasoning so is (1).

The apparent logical distinction is explained by the fact that (2) seems on first glance to be stronger than (1). But the appearance is the result of not immediately recognizing that both propositions are logical falsehoods.


Aidan said...

My copy of Rescher's at the office, so I can't check just now. I'm not seeing how there are no logical facts involved in showing (i) - that there are more truths about the rationals than things we could know about them. In what way is it a logical fact that we could as most think a countable number of propositions? (Obviously I agree that this is in fact the case! But what's the case for holding this to be so as a matter of logic?).

Secondly, in her review of Kvanvig's book Carrie questions his contention that there's a logical equivalence between (1) and (2), because of the reliance on factivity and distribution. If there's something to that thought, we might worry you've explained too much; you've explained a logical equivalence where there isn't one. (Sorry if this is all too inchoate to be of any use. But I guess we're blogging after all, so that's always a risk).

Joe said...

If factivity and distributivity are not logically necessary but rather are metaphysically necessary, then that's fine. The problem and the corresponding required explanation can be reformulated, mutatis mutandis. The problem becomes: why is it that an apparently plausible brand of anti-realism stands or falls with an obviously false brand of idealism. The corresponding answer becomes: they are both metaphysically impossible.

Notice that the shift from logical to metaphysical modality facilitates an answer to your first question. The task now is to argue that it is metaphysically impossible (rather than logically impossible) for humans to know an uncountable number of propositions.

Bryan Frances said...

Hi Joe,

I enjoy your blog. Berit's too.

I don't see why we can't know uncountably many truths. Why can't I know uncountably many truths about the uncountably many rational numbers? It seems to me that I actually know uncountably many truths (setting aside radical skepticism, which I take much more seriously than, well, just about anybody these days).

Also, one might say that due to semantic indecision facts when I seem to know just one thing, e.g., that the cat is on the mat, I actually know uncountably many truths because uncountably many truths are of maximally good fit with my use of 'The cat is on the mat'. Well, I am not sure how to get to uncountably many, but maybe we can if space is sufficiently dense?

Aidan said...

Aren't the rationals countable?

Bryan Frances said...

oops. use irrationals.

joe said...

ditto for my post.

Andrew Bacon said...

Hi - I'm afraid I'm a bit of a newbie to philosophy (I'm only an undergrad) so you'll have to bear with me.

I can see why the metaphysical possibility of knowing uncountably many truths all in one go seems implausible. But it seems to me we only need to show that for any particular truth, that it is possible to know that truth. Why is that implausible?

Perhaps you could argue, supposing I am functionally equivalent to some Turing machine T, that T can only compute countably many functions out of cantor space (whichever possible world its in). But surely it would be possible that I change which Turing machine I implement (say I got a bash on the head), technically I could cover all functions this way. Or perhaps you could argue that there are only countably many sentences, but similarly why couldn't it be possible that I learn a new language which, although only expressing countably many truths, expresses a *different* countable subset of all the truths - enough to cover the truth I want.

Richard Zach said...

I think I'm with Andrew on the question of how many knowable truths there are. I seem to recall--but couldn't find a relevant passage--that Chihara makes this point somewhere. In Chihara's system (Ontology and the Viscious Circle Principle, Constructibility and Mathematical Existence), the basic primitive notion is a constructibility quantifier for open sentences ("It is possible to construct an open sentence..."), and you have a similar question about how many open sentences it is possible to construct. Maybe the point is already in Quine and Goodman's "Steps towards a constructive nominalism"?