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Sunday, September 24, 2006

A New Solution to the Knowability Paradox

MOVING TO FRONT FROM SEP. 7

Joe Salerno has an interesting post on the knowability paradox over at Knowability. The knowability paradox is this (this is Jon Kvanvig's take on it). Suppose, for reductio, that all truths are knowable but that not all truths are known. Then there is a truth p, such that p & p is unknown. This truth is knowable. So, assuming that 'know' distributes over conjunction, it is possible that (p is known and it is known that p is unknown). So by the factivity of 'know', it is possible that (p is known and p is unknown). Contradiction. So, it is a theorem that if all truths are knowable, then all truths are known (assuming classical logic). But we also know that if all truths are known, then all truths are knowable. So, it is a theorem of classical (epistemic) logic that all truths are knowable iff all truths are known. QED.

The equivalence is prima facie puzzling. But, Joe argues, Nicholas Rescher's Epistemic Logic (2005) gives us a simple reason to believe that it is a logical truth that there are more truths than knowables. We can think only a countable number of propositions but there are uncountably many truths. So, there is nothing puzzling about the equivalence. Both sides are logical falsehoods.

32 comments:

Mike said...

But wouldn't that conclusion entail that no possible agents could know uncountably many truths? Otherwise the biconditional is not flanked by logical falsehoods. But how do we know that no possible agent could know uncountably many truths?

Brit said...

Hi Mike,
The knowability principle says (roughly) that if p, then it is possible for someone to know that p.

So you're right that if the 'one' in 'someone' may quantify over beings that are quite different from us, then we cannot rule out the possibility that all truths are known.

However, I suspect an anti-realist would impose restrictions on the sorts of beings 'someone' could quantify over. Anti-realists who assent to the knowability principle think that there cannot be truths that we cannot know.

I think what Joe is suggesting is that such a requirement is implausible, given our actual constitution.

I want to raise a related concern, however. The truths in question do not have to be known by exactly one being. All that is required is that for any truth p, some being in the world in question knows p. I wonder whether that is ruled out on logical grounds (broadly construed)? Is it impossible for there to be uncountably many human beings?

Mike said...

Nice. Uncountably many human beings each knowing uniquely one proposition (knowing many others non-uniquely) would do it. Why couldn't there be uncountably many human beings given a history of uncountable length? Or a history of finite length with uncountably many incarnations?

Joe said...

mike and berit,

i'm enjoying your conversation very much. i think that it is inconsistent with the current state of the world that there might be an uncountably infinite number of humans or an uncountably infinite number of propositions thought. Even if it turns out that humans could (as a group or individually) reproduce a countably infinite number of times or think a countably infinite number of propositions, we could never have a situation in which there are an uncountably infinite number of humans or propositions-thought-by-humans. For even if there were right now a countably infinite number of humans, we would get at most a countably infinite number of future humans and at most a countably infinite number of future thoughts-by-humans. And that is because multiplying countable infinities by countable infinities yields at most a countable infinity.

Aidan said...

"However, I suspect an anti-realist would impose restrictions on the sorts of beings 'someone' could quantify over. Anti-realists who assent to the knowability principle think that there cannot be truths that we cannot know.

I think what Joe is suggesting is that such a requirement is implausible, given our actual constitution."

Isn't this going to collapse anti-realism into (a perhaps generalised form of) strict-finitism? The usual brand of anti-realism's surely not going to want restrict decidability or knowability by contingent limitations of memory, available time, mental capacities, etc.

On a different note, there are also some tricky issues about what we want to count as a belief here. I've met people who hold that the deductive consequences of all your occurent beliefs are also beliefs you have. So if your notion of what counts as a belief is as liberal as that, you might hold that one particular person could know an uncountable number of truths in virtue of knowing some axiomatic theory that generates those truths as theorems. But I'm guessing most people will join me in wanting a notion of belief that's more, um, involved than that.

Joe Salerno said...

All of this is worth thinking about.

Mike said...

"we *could never have* a situation in which there are an uncountably infinite number of humans or propositions-thought-by-humans. For even if there were right now a countably infinite number of humans, *we would get* at most a countably infinite number of future humans and at most a countably infinite number of future thoughts-by-humans." (my emphasis).

Joe, I don't see how you arrive at what we *could have* from what *we would get*. Why not consider an uncountably large multiverse containing uncountably many human beings inhabiting separate cosmoi or universes? Of course we can't "get there" from here (supposing that there aren't such universes) but theyv do seem possible. Why assume that the only relevant worlds are those we could physically (i.e., w/o violating physical laws, etc.) arrive at from here? Restricting worlds in this way seems to make the paradox too dependent on controversial metaphysical assumptions.

joe said...

Hi Mike,

You ask, "Why assume that the only relevant worlds are those we could physically (i.e., w/o violating physical laws, etc.) arrive at from here?"

I assume so for the reason that Berit articulated earlier---viz., "Anti-realists who assent to the knowability principle think that there cannot be truths that we cannot know."

Julien Murzi said...

Here's one thing Kvanvig once told me. Of course the realist will think that both WVER --- the standard formulation of the Knowability Principle --- and SVER --- the absurd claim that we get by Fitch's reasoning --- are both metaphysically false. Yet, what's really troubling, is that we can *derive* the latter from the former. Mutatis mutandis, it *may* well be that there are uncountable many other parallel universes. But, arguably, this would be a metaphysically contingent fact. And, as long as it is *possible* that there be only one universe and that all truths are knowable and some truths are unknown, the Church-Fitch result is troubling.

In a paper forthcoming in Joe's anthology, Grahm Priest suggests the following example. Gabriel is an (arc)angel, but he is not omniscient. Thus, for some p,

(i) p and Gabriel will never know that p.

However, Gabriel can always ask God about the truth-value of any proposition whatsoever, and God will be kind enough to give him the correct answer. So,

(ii) all truths can be known by Gabriel.

According to the Church-Fitch reasoning, the scenario above is impossible. For

(iii) all truths are knowable by Gabriel iff all truths will be known by him, soon or later.

If we take the conjunction of (i) and (ii) to be possible, then, it seems, there should be something wrong with (iii).

Mike said...

Joe,

Who do you think inhabits the possible multiverse? I am talking about us and, of course many others like us. Or, if you like, counterparts of us and many others like us. So it is possible that *we* (humans) know every truth. Supposing every world is accessible from every world, that seems to count against the anti-realist claim that there is no world in which it is impossible that we know some truths or, as you put it, "there cannot be truths that *we* cannot know". Maybe you're doubting that there is such a world. But I can't offhand see why. Maybe you're doubting that every world is accessble from every world. But that's a sort of familiar S5 assumption. So I'm not sure where this goes wrong.

joe said...

Thanks Mike. These remarks are helpful. I think we should modify the reason. So,

"Why assume that the only relevant worlds are those we could physically (i.e., w/o violating physical laws, etc.) arrive at from here?"

For the reason that the anti-realists who assent to the knowability principle think that there cannot be truths that the actual world will not permit us to recognize.

Mike said...

Joe, this is difficult to understand,

". . .there cannot be truths that the actual world will not permit us to recognize".

What do you mean 'the actual world will not permit us..'? How does the actual world permit or not permit something? You might have another modal claim in mind. Something like this,

'necessarily, there are no truths-in-@ that we do not recognize-in-@'

But then a truth-in-@ p necessarily has the world-indexed property of not being recognized in @ iff. p actually has the property of not being recognized in @. Modalities collapse. I'm guessing that can't be the anti-realist claim either. So what then is the precise formulation?

Andrew Bacon said...

Regarding the possibility of uncountably many humans, there seems to be a much more straight forward reason to see why this is impossible. Every human occupies a region of space of non-zero volume. Assuming space is like the real numbers cubed, each region will contain a rational point (a theorem of real analysis), therefore there must be countably many humans (or extended objects for that matter). Of course this argument must ignore exotic models of space-time, so I'm not totally convinced by it.

I recently said this on Joe's blog, but the discussion seems to be concentrating on whether there's a possible world in which someone knows uncountably many truths. Surely the claim we need to show is that for any truth (of the uncountably many truths) there is a possible world in which someone knows that truth. This isn't so clearly false.

Andrew Bacon said...

Aidan said: "I've met people who hold that the deductive consequences of all your occurent beliefs are also beliefs you have. So if your notion of what counts as a belief is as liberal as that, you might hold that one particular person could know an uncountable number of truths in virtue of knowing some axiomatic theory that generates those truths as theorems."

I think there should only be countably many deductive consequences of an axiomatic theory (provided we have a finite language).

Aidan said...

Ok, thanks, I wasn't sure of the details.

Mike said...

Andrew,
Interesting argument against uncountably many humans. What about the possiblity of uncountably many instantaneous humans rather than enduring humans?
Supposing that time is dense and each human exists for exactly an instant.

Andrew Bacon said...

If I'm understanding you correctly, you're asking about objects that are temporally one instant wide. I'm not sure what qualifies such things as humans, since they are neither born nor die, they don't think, eat or sleep and importantly I can't see how they could be bearers of knowledge.

Either way, I think there are ways to get round it. Imagine a world with 4 spatial dimensions and a temporal dimension. Then you could have uncountably many copies of our world (say) spread out along the extra dimension getting you all the humans, in their 4D glory, that you need.

Mike said...

Right, yes. The multiverse solution was also suggested above.

Andrew Bacon said...

Ah yes so you did. What about this:

Suppose I know everything, then for any truth p I know p, Kp, KKp, KKKp, KKKKp... Supposing we grant there is nothing wrong with knowing countably many truths or even uncountably many truths, this poses no problems. But similarly for every *set*, X, of truths there is another truth: K(X) - that I know all of the members of X are true. Surely this *is* a problem because of Cantor's theorem (we have as many truths as sets of truths).

To give an example suppose there *were* uncountably many humans. Say Aleph-1 humans each knowing a different proposition. Then there are 2^(Aleph-1) truths about what each set of humans know. So there are more truths than knowns.

This argument has the same sort thrust (that there are more truths than known truths) except I think it fairs better since it doesn't matter how many truths can be known at once. (Also what about sentences like 'this sentence is unknown', if you believe this expresses a proposition at all, then it must be necessarily unknown.)

Mike said...

"But similarly for every *set*, X, of truths there is another truth: K(X) - that I know all of the members of X are true. Surely this *is* a problem because of Cantor's theorem"

It is a problem under the assumption that knowledge iterates in the way you describe for human beings. But no human being--probably no possible human being--that Kp also K1000p or even K100p. So there is not going to be a proposition K100 to know at iteration K101.
You might be talking about an omniscient knowers. Patrick Grim has the same sort of argument about omniscient knowledge.

Andrew Bacon said...

I'll have to look the Grim argument up.

"But no human being--probably no possible human being--that Kp also K1000p or even K100p."

Does this not support the impossibility of all truths being known? If you use 'is known by someone that' instead of K we get similar results (it's not just a problem for one person knowing everything).

Also I don't think it really matters how many times you can iterate K for the argument:

Suppose there are kappa known truths. Then there will be another 2^kappa true propositions. (For each set of known truths we take the proposition to be the set of worlds in which those truths are known.) Therefore there are always more truths than knowns.

Mike said...

"But no human being--probably no possible human being--that Kp also K1000p or even K100p."

Does this not support the impossibility of all truths being known?
-------
No. If at iteration 100, it is not true that K100p, then there is no truth to be known at iteration K101. In short, 'K100p' is false, so it is not among the truths that must be known (in order ot know all truths).

Mike said...

Andrew, you write,
"Suppose there are kappa known truths. Then there will be another 2^kappa true propositions"

I understand that for the set of known propositions S there is a larger power set of propositions, but I'm not sure how you arrive at the conclusion that there are more true propositions to be known. What are the new, unknown propositions in the power set?

Andrew Bacon said...

Re your first point: I was thinking in terms of classical logic, if you're going to run some kind of Sorites argument then sure.

As for the second question, however big the set of known truths is, there is always a bigger set of truths (if you accept my reasoning - or what that your issue?). If there are more truths than known truths then there must be some unknown truths.

Mike said...

I agree that for any finite or infinite set S of propositions p (let those be just the propositions that are known) there is a larger set P(S) of propositions. Right. What I'm not sure about is what the new, unknown propositions are. The power set does not include any new, unknown, propositions. It only contains more and new elements. None of those elements is a new proposition.

Andrew Bacon said...

Ok, I think I understand now. P(S) isn't a set of propositions, it's a set of sets of propositions. The *new* propositions are constructed this way:

let X be in P(S) (so X is a set of known truths) let p be the set of worlds in which the known truths include X's elements. So each element of P(S) has its own unique proposition p [*]. Each p is true since the corresponding set of truths for p, X, is included among the known truths in this world.

So each element of P(S) has its own true proposition. I hope that made sense!

[*] I've omitted showing its unique.

thepoetryman said...

Story of my life... More truths than knowables.
(Or in my case would it be, more knowables than unknown truths?) Great! Add another to both categories! My brain hurts...

Mike said...

"So each element of P(S) has its own unique proposition p [*]. "

So the set, say, {p, q} is a proposition under your description because 'p' and 'q' are themselves just sets of possible worlds and so {p,q} is the set of sets worlds at which 'p' is true and 'q' is true or, say (to keep it simple) {{w},{w'}}. But the set {{w},{w'}} is also not a proposition. It's a set of two propositions, each of which (by hypothesis) is a known proposition. Isn't that right? So there is no new, unknown proposition yet, I don't think.

Andrew Bacon said...

I'm not sure where I said that a set of propositions is also a proposition! I'm treating a proposition as a set of worlds - a set of propositions is just a set of sets of worlds. I'm saying *given* a set of known propositions there is a further true proposition about them. So in your example suppose p and q are known truths. Then for {p, q} we make a new proposition, r, to be the set of worlds in which p and q are known. r is true since the actual world belongs to r (because in the actual world p and q are known). Roughly r says 'the elements of {p, q} are known' (but phrasing it in the way I did prevents the proposition being about sets). I hope this helps - I'm not sure I can add much more to what I've already said.

Mike said...

"Then for {p, q} we make a new proposition, r, to be the set of worlds in which p and q are known. r is true since the actual world belongs to r (because in the actual world p and q are known). Roughly r says 'the elements of {p, q} are known'"

No, that helps. r is the result of some operation on {p, q}. When you make the new proposition r, are you taking the intersection of p and q (i.e., the worlds where the conjunction 'p & q' is true and known--which may not include the actual world) or are you taking the union of worlds where p is true and q is true (where the disjunction is known) or are you taking the worlds where p is known by someone and q is known by someone, but p & q might well be known by no one, or something else? r seems clearly the result of some operation on {p, q}, but I'm not entirely sure which. Thanks.

Andrew Bacon said...

Right, I'll try and write the operation a bit more specifically. Let X be any set of known propositions, let W be the class of worlds and we define sigma(X), the operation in question, as:

sigma(X) := {w in W | for all p in X, p is known (and true) in w}

Andrew Bacon said...

I don't know if that will make any difference - it's just another way of describing what I said above.