According to the knowability principle, if a proposition is true then it is knowable. Therefore, if a proposition is not knowable then it is not true. — Michael

It seems to me though, that the knowability principle ought to apply equally to 'unknown truths' and 'unknown falsehoods'. A false proposition is the mirror image of a true one: its negation.

If p --> possibility of Kp, then non p --> possibility of Knonp

In order to get this paradox you do have to think of truth as something that can be taken as universal, not something contingent on a perspective.

If you follow Berkeley on "to be is to be perceived," (at least as far a knowledge is concerned) then I don't think you have an issue. The truth of propositions like "no one knows that Theseus is standing" cannot be perceived, as the perception of said truth entails that the knower does, in fact, know that Theseus is standing (the paradox in a universalist view). But if to be is to be perceived then this imperceivable "truth" isn't true, since truths presumably cannot lack being.

This would indeed entail that "all truths are known," but rather than being a paradox it is simply trivial, a result of the ontology.

For people who don't buy into those sorts of Berkelean arguments about being this might seem facile, but consider that, if being can exist outside perception, that would entail that you're committed to "truths that cannot be perceived." But if you have truths that cannot be perceived then clearly "all truths are knowable," cannot obtain. The difficulty I see for this position is this: what difference can any necessarily unknowable truth ever make to anyone? It seems like a totally extraneous ontological entity that can't do any lifting.

Now, Berkeley would say all truths are known because God knows them (God is always a big help at resolving issues). Hegel would have the paradox driving the engine of the dialectical and progress towards the Absolute. In a Hegelian system the two truths result in a new entity, a world where "no one knew (past tense)" that Thesus is standing, but things have progressed and now someone does know -> being into becoming. The being of both truths creates a contradiction, the becoming of our world has one proposition pass into a present tense. This could be formalized nicely, but instead we're more likely to get a page long run on sentence about how this is the progression of Spirit (or some shit like that, smart guy, not the easiest style). These sorts of contradictions then are what drive the process of becoming that we exist in, as opposed to static "being" which is also a contradiction.


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On a side note: You can get the same paradox to show up with truthmakers thrown in:

-Theseus is standing. (Truthmaker: Theseus standing)
-No one knows Theseus is standing. (Truthmaker: everyone's lack of knowledge of the fact that Theseus is standing, presumably Theseus as well, perhaps he is asleep)
-Persumably, knowledge can only be of true things. "No one knows the Earth is flat," would not cause this paradox if the world is actually round.
-Thus, the truthmaker for "no one knows Theseus is standing" relies on the very same truthmaker as "Theseus is standing," plus an added truthmaker about the state of knowledge relative to said truthmaker amongst all entities. The paradox emerges from this sharing of a single truthmaker.

Just a different way to view the same problem but I think some may find it more intuitive.

You can see how this isn't an issue with a Berkeley inspired system because the truthmaker for the thing no one knows about doesn't exist (granted, a sleeping man is a bad example here because people arguably still have perceptions while asleep).

Isn't the unknown truth "p & ~Kp" both knowable and unknowable, according to the argument? — Luke

No. Line 3 of the SEP proof asserts that "p & ~Kp" is knowable, i.e., "<>K(p & ~Kp)". "<>K(p & ~Kp)" is then subsequently proved to be false. Therefore "p & ~Kp" is not knowable. As the comment after Line 3 says:

However, it can be shown independently that it is impossible to know this conjunction. Line 3 is false. — 2. The Paradox of Knowability - SEP

No. Line 3 of the SEP proof asserts that "p & ~Kp" is knowable, i.e., "<>K(p & ~Kp)". "<>K(p & ~Kp)" is then subsequently proved to be false. Therefore "p & ~Kp" is not knowable. — Andrew M

In that case there would be no contradiction, but as the SEP proof asserts:

Line 9 contradicts line 3. So a contradiction follows from KP and NonO. — 2. The Paradox of Knowability - SEP

In that case there would be no contradiction, but as the SEP proof asserts:

Line 9 contradicts line 3. So a contradiction follows from KP and NonO.
— 2. The Paradox of Knowability - SEP — Luke

The contradiction means that one of the premises is false (KP or NonO). Not that "p & ~Kp" is both knowable and unknowable.

If KP is false, "p & ~Kp" can be true but not knowable. If NonO is false, "p & ~Kp" is never true and so also not knowable.

The contradiction means that one of the premises is false (KP or NonO). Not that "p & ~Kp" is both knowable and unknowable. — Andrew M

But there is no contradiction unless “p & ~Kp” is both knowable and unknowable.

But there is no contradiction unless “p & ~Kp” is both knowable and unknowable. — Luke

Fitch's paradox shows that a contradiction follows from KP and NonO. Per the law of non-contradiction, contradictions are false. Thus it's false that "p & ~Kp" is both knowable and unknowable. So we need to reject at least one of KP or NonO, not conclude that the contradiction is true.

The reason is that knowing "p & ~Kp" would entail knowing p and also not knowing p which is impossible.. — Andrew M

That still seems wrong to me. The proposition is an assumption or stipulation: let's assume or stipulate that p and that we don't know p. There doesn't seem to be any problem with that until what seems like the absurd idea of "knowing" (the truth of, presumably) that proposition is introduced.

The alternative I proposed:

Is the truth of the proposition that there are unknowable propositions itself unknowable? We might want to say that it is, because if there are unknowable propositions then we could never know there are, just because they are unknowable.

But then it would follow that there is at least one unknowable truth, that it is unknowable as to whether there are unknowable truths; and that is a contradiction, because it would also follow that we know that there is at least one unknowable truth. — Janus

Does seem to show that we do know that there is at least one unknowable truth; that it is unknowable as to whether there are unknowable truths, although I was wrong above to say that is a contradiction, because we are not knowing an unknowable truth but the knowable truth that there is at least one unknowable truth.

Fitch's paradox shows that a contradiction follows from KP and NonO. — Andrew M

If the contradiction is not that “p & ~Kp” is both knowable and unknowable, then what is the contradiction?

Line 9 contradicts line 3. So a contradiction follows from KP and NonO. — 2. The Paradox of Knowability - SEP

That still seems wrong to me. The proposition is an assumption or stipulation: let's assume or stipulate that p and that we don't know p. — Janus

It's not merely an assumption or stipulation though, it's the justifiable proposition that there is some particular truth that isn't presently known. That can be anything from Goldbach's conjecture to whether there's any milk left in the fridge (assuming no-one knows that).

There doesn't seem to be any problem with that until what seems like the absurd idea of "knowing" (the truth of, presumably) that proposition is introduced. — Janus

The consequence, though, is that either the knowability principle or non-omniscience has to be given up. That's a problem for philosophical positions that assume those two principles. From SEP:

Timothy Williamson (2000b) says the knowability paradox is not a paradox; it’s an “embarrassment”––an embarrassment to various brands of antirealism that have long overlooked a simple counterexample. — Fitch’s Paradox of Knowability - SEP

The alternative I proposed:

Is the truth of the proposition that there are unknowable propositions itself unknowable? We might want to say that it is, because if there are unknowable propositions then we could never know there are, just because they are unknowable.

But then it would follow that there is at least one unknowable truth, that it is unknowable as to whether there are unknowable truths; and that is a contradiction, because it would also follow that we know that there is at least one unknowable truth.
— Janus

Does seem to show that we do know that there is at least one unknowable truth; that it is unknowable as to whether there are unknowable truths, although I was wrong above to say that is a contradiction, because we are not knowing an unknowable truth but the knowable truth that there is at least one unknowable truth. — Janus

I agree with your conclusion, but not your argument. First, we already know there are unknowable truths via Fitch's proof (and that we're not omniscient). Second, if we didn't have that proof (or others that I may not be aware of), then we wouldn't know whether there were unknowable truths or not.

If the contradiction is not that “p & ~Kp” is both knowable and unknowable, then what is the contradiction? — Luke

That's the contradiction. However it's not true that a proposition can be both knowable and unknowable is it?

That's the contradiction. However it's not true that a proposition can be both knowable and unknowable is it? — Andrew M

Right, but neither should the contradiction imply that “p & ~Kp” is necessarily unknowable. If the contradiction is false, then “p & ~Kp” is either knowable or unknowable.

If we accept that an unknown truth is knowable, that seems almost trivially true.

It is only if we reject that triviality and accept that an unknown truth is unknowable that the seemingly absurd result follows that all truths are known.

But upon reflection, it doesn’t seem so absurd. The reason it would be impossible to come to know an unknown truth is because there are no further unknown truths to know; because all truths are (already) known.

I still cannot get the move from unknown truth to unknowable truth in the argument.

In any case:

Second, if we didn't have that proof (or others that I may not be aware of), then we wouldn't know whether there were unknowable truths or not. — Andrew M

That may be true, but if it is unknowable as to whether there are unknowable truths, which seems easy enough to show, then we know there is an unknowable truth, no?

What is knowable or unknowable in Fitch’s proof is not an unknowable truth, but an unknown truth.

But how does it follow that an unknown truth leads to the conclusion that there is an unknowable truth. I don't know, may I'm just not bright enough for this argument...

As I said in my post above to Andrew, one reason that an unknown truth would be unknowable (or impossible to know) is if all truths were already known and there were no unknown truths.

But if there were no unknown truths, wouldn't it then follow that there would be no unknowable truths? In any case, that is not how the argument gets from unknown to unknowable is it?

Hence, the statement "p is an unknown truth" cannot be both known and true at the same time. Therefore, if all truths are knowable, the set of "all truths" must not include any of the form "something is an unknown truth"; thus there must be no unknown truths, and thus all truths must be known. — Fitch's paradox of knowability

There's a jump to 'there are no unknown truths'. You've gone from a specific situation where a certain unknowln proposition is (somehow) known to be true. That is the problematic situation. The specific case. The generalization that there are no unknown truths is not supported by the problems of assertiing Assertion X, which we do not know since it is unknown is true. In the general case no one is claiming to know that any specific unknown truth is true. What we do is go by the experience that we find out new true things and there are likely more, but we, by definition, do not know what these are and can make no claims about any specific unknown truth.

But if there were no unknown truths, wouldn't it then follow that there would be no unknowable truths? — Janus

Possibly. What's your reasoning?

In any case, that is not how the argument gets from unknown to unknowable is it? — Janus

The move from unknown to unknowable is given in the "independent result" in lines 4-9 of the SEP proof. The logic of that reductio argument is beyond my understanding, and I would welcome someone to explain it. However, I don't dispute its conclusion.

Possibly. What's your reasoning? — Luke

Actually, it doesn't follow. All knowable truths could be known with only unknowable truths left. But then surely new truths are arising every moment, so it seems absurd to think that there could be no unknown truths; we (collectively) would have to be constantly up to the minute.

The move from unknown to unknowable is given in the "independent result" in lines 4-9 of the SEP proof. The logic of that reductio argument is beyond my understanding, and I would welcome someone to explain it. However, I don't dispute its conclusion. — Luke

Yeah, I don't comprehend it either, as I said, but I also accept the conclusion (although not on account of the "paradox") that there must be unknowable truths.

Yeah, I don't comprehend it either, as I said, but I also accept the conclusion (although not on account of the "paradox") that there must be unknowable truths. — Janus

As I understand it, the conclusion of the independent result is not that there must be unknowable truths. The conclusion of the independent argument is that it is impossible to know an unknown truth. It follows from this in the SEP proof that there does not exist an unknown truth (at line 10) and that all truths are known (at line 11).

That's the contradiction. However it's not true that a proposition can be both knowable and unknowable is it?
— Andrew M

Right, but neither should the contradiction imply that “p & ~Kp” is necessarily unknowable. — Luke

It does imply that. If the independent result (from Lines 4 to 9) doesn't convince you, can you come up with a concrete instance where “p & ~Kp” can be known? See also the example below.

I still cannot get the move from unknown truth to unknowable truth in the argument. — Janus

Let me try a concrete example. Suppose there is milk in the fridge and no-one knows there is.

It's thus true that there's milk in the fridge and no-one knows there is.

That true statement is unknowable. Why? Because anyone coming to know that there's milk in the fridge (say, by looking) would render the statement false (since the second conjunct would be false). The statement doesn't change from an unknown truth to a known truth. It changes from an unknown truth to a known falsity.

That's it. If one holds that all truths are knowable then Fitch's proof requires that they either change their position (i.e., reject that all truths are knowable) or, else, hold that all truths are known (i.e., reject non-omniscience).

That may be true, but if it is unknowable as to whether there are unknowable truths, which seems easy enough to show, ... — Janus

We do know that there are unknowable truths, as the above example demonstrates.

Correct me if I'm wrong, but doesn't this only hold if you take as a premise that "all truths are knowable." The issue is the existence of an unknown truth that cannot be known, as its being known entails it no longer being true.

But if you accept that there are unknowable truths then you're not in any difficulty. So, this seems to me like a potentially major problem for verificationalism or versions of epistemology where truth is actually about attitudes and beliefs (but not all of such systems, I think Bayesian systems escape unscathed), yet not much of a problem for other systems.

I am honestly flummoxed by SEP's list of systems that would be imperiled by this. Berkley seems like he can get out of this easily due to the fact that the unperceived truth of p doesn't exist, and anyhow God definitionally knows all truths for him already. Peirce’s system is also an odd one on the list. The "end of inquiry" would presumably be once we know the truth of p's referent, and so the paradox of one truth passing away as another is recognized is just part of the pragmatic process of gaining knowledge. I'm not even sure logical positivism is hit that hard. After all, it was the basis for the Copenhagen Interpretation of Quantum Mechanics, which very much supposed unknowable truths. The conclusion was simply that statements about the true absolute position/velocity of a particle were meaningless. Copenhagen is generally criticized for being incoherent, but this is because it creates a totally arbitrary boundary between quantum systems and classical ones not because it discounts statements about unmeasurable values.

According to the knowability principle, a statement is true if it can be known to be true, — Michael

Is "if" in the wrong place, or does it just need an "only"?

Cheers, I'll take another look.

It's thus true that there's milk in the fridge and no-one knows there is.

That true statement is unknowable. Why? Because anyone coming to know that there's milk in the fridge (say, by looking) would render the statement false (since the second conjunct would be false). The statement doesn't change from an unknown truth to a known truth. It changes from an unknown truth to a known falsity. — Andrew M

You seem to be saying that the truth of the statement "It's true that there's milk in the fridge and no-one knows there is" is unknowable, which seems reasonable, since I don't know there's milk in the fridge unless I open it but then if I do that someone knows there is milk in the fridge. But when I open the fridge I know (excluding weirdness like the milk coming to be there only when I looked) that the statement was true before I looked. So, again, there seems to be a time element involved.

If I go down the 'weirdness' rabbit hole and say that when I look and see the milk I still don't know that the milk had been there prior to my looking, then all bets are off.

Suppose there is milk in the fridge and no-one knows there is.

It's thus true that there's milk in the fridge and no-one knows there is.

That true statement is unknowable. Why? Because anyone coming to know that there's milk in the fridge (say, by looking) would render the statement false (since the second conjunct would be false). The statement doesn't change from an unknown truth to a known truth. — Andrew M

Aye, there's the rub. If a truth is knowable, then it can come to be known; that is, it can change from being unknown to being known. However, as you note, the statement "p & ~Kp" does not (and cannot) change from being unknown to being known. Therefore, the starting suppositions make it impossible for an unknown truth to become a known truth. The starting suppositions give the impression that all truths are knowable and that we should be able to come to know an unknown truth. But if "p & ~Kp" cannot possibly change from being unknown to being known, then of course it is unknowable: it's a rigged game from the outset. It follows only from this logical impediment that it is impossible to know an unknown truth, that no truths are knowable, and that all truths are known. These conclusions can safely be ignored, however, given that the confidence trick does not allow for an unknown truth to become a known truth.

Therefore, the starting suppositions make it impossible for an unknown truth to become a known truth. — Luke

No it doesn't.

"There are 163 coins in the jar" was an unknown truth before someone counted, and then it became a known truth.

But if "p & ~Kp" cannot possibly change from being unknown to being known, then of course it is unknowable: it's a rigged game from the outset. — Luke

I don't know what you mean by it being "rigged". It just shows that the knowability principle is wrong. Some truths are, in fact, unknowable.

"There are 163 coins in the jar" was an unknown truth before someone counted, and then it became a known truth. — Michael

To borrow @Andrew M's example:

Suppose there are 163 coins in the jar and no-one knows there is.

It's thus true that there's 163 coins in the jar and no-one knows there is.

That true statement is unknowable. Why? Because anyone coming to know that there's 163 coins in the jar (say, by counting) would render the statement false (since the second conjunct would be false). The statement doesn't change from an unknown truth to a known truth. It changes from an unknown truth to a known falsity.

I don't know what you mean by it being "rigged". It just shows that the knowability principle is wrong. Some truths are, in fact, unknowable. — Michael

I mean that the unknown truth "p & ~Kp" of NonO cannot possibly become a known truth. If that is impossible from the outset, then so is knowability.

But if you accept that there are unknowable truths then you're not in any difficulty. — Count Timothy von Icarus

The result of the argument seems to be that all unknown truths are unknowable, as there is no unknown truth of the form "p & ~Kp" that can change into a known truth or that can become known. That all unknown truths are unknowable is just as absurd as the result that all truths are known.