↪Olivier5
I wasn't objecting to Fitch there. Just giving an example of bewitchment of language leading to metaphysical conclusions. — Tate

Right. Thanks for the clarification.

We can escape Fitch by just saying we don't know if the status of Riemann's hypothesis is knowable.

Yes, that's perfectly fine.

My solution is to add the time variable to the formalism. Knowledge evolves over time.

So I would write:

Suppose p is a sentence that is an unknown truth; that is, the sentence p has been proposed, it is true, but it is not known that p is true. In such a case, the sentence "the sentence p is an unknown truth" is true today; and, if all truths are knowable, it should be possible one day to know learn that "p was an unknown truth" up untill that day.

Yes.

I saw this, but I don't see a cogent argument in this:
"The problem is that according to the knowability principle, if "the box is empty" is true and we don't know that it's true then it's possible to know that "the box is empty" is true and that we don't know that it's true, which is a contradiction, and that if "the box is not empty" is true and we don't know that it's true then it's possible to know that the "the box is not empty" is true and that we don't know that it's true, which is a contradiction."

If "the box is empty is true" and we don't know that it is true it does not follow that it's possible to know that "the box is empty is true" and that we don't know that it's true, at the same time. We don't know that it's true, but we may come to know that it's true, and if we come to know that it's true, it will no longer be the case that we don't know that it's true; and hence there is no contradiction. Am I missing something? I'm finding it impossible to see why anyone would think there is a paradox here. If I am missing something it should be explainable, no?

Suppose there is some statement t that is true AND no-one knows that t is true (say, Goldbach's conjecture or its negation). That conjunctive statement is itself true but unknowable. — Andrew M

We don't know if that statement is true, though; someone might know but isn't telling, so it's truth is merely being stipulated. It is unknown whether anyone knows the truth of Golbach's conjecture, but not unknowable, because someone may demonstrate that they know that it is true or false.

If "the box is empty is true" and we don't know that it is true it does not follow that it's possible to know that "the box is empty is true" and that we don't know that it's true, at the same time. — Janus

It does according to the knowability principle: if a proposition is true then it is possible to know that the proposition is true (p → ◊Kp).

a) "the box is empty" is true and we don't know that it's true

The above is a proposition which, if true, entails that it is possible to know that it's true (a → ◊Ka).

It does according to the knowability principle: if a proposition is true then it is possible to know that the proposition is true.

1. "the box is empty" is true and we don't know that it's true

The above is a proposition which, if true, entails that it is possible to know that it's true. — Michael

I'm sorry, but I don't see why "1.", if it is true, entails that it is possible to know that it is true. In other words, we don't know whether the knowability principle is itself true, but we do know that we don't know everything. The stumbling block in the argument, for me, remains the fact that time is apparently being ignored, and it is that ignore-ance that creates the apparent paradox, as far as I can tell. I am very open to being corrected, but no one seems able to explain what it is that I'm purportedly missing.

I'm sorry, but I don't see why "1.", if it is true, entails that it is possible to know that it is true. — Janus

Because that's what the knowability principle says. If some proposition p is true then it is possible to know that proposition p is true, and in this case:

p. "the box is empty" is true and we don't know that it's true

Because that's what the knowability principle says. If some proposition p is true then it is possible to know that proposition p is true, and in this case:

p. "the box is empty" is true and we don't know that it's true — Michael

OK, assuming the knowability principle is itself true, the case doesn't contradict it anyway, because it says that ""the box is empty" is true and we don't know that it's true" not ""the box is empty" is true and we can't know that it's true".

OK, assuming the knowability principle is itself true, the case doesn't contradict it anyway, because it says that ""the box is empty" is true and we don't know that it's true" not ""the box is empty" is true and we can't know that it's true". — Janus

But there are two parts to proposition p:

1) "the box is empty" is true
2) we don't know that "the box is empty" is true

If we know part 1) then we can't know part 2) and vice-versa. Therefore it's impossible to know that proposition p is true. But if it's impossible to know that proposition p is true then, according to the knowability principle, proposition p isn't true.

As I see it though the proposition is disjointed because we don't know 1) we are merely stipulating it or imagining it is the case. And there would be no contradiction unless we make the mistake of thinking that we are not merely stipulating 1) but knowing it.

What about this: 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.

As I see it though the proposition is disjointed because we don't know 1) we are merely stipulating it or imagining it is the case. And there would be no contradiction unless we make the mistake of thinking that we are not merely stipulating 1) but knowing it. — Janus

We know that one of these must be true, as per the law of excluded middle (and assuming for the sake of argument that we don't know whether or not the box is empty):

a) the box is empty and we don't know that it's empty
b) the box is not empty and we don't know that it's not empty

But we can never know either to be true because that would be a contradiction. We can't know that the box is empty and that we don't know that the box is empty, therefore we can't know a. We can't know that the box is not empty and that we don't know that the box is not empty, therefore we can't know b.

However, either a or b must be true. Therefore, either a or b is an unknowable truth. And if either a or b is an unknowable truth then the knowability principle is false.

is the truth of the proposition that there are unknowable propositions itself unknowable? — Janus

No. Both a) and b) are known to be unknowable propositions.

I see. If I'm a verificationist, then I can be accused of saying that the human race knows all (not that any individual does.) — Tate

Yes, or else that the verificationist holds a contradictory view.

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

But since I haven't ruled out the expansion of human knowledge, I should show up as reasonable. — Tate

Yes.

I think Wikipedia is talking about truth anti-realism in the second case (not idealism). Deflationary accounts of truth are apt to be anti-realist, redundancy and so forth. — Tate

OK, though from SEP again:

As such the proof does the interesting work in collapsing moderate anti-realism into naive idealism. — Fitch’s Paradox of Knowability - SEP

Suppose there is some statement t that is true AND no-one knows that t is true (say, Goldbach's conjecture or its negation). That conjunctive statement is itself true but unknowable.
— Andrew M

We don't know if that statement is true, though; someone might know but isn't telling, so it's truth is merely being stipulated. It is unknown whether anyone knows the truth of Golbach's conjecture, but not unknowable, because someone may demonstrate that they know that it is true or false. — Janus

Goldbach's conjecture was just an example. The point is that if there is any unknown truth (i.e., if we are not collectively omniscient), then there is also a related unknowable truth.

But we can never know either to be true because that would be a contradiction. — Michael

Yes, I agree, but we do know that one of them is true, we just can't know which one without chaging the state of the game.

is the truth of the proposition that there are unknowable propositions itself unknowable? — Janus


No. Both a) and b) are known to be unknowable propositions. — Michael

Read again; I wasn't referring to a) or b).

Goldbach's conjecture was just an example. The point is that if there is any unknown truth (i.e., if we are not collectively omniscient), then there is also a related unknowable truth. — Andrew M

I'm still not getting it from that angle but I think this shows that there is at least one unknowable truth:

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.

I think Wikipedia is talking about truth anti-realism in the second case (not idealism). Deflationary accounts of truth are apt to be anti-realist, redundancy and so forth.
— Tate

OK, though from SEP again:

As such the proof does the interesting work in collapsing moderate anti-realism into naive idealism.
— Fitch’s Paradox of Knowability - SEP — Andrew M

I don't know what kind of anti-realism the SEP is talking about.

Do you?

I think this or a form of it is the obvious solution. If I imagine a database of all possible propositions, with a truth value column, I can just as well imagine duplicates of many propositions with them being differentiated by a timestamp column. This would allow you to have the set of all true propositions without timing becoming a source of contradiction.

But you can also look at the truth of a thing being something progressive through time. True propositions about the thing sprout up and die away with time. So, the truth of a tree is the acorn, the sappling, the tree, and the mature branch that yields another acorn, all together. "The flower does not refute the bus."

Or, with more detail at the risk of being more convoluted:

Philosophy, on the contrary, does not deal with a determination that is non-essential, but with a determination so far as it is an essential factor. The abstract or unreal is not its element and content, but the real, what is self-establishing, has life within itself, existence in its very notion. It is the process that creates its own moments in its course, and goes through them all; and the whole of this movement constitutes its positive content and its truth. This movement includes, therefore, within it the negative factor as well, the element which would be named falsity if it could be considered one from which we had to abstract. The element that disappears has rather to be looked at as itself essential, not in the sense of being something fixed, that has to be cut off from truth and allowed to lie outside it, heaven knows where; just as similarly the truth is not to be held to stand on the other side as an immovable lifeless positive element. Appearance is the process of arising into being and passing away again, a process that itself does not arise and does not pass away, but is per se, and constitutes reality and the life-movement of truth. The truth is thus the bacchanalian revel, where not a member is sober; and because every member no sooner becomes detached than it eo ipso collapses straightway, the revel is just as much a state of transparent unbroken calm. Judged by that movement, the particular shapes which mind assumes do not indeed subsist any more than do determinate thoughts or ideas; but they are, all the same, as much positive and necessary moments, as negative and transitory. In the entirety of the movement, taken as an unbroken quiescent whole, that which obtains distinctness in the course of its process and secures specific existence, is preserved in the form of a self-recollection, in which existence is self-knowledge, and self-knowledge, again, is immediate existence.

Whew...

:up: Thanks.

Historical examples of such theories arguably include Michael Dummett’s semantic antirealism (i.e., the view that any truth is verifiable), mathematical constructivism (i.e., the view that the truth of a mathematical formula depends on the mental constructions mathematicians use to prove those formulas), Hilary Putnam’s internal realism (i.e., the view that truth is what we would believe in ideal epistemic circumstances), Charles Sanders Peirce’s pragmatic theory of truth (i.e., that truth is what we would agree to at the limit of inquiry), logical positivism (i.e., the view that meaning is giving by verification conditions), Kant’s transcendental idealism (i.e., that all knowledge is knowledge of appearances), and George Berkeley’s idealism (i.e., that to be is to be perceivable). — SEP

I don't see why we couldn't fix that using indexicals. So i don't see this as a solution.

Goldbach's conjecture was just an example. The point is that if there is any unknown truth (i.e., if we are not collectively omniscient), then there is also a related unknowable truth.
— Andrew M

I'm still not getting it from that angle — Janus

OK, though it's not clear to me what you are objecting to.

but I think this shows that there is at least one unknowable truth:

Is the truth of the proposition that there are unknowable propositions itself unknowable? — Janus

No. False propositions are unknowable in the sense that you can't know what is false. And, in the absence of omniscience, Fitch's paradox shows that there are true propositions that are unknowable. That demonstration is how we know that there are unknowable truths.

I don't know what kind of anti-realism the SEP is talking about.

Do you? — Tate

↪Tate

Historical examples of such theories arguably include Michael Dummett’s semantic antirealism (i.e., the view that any truth is verifiable), mathematical constructivism (i.e., the view that the truth of a mathematical formula depends on the mental constructions mathematicians use to prove those formulas), Hilary Putnam’s internal realism (i.e., the view that truth is what we would believe in ideal epistemic circumstances), Charles Sanders Peirce’s pragmatic theory of truth (i.e., that truth is what we would agree to at the limit of inquiry), logical positivism (i.e., the view that meaning is giving by verification conditions), Kant’s transcendental idealism (i.e., that all knowledge is knowledge of appearances), and George Berkeley’s idealism (i.e., that to be is to be perceivable).
— SEP — Banno

:up:

But Kant does allow truths that are unknowable: how things in themselves really are. Why is he on the list?

Thanks!

OK, that seems fine: so it is possible to know there is an unknown truth; that does not mean it is possible to know an unknown truth (which would be a contradiction) but that it is possible to know that there is an unknown truth (which is not a contradiction). — Janus

Right. I think my failure to note this distinction may have caused some issues earlier in the discussion.

…in the absence of omniscience, Fitch's paradox shows that there are true propositions that are unknowable. — Andrew M

In an attempt to justify the scare quotes in the title of the OP, I will explain why I find the results of Fitch’s argument unsurprising.

As noted in my penultimate post, a contradiction arises from the combination of the knowability principle and the non-omniscience principle.

If we reject the non-omniscience principle and retain the knowability principle, it follows from Fitch’s argument that all truths are not only knowable but known. This is unsurprising given our omniscience!

If we reject the knowability principle and retain the non-omniscience principle, it follows from Fitch’s argument that there is not only an unknown truth but an unknowable truth. This is unsurprising as it prevents our omniscience! It is also unsurprising given that not all truths can be known!

Goldbach's conjecture was just an example. The point is that if there is any unknown truth (i.e., if we are not collectively omniscient), then there is also a related unknowable truth.
— Andrew M

I'm still not getting it from that angle — Janus


OK, though it's not clear to me what you are objecting to. — Andrew M

I'm not strictly objecting to anything. I'm just not seeing how it follows from there being unknown truths, that there are unknowable truths.

As I pointed out with my example we know that it is unknowable as to whether there are unknowable truths, because we can never be sure that there are not unknowable truths. But then I've just said that it it is knowable that it is unknowable as to whether there are unknowable truths, from which it seems to follow, paradoxically that we do know there are unknowable truths, or at least one, at any rate.

I think this or a form of it is the obvious solution. If I imagine a database of all possible propositions, with a truth value column, I can just as well imagine duplicates of many propositions with them being differentiated by a timestamp column. This would allow you to have the set of all true propositions without timing becoming a source of contradiction. — Count Timothy von Icarus

Yes. Fitch is just poor logical formalism (or poor English) passing for a paradox. @Janus and others made the same point.

I must admit I did not understand the rest of your post... :worry:

I'm just not seeing how it follows from there being unknown truths, that there are unknowable truths. — Janus

The two principles of Fitch's argument are that all truths are knowable (Knowability Principle - KP) and that there is an unknown truth (Non-Omniscience Principle - NonO). If we take the unknown truth (of NonO) to be one of the knowable truths (of KP), then it follows that an unknown truth is knowable. However, it can also be independently proven that an unknown truth is unknowable. This contradiction leads us to reject either KP or NonO. If we reject NonO, then it follows that all truths are known. If we reject KP, then it follows that there is an unknowable truth. Hope this helps.

But Kant does allow truths that are unknowable: how things in themselves really are. Why is he on the list? — Tate

Perhaps see here (bold mine):

It is often claimed that anti-realism is a form of transcendental idealism or that Kant is an anti-realist.[1] It is also often claimed that anti-realists are committed to some form of knowability principle to the effect that all truths (or at least all truths of a certain class) are knowable and that such principles have problematic consequences.[2]
...
In §1.1, I present evidence that suggests Kant is indeed committed to a knowability principle and I show that a Fitch-Church style proof can be constructed on this basis. Kant does not think that all truths whatsoever are knowable, but it can seem as though he is committed to the claim that all empirical truths are knowable, and on moderate background assumptions this entails that no empirical truth is unknown. — Kant, the Paradox of Knowability, and the Meaning of ‘Experience’ - Andrew Stephenson

I'm not strictly objecting to anything. I'm just not seeing how it follows from there being unknown truths, that there are unknowable truths. — Janus

Because Alice can (speculatively) say of an unknown truth, t, that "t is true and no-one knows that t is true".

Alice's statement will, in turn, be an unknown truth. While someone could come to know that t is true, no-one could come to know that Alice's statement is true. (Though one could potentially come to know that Alice's statement was true in the past, but not now.)

As I pointed out with my example we know that it is unknowable as to whether there are unknowable truths, — Janus

But we don't know that, since it is false. We instead know, per Fitch's paradox, that there are unknowable truths.