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

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.

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

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.

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

p is an unknown truth. "p & ~Kp" asserts that p is an unknown truth. p is true and knowable. — Andrew M

Fair enough.

"p & ~Kp" is true but not knowable. — Andrew M

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

My thinking was that p is just a true proposition and "p & ~Kp" represents that it is an unknown truth. You now appear to be saying that it is this unknown truth which follows from the argument as unknowable:
— Luke

p is the unknown truth and that is expressed by the above conjunction. The conjunction itself is unknowable. — Andrew M

If the unknown truth is expressed by "p & ~Kp", then it is not expressed by "p". The unknown truth expressed by "p & ~Kp" is equivalent to your "t":

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

If the unknown truth "t" is equivalent to the expression "p & ~Kp", then what Alice can (speculatively) say of the unknown truth, t, (via substitution) is that ""p & ~Kp" is true and no-one knows that "p & ~Kp" is true." This would make "p & ~Kp" knowable, but you have told me that:

The conjunction itself is unknowable. — Andrew M

This is why I said in my initial response that:

I would have thought that it was the unknown truth (of NonO) [i.e. "p & ~Kp"] that becomes unknowable upon the rejection of the knowability principle, rather than a [i.e. Alice's] statement regarding the unknown truth. — Luke

I would have thought that it was the unknown truth (of NonO) that becomes unknowable upon the rejection of the knowability principle, rather than a statement regarding the unknown truth.
— Luke

It's the latter. In the SEP proof, line 1 asserts that p is an unknown truth. Line 3 asserts that it is possible to know the conjunction from line 1. Finally, line 3 is shown to be false. The essential point here is that p and "p & ~Kp" are different statements - the former is unknown (but potentially knowable), the latter is unknowable. — Andrew M

My thinking was that p is just a true proposition and "p & ~Kp" represents that it is an unknown truth. You now appear to be saying that it is this unknown truth which follows from the argument as unknowable:

The essential point here is that p and "p & ~Kp" are different statements - the former is unknown (but potentially knowable), the latter is unknowable. — Andrew M

Whereas, you previously said that it was Alice's statement about the unknown truth which becomes unknowable.

it's showing that if we accept the knowability principle then all truths are known. — Michael

Is the knowability principle that 'all truths are known'? No.

Neither is NonO that 'there is an unknowable truth'.

What follows from the knowability principle being denied has nothing to do with Fitch's paradox. — Michael

I find it epistemologically interesting that if we reject NonO then all truths are not only knowable but known, and if we reject KP then there is not only an unknown but an unknowable truth. These both follow from Fitch's argument, so I wouldn't say it has nothing to do with it. Is it wrong to have an interest and be curious about the argument?

Assume that John argues that an omniscient God exists and that we have free will. Jane provides an argument to show that if an omniscient God exists then we don't have free will.

You then want to know what follows from an omniscient God not existing, which has nothing to do with Jane's argument. — Michael

So? Maybe I'm curious to know whether we have free will.

I don't see why "p & ~Kp" is unknowable.
— Luke


Because knowing it renders it false. — unenlightened

Yes, my mistake. I mistook @Andrew M to be saying that "p & ~Kp" stands for an unknowable truth.

Then just reject the knowability principle.I don't understand the problem. — Michael

There wasn't a problem.

As per Banno's summary of the argument:

Hence there is a contradiction between KP and NonO. They cannot both be true.

So someone who maintains that KP is true must deny NonO - they admit omniscience.

Hence, if all truths are knowable, everything is known. — Banno

The above describes what follows when NonO is denied. But given the contradiction between KP and NonO, KP could also be denied. I am merely interested, for the sake of symmetry or completeness, to see what follows if KP is denied. What follows is that there is an unknowable truth. A further discussion about unknowability also occurred when Janus asked how we get from an unknown to an unknowable truth in the argument.

And Luke is not the only one.

Folks, in outline, the SEP proof works as follows:

Part 1
Assuming KP and NonO, we derive line (3)

Part 2
Assuming A,B,C,& D, we derive Line (9)

Conclusion:
Line (9) contradicts line (3);

hence, one of the assumptions here is wrong.
Or we need an alternative logic.

A,B,C,D are unassailable (I'm sure that won't stop someone here making the attempt...)

Hence there is a contradiction between KP and NonO. They cannot both be true.

So someone who maintains that KP is true must deny NonO - they admit omniscience.

Hence, if all truths are knowable, everything is known. — Banno

How is that any different to what I said here and here?

I think you misunderstand Fitch's paradox. It is a reductio ad absurdum against the knowability principle. So, Fitch's paradox is literature that speaks to the rejection of the KP side. — Michael

The argument may have implications for KP, but what is presented in the SEP article is what follows from rejecting the NonO principle (my emphasis):

Line 9 contradicts line 3. So a contradiction follows from KP and NonO. The advocate of the view that all truths are knowable must deny that we are non-omniscient:
(10)¬∃p(p∧¬Kp).

And it follows from that that all truths are actually known:
(11)∀p(p→Kp). — SEP article

As Banno says (despite accusing me of getting it wrong):

...someone who maintains that KP is true must deny NonO - they admit omniscience. — Banno

And besides, I find it logically interesting to consider the rejection of each side. Not to mention that @Janus raised a question about unknowability which follows from rejecting the KP side instead of the NonO side.

The essential point here is that p and "p & ~Kp" are different statements - the former is unknown (but potentially knowable), the latter is unknowable. — Andrew M

The SEP article states:

Let K be the epistemic operator ‘it is known by someone at some time that.’ — SEP article

Doesn't "~Kp" therefore mean that "it is not known by someone at some time that'? That is, p is unknown.

I don't see why "p & ~Kp" is unknowable.

Moreover, "p & ~Kp" is the conjunction of the non-omniscience principle, which looks like what the SEP calls an unknown (not an unknowable) truth:

And suppose that collectively we are non-omniscient, that there is an unknown truth:
(NonO) ∃p(p∧¬Kp) — SEP article

It is only once the knowability principle is rejected that there is an unknowable truth.

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. — Andrew M

Someone could come to know the unknown truth, t, but no-one could come to know Alice's statement about t is true? Couldn't Alice come to know that their statement is true, at least? What do you make of @Michael's earlier claims in this discussion regarding the Riemann hypothesis and its being an unknown truth that it is correct (or else an unknown truth that it is incorrect)? Can't we all come to know the truth of Michael's statement(s)?

I would have thought that it was the unknown truth (of NonO) that becomes unknowable upon the rejection of the knowability principle, rather than a statement regarding the unknown truth. The SEP article appears to show only the rejection of the NonO side of things. Do you know of any literature that speaks to the rejection of the KP side?

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.

…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!

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.

I don't see how it follows from the fact that we know (if we do know) there are unknown truths that an unknown truth is knowable — Janus

It isn't that we do know there are unknown truths, it is that it is possible to know there is an unknown truth. If it is possible to know, then it is knowable. These terms are simply synonymous.

A reminder here that this comes from combining the two starting principles, KP and NonO.

the fact that there are unknown truths (if there are) is not itself an unknown truth (if it is known). — Janus

No, but why do you think it should be?

Can you lay out the argument clearly in plain English? — Janus

I'll have a go. It might not be correct (or helpful) but maybe others can chime in to correct and clarify.

Suppose both of these principles:

All truths are knowable (the knowability principle)
We are non-omniscient; there is an unknown truth (the non-omniscience principle)

Combine these principles:

If one of all of the knowable truths (KP) is that we are non-omniscient or that there is an unknown truth (NonO) - in other words, if it is possible to know that there is an unknown truth - then it follows that an unknown truth is knowable.

However, it can be independently shown that an unknown truth is unknowable.

Given the contradiction that an unknown truth is both knowable and unknowable, one of the starting principles (KP or NonO) must be rejected.

However, if we reject the non-omniscience principle (which says that there is an unknown truth) such that there are no unknown truths, then it follows that not only are all truths knowable, but all truths are in fact known.

On the other hand, if we reject the knowability principle (which says that all truths are knowable) such that not all truths are knowable, then it follows that not only is there an unknown truth, but there is an unknowable truth.

Or, as the archived SEP article puts it:

The paradox of knowability is a logical result suggesting that, necessarily, if all truths are knowable in principle then all truths are in fact known. The contrapositive of the result says, necessarily, if in fact there is an unknown truth, then there is a truth that couldn't possibly be known [i.e. an unknowable truth]. — Archived SEP article

I take it all truths are known implies that no truths are knowable (because they are known)?
— Luke

In fact the opposite: Kp → ◊Kp. — Michael

But, according to the independent argument, starting with the assumption K(p ∧ ¬Kp) leads to the conclusion ¬◊K(p ∧ ¬Kp). That is, if the conjunction is known, then the conjunction is not knowable.

Just a thought.

At the beginning of the discussion @Agent Smith made reference to Meno's paradox, and I think there could be an interesting parallel to Fitch's. An ambiguity is noted wrt Meno's paradox:

Suppose Tom wants to go to the party, but he doesn't know what time it begins. Furthermore, he doesn't even know anyone who does know. So he asks Bill, who doesn't know when the party begins, but he does know that Mary knows. So Bill tells Tom that Mary knows when the party begins. Now Tom knows something, too—that Mary knows when the party begins.

So Tom knows what Mary knows (he knows that she knows when the party begins). Now consider the following argument:

Tom knows what Mary knows.
What Mary knows is that the party begins at 9 pm.
What Mary knows = that the party begins at 9 pm.
Therefore, Tom knows that the party begins at 9 pm.

What is wrong with this argument? It commits the fallacy of equivocation.

In (A), “what Mary knows” means what question she can answer. But in (B) and (C), “what Mary knows” means the information she can provide in answer to that question. — Meno's paradox ambiguity

I wonder whether the same/similar type of ambiguity applies to your Riemann hypothesis examples. You can say (about unknown truths) which set of statements are truth apt, but not which statements are true. In other words, you can know of unknown truths. but you cannot know them (or which of them) to be true.

Technically speaking Fitch's argument shows that the knowability principle entails that all truths are known — Michael

Known by everyone always, or known only by someone at some time? I take it all truths are known implies that no truths are knowable (because they are known)? But if they are known only by someone at some time, would that imply they can be knowable by others, in order to save KP?

However, it's a fact that some truths aren't know. Either "the Riemann hypothesis is correct" or "the Riemann hypothesis is not correct" is one such truth that isn't know. — Michael

So there are unknown truths? Are they knowable?

This is what I am denying, since if an unknown truth becomes known, then it is not an unknown truth.

Therefore, the knowability principle fails. — Michael

Not according to Fitch's argument.

If there are no unknown truths then all truths are known. — Michael

Also, if there are no unknown truths, then only known truths are known.

However, in order to show otherwise, you would need to demonstrate that some unknown truth can be known.
— Luke

No, I need to show that there are no unknown truths, which is what Fitch's paradox does; see above. — Michael

In order to disprove my claim, which is that the argument demonstrates that only known truths are known, then you would need to show that there are no unknown truths? Doesn't that just support my claim? If there are no unknown truths then only known truths are known.

You seem to want to draw from Fitch's conclusion that not only are known truths known, but also that unknown truths are known, such as that (e.g.) "the Riemann hypothesis is correct". I don't draw this absurd conclusion from the argument.

Yes, either "the Riemann hypothesis is correct" is an unknown truth or "the Riemann hypothesis is not correct" is an unknown truth. — Michael

You disagreed with my claim that the argument implies only that known truths are known. However, in order to show otherwise, you would need to demonstrate that some unknown truth can be known. Since you do not know which one of the above statements is true, then you have not demonstrated knowledge of an unknown truth.

Which is a false interpretation. I've explained the logic several times. — Michael

I don't believe that you have.

Is "either the Riemann hypothesis is correct or the Riemann hypothesis is not correct" a known truth or an unknown truth? You've said that that's a known truth, but you've also used this to argue that not all truths are known.

On the other hand, it is unknown which one is true, so you cannot claim that one of them is a known truth.

And yet we don't know which of "the Riemann hypothesis is correct" and "the Riemann hypothesis is not correct" is true, but one of them must be. Therefore not all truths are known. — Michael

Well, I'm saying that the argument implies only that known truths are known, which excludes knowing unknown truths. The independent argument given in the SEP article shows that it is impossible to know unknown truths.

Yes, my mistake. It is the substitution of NonO into KP which is the problem. These principles combine to imply that an unknown truth is knowable. However, the independent argument shows that it is impossible to know an unknown truth. Therefore, NonO is rejected and hence all truths must be known.

According to logic, known truths and unknown truths forever stay that way. Otherwise, we could allow for an unknown truth to become known, but then it would no longer be an unknown truth.

In other words, @unenlightened was right.

The implication for the argument remains what I said earlier: no unknown truths can be known and only known truths must be known. That still doesn't seem very omniscient to me, given what we know.

So either every true proposition is known to be true (abandon non-omniscience) or for some true propositions it is not possible to know that they are true (abandon knowability principle). — Michael

What I'm trying to say is that we can abandon the principle of non-omniscience (as given) without implying that all (known and unknown) truths must be known. I believe that all the argument implies is that only known truths must be known; or, more to the point, that no unknown truths can be known.

The principle of non-omniscience implies that there are unknown truths which are or can be known. This is simply a contradiction in terms. If a truth is known then it cannot be unknown, and if a truth is unknown then it cannot be known (per modal principle D in the SEP article). If a truth becomes known then it is no longer unknown. To repeat: no unknown truths can be known and only known truths must be known.

I'm not following your argument. — Banno

The crux of my argument is that "Kp" conflates the knowledge that:

(a) p (where "p" represents a meaningful proposition); and
(b) p is true

These are both entailed by "Kp".

Note that this is the same distinction that you emphasised earlier between knowing a sentence (e.g. "There is a teapot in orbit around Jupiter") and knowing the truth of that sentence.

Hence, "¬Kp" could mean either that:

(a) p (the meaningful proposition) is unknown; or
(b) p is true is unknown.

Upon further reflection, and thanks in large part to the responses from @Michael, I believe that I am disputing the non-omniscience supposition of the argument:

And suppose that collectively we are non-omniscient, that there is an unknown truth:

(NonO) ∃p(p∧¬Kp) — SEP article on Fitch's paradox

However, I do not claim omniscience. Instead, I would argue that truth implies knowledge. This is the conclusion of the argument, after all: for all p, if p is true, then it is known that p is true. The reason that the (NonO) statement is false is because p is true implies p is known, so there cannot be any p for which p is true and p is unknown. The reason that p is true implies p is known is because p cannot be true without knowing the meaningful proposition represented by p. Again, this results from the equivocation over the meaning of p and the truth of p.

So what problem are you trying to raise? — Bartricks

See the OP and the rest of the discussion.

Do you think the knowability thesis has some prima facie plausiblity? — Bartricks

I can see no reason why any true statement might be unknowable. Let's agree to disagree.

I don't think so. Logic deals in sentences, not meanings of sentences, whatever they are. — Banno

The meaning of a sentence is irrelevant to its truth value?

That thesis is demonstrably false. I am demonstrating its falsity by providing you with examples of truths that, if true - and it's metaphysically possible that they are - could not be known. — Bartricks

My point is that you don't know whether those statements are true or not; they are only possibly true statements. Therefore, they cannot be used to disprove the claim that all true statements are, in principle, knowable. The knowability thesis is not about possibly true statements. You are claiming that if those statements are true, then not all true statements are knowable. That's a big IF. Unless you can show that they are true, then you have not disproven the knowability thesis.

That proposition might be true. Assume it is. — Bartricks

The knowability thesis is that all truths (i.e. all true statements) are, in principle, knowable.

In order to disprove this, you want me to assume something that might or might not be true? The knowability thesis is about true statements only. If you want to disprove it then use a true statement. You can't just assert that some true statements are unknowable.

Yes. I gave you some examples of such unknowable truths. — Bartricks

But you cannot justify that they are true. Neither can you justify that "there are no justifications" is true.

So, there's a case of a true proposition that I believe to be true and that is not known. — Bartricks

I thought you were making a point about (not) all truths being knowable?

Knowing that P could equally mean knowing the meaning of the sentence. If you don't know the meaning of the sentence then neither can you know that P is true. That is the reason why P's knowability implies P must be known. The truth value of P is beside the point and merely gets conflated with the meaning of P.

Yes. So? That a proposition is true does not entail that it is known to be. — Bartricks

But you said that it was true. You both know and don't know that it's true?