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

Yes.

If we reject the knowability 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! — Luke

Yes. Though note there is nothing in Fitch's argument that precludes humans from coming to know the unknown (but knowable) truths.

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?

~K(p & ~Kp) $\to$ □~K(p & ~Kp) $\to$ ~◇K(p & ~Kp) $\to$ ~$\exists$p(p & ~Kp) $\to$ $\forall$p(p $\to$ Kp)

Knowability principle (modal logic variant): p $\to$ ◇Kp

Non-O: $\exists$p(p & ~Kp)

Instantiation of Non-O: p & ~Kp

Input p & ~Kp in the Knowability principle and we get: (p & ~Kp) $\to$ ◇K(p & ~Kp)

Compare the two bolded + underlined statements (vide infra).
~◇K(p & ~Kp) contradicts ◇K(p & ~Kp)

In other words, Fitch's argument is rather interesting in that the reductio ad absurdum argument is itself a reductio ad absurdum argument. A Zen moment for me!

Do you know of any literature that speaks to the rejection of the KP side? — Luke

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. Fitch is saying "if you accept the knowability principle then this implausible conclusion follows, therefore we must reject the knowability principle."

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 [her] statement is true, at least? — Luke

No.

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)? — Luke

Perhaps we can - who knows? But they are not the unknowable truths that Fitch's paradox expresses.

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.

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

Yep. 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.

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.

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.

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?

And yet you asked for literature rejecting "the KP side".

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

...and yet we do not know everything, and hence must reject KP.

The purpose here is to show that such versions of antirealism as accept KP are committed to an unacceptable conclusion, andhence we must reject KP.

The whole of the literature is the rejection of KP...

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. — Luke

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

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

Because knowing it renders it false.

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.

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.

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. — Luke

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

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.

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.

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

Then it's a topic for another discussion, not this one.

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. — Luke

This is where you're misunderstanding Fitch's paradox. It isn't showing that if we reject the non-omniscience principle then all truths are known or that if we reject the knowability principle then some truths are unknowable; it's showing that if we accept the knowability principle then all truths are known.

That a rejection of the non-omniscience principle entails that all truths are known is a truism, and that a rejection of the knowability principle entails that some truths are unknowable is a truism. This has nothing to do with Fitch's paradox.

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'.

I see. That makes sense. If I say that truth only has a social function, then there are no unknowable truths, and I would be comfortable saying all truths are known. Fitch's target is trying to do more with truth. That's interesting, thanks.

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.

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. — Luke

Yes, which is what I said above ("the former is unknown").

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

Because that's what the proof shows. "<>K(p & ~Kp)" (line 3 in the SEP proof) is proved to be false.

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

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

No, p is the unknown truth. The above conjunction asserts that about p (i.e., that p is true and that p is not known).

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.

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. — Luke

They say the same thing. Alice's statement is "p & ~Kp".

I see. That makes sense. If I say that truth only has a social function, then there are no unknowable truths, and I would be comfortable saying all truths are known. Fitch's target is trying to do more with truth. That's interesting, thanks. — Tate

:up:

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

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": — Luke

To clarify, p is the unknown truth and that p has the characteristics of being unknown and true is expressed by the conjunction "p & ~Kp".

So to summarize:

p is an unknown truth. "p & ~Kp" asserts that p is an unknown truth. p is true and knowable. "p & ~Kp" is true but not knowable.

p is equivalent to my earlier t. "p & ~Kp" is equivalent to my earlier "t is true and no-one knows that t is true".

Hope that clears it up.

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?

I'm not sure if I get your meaning. Indexicals could certainly preform the function of a timestamp by fixing a proposition's referent as well. Why is that not a solution?

As noted in my earlier post, I think the problem here is deeply rooted to one's ontology and one's conception of time.

For eternalists, this does seem like a problem, but a referent to the time the proposition refers to seems like it would resolve the issue.

For presentists, I'm not sure if a contradiction ever actually exists. Only propositions about the present can be true.

Moreover, very bare ontologies would have it that only a very small set of all possible semantic propositions are actually meaningful, and would exclude these examples anyhow in favor of a binary representation of what most people would call the "physical world." It seems like the paradox needs certain assumptions unless I am missing something.

By the way, this is an area where I think formalism might be making things less clear because you're not creating a definition for what a set of all truths entails. For this set to be defined, you surely have to decide if the past or future exists or not.

The paradox reminds me a bit of problems in physics around information and entropy (which is really the truth value of propositions about the configuration of particles if you think about it). We have elaborate statistical ways of knowing about systems based on the possible configurations given X,Y, etc.

But, in reality, none of these "possible" configurations are actually possible aside from the one that actually obtains. The entire intellectual apparatus is based on a finite being's ability to know X about Y (the same can arguably be said for epistemology). Now in physics, we can throw out paradoxes that result from infinite information, such as Maxwell's Demon, by simply pointing out that said demon violates the laws of physics by needing to collect potentially infinite information to do his thing, and such infinite information cannot physically exist. But in the world of epistemology we can talk about sets of all true propositions (something also potentially infinite).

Perhaps there is a similar issue here where we are attempting to define truth from an absolute perspective, when really it is about information X can have about Y, as it has to be in physics.

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 learn that "p was an unknown truth" up untill that day. — Olivier5

a. "p" is an unknown truth
b. "p" was an unknown truth

These are not the same proposition.

According to the knowability principle, if a proposition is true then it is possible to know that proposition. Therefore, if a is true then it is possible to know a. You've only argued that we can know b. Knowing b is not the same as knowing a.

A proposition can be true within a certain period of time and false outside of that period. For instance the proposition: 'Now summer is back' is true at the start of summer; 'Socrates is alive and well' was true untill what? 400BC? Then it became false after his death.

Your proposition a. can likewise be true untill such a time when it becomes false.

Your proposition a. can likewise be true untill such a time when it becomes false. — Olivier5

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.

As Fitch's paradox shows, a isn't knowable. Therefore, according to the knowability principle, a isn't true.