Wouldn't it then just be "it might be known that there are truths that are not known" rather than " It might be known that there is a truth that is not known" ? Is there a salient difference? — Janus

I'm not sure if I'm following you, but I'm seeing a problem with this:

(2) If there is a truth that is not known, then it might be known that there is a truth that is not known
....(sub (1) into KP) — Banno

It seems to me that 'an unknown truth' cannot legitimately be formalised as p but only as (p or ~p) Is that right? does it make sense?

That is to say that I know that there is an umpteenth digit of pi, and can say so, but I cannot say that any particular one of the statements p0 -p9 is true, but only one of all of them. that is what it means for the truth to be unknown.

It seems to me that 'an unknown truth' cannot legitimately be formalised as p but only as (p or ~p) Is that right? does it make sense? — unenlightened

The formal definition is ∃p(p ∧ ¬Kp): there exists some proposition p that is true and not known to be true.

For example, either "the Riemann hypothesis is correct" is true and not known to be true or "the Riemann hypothesis is not correct" is true and not known to be true, and so either "the Riemann hypothesis is correct" is an unknown truth or "the Riemann hypothesis is not correct" is an unknown truth.

It seems to me that 'an unknown truth' cannot legitimately be formalised as p but only as (p or ~p) Is that right? does it make sense? — unenlightened

Sounds interesting. Does the proof do this?

Seems to me it uses (~Kp) for "p is not known".

(p v ~p) is p is true or false, not known or unknown.

. if the Riemann hypothesis is true and we don't know that the Riemann hypothesis is true then it is possible at some point in the future to know that the Riemann hypothesis is true and that we don't didn't know today that the Riemann hypothesis is was true — Michael

Fixed.

There exists some proposition p that is both true and not known to be true — Michael

Yes. I am questioning the legitimacy of that. It seems to be stating a contradiction by asserting p and claiming it to be unknown. If I substitute (p0 or p1 ... or p9) then it is not unknown, but on the contrary that is what is known.

I am questioning the legitimacy of that. — unenlightened

But that's the non-omniscience principle? Without it we must accept that every true proposition is known to be true – which is what Fitch's paradox shows follows from the knowability principle.

It seems to be stating a contradiction by asserting p and claiming it to be unknown. — unenlightened

It's not a contradiction to say "there is intelligent alien life but I don't know that there is." Such a statement is possibly true.

But that's the non-omniscience principle? Without it we must accept that every true proposition is known to be true – which is what Fitch's paradox shows follows from the knowability principle. — Michael

I don't think so. I think the principle needs to be formalised differently, as I indicated.

It's not a contradiction to say "there is intelligent alien life but I don't know that there is." Such a statement is possibly true. — Michael

I think it is a contradiction, because it asserts something and denies that it is known. "Either there is intelligent alien life or there isn't, but I don't know which." -- that makes sense.

I don't think so. I think the principle needs to be formalised differently, as I indicated. — unenlightened

That doesn't work. p ∨ ¬p just means "p is true or p is false" and says nothing about what we know.

For example: either my name is Michael or my name is not Michael. This statement is true, but doesn't say that I don't know my name.

I think it is a contradiction, because it asserts something and denies that it is known. "Either there is intelligent alien life or there isn't, but I don't know which." -- that makes sense. — unenlightened

We are able to assert things we don't know. We can make arguments with premises we either don't know or believe to be false, e.g.:

1. My name is Andrew
2. If my name is Andrew then my name is not Michael
3. My name is not Michael

The argument is valid.

1. There is intelligent alien life
2. If there is intelligent alien life then humans are not the only intelligent life
3. Humans are not the only intelligent life

The argument is valid.

Can we stick with the umpteenth digit of pi, instead of the names thing?

So my suggestion is that the non-omnicience principle should go something like:

(p or ~p) and ~Kp and ~K~p.

Can you work with that a little and see how it goes? (My formal logic is fifty years faded)

(p or ~p) and ~Kp and ~K~p.

Can you work with that a little and see how it goes? — unenlightened

1. (p ∨ ¬p) ∧ ¬Kp ∧ ¬K¬p
2. (p ∧ ¬Kp ∧ ¬K¬p) ∨ (¬p ∧ ¬Kp ∧ ¬K¬p)
3. (p ∧ ¬Kp) ∨ (¬p ∧ ¬K¬p)

4. q → ◇Kq (knowability principle)

5. p ∧ ¬Kp → ◇K(p ∧ ¬Kp) (contradiction)
6. ¬p ∧ ¬K¬p → ◇K(¬p ∧ ¬K¬p) (contradicton)

Right. Epic fail, unenlightened.

3. (p ∧ ¬Kp) — Michael

3. (p ∧ ¬Kp) ∨ (¬p ∧ ¬K¬p) — Michael

In fact from this I'm pretty sure it follows that ∃q(q ∧ ¬Kq), so we're back to the initial formalism.

In fact from this I'm pretty sure it follows that ∃q(q ∧ ¬Kq) — Michael

Well if I am forced to say that because we are not omniscient, there are things we cannot know, I might be able to live with that, at a pinch.

Wait, it doesn't say that, though, it says there is something we don't know, Sorry, brain overheating and I am confused between unknown and unknowable. Need to lie down in a darkened room.

Fitch's paradox shows that if all truths are knowable then all truths are known. Some truths aren't known, therefore some truths aren't knowable.
— Michael

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

The basic idea is that if there's a truth that isn't known then that implies a related truth that isn't knowable.

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. Why? Let's assume that someone comes to know that the conjunctive statement is true. That implies that they know that t is true. But that then renders the second conjunct false. The conjunctive statement is therefore false and so not known to be true, which contradicts our initial assumption. So it's not possible to know that the conjunctive statement is true. It's an unknowable truth.

The only way to avoid such unknowable truths is for there to be no unknown truths (i.e., for all truths to be known). That is, for all for truths to be knowable implies that all truths be known.

The only way to avoid such unknowable truths is for there to be no unknown truths (i.e., for all truths to be known). That is, for all for truths to be knowable implies that all truths be known. — Andrew M

Why not just accept unknowable truths?

Why not just accept unknowable truths? — Tate

Well, one might prefer to think they are omniscient. ;-)

More seriously, presumably people who have considered Fitch's paradox do accept that. But from Wikipedia:

The paradox is of concern for verificationist or anti-realist accounts of truth, for which the knowability thesis is very plausible,[1] but the omniscience principle is very implausible. — Fitch's paradox of knowability - Wikipedia

More seriously, presumably people who have considered Fitch's paradox do accept that. But from Wikipedia:

The paradox is of concern for verificationist or anti-realist accounts of truth, for which the knowability thesis is very plausible,[1 — Andrew M

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

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.

Deflationary accounts of truth are apt to be anti-realist — Tate

Not according to many here.

Not according to many here. — Michael

A typical deflationist will say that truth only serves a social function. Is someone disagreeing with that?

Perhaps I am missing something, but isn't this problem resolved by declaring that propositions are fixed implicitly in the time of their utterance (unless otherwise modified)?

This same sort of issue shows up in Aristotle with talk about the true/false values of past truths that no longer maintain. For example, "the Colossus of Rhodes is standing," is a proposition that had a truthmaker at one point, but that truthmaker disappeared when the statue fell over. Likewise, the proposition that "p is an unknown truth," is a negative claim about knowledge, and so it has a corresponding falsemaker that ceases to maintain when the truth of p is discovered. By allowing truth values to change over time you solve the problem.

People were unhappy with this storm of changing truth values, but it has, perhaps, been rectified with the idea of possible worlds. We start off with a set of all possible worlds, all those that aren't logically contradicted. As time progresses, the number of worlds consistent with actual events is winnowed down, and so changing truth values is really just the winnowing of possible worlds. Although, I'm not sure you even need possible worlds, you could also just have a set of all truths that has a time stamp on when a proposition was uttered.

It seems to me that the problem only holds up under a narrow set of assumptions. Let's look at how it fares under a few possible viewpoints:

1. Presentism holds that the past and future lack existence. In this case, the unknown truth could be in the set of all truths but it would cease to be as soon as someone knows about p. But really though, the past doesn't actually exist, so the set of all truths never has the contradictory overlap, fixing our problem.

2. This is no issue for eternalism as the future is as existent as the past or present (e.g., block time universes), so if p is ever known, it is not an unknown truth, since the future already exists.

3. For many forms of actualism (i.e. actual occurrences exist, modal truths do not) it seems like this is just the regular occurrence of actual events narrowing the horizon of all possible worlds consistent with actual true propositions. So the "p being an unknown truth" worlds just get shifted from the possible to impossible side of our possible worlds ledger.

There is also the information theoretic approach in which the primary ontological entity is information, that is, propositions. But many of these propositions are "derived" propositions. The only fundemental propositions are about fundemental particles/field excitations as related to each other in space and time. In this view, seeming contradictions are just the result of error and data compression. Broad, high level, derived propositions are multiply realizable because they are compressing information and dropping a lot of it. But in reality, this isn't causing contradictions, the problem is simply that multiple informational microstates are consistent with the truth value of a single macro-proposition.

Thankfully, information has this protean character where it can take multiple forms, and reencoding of information (with relative amounts of compression and error) in forms of self-similarity at different scales (fractal recurrence) allows us to make these derived propositions with some degree of accuracy, but we shouldn't take derived propositions as having ultimate truth values in terms of contradiction as they are multiply realizable.

But in these systems, you're also still talking about truth values given a certain slice of time (generally, most I've seen tend to be actualist or eternalism).

See here for one such discussion we had.

A typical deflationist will say that truth only serves a social function. Is someone disagreeing with that? — Tate

I would think that truth has an important psychological dimension; we (minds) cannot exist without a concept of truth. Even folks who think they have 'deflated truth' do in actual fact believe that it is true that they have deflated truth. They don't usually believe that they have come to some social agreement to pretend that truth was deflated.

Even folks who think they have 'deflated truth' do in actual fact believe that it is true that they have deflated truth — Olivier5

Right. They just don't believe that "is true" adds anything except emphasis.

Is that true, though?

Is that true, though? — Olivier5

Of course. I think you're misunderstanding. A deflationist does not have a problem with using the word "true" in the normal way. She just resists piling unwarranted projects on top of that normal usage.

She would say we shouldn't be bewitched by
language.

She would say we shouldn't be bewitched by language. — Tate

A most bewitching Wittgensteinian proposition... both recognising the power that a language holds upon individuals speaking it, warning them against it in fact, but then offering no practical method to deliver them from the spell of their spell.

I will agree that one needs to use words with care, that they are 'treacherous' in some sense. But there are solutions, such as the pragmatic, instrumentalist approach: any given problem will require a certain degree of precision in language for its resolution, much beyond which it is useless to go. That's similar to the standard practice in math and physical sciences.

Its downside is of course that its very practicality misses on the creative, poetic and polysemic virtues of language. The bewitching can be a feature, in an explorative way. But then, even Witty's defiance towards language is perhaps missing on that, on the accidental creativity of the bewitching.

Anyway, this is an aside.

A deflationist does not have a problem with using the word "true" in the normal way. She just resists piling unwarranted projects on top of that normal usage. — Tate

Would you have examples of these unwarranted projects?

Would you have examples of these unwarranted projects? — Olivier5

If truth is a property of statements, talk of "unknown truths" might give us unstated statements. Not good.

If truth is a property of statements, talk of "unknown truths" might give us unstated statements. Not good. — Tate

Excellent. I made a similar remark on a related thread sometime back, about yet unproposed propositions.

Although I think @Michael avoids this specific objection to Fitch by chosing as an example a mathematical hypothesis already stated (Riemann's) but whose truth value is yet unknown.

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

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

The anti-realist Wikipedia mentioned would just say we're using figures of speech when we say that.