There exists a being x and a time t such that x knows at t that proposition p is true: ∃x∃t(Kxtp)

1. p → ◊∃x∃t(Kxtp)
2. p ∧ ¬∃x∃t(Kxtp)
3. ◊∃x∃t(Kxt(p ∧ ¬∃x∃t(Kxtp)))

4. ∃x∃t(Kxtp ∧ Kxt(¬∃x∃t(Kxtp)))

There exists a being x and a time t such that x knows at t that proposition p is true and knows at t that there doesn't exist a being x and a time t such that x knows at t that proposition p is true. This is a contradiction. Therefore 3 is false. Therefore either 1 or 2 is false.

Admittedly this doesn't entail that every true statement is currently known to be true, only that every true statement is known to be true at some point, but that might also be an undesirable conclusion. It's possible that the Riemann hypothesis is never proved nor disproved.

See here for an explanation in ordinary language.

This is true, but Fitch's paradox is self-referential. Actually, after looking at it more, including SEP, I'm not sure it is. It seems more like a tautology, or at least a trivial statement, a language game. Calling a particular statement a truth means the same thing as saying it is true. If I know something is true, it isn't unknown. — T Clark

Where's the language game here?

1. p → ◊Kp (knowability principle)
2. q ≔ the Riemann hypothesis is correct
3. r ≔ the Riemann hypothesis is not correct
4. q ∨ r (law of excluded middle)
5. ¬Kq ∧ ¬Kr (whether or not the Riemann hypothesis is correct is not known)

6. (q ∧ ¬Kq) ∨ (r ∧ ¬Kr) (from 4 and 5)
7. q ∧ ¬Kq → ◊K(q ∧ ¬Kq) (from 1)
8. r ∧ ¬Kr → ◊K(r ∧ ¬Kr) (from 1)
9. ◊K(q ∧ ¬Kq) ∨ ◊K(r ∧ ¬Kr) (from 6, 7, and 8)

10. K(s∧ ¬Ks) (assumption)
11. K(s ∧ t) ⊢ Ks ∧ Kt (knowing a conjunction entails knowing each of the conjuncts)
12. Ks ∧ K¬Ks (from 10 and 11)
13. Kt ⊢ t (knowledge entails truth)
14. Ks ∧ ¬Ks (from 12 and 13)

14 is a contradiction, therefore 10 isn't possible, therefore 9 is false, therefore either 1 or 5 is false.

By everyone always, or by someone at some time? — Luke

By someone at some time.

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

In fact the opposite: Kp → ◊Kp.

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? — Luke

No, because if you address the formal logic of the argument you will see that it entails a contradiction:

a. p → ◊Kp (knowability principle)
b. p ∧ ¬Kp (some proposition that is true but not known to be true)
c. b → ◊Kb (apply the knowability principle to b)
d. p ∧ ¬Kp → ◊K(p ∧ ¬Kp) (substitute in the terms of b)

However, K(p ∧ ¬Kp) is a contradiction, and so isn't possible, as shown below:

e. K(p ∧ ¬Kp) (assumption)
f. K(p ∧ q) ⊢ Kp ∧ Kq (knowing a conjunction entails knowing each of the conjuncts)
g. Kp ∧ K¬Kp (from e and f)
h. Kp ⊢ p (knowledge entails truth)
i. Kp ∧ ¬Kp (from g and i)

i is a contradiction. We cannot know that p is true and not know that p is true. Therefore d is false. Therefore either a (the knowability principle) or b (there is some unknown truth) is false.

So there are unknown truths? — Luke

In reality, yes. However, Fitch's paradox shows that the knowability principle entails that there are no unknown truths. That's why Fitch's paradox shows that the knowability principle is false.

Not according to Fitch's argument. — Luke

Technically speaking Fitch's argument shows that the knowability principle entails that all truths are known. This conclusion is then a reductio ad absurdum to disprove the knowability principle, given that there are unknown truths.

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

OK. But it's still the case that the argument shows that, given the knowability principle, all truths are known.

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

Therefore, the knowability principle fails.

If there are no unknown truths then only known truths are known. — Luke

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

Someone, somewhere, at some point in time has some knowledge. — Olivier5

OK. This has nothing to do with Fitch's paradox.

You disagreed with my claim that the argument implies only that known truths are known. — Luke

The argument shows that if we assume p → ♢Kp then p → Kp follows.

Kp → Kp is a truism that doesn't need Fitch's paradox to prove.

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.

To be known is NOT a quality intrinsic to things, therefore 'an unknown truth' or a 'known truth' have no clear meaning. — Olivier5

By "known truth" I mean "a proposition that someone knows to be true" and by "unknown truth" I mean "a proposition that no-one knows to be true."

Is "either the Riemann hypothesis is correct or the Riemann hypothesis is not correct" a known truth or an unknown truth? — Luke

A known truth.

You've said that that's a known truth, but you've also used this to argue that not all truths are known. — Luke

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

On the other hand, it is unknown which one is true — Luke

Which is precisely the point. Fitch's paradox entails that we do know which one is true. Given that we don't know which one is true me must reject the knowability principle.

Well, I'm saying that the argument implies only that known truths are known — Luke

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

Therefore, NonO is rejected and hence all truths must be known. — Luke

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.

The non-omniscience principle is the principle that there is some proposition p that is true and that we don't know to be true. Either "the Riemann hypothesis is correct" or "the Riemann hypothesis is not correct" is one such proposition (as per the law of excluded middle, and given that the Riemann hypothesis has neither been proven nor disproven). Fitch's paradox shows that if a proposition p is true iff it is possible to know that p is true then it follows that either we know that "the Riemann hypothesis is correct" is true or we know that "the Riemann hypothesis is not correct" is true; that there is no proposition p that is true and that we don't know to be true.

And someone else at another time would have a different knowledge. So there's no such thing as 'a truth known', or 'a truth unknown', in the absolute. It all depends on who does the knowing and when. — Olivier5

This has no bearing on Fitch's paradox.

Known by whom, and when? — Olivier5

Us, now.

What if the Riemann hypothesis is false? Then we do not reject 1. It is not enough that we don't know whether p is true; it must also be true. "p" means/entails "p is true". This is where the equivocation lies. — Luke

p is "the Riemann hypothesis is true". q is "the Riemann hypothesis is false". Either p or q is true and neither p nor q is known to be true. Therefore, either p∧¬Kp or q∧¬Kq. Then applying the knowability principle, either ◊K(p∧¬Kp) or ◊K(q∧¬Kq). Both are contradictions.

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).

¬Kp could mean that we don't know the content/meaning of p and/or that we don't know the truth of p; that we don't know the Riemann hypothesis and/or that we don't know that it is true. — Luke

In the context of Fitch's paradox it means that we don't know the truth of p.

it makes no sense to say that a proposition no one knows about is true — Olivier5

As I said to Luke, this isn't what Fitch's paradox is (necessarily) saying. It's saying that there is some proposition that is not known to be true. That's not the same thing. For example, the Riemann hypothesis is not known to be true. The paradox can be applied to this single proposition (see my next comment).

Under idealism something needs to hold all thought or 'reality' together for us to have regularities — Tom Storm

Does it? On the opposite view, what is needed to hold all material things together for them to have regularities?

Sometimes I use the (fuzzy) term "common logic" — Alkis Piskas

The fuzzy term "common logic" but not the term "fuzzy logic"?

Sorry, couldn't help myself. :wink:

However, this seems a well reasoned reassurance:

Why Other Fundamental Rights Are Safe (At Least for Now)

It's also clear that Griswold, Obergefell, Lawrence, & Loving should be codified — Maw

Yeah. Got to get rid of the filibuster rule first. :up: — 180 Proof

The problem with that is that it's relatively easy to overturn a law. Even if the Democrats are able to pass a federal law to protect these rights, when the Republicans are next the majority they'll just repeal it. Such rights need to be constitutional rights.

Yes, his solution is probably the one I'm most partial to. See here.

I think it's trivially true that the knowabilty principle cannot apply to propositions about our own knowledge — Isaac

Yes, I was considering the same sort of thing. I think this kind of self-referential knowledge is victim to the same problems as other self-referential knowledge/truth claims like the Liar Paradox. Technically speaking, if all meaningful propositions have a truth value and if "this statement is false" is a meaningful proposition then we have a contradiction. But is it really a problem to say that all meaningful propositions except propositions like "this statement is false" have a truth value? Or is that special pleading?

Perhaps we can say (as me and @Banno discussed in the other thread) that empirical truths are subject to the knowability principle, but that the truth of self-referential knowledge claims, counterfactuals, predictions, mathematics, etc. work differently?

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. — Luke

I think you're the only one guilty of equivocation here. In the context of the argument, Kp means "it is known that the statement p is true". It does not mean "the statement p is known of" or "the meaning of statement p is known".

the problem seems trivially solved by saying that some proposition exists for which it is not possible to know the truth. — Isaac

Then that's a denial of the knowability principle. The problem is that if you insist on the knowability principle then the only other way to avoid a contradiction is to deny the non-omniscience principle (i.e. to accept that every true proposition is known to be true).

It doesn't make any difference expressed in notation. 3 does not follow from 1 and 2. — Isaac

Maybe if I make it clearer you can see:

1. p→◊Kp (knowability principle)
2. q ≔ p∧¬Kp (define q as something that is true but not known to be true)
3. q→◊Kq (apply the knowability principle to q)
4. p∧¬Kp→◊K(p∧¬Kp) (substitute in the definition of q)

But I'm saying it is possible to know that the RH is true (just not at the same time as knowing that we don't know it's true). In other words, it is generally possible to know that the RH is true (your 1), but not in all circumstances (ie not whilst your 2 is the case). The fact that there exists a circumstance under which something is impossible, doesn't mean that that something is impossible in general. — Isaac

1. p→◊Kp
2. ¬Kp
3. p∧¬Kp→◊K(p∧¬Kp)

The logic is straightforward and results in a contradiction.

Therefore 2 could be one of the cases where it is not possible to know that the Riemann hypothesis is true despite it being true. — Isaac

The knowability principle is the principle that a statement is true if and only if it is possible to know that the statement is true. If it is not possible to know that the Riemann hypothesis is true despite it being true then the knowability principle is refuted.

That's what I'm disputing about the argument. — Luke

Then I will offer a specific example of p:

1. if the Riemann hypothesis is true then it is possible to know that the Riemann hypothesis is true
2. we don't know that the Riemann hypothesis is true
3. if the Riemann hypothesis is true and we don't know that the Riemann hypothesis is true then it is possible to know that the Riemann hypothesis is true and that we don't know that the Riemann hypothesis is true

The conclusion is a contradiction, and so we must reject either 1 or 2.

It's not the truth value of p which is unknown, because we know that p is true. — Luke

We don't know that p is true in this case.

I think the argument implies that every known true statement is known to be true, As I stated in the OP, this excludes all unknown statements and statements with unknown truth values. — Luke

No, it shows that every true statement is known to be true. I explained this here. I'll try to be even clearer now:

1. if p is true then it is possible to know that p is true
2. the truth value of p is unknown
3. if p is true and the truth value of p is unknown then it is possible to know that p is true and that the truth value of p is unknown (from 1)

3 is a contradiction. I can't know that p is true and know that the truth value of p is unknown. It must be one or the other. Therefore we must reject either 1 or 2.

If it is possible to know that p is true, then we must know that p (is true) — Luke

Yes, and as the knowability principle is the principle that p is true if it is possible to know that p is true it then follows from what you say here that every true statement is known to be true. That's Fitch's paradox.

it's that we don't know the statement that is true — Luke

No it isn't. The non-omniscience premise of the argument is that there is some statement p that is not known to be true. We might very well know of the statement, and what it means, just not its truth value. "The box is empty" is one such example. I know of it, I know what it means, but I don't know if it's true. However, the knowability principle entails that if it is true then I know that it is true, which contradicts the fact that I don't know if it's true.

I don't reject the knowability principle. On what grounds would you? — Luke

On the grounds that we can't know both that p is true and that we don't know that p is true. That's a contradiction.

I would say that we (now) know both of these statements, particularly since you have stated them. — Luke

But we don't know which of the statements is true, which means that we must reject the knowability principle.

The argument says that if it is possible to know a true p, then we must know that p is true. — Luke

The argument is that if it is possible to know that p is true then we must know that p is true.

I don't see how that addresses the paradox.

Assuming the law of non-contradiction and the law of excluded middle, either "the box is empty" is true or "the box is not empty" is true. According to the knowability principle, a statement is true if it can be known to be true, and so either we can know that "the box is empty" is true or we can know that "the box is not empty" is true. Now assume that we don't know which of the two is true. From this, either "the box is empty" is true and we don't know that it's true or "the box is not empty" is true and we don't know that it's true.

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.

Given this contradiction we must either reject the knowability principle or accept that we know which of "the box is empty" and "the box is not empty" is true. And we must do this for every statement and its negation. Therefore if we insist on the knowability principle then we must accept that every true statement is known to be true.

However, something must separate the two things from each other, or else they would be only one thing. And, the logic of mathematics would be rendered useless in that way, as well. As I explained above, that which separates them cannot be a third thing. Therefore we need to employ a dualism to understand the existence of independent things. Aristotle resolved this type of logical dilemma with hylomorphism, a type of dualism. — Metaphysician Undercover

I don't understand what this is saying at all.

Less formally the impulse is that if idealism is true, and hence only minds and mental phenomena exist, then all that can be true must be apparent to a conscious mind. — Banno

I think it more accurate to say that if idealism is true then all true statements of the form "p exists" is apparent to a conscious mind, which doesn't require Fitch's paradox to show as it seems to be quite explicit in the idealist's position. — Michael

Actually, thinking on it more, even this might not be correct. Consider the statement "there exists more than one mind" (or even the more specific "there exist n minds"). Such a statement is about minds and mental phenomena, unproblematically has a truth value (unlike counterfactuals, predictions, and mathematics), but can be true even if it isn't apparent to a conscious mind.

Less formally the impulse is that if idealism is true, and hence only minds and mental phenomena exist, then all that can be true must be apparent to a conscious mind. — Banno

I think it more accurate to say that if idealism is true then all true statements of the form "p exists" are apparent to a conscious mind, which doesn't require Fitch's paradox to show as it seems to be quite explicit in the idealist's position.

The problem with the more ambiguous conclusion that all true statements are apparent to a conscious mind is that without clarification it would appear to cover such statements as statements about the future, counterfactuals, mathematics, and so on, which personally I believe can be true even if they do not correspond to some entity that exists (e.g. a counterfactual can be true even without actual parallel worlds, predictions can be true even if eternalism isn't the case, mathematics can be true even if mathematical anti-realism is correct, and so on).