I guess this intriguing paradox has something to do with Meno's paradox:

Either I know what I'm inquiring about or I don't know what I'm inquiring about.

If I know what I'm inquiring about then inquiry is unnecessary.

If I don't know what I'm inquiring about then inquiry is impossible.

Ergo,

Either inquiry is unnecessary Or inquiry is impossible.

The other point that's up for discussion is that somewhere in Fitch's argument, K(P) $\to$ KP where K(P) means P is knowable and KP means Known that P. Feels like an illegal move to me. An example: I know that calculus is knowable, but that hasn't helped me at all, I haven't the slightest clue what calculus is about. :snicker:

Thanks for your responses.

The other point that's up for discussion is that somewhere in Fitch's argument, K(P) → KP where K(P) means P is knowable and KP means Known that P. Feels like an illegal move to me. — Agent Smith

I'm not arguing along these lines, but I would be interested in an argument for it.

An example: I know that calculus is knowable, but that hasn't helped me at all, I haven't the slightest clue what calculus is about. :snicker: — Agent Smith

I'm not sure I would agree. As the WIki article notes, p is a sentence or a proposition. Such sentences are typically truth apt. I don't consider a field of study, such as calculus, to fit the bill of a truth-apt proposition.

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.

Fitch's "paradox" of knowability — Luke

How is this any different than the liar's sentence: "This sentence is false?" It's a grammatically correct sentence that no one would ever speak in real life. Or can you think of a reason for anyone but a philosopher, a13-year-old boy, or a 13-year-old philosopher to say or write it. We've discussed that many times here on the forum. My conclusion - self-referential "paradoxes" are just word games with no intellectual or philosophical significance.

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. — Michael

I would say that we (now) know both of these statements, particularly since you have stated them. However, Fitch's argument speaks only of our knowledge - or lack thereof - of true statements.

The argument says that if it is possible to know a true p, then we must know that p is true. I argue that this is not due to our knowledge of p's truth, but due to our knowledge of p: If it is possible to know a true p, then we must know what p states. That is, my point is that the argument equivocates on knowledge of p (i.e. knowing what p states) and knowledge of p's truth value. Btw, I don't disagree with the argument's conclusion. I just don't see it as implying that we must know the truth of any proposition. The argument refers only to those propositions that are true in the first place.

Therefore if we insist on the knowability principle then we must accept that every true statement is known to be true. — Michael

I'm curious: do you reject the knowability principle or do you believe that the argument's conclusion endows us with knowing the truth value of every proposition? I don't reject the knowability principle. On what grounds would you?

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 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.

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

That's not Fitch's argument, which assumes the truth of p. It's not that we don't know which statement is true (and which is false); it's that we don't know the statement that is true. So, which of those statements (about the box) is true?

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.

However, the knowability principle entails that if it is true then I know that it is true, which contradicts the non-omniscience premise. — Michael

My view is that if it is a true statement, then it cannot be unknown that it is a true statement (see the Wiki quote again). And that's because in order for it to be possible to know that it is true, we must first know the statement and what it means. If it is possible to know that p is true, then we must know that p (is true), And the truth of the statement is presupposed.

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.

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. — Michael

I think the argument implies that every known true statement is known to be true:

,,,as soon as we know "p is an unknown truth", we know that p is true, rendering p no longer an unknown truth, so the statement "p is an unknown truth" becomes a falsity. Hence, the statement "p is an unknown truth" cannot be both known and true at the same time. Therefore, if all truths are knowable, the set of "all truths" must not include any of the form "something is an unknown truth"; thus there must be no unknown truths, and thus all truths must be known. — Fitch's paradox of knowability

As I said in the OP, this excludes all unknown statements and statements with unknown truth values.

Logic is really bad at doing time. Truths have to be eternal. That p is an unknown truth is unknowable until p is known, and then it is not an unknown truth. the difficulty arises because knowability implies time.

Suppose p is a sentence that is an unknown truth — Fitch's paradox of knowability

This is the heart of darkness - suppose we know something that we suppose we do not know. "the 79 squillionth decimal iteration of pi is a '2'." Well do we know or don't we? Make up your mind, Fitch. The digit is knowable, but 'that it it 2' is knowable only if it happens to be 2, which we don't know. p0, p1... p9 - one of them is an unknown truth, and the others are unknown falsehoods.

Suppose what you cannot even in principle know... arrive at a paradox... everyone gasps at your cleverness.

As one of the ol’ muppet dudes on the balcony says to the other.....BRILLIANT!!!

Or.....how to take the pristine condition of human reason, and turn it against itself.

Logic is really bad at doing time. Truths have to be eternal. That p is an unknown truth is unknowable until p is known, and then it is not an unknown truth. the difficulty arises because knowability implies time. — unenlightened

I agree with you completely on this.

This is the heart of darkness - suppose we know something that we suppose we do not know. "the 79 squillionth decimal iteration of pi is a '2'." Well do we know or don't we? Make up your mind, Fitch. The digit is knowable, but 'that it it 2' is knowable only if it happens to be 2, which we don't know. p0, p1... p9 - one of them is an unknown truth, and the others are unknown falsehoods. — unenlightened

That we do and/or do not know something is not about the same sort of temporal possibility/knowability that you describe above. To "suppose we know something that we suppose we do not know" just seems like a pure contradiction.

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.

2. the truth value of p is unknown — Michael

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

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.

Suppose p is a sentence that is an unknown truth; that is, the sentence p is true, but it is not known that p is true. — Fitch's paradox of knowability

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

That's what I'm disputing about the argument. This is the equivocation I'm talking about.

Must get to bed. I'll respond further tomorrow.

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.

Proposition 1 only says that it is possible to know that the Riemann hypothesis is true. It doesn't state that it is always possible.

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.

You'd need 1 to be "so long as the Riemann hypothesis is true then it is always possible to know that the Riemann hypothesis is true". Otherwise you're left without 3 necessarily following.

Ok.

What are your views on K(P) $\to$ KP?

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.

If it is not possible to know that the Riemann hypothesis is true despite it being true then the knowability principle is refuted. — Michael

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.

I think calling something an "unknown truth" is a fallacy or just wrong, since truth is that which is in accordance with fact or reality. So, either we know that something is true or false or we cannot say anything about its truthness or falseness.

Then, "if all truths are knowable" is meaningless because truth is something known by definition!
Besides that, it is also an arbitrary assumption or hypothesis that looks like being used to serve supporting the above mentioned fallacy or wrong statement.

Therefore, I consider the whole construct as unfounded.

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.

That does not appear anywhere in the argument.