You're avoiding the questions requesting the definition of the terms you're using but fail to provide any.

What does it mean to be omniscient vs non-omniscient? Don't you have to define knowledge to make sense of that distinction?

Does being non-omniscient mean that we know nothing or that we don't know everything? If the latter then how do we know that what we do know is true? If the former then knowledge is meaningless. — Harry Hindu

We address the problem in formal logic. We start with the two premises that the anti-realist accepts:

Knowability principle
∀p(p → ◊Kp)

Non-omniscience principle
∃p(p ∧ ¬Kp)

We then apply the accepted rules of inference to derive the conclusion:

All truths are known
∀p(p → Kp)

So the anti-realist must reject either the knowability principle or the non-omniscience principle.

Fitch isn't interested in a drawn out debate on what the anti-realist means by knowledge; he's only interested in the internal consistency of their position. So whatever it is they mean by knowledge he shows that their position entails that all truths are known. The anti-realist then has to either accept that or abandon their knowability principle.

Is (2) both true and false? — Luke

No. It was true before we knew 1 and false after.

Is (3)? — Luke

No. It was true before we knew 1 and false after.

A is an assertion of knowledge — Harry Hindu

In practice it may be that asserting a proposition implies that one believes one's assertion (see Moore's paradox), but in formal logic there is a distinction between asserting that a proposition is true and asserting that a proposition is known to be true.

Regardless, your comments have nothing to do with Fitch's paradox. The non-omniscience principle states that ∃p(p ∧ ¬Kp). However, ¬◇K(p ∧ ¬Kp). Therefore, ¬(p → ◇Kp). The knowability principle is false.

I don't understand what your comments have to do with anything. If we accept the non-omniscience principle then there is some p which is true and not known to be true, so we address:

1. p is true, and
2. p is not known to be true,
3. therefore, p is true and p is not known to be true

It's not possible to know 3, therefore the knowability principle is false.

We don't need to know a real example of p for the logic to work.

Why was A stated in the first place? How us possible to positively assert that which is not known? — Harry Hindu

I might believe it to be so? e.g. intelligent alien life exists, the real part of every nontrivial zero of the Riemann zeta function is 1/2, and it will rain tomorrow.

But to be more formal, it follows from the non-omniscience principle ∃p(p ∧ ¬Kp) that there is some p such that:

1. p is true, and
2. p is not known to be true

We might not know what specific p satisfies this criteria, but that's irrelevant. It is not possible to know p ∧ ¬Kp and so therefore the unrestricted knowability principle is false.

If knowing 2 makes 1 unknown, then how is 1 knowable?

That is, if 'the cat is on the mat' is true (as a result of 1) AND unknown (as a result of 2), because of the relationship between 1 and 2, then how can 1 be knowable? — Luke

It's knowable because we can look for the cat and see it to be on the mat. In doing so, what was once an unknown truth (1) is now a known truth and what was once a known truth (2) is now a known falsehood. And what was once an unknown truth (3) is now a known falsehood.

3 can never be a known truth.

Why is 1 an unknown truth? It could equally be a known truth. — Luke

It’s an unknown truth because 2 says so. Do you not understand than an argument can have more than one premise? Your reasoning here is ridiculous.

If p is an unknown truth, then it is represented by "p ∧ ¬Kp".

It's that simple. — Luke

You just don't understand symbolic logic, so address the argument in natural language.

1. the cat is on the mat
2. it is not known that the cat is on the mat
3. the cat is on the mat and it is not known that the cat is on the mat

1 is an unknown truth but is knowable
2 is a known truth
3 is an unknown truth and is not knowable

Likewise.

Every truth ("p") is either known ("p & Kp") or unknown ("p & ~Kp"). There are no other known or unknown truths.

Your mistake (and mine, too, previously) is in thinking that a truth either mentions that it is unknown or does not. However, the expression "p & ~Kp" does not "mention" that it is unknown. Instead "p & ~Kp" represents that p is true AND unknown; "p" represents only that p is true; and "p & Kp" represents that p is true AND known. This accounts for all known and unknown truths.

If there is some other way to express that p is both true AND unknown, then I welcome you to provide that expression. — Luke

p means "the cat is on the mat"
¬Kp means "it is not known that the cat is on the mat"
p ∧ ¬Kp means "the cat is on the mat and it is not known that the cat is on the mat"

p is an unknown truth but is knowable
¬Kp is a known truth
p ∧ ¬Kp is an unknown truth and is not knowable

It's that simple.

It is unknowable that p is true and that somebody knows p is true? Why is it unknowable?

You claim that "p" can be unknown and knowable.

But if all truths are expressible as 1. and 2. above, then what other "p" is there? Where is this knowable unknown truth? — Luke

a the cat is on the mat
b nobody knows that the cat is on the mat

Both a and b are true. This means that, even though a doesn't say so about itself, a is an unknown truth. Compare with:

c Michael is a man
d Michael is 34 years old

Even though c doesn't say so, it is about a 34 years old. When presented with both c and d it doesn't make sense to say that Michael's age is ambiguous because c doesn't say anything about Michael's age. It doesn't matter what c says about Michael's age because d provides that information.

And by the same token, it doesn't matter what a says about whether or not it is known that the cat is on the mat (it says nothing about knowledge) because b provides that information.

So with that in mind, given the truth of b it then follows that a is an unknown truth even though a doesn't refer to itself as being unknown.

Now, it is possible to know a and it is possible to know b, but as Fitch's paradox shows, it isn't possible to know the conjunction a ∧ b even though the conjunction a ∧ b is true, thereby showing that the (unrestricted) knowability principle is false (there is at least one truth that is impossible to know).

Contraceptive failure rates are negligible and most pregnancies that are aborted are not due to this reason. It shows (some) women have no respect for life.

As for rape pregnancies, what are the stats on that? I'm fairly certain that only a handful of abortion requests are for rape pregnancies. — Agent Smith

Who cares about the rate? They happen, and so abortions should be allowed to account for them.

The availability of cheap contraceptives implies that abortion isn't necessary for responsible women of child-bearing age. If I don't want an omelette, I shouldn't break an egg. To break an egg, make an omelette and then throw it away is being mean, not only to the egg, but to yourself as well. — Agent Smith

Contraceptives fail. Rape happens.

I was simply amazed at why women would not care about being thought of as murderers (even if that were to be false) when they could easily nip the problem in the bud by taking pills/asking their partners to use condoms/etc.? — Agent Smith

Did you not read what I said? Contraceptives fail. Rape happens.

If all truths can be expressed as either:

1. p ∧ Kp [known]; or
2. p ∧ ¬Kp [unknown]

Then which of these are knowable? — Luke

1 is knowable.

But this doesn't address what I said before. You clearly just don't understand logic.

Women can prevent pregnancies by using contraceptives of which there's a wide variety, but yet they get pregnant and then wanna tread the fine line between murder and freedom by seeking abortions. If it were possible to avoid giving people the impression that one is a murderer (by having an abortion), why would you ever put yourself in the situation where you would, for certain, be conflated as one? — Agent Smith

Contraceptives fail. Rape happens.

Also, there is no fine line between abortion and murder.

The quashed Roe vs. Wade ruling doesn't prohibit contraception. — Agent Smith

True, but Thomas' opinion questions the legitimacy of the Griswold case which ruled that the use of contraceptives is a constitutional right. It's possible that the Supreme Court could later overturn Griswold, allowing States to outlaw contraception.

This removes the ambiguity of your unknown truth expressed merely as "p". — Luke

It's not ambiguous because of the second premise:

a) p
b) p is unknown

That's either making it ambiguous again (if "p" can be either known or unknown) — Luke

It doesn't make it ambiguous. b is a second (true) proposition that asserts that p is unknown.

To repeat an example I gave earlier:

1. "the cat is on the mat" is true
2. "the cat is on the mat" is written in English

Is it ambiguous whether or not "the cat is on the mat" is written in English? No; it's explicitly stated in 2. So then apply the same understanding to:

3. "the cat is on the mat" is true
4. "the cat is on the mat" is not known to be true

I thought you said "p" could either be known or unknown? — Luke

It can, but Fitch's paradox takes an example of an unknown truth to show what follows.

Or we write it as:

a) p
b) ¬Kp

a is knowable, b is not knowable, a ∧ b is not knowable.

It's really straightforward logic. Fitch et al. know what they're talking about. You haven't found some fundamental flaw with their reasoning.

How do we get to K(p & ~Kp) → Kp & ~Kp? — Agent Smith

It's in the article.

(A) K(p ∧ q) ⊢ Kp ∧ Kq
(B) Kp ⊢ p

1. K(p ∧ ¬Kp) Assumption [for reductio]
2. Kp ∧ K¬Kp from 4, by (A)
3. Kp ∧ ¬Kp from 5, applying (B) to the right conjunct

3 is a contradiction so 1 isn't possible, so ◇K(p ∧ ¬Kp) is false.

The knowability principle: p → Kp.

1. K = Knowable — Agent Smith

No. It's p → ◇Kp.

◇ is the symbol for "it is possible that".

We cannot know both a) and b) means that we cannot come to know an unknown truth. — Luke

No it doesn't.

As I've said before, I just don't know how to explain this to you any more clearly than I already have.

3 only says that p is true, not that it is true and unknown. — Luke

I know. But as I said above, a statement doesn't need to state every fact about itself.

Your claim that I am quoting is written in English, even though it doesn't say so about itself. Your claim that I am quoting contains 44 letters, even though it doesn't say so about itself.

So we can do it as two propositions:

a) p
b) p is unknown

p is an unknown truth. When we come to know a we no longer know b (because b is false). And so c can never be known:

c) p and p is unknown

You claim that we can know "p" even though we can't know "p & ~Kp". But that implies that we can't come to know anything that is unknown to be true. — Luke

No, it doesn't. Imagine these two propositions:

1. "the cat is on the mat" is true
2. "the cat is on the mat" is true and is written in English

To represent these in symbolic logic we would do something like:

1. p
2. p ∧ E(p)

Even though 1 doesn't say anything about p being written in English, p is in fact written in English. Just look at the previous sentence; it is written in English even though it doesn't say it about itself. A proposition doesn't need to state every fact about itself.

And the same with unknown truths:

3. p
4. p ∧ ¬Kp

Even though 3 doesn't say anything about p being unknown, p is in fact unknown. We can come to know 3, in which case an unknown truth has become a known truth. But we can never know 4 as that would be a contradiction.

The logic of Fitch's proof absurdly implies that an unknown truth cannot become known. — Luke

It doesn't. I thought we went over this? You seemed to understand it here:

I get it now. Unknown truths can either mention they are unknown or not mention they are unknown. Only the former are unknowable. — Luke

I accept that. But it is only wrong in the sense that one cannot both know the proposition and know that it is unknown. Knowing it negates its being unknown. If it's known then you cannot know it to be unknown. — Luke

Yes, that's exactly the point. It is true but can't be known. Therefore, the (unrestricted) knowability principle is false.

I accept that the problematic statement (form) "p & ~Kp" is inconsistent. My only qualification is that it's a kind of logical loophole that doesn't really affect knowability. I accept that it's unknowable, but it's also trivial. If I know something then I can't also know that it's unknown. Okay, so what? — Luke

Then the claim that if a proposition is true then it is knowable is wrong. One must instead claim, as Tennant does, that if a Cartesian proposition is true then it is knowable.

That's just not true, though; they each reflect light at wavelengths closer to each other than objects of other colours do compared to them, and consequently they look more similar to each other in terms of colour than objects of other colours do compared to them. The first is a material condition of the second and the second is the reason we refer to both as being grey, for our use of the word "grey". — Janus

So light with a wavelength of 650nm is the same colour as light with a wavelength of 651nm because they're very similar wavelengths?

Or if you prefer, abstract objects do not exist. — Banno

You've talked before about unspoken propositions. How does that square with abstract objects not existing? Does it just mean that we can say things that haven't been said before?

Assuming the above conditions, how can we be confident we are not Boltzmann brains? — hypericin

We can't be, that's the problem.

If you want to say that the scribbles are not the cat on the mat, that is trivial and useless to the conversation — Harry Hindu

Not when you're arguing against certain brands of anti-realism which deny the "trivial" distinction that realists take for granted.

As I pointed out before, the map is part of the territory, not separate. — Harry Hindu

Well, I have a map of the United States which is definitely not a part of the United States.

If the ones that are using the term, "separate" don't mean it literally, then they don't really mean that language is separate from the world, then what is it they do mean? — Harry Hindu

Exactly what I said before; the utterance "the cat is on the mat" is separate to the cat being on the mat.

There is nothing special about words in this regard that would make one think that they are separate from the world. — Harry Hindu

And again, you're just being too literal in your reading. When others talk about a distinction between language and the world understand it as your oft-quoted distinction between a map and the territory.

It makes no sense to say that language is in the world but separate from the world. — Harry Hindu

I think you're being too literal in your reading. They're just saying that the utterance "the cat is on the mat" is not the cat being on the mat. What is said is not what is being talked about.

I’ve heard this many times. Where on earth do you get the idea that it is the far right in the U.S. that believes truth is something made up? — Joshs

Kellyanne Conway denies Trump press secretary lied: 'He offered alternative facts'

If the only unknowable truths are that 'p is true and no one knows that p is true', then that's merely a quirk of logic that has little effect on substantive knowability. — Luke

Then read up on Tennant’s and Dummett’s responses. They’re in that SEP article. Tennant’s is the simplest:

Tennant (1997) focuses on the property of being Cartesian: A statement p is Cartesian if and only if Kp is not provably inconsistent. Accordingly, he restricts the principle of knowability to Cartesian statements. Call this restricted knowability principle T-knowability or TKP:

(TKP) p→◊Kp, where p is Cartesian.

Notice that T-knowability is free of the paradoxes that we have discussed. It is free of Fitch’s paradox and the related undecidedness paradox.

I get it now. Unknown truths can either mention they are unknown or not mention they are unknown. Only the former are unknowable. Since there is at least one unknowable truth then we must reject KP. — Luke

Yes.

However, my point is that we can safely ignore these unknowable truths since they can be re-written without self-reference; the unknown truths on which they are based can be re-written such that they do not mention they are unknown. If the only unknowable truths are those that mention they are unknown, then there is no loss of information or knowledge which comes from expressing these unknown truths as “p” instead of “p & ~Kp”. — Luke

Regardless of the symbols you use to express the proposition, it is impossible to know that the cat is on the mat and that nobody knows that the cat is on the mat.

What’s the issue with just accepting that some truths are unknowable?

Are you saying that we can change the expression of the unknown truth in Fitch’s proof to “p” instead of “p & ~Kp”? — Luke

I'm sorry but I just don't know how to fix your confusion. I've tried my best.