Then

(p & Kp) (only) $\to$ ◇K(p & Kp).

How do we get to K(p & ~Kp) $\to$ Kp & ~Kp?

:brow:

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.

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

Don't leave it ambiguous then. If truths are either known or unknown, then this can be expressed as:

1. p ∧ Kp; or
2. p ∧ ¬Kp

1. is knowable. 2 is unknowable. I imagine you will find that the paradox occurs for all unknown truths.

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.

Or we write it as:

a). p
b). ¬Kp — Michael

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

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.

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

Okay, but I removed the ambiguity by expressing known and unknown as:

1. p ∧ Kp; or
2. p ∧ ¬Kp — Luke

To which you said "we write it as":

a). p
b). ¬Kp — Michael

That's either making it ambiguous again (if "p" can be either known or unknown), or refuting what you said earlier (if "p" represents "p is known").

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

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

Then you misunderstood that I was expressing both known and unknown truths.

Don't leave it ambiguous then. If truths are either known or unknown, then this can be expressed as:

1. p ∧ Kp; or
2. p ∧ ¬Kp — Luke

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

Then:

1. is knowable. 2 is unknowable. I imagine you will find that the paradox occurs for all unknown truths. — Luke

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

I removed the ambiguity — Luke

:snicker: So it was you all along!

What if one person knows the proposition as true and another knows it as false? Is it 'known' then? — Olivier5

Can't know what isn't so. From Fitch's proof:

Second, knowledge entails truth.
...
(B) Kp ⊢ p — 2. The Paradox of Knowability - SEP

Fitch is easily solved by noting that knowledge evolves over time. Lamest paradox ever. — Olivier5

Noting that knowledge evolves over time doesn't help those theories that depend on the knowability principle.

Fitch’s paradox of knowability (aka the knowability paradox or Church-Fitch Paradox) concerns any theory committed to the thesis that all truths are knowable. — Fitch’s Paradox of Knowability - SEP

If there is milk in the fridge and no-one knows there is, is the statement "there is milk in the fridge and no-one knows there is" true?
— Andrew M

According to logic, if it is true and unknown that there is milk in the fridge, then it can never become known. — Luke

I'm not sure how that answers the question above. The point is that the statement above is a counterexample to various antirealist theories.

What’s the paradox? Timothy Williamson (2000b) says the knowability paradox is not a paradox; it’s an “embarrassment”––an embarrassment to various brands of antirealism that have long overlooked a simple counterexample. — Fitch’s Paradox of Knowability - SEP

It's not ambiguous because of the second premise:

a) p
b) p is unknown — Michael

Our dispute is over your claim that there are knowable unknown truths.

If all truths can be expressed as either:

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

Then which of these are knowable?

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.

If all truths can be expressed as either:

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

Then which of these are knowable? — Luke

None of them are knowable, but p is knowable. — Michael

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?

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).

I have explained this to you so many times. I'll try one more time. If you still don't understand then I give up. — Michael

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.

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.

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.

It's that simple. — Michael

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

It's that simple.

Can't know what isn't so. — Andrew M

Since we don't have access to the registry of things that are, how is one to ascertain that "P is known", as opposed to "persons A, B and C believe that P is true, while person D may disagree"?

In other word, the concept of knowledge is mistreated here, cheapened, overly simplified when made an absolute. Knowledge is not something that exists objectively out there. It's something that people do.

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

You just don't understand symbolic logic, — Michael

p means "the cat is on the mat"
¬Kp means "it is not known that the cat is on the mat" — Michael

What does "p ∧ ¬Kp" represent if not that the cat is on the mat AND that it is not known that the cat is on the mat?

"p" does not represent that p is true and unknown; only that p is true.

1. the cat is on the mat

1 is an unknown truth but is knowable — Michael

Why is 1 an unknown truth? It could equally be a known truth. I have removed this ambiguity in my post above, yet you continue to ignore it.

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.

Do you not understand than an argument can have more than one premise? — Michael

I didn't realise that they were premises; I thought they were unrelated statements.

It’s an unknown truth because 2 says so. — Michael

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?

This would mean that "p ∧ ¬Kp" is knowable.

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.

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. — Michael

It seems to me that b renders a as a meaningless string if scribbles.

If no one knows the cat is on the mat then from from where does A follow? Why was A stated in the first place? How is it possible to positively assert that which is not known?

We could go on ad infinitium with

c no one knows that know one knows the cat is on the mat
d no one knows that no one knows that no one knows the cat is on the mat

etc.
With each subsequent statement rendering the prior statement as useless.

The question is, what is knowing? How does knowledge relate to truth? Have you ever claimed to know something and later found it was not true?

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.

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

But one has reasons to believe alien life exists and that it will rain tomorrow. What reasons does one have to know that know one knows alien life exists or that it will rain tomorrow?

And then we can always cancel out the prior statement with a subsequent statement that no knows the prior statement is true. What prevents sliding down the slippery slope? Have you ever claimed to know something and found that it was not true?

How is belief different than knowledge?

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.