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.

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.

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.

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.

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'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?

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

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").

Or we write it as:

a). p
b). ¬Kp — Michael

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

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.

No it doesn't. — Michael

And suppose that collectively we are non-omniscient, that there is an unknown truth:
(NonO) ∃p(p∧¬Kp).

If this existential claim is true, then so is an instance of it:
(1) p∧¬Kp. — Fitch's proof

Isn't that unknowable?

So we can do it as two propositions:

a) p
b) a is unknown

p is an unknown truth. When we come to know a we no longer know b. — Michael

Yes, that's the reason that we can't know both a) and b). Again, I'm not disputing the logic, only its implications.

We cannot know both a) and b) means that we cannot come to know an unknown truth. Which is absurd.

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 an unknown truth. — Michael

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

I thought we went over this? — Michael

We did, but I didn't realise then, and wasn't making the point then, that the issue was with logic and not with knowability.

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. That's surely a problem - not just for knowability but for everyday reason. Isn't it? That's just as absurd as the result of Fitch's proof that 'all truths are known'.

I'm saying that we can retain knowability by acknowledging that logic cannot account for any changes from a truth being unknown to its being known. This failing of logic creates the paradox. The paradox dissolves in everyday reason where we obviously can come to know unknown truths.

"p & ~Kp" is sometimes true. There have been plenty of examples in this thread. — Andrew M

You're right. I meant to say that it is never known to be true.

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.

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

I accept that, according to the logic, "p & ~Kp" is unknowable. However, I don't think this is an issue for knowability, but an issue for logic.

"p & ~Kp" is supposed to represent an unknown truth. The logic of Fitch's proof absurdly implies that an unknown truth cannot become known. The problem, as I have stated in several recent posts, is the conjunct of ~Kp. But that is only a problem in logic, not a problem in reality. In reality, coming to know that p is true means that it has become known and is no longer unknown, not that we impossibly know both that p is true and that p is unknown to be true. Logic holds one set of truths to be eternally known and the other to be eternally unknown, and those sets can never change. But in reality, those known and unknown truths are not eternal and do change; what is unknown can become known.

Then the claim that if a proposition is true then it is knowable is wrong. — Michael

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.

Then read up on Tennant’s and Dummett’s responses. They’re in that SEP article. — Michael

From the little I've read, they seem to be looking to qualify the theory in some way (as Andrew put it). For example:

A statement p is Cartesian if and only if Kp is not provably inconsistent. — SEP article

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?

Since no-one would ever plausibly agree that "p & ~Kp" is true, does it follow that it is never true? Presumably not, and so the theory either needs to be rejected or else qualified in some way. — Andrew M

Surely it is never true. If a statement is known to be true, then it cannot also be unknown to be true ("by somebody at some time"). Which is what the independent result tells us.

It's a trick of logic. Every "p" remains knowable, but not when put into a conjunction with "~Kp". Therefore, it cannot be known both that p is true and p is unknown to be true. That's just word play (or logic play) which does not affect every (other) "p" being knowable.

The same could be done for other propositional attitudes. For example, desires (D):

D(p & ~Dp) - someone at some time has the desire that 'p is true and nobody desires that p is true'. Is this undesirable?

Or beliefs:

B(p & ~Bp) - someone at some time has the belief that 'p is true and nobody believes that p is true'. Is this Moore's paradox?

It's like a liar paradox for propositional attitudes. But less paradoxical and more nefarious.

Just don’t set it too low, in case we need one more.

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

It just seems counterintuitive to me that any unknown truths should be unknowable in priniciple. 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. It is still knowable that p is true. The only reason we cannot know 'p is true and no one knows that p is true' is because knowing the first conjunct would falsify the second. I don't see why this should be "of concern for verificationist or anti-realist accounts of truth", as the WIkipedia article states.

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.

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”.

No, that's just changing the subject. There are unknowable truths regardless of whether there's a proof about them. — Andrew M

You want to disregard Fitch's proof, but I'm the one changing the subject?

Either an unknown truth is expressed as “p & ~Kp” and it follows that we must reject KP because some/all unknown truths are unknowable, or else an unknown truth is expressed as "p" and it follows that we need not reject KP because all truths are knowable.

The result of Fitch's proof is that some truths are unknowable. However, if one of its suppositions brackets off and excludes those unknown truths that do not mention they are unknown, then that leaves most unknown truths as knowable.

You want to say, in essence, that Fitch's proof affects only those unknown truths that mention they are unknown. Fine. There are some unknown truths which are unknowable, and it is only those unknown truths which mention that they are unknown. But unless it is necessary for an unknown truth to mention that it is unknown, then all truths are knowable.

Is there a reason why an unknown truth must mention that it is unknown, or can any unknown truth be expressed as "p"? If we can simply re-express the unknown truths of Fitch's proof such that they do not mention that they are unknown, then all truths are knowable. If this re-expression is possible, then knowing these truths is possible.

It doesn't. That information is part of the context. The statement doesn't mention it. It also doesn't mention a host of other things, such as whether it's lite or full cream milk, whether it's in Alice's fridge or Bob's fridge, and so on. — Andrew M

Then we can simply express the unknown truth in Fitch’s proof as “p” and the problem goes away: there are no unknowable truths.

EDIT: Does Fitch’s proof allow for some unknown truths to be expressed as “p” and others to be expressed as “p & ~Kp”?

Sure, just don't mention it's unknown. So instead of "p & ~Kp", that would be "p". — Andrew M

How does that express that it is unknown?

No, whether a statement is unknowable or not is conditional on the content of the statement. As Michael is pointing out, regular statements that don't mention that they're not known can be known. — Andrew M

So is there a way to express an unknown truth in logical notation without mentioning that it is unknown?

Both are true. And even though the third proposition doesn't express it, nobody knows that the cat is on the mat (as explained by the fourth proposition). — Michael

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

I'm asking you how else "p is unknown" could be expressed in logical notation - other than as "p ∧ ¬Kp", and other than as your mere assurance outside of logical notation that p is unknown.
— Luke

p ∧ ¬Kp is how you express it.

The problem is that you seem to go from "p ∧ ¬Kp" is unknowable to "p" is unknowable. And that just doesn't follow. — Michael

Please tell me where I am going wrong here:

The unknown truth that is the number of coins in the jar is expressed as: p ∧ ¬Kp

It is impossible to know the unknown truth: p ∧ ¬Kp

Therefore, it is impossible to know the unknown truth that is the number of coins in the jar.

That p ∧ ¬Kp is unknowable isn't that p is unknowable. — Michael

I'm asking you how else "p is unknown" could be expressed in logical notation - other than as "p ∧ ¬Kp", and other than as your mere assurance outside of logical notation that p is unknown.

It is not possible to know any proposition of the form "p & ~Kp", which means that all unknown truths (expressed in this way, at least) are unknowable.
— Luke

This is where you have a fundamental misunderstanding that I don't know how to explain to you. Maybe like this?

a) p
b) a is not known to be true

Both a and b are true. Neither a nor b are known to be true. It is possible to know a but not possible to know b. — Michael

I understand the conjunction. I don't see how this contradicts what I said.

I asked how you would express (in logical notation) that it was unknown.
— Luke

2 does that. — Michael

2 (when expressed as "p & ~Kp") is unknowable, which means that so is the number of coins in the jar.

I don't understand what you're asking for here. The argument simple shows that if you take the knowability principle and the non-omniscience principle as premises then it in fact follows that the non-omniscience principle is false. It is then up to the reader to decide whether to accept that the non-omniscience principle is false or to reject the knowability principle.

So why can't you just accept that the knowability principle is wrong? Some truths are, in fact, unknowable. — Michael

It is not possible to know any proposition of the form "p & ~Kp", which means that all unknown truths (expressed in this way, at least) are unknowable. An unknown truth cannot become a known truth, and vice versa. The result of the argument is therefore that all (known) truths are known and all unknown truths are unknowable, and never the twain shall meet. The conclusion is not a failure of KP, but a failure of logic.

On the one hand, you want me to accept the argument's implication that there is at least one unknowable truth, and that therefore KP must be rejected.

On the other hand, you do not accept the argument's implication that we cannot come to know mundane unknown truths such as the number of coins in a jar.

Therefore, the number of coins in the jar remains unknowable.
— Luke

It isn't. We can count the coins and then we will know how many coins are in the jar. — Michael

Then this should be able to be expressed in the argument. If it cannot be expressed in the argument, then it is not a failure of the knowability principle, but a failure of logic. Otherwise, accept the logic and the number of coins in the jar is unknowable.

1. does not express that it is unknown
— Luke

Which is why it is possible to know it. — Michael

I asked how you would express (in logical notation) that it was unknown.

1. p
2. p ∧ ¬Kp

Assume p is true. Both 1 and 2 are true. Neither 1 nor 2 are known to be true. 1 can be known to be true. 2 can't be known to be true. — Michael

1. does not express that it is unknown
2. expresses that it is unknown, but it is unknowable.

Therefore, the number of coins in the jar remains unknowable.

I mean that the unknown truth "p & ~Kp" of NonO cannot possibly become a known truth. If that is impossible from the outset, then so is knowability.
— Luke

No it isn't. There are some things which are unknown truths which can become known, e.g. the number of coins in a jar. — Michael

Presumably, the unknown truth of the number of coins in a jar is not expressed as "p & ~Kp", since this is unknowable. So how would you express the unknown truth about the number of coins in a jar?

But if you accept that there are unknowable truths then you're not in any difficulty. — Count Timothy von Icarus

The result of the argument seems to be that all unknown truths are unknowable, as there is no unknown truth of the form "p & ~Kp" that can change into a known truth or that can become known. That all unknown truths are unknowable is just as absurd as the result that all truths are known.

I don't know what you mean by it being "rigged". It just shows that the knowability principle is wrong. Some truths are, in fact, unknowable. — Michael

I mean that the unknown truth "p & ~Kp" of NonO cannot possibly become a known truth. If that is impossible from the outset, then so is knowability.

"There are 163 coins in the jar" was an unknown truth before someone counted, and then it became a known truth. — Michael

To borrow @Andrew M's example:

Suppose there are 163 coins in the jar and no-one knows there is.

It's thus true that there's 163 coins in the jar and no-one knows there is.

That true statement is unknowable. Why? Because anyone coming to know that there's 163 coins in the jar (say, by counting) would render the statement false (since the second conjunct would be false). The statement doesn't change from an unknown truth to a known truth. It changes from an unknown truth to a known falsity.

Suppose there is milk in the fridge and no-one knows there is.

It's thus true that there's milk in the fridge and no-one knows there is.

That true statement is unknowable. Why? Because anyone coming to know that there's milk in the fridge (say, by looking) would render the statement false (since the second conjunct would be false). The statement doesn't change from an unknown truth to a known truth. — Andrew M

Aye, there's the rub. If a truth is knowable, then it can come to be known; that is, it can change from being unknown to being known. However, as you note, the statement "p & ~Kp" does not (and cannot) change from being unknown to being known. Therefore, the starting suppositions make it impossible for an unknown truth to become a known truth. The starting suppositions give the impression that all truths are knowable and that we should be able to come to know an unknown truth. But if "p & ~Kp" cannot possibly change from being unknown to being known, then of course it is unknowable: it's a rigged game from the outset. It follows only from this logical impediment that it is impossible to know an unknown truth, that no truths are knowable, and that all truths are known. These conclusions can safely be ignored, however, given that the confidence trick does not allow for an unknown truth to become a known truth.

Yeah, I don't comprehend it either, as I said, but I also accept the conclusion (although not on account of the "paradox") that there must be unknowable truths. — Janus

As I understand it, the conclusion of the independent result is not that there must be unknowable truths. The conclusion of the independent argument is that it is impossible to know an unknown truth. It follows from this in the SEP proof that there does not exist an unknown truth (at line 10) and that all truths are known (at line 11).

But if there were no unknown truths, wouldn't it then follow that there would be no unknowable truths? — Janus

Possibly. What's your reasoning?

In any case, that is not how the argument gets from unknown to unknowable is it? — Janus

The move from unknown to unknowable is given in the "independent result" in lines 4-9 of the SEP proof. The logic of that reductio argument is beyond my understanding, and I would welcome someone to explain it. However, I don't dispute its conclusion.

As I said in my post above to Andrew, one reason that an unknown truth would be unknowable (or impossible to know) is if all truths were already known and there were no unknown truths.