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.

:up:

Not the best thread, this one. — Banno

Not the best comment, this, o pompous one.

playing to my audience.

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.

Fulfilling what you imagine are virtual expectations re your persona?

Ironically, this is relevant given empiricism tells us that knowing a quantum object's velocity makes its position unknowable and vice versa. Another point for unknowable truths. It's all quantum! (...you brought this on yourself).

(...you brought this on yoursel — Count Timothy von Icarus

And the rest of us. :cry:

Succsessful invocations surely merit a ban for witchcraft?

In my defence, it's simple stats. The chance of some engineer/science teacher claiming that the answer to some philosophical question is quantum is proportional to the square of the number of posts. By page eleven it is almost certain.

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? — Luke

The knowability principle is like the proposition that all swans are white. When someone discovered that some swans were black, then that refuted the original proposition. Regardless, the original proposition was false independent of that discovery.

Similarly "p & ~Kp" was a counterexample to the knowability principle before Fitch ever formulated his proof.

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. — Luke

For an example of why the counterexample matters, consider Peirce’s pragmatic theory of truth, i.e., that truth is what we would agree to at the limit of inquiry. 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.

Has someone explained what they mean by "knowing a proposition" yet? Does it mean just being aware of the proposition, or knowing it to be true?

If the latter, please note that in practice it is often extremely hard to prove that some proposition is true, beyond any doubt. We almost never 'know X to be positively true'. What we do instead is eliminate theories that are proven false.

So from a pure epistemic view point, the knowability principle is false because contradicted by day-to-day experience, and by our knowledge that we know very little. That'd be why most examples given on this thread are mathematical, as the only domain of knowledge where certainty applies.

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.

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?

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.

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.

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.

Has someone explained what they mean by "knowing a proposition" yet? Does it mean just being aware of the proposition, or knowing it to be true? — Olivier5

It means to know that something is true, e.g., that it is raining (say, as a consequence of looking out the window).

If the latter, please note that in practice it is often extremely hard to prove that some proposition is true, beyond any doubt. We almost never 'know X to be positively true'. What we do instead is eliminate theories that are proven false.

So from a pure epistemic view point, the knowability principle is false because contradicted by day-to-day experience, and by our knowledge that we know very little. That'd be why most examples given on this thread are mathematical, as the only domain of knowledge where certainty applies. — Olivier5

Mathematical certainty isn't required for the ordinary use of "know". However it does require a higher bar then mere opinion or guesswork (i.e., there need to be good reasons, or evidence, or justification for making knowledge claims). But the knowability principle is false not because we don't know some things, but because we can't know some things (i.e., propositions of the form "p & ~Kp").

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. — Luke

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

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. — Luke

That's right. But "<>K(p & ~Kp)" (which is never true) is a different proposition to "p & ~Kp" (which can be true).

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. — Luke

It's not "word play" if one's theory of truth depends on the knowability principle being true. Consider again Peirce’s pragmatic theory of truth, i.e., that truth is what we would agree to at the limit of inquiry. 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? According to Peirce's theory, it isn't true. But that's mistaken.

It means to know that something is true, e.g., that it is raining (say, as a consequence of looking out the window). — Andrew M

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

But the knowability principle is false not because we don't know some things, but because we can't know some things (i.e., propositions of the form "p & ~Kp"). — Andrew M

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

But yes, there are many things we cannot know, such as the things in themselves, as Kant explained, or whether it rained on a given site 36,785,477 years, 278 days and 4 hours ago, or what your wife thinks.

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.

"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.

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 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.

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.

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.

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

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.

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.

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?

There's a problem:

The knowability principle: p $\to$ Kp.

1. K = Knowable

p is true and p is unknown: p & ~Kp

We know that p is true and p is unknown: K(p & ~Kp)

2. K = Know(n)

Inconsistency in the meaning of symbol K (compare 1 and 2).

The knowability principle: p → Kp.

1. K = Knowable — Agent Smith

No. It's p → ◇Kp.

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