What’s the issue with just accepting that some truths are unknowable? — Michael
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
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.
(...you brought this on yoursel — Count Timothy von Icarus
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
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
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
Then read up on Tennant’s and Dummett’s responses. They’re in that SEP article. — Michael
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? — Luke
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. — Luke
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
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
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
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
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 means to know that something is true, e.g., that it is raining (say, as a consequence of looking out the window). — Andrew M
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
Yes, that's exactly the point. It is true but can't be known. Therefore, the (unrestricted) knowability principle is false. — Michael
"p & ~Kp" is sometimes true. There have been plenty of examples in this thread. — Andrew M
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
The logic of Fitch's proof absurdly implies that an unknown truth cannot become known. — Luke
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
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
3 only says that p is true, not that it is true and unknown. — Luke
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
The knowability principle: p → Kp.
1. K = Knowable — Agent Smith
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