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. — Luke
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? — Luke
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
Therefore, the number of coins in the jar remains unknowable. — Luke
1. does not express that it is unknown — Luke
2. expresses that it is unknown, but it is unknowable. — Luke
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
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. — Luke
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. — Luke
I asked how you would express (in logical notation) that it was unknown.
— Luke
2 does that. — Michael
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. — Luke
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. — Luke
That p ∧ ¬Kp is unknowable isn't that p is 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.
— 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'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
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. — Luke
You seem to be saying that the truth of the statement "It's true that there's milk in the fridge and no-one knows there is" is unknowable, which seems reasonable, since I don't know there's milk in the fridge unless I open it but then if I do that someone knows there is milk in the fridge. But when I open the fridge I know (excluding weirdness like the milk coming to be there only when I looked) that the statement was true before I looked. — Janus
So, again, there seems to be a time element involved.
If I go down the 'weirdness' rabbit hole and say that when I look and see the milk I still don't know that the milk had been there prior to my looking, then all bets are off. — Janus
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. — Luke
Therefore, the starting suppositions make it impossible for an unknown truth to become a known truth. — Luke
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. — Luke
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? — Luke
How does that express that it is unknown? — Luke
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. — Luke
EDIT: Does Fitch’s proof allow for some unknown truths to be expressed as “p” and others to be expressed as “p & ~Kp”? — Luke
No, that's just changing the subject. There are unknowable truths regardless of whether there's a proof about them. — Andrew M
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
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
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