If we reject the non-omniscience principle, it follows from Fitch’s argument that all truths are not only knowable but known. This is unsurprising given our omniscience! — Luke
If we reject the knowability principle, it follows from Fitch’s argument that there is not only an unknown truth but an unknowable truth. This is unsurprising as it prevents our omniscience! It is also unsurprising given that not all truths can be known! — Luke
Because Alice can (speculatively) say of an unknown truth, t, that "t is true and no-one knows that t is true".
Alice's statement will, in turn, be an unknown truth. While someone could come to know that t is true, no-one could come to know that Alice's statement is true. — Andrew M
Do you know of any literature that speaks to the rejection of the KP side? — Luke
Someone could come to know the unknown truth, t, but no-one could come to know Alice's statement about t is true? Couldn't Alice come to know that [her] statement is true, at least? — Luke
What do you make of Michael's earlier claims in this discussion regarding the Riemann hypothesis and its being an unknown truth that it is correct (or else an unknown truth that it is incorrect)? Can't we all come to know the truth of Michael's statement(s)? — Luke
I would have thought that it was the unknown truth (of NonO) that becomes unknowable upon the rejection of the knowability principle, rather than a statement regarding the unknown truth. — Luke
However, it can be shown independently that it is impossible to know this conjunction. Line 3 is false. — 2. The Paradox of Knowability - SEP
The essential point here is that p and "p & ~Kp" are different statements - the former is unknown (but potentially knowable), the latter is unknowable. — Andrew M
Let K be the epistemic operator ‘it is known by someone at some time that.’ — SEP article
And suppose that collectively we are non-omniscient, that there is an unknown truth:
(NonO) ∃p(p∧¬Kp) — SEP article
I think you misunderstand Fitch's paradox. It is a reductio ad absurdum against the knowability principle. So, Fitch's paradox is literature that speaks to the rejection of the KP side. — Michael
Line 9 contradicts line 3. So a contradiction follows from KP and NonO. The advocate of the view that all truths are knowable must deny that we are non-omniscient:
(10)¬∃p(p∧¬Kp).
And it follows from that that all truths are actually known:
(11)∀p(p→Kp). — SEP article
...someone who maintains that KP is true must deny NonO - they admit omniscience. — Banno
And Luke is not the only one.
Folks, in outline, the SEP proof works as follows:
Part 1
Assuming KP and NonO, we derive line (3)
Part 2
Assuming A,B,C,& D, we derive Line (9)
Conclusion:
Line (9) contradicts line (3);
hence, one of the assumptions here is wrong.
Or we need an alternative logic.
A,B,C,D are unassailable (I'm sure that won't stop someone here making the attempt...)
Hence there is a contradiction between KP and NonO. They cannot both be true.
So someone who maintains that KP is true must deny NonO - they admit omniscience.
Hence, if all truths are knowable, everything is known. — Banno
Hence, if all truths are knowable, everything is known. — Banno
And besides, I find it logically interesting to consider the rejection of each side. Not to mention that Janus raised a question about unknowability which follows from rejecting the KP side instead of the NonO side. — Luke
I don't see why "p & ~Kp" is unknowable. — Luke
Then just reject the knowability principle.I don't understand the problem. — Michael
Hence there is a contradiction between KP and NonO. They cannot both be true.
So someone who maintains that KP is true must deny NonO - they admit omniscience.
Hence, if all truths are knowable, everything is known. — Banno
But given the contradiction between KP and NonO, KP could also be denied. I am merely interested, for the sake of symmetry or completeness, to see what follows if KP is denied. — Luke
What follows from the knowability principle being denied has nothing to do with Fitch's paradox. — Michael
Assume that John argues that an omniscient God exists and that we have free will. Jane provides an argument to show that if an omniscient God exists then we don't have free will.
You then want to know what follows from an omniscient God not existing, which has nothing to do with Jane's argument. — Michael
So? Maybe I'm curious to know whether we have free will. — Luke
I find it epistemologically interesting that if we reject NonO then all truths are not only knowable but known, and if we reject KP then there is not only an unknown but an unknowable truth. These both follow from Fitch's argument, so I wouldn't say it has nothing to do with it. — Luke
I would have thought that it was the unknown truth (of NonO) that becomes unknowable upon the rejection of the knowability principle, rather than a statement regarding the unknown truth.
— Luke
It's the latter. In the SEP proof, line 1 asserts that p is an unknown truth. Line 3 asserts that it is possible to know the conjunction from line 1. Finally, line 3 is shown to be false. The essential point here is that p and "p & ~Kp" are different statements - the former is unknown (but potentially knowable), the latter is unknowable. — Andrew M
The essential point here is that p and "p & ~Kp" are different statements - the former is unknown (but potentially knowable), the latter is unknowable. — Andrew M
The essential point here is that p and "p & ~Kp" are different statements - the former is unknown (but potentially knowable), the latter is unknowable.
— Andrew M
The SEP article states:
Let K be the epistemic operator ‘it is known by someone at some time that.’
— SEP article
Doesn't "~Kp" therefore mean that "it is not known by someone at some time that'? That is, p is unknown. — Luke
I don't see why "p & ~Kp" is unknowable. — Luke
Moreover, "p & ~Kp" is the conjunction of the non-omniscience principle, which looks like what the SEP calls an unknown (not an unknowable) truth: — Luke
My thinking was that p is just a true proposition and "p & ~Kp" represents that it is an unknown truth. You now appear to be saying that it is this unknown truth which follows from the argument as unknowable: — Luke
The essential point here is that p and "p & ~Kp" are different statements - the former is unknown (but potentially knowable), the latter is unknowable.
— Andrew M
Whereas, you previously said that it was Alice's statement about the unknown truth which becomes unknowable. — Luke
My thinking was that p is just a true proposition and "p & ~Kp" represents that it is an unknown truth. You now appear to be saying that it is this unknown truth which follows from the argument as unknowable:
— Luke
p is the unknown truth and that is expressed by the above conjunction. The conjunction itself is unknowable. — Andrew M
Because Alice can (speculatively) say of an unknown truth, t, that "t is true and no-one knows that t is true". — Andrew M
The conjunction itself is unknowable. — Andrew M
I would have thought that it was the unknown truth (of NonO) [i.e. "p & ~Kp"] that becomes unknowable upon the rejection of the knowability principle, rather than a [i.e. Alice's] statement regarding the unknown truth. — Luke
p is the unknown truth and that is expressed by the above conjunction. The conjunction itself is unknowable.
— Andrew M
If the unknown truth is expressed by "p & ~Kp", then it is not expressed by "p". The unknown truth expressed by "p & ~Kp" is equivalent to your "t": — Luke
In such a case, the sentence "the sentence p is an unknown truth" is true today; and, if all truths are knowable, it should be possible one day to learn that "p was an unknown truth" up untill that day. — Olivier5
Your proposition a. can likewise be true untill such a time when it becomes false. — Olivier5
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