Suppose p is a sentence that is an unknown truth; that is, the sentence p is true, but it is not known that p is true. In such a case, the sentence "the sentence p is an unknown truth" is true; and, if all truths are knowable, it should be possible to know that "p is an unknown truth". But this isn't possible, because as soon as we know "p is an unknown truth", we know that p is true, rendering p no longer an unknown truth, so the statement "p is an unknown truth" becomes a falsity. Hence, the statement "p is an unknown truth" cannot be both known and true at the same time. Therefore, if all truths are knowable, the set of "all truths" must not include any of the form "something is an unknown truth"; thus there must be no unknown truths, and thus all truths must be known. — Fitch's paradox of knowability
The other point that's up for discussion is that somewhere in Fitch's argument, K(P) → KP where K(P) means P is knowable and KP means Known that P. Feels like an illegal move to me. — Agent Smith
An example: I know that calculus is knowable, but that hasn't helped me at all, I haven't the slightest clue what calculus is about. :snicker: — Agent Smith
Fitch's "paradox" of knowability — Luke
Given this contradiction we must either reject the knowability principle or accept that we know which of "the box is empty" and "the box is not empty" is true. — Michael
Therefore if we insist on the knowability principle then we must accept that every true statement is known to be true. — Michael
I would say that we (now) know both of these statements, particularly since you have stated them. — Luke
The argument says that if it is possible to know a true p, then we must know that p is true. — Luke
But we don't know which of the statements is true, which means that we must reject the knowability principle. — Michael
it's that we don't know the statement that is true — Luke
However, the knowability principle entails that if it is true then I know that it is true, which contradicts the non-omniscience premise. — Michael
If it is possible to know that p is true, then we must know that p (is true) — Luke
Yes, and as the knowability principle is the principle that p is true if it is possible to know that p is true it then follows from what you say here that every true statement is known to be true. — Michael
,,,as soon as we know "p is an unknown truth", we know that p is true, rendering p no longer an unknown truth, so the statement "p is an unknown truth" becomes a falsity. Hence, the statement "p is an unknown truth" cannot be both known and true at the same time. Therefore, if all truths are knowable, the set of "all truths" must not include any of the form "something is an unknown truth"; thus there must be no unknown truths, and thus all truths must be known. — Fitch's paradox of knowability
Suppose p is a sentence that is an unknown truth — Fitch's paradox of knowability
Logic is really bad at doing time. Truths have to be eternal. That p is an unknown truth is unknowable until p is known, and then it is not an unknown truth. the difficulty arises because knowability implies time. — unenlightened
This is the heart of darkness - suppose we know something that we suppose we do not know. "the 79 squillionth decimal iteration of pi is a '2'." Well do we know or don't we? Make up your mind, Fitch. The digit is knowable, but 'that it it 2' is knowable only if it happens to be 2, which we don't know. p0, p1... p9 - one of them is an unknown truth, and the others are unknown falsehoods. — unenlightened
I think the argument implies that every known true statement is known to be true, As I stated in the OP, this excludes all unknown statements and statements with unknown truth values. — Luke
Suppose p is a sentence that is an unknown truth; that is, the sentence p is true, but it is not known that p is true. — Fitch's paradox of knowability
That's what I'm disputing about the argument. — Luke
Therefore 2 could be one of the cases where it is not possible to know that the Riemann hypothesis is true despite it being true. — Isaac
If it is not possible to know that the Riemann hypothesis is true despite it being true then the knowability principle is refuted. — Michael
But I'm saying it is possible to know that the RH is true (just not at the same time as knowing that we don't know it's true). In other words, it is generally possible to know that the RH is true (your 1), but not in all circumstances (ie not whilst your 2 is the case). The fact that there exists a circumstance under which something is impossible, doesn't mean that that something is impossible in general. — Isaac
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