Talk:Fitch's paradox of knowability

message: Does it mean knowable with all properties or partially knowable? This is unclear and therefore IMHO it is a wishy washy empty logical statement that is of no use.
It starts with the definition K p  : p is knowable
what is a definition that is not negatable:
Not K p
would mean that p is not knowable. But Not K p is an expression describing p, so it *is* knowable to the extend that it is unknowable, what is an intrinsic contradiction. This is due to the fact that the definition K p is not restricted to a certain class of objects, but refers to the global aspect of knowability that includes those very expressions and definitions. It is simply impossible to talk or reflect about unknowables without hitting such recursive contradictions. 
IMHO such problems are of the same class as the problem of the S: "set of all sets that do not contain themselves", that caused so much trouble for Russell.
-- Rainer Dickermann  —Preceding unsigned comment added by 117.96.138.64 (talk) 10:41, 24 November 2008 (UTC)
thread_title:  How exactly is 'being knowable' defined? 

message: The following text has been added by various contributors:
:A concise statement of Fitch's paradox is: "It is impossible for all truths to be knowable unless all truths are known, because the fact of an unknown truth existing is unknowable."
:As a simple example, suppose that your friends are considering throwing you a surprise party.  It is then impossible for you to even know that the statement "My friends are throwing me a surprise party" is true; if you ever did know that, then the party would no longer be a surprise, and the statement would become false.  Thus, you can know that this statement is false, but it's impossible for you to ever know that it is true.  In spite of this, it is perfectly possible for that statement to be true; all of your friends can know about it and it will remain true as long as you don't know about it.  
:That example covered the knowledge of only one person, but it can be taken further.  There is an old children's riddle: "What was the tallest mountain in the world before Mount Everest was discovered?"  The answer is "Mount Everest", because Mount Everest still existed - and was still, in fact, the tallest mountain in the world - even before it was discovered; we just didn't know it.  
:Thus, prior to the discovery of Mount Everest, the statement "Mount Everest (or, to be pedantic, the mountain that would later be named Mount Everest) is the tallest mountain in the world, but nobody knows it" was perfectly true.  However, just as with the "surprise party" statement, it was impossible for anyone to know that it was true - because as soom as anyone knows it, it becomes false.
:The paradox, therefore, is that if we assume that there are some statements that are true which we are not yet aware of - undiscovered scientific principles, information about the future, secrets yet to be revealed, and similar - then for any such statement, the fact that "(the statement) is true, but we don't know it" is true, but it's impossible for us to ever know it.  Thus, there must be some true statements which are unknowable, and thus not all truths are knowable.  This is a very uncomfortable conclusion, but the only escape - assuming that we already know all there is to know, so that no statements of the form above can exist and be true - is even more uncomfortable.
The above text is not about Fitch's paradox.  It is perhaps about the paradox of the knower, which is quite different. --- Charles Stewart 19:48, 18 August 2005 (UTC)
thread_title: Lengthy deleted text

message: This section includes a tantalizing development, but without much elaboration. It replaces the premise in (C) that all truths are knowable (p → LKp) with the tautologically undeniable "all knowable unknown truths can be known":
 (C')	∃x((x & ¬Kx) & LKx) & LK((x & ¬Kx) & LKx) = changing the x's into p's like the rest of the article we get:
 (C')    ∃p((p & ¬Kp) & LKp) & LK((p & ¬Kp) & LKp)
Maybe we could expand this from line 9, because as complex as it seems, it's actually very simple. You don't need line 8 really here because the rule incorporates it since Rule (C') does not relate to all truths but only to knowable truths, but let's presume it so:
-if ∃p((p & ¬Kp) & LKp) & LK((p & ¬Kp) & LKp) is existentially true, then an instance is true, so:
     
  9. ((p & ¬Kp) & LKp) & LK((p & ¬Kp) & LKp)
 10. ((p & ¬Kp) & LKp) & LK(p & ¬Kp) & LK(LKp)   from line 9 by Rule (B)
 11. LK(p & ¬Kp)                                 from line 10 by conjunction elimination
and then lines 12-13 here are identical to lines 10-11 in the article, concluding all knowable truths must be known instead of all truths must be known (making them all knowable): even though p represents simply a "truth" not a "knowable truth" - because (C') replaces (C) (or else we're just assuming (C) again). Line 11 can't be denied anyway, otherwise one is arguing it's impossible to ever learn an unknown truth. In fact, (C') can be reduced to: LK((p & ¬Kp) & LKp) - it's possible to know an unknown, but knowable truth.
Bonus, I want to make the following personal observation/contribution. One could allege the paradox of semantic or conceptual trickery when assuming in Line 6 the opposite of Line 1 which led to a contradiction. If we expand line 6 the way we did line 1, however, we don't get a contradiction:
 6. ¬K(p & ¬Kp)
 7. ¬Kp & ¬K¬Kp - From line 6 by rule (B)
 8. ¬K¬Kp       - From line 7 by conjunction elimination
The question becomes, does ¬K¬Kp = K(Kp) which equals Kp? No! If you think about it, if I don't know that I don't know a truth, that doesn't mean I know it. It's like not knowing a question or its answer and concluding I know both. This can be confirmed if we realize that if I don't know a truth, it is still a truth, then we add this rule:
 (E) ¬Kp → p . If we substitute ¬Kp for p, we get ¬K¬Kp → ¬Kp:
  9. ¬Kp & ¬Kp    - Line 7 by rule (E)
 10. ¬Kp          - Line 9, conjunction elimination
So unlike Line 1, Line 6 doesn't lead to a contradiction. If one objects to the idea that not knowing a truth makes it a truth, then we can modify it to:
 (E') to ¬Kp → p v ¬p  - If I don't know a truth it is either a truth or untruth.
Then we get, ¬K¬Kp → ¬Kp v ¬¬Kp:
  9. ¬Kp v ¬¬Kp        - From line 8 by rule (E')
 10. ¬Kp v Kp          - negation elimination
 11. ¬Kp & (¬Kp v Kp)  - Line 7 and Line 10
 12. (¬Kp & ¬Kp) v Kp  - Associative Property
 13. ¬Kp v Kp          - Identity
Again, no contradiction in itself nor with the conclusion from rule (E), showing the negation of Line 1 in Line 6 is valid. Additionally, one may not propose ¬Kp → ¬p, that not knowing a truth makes it an untruth (which would lead to a contradiction), if one cannot propose ¬Kp → p.
Cornelius (talk) 16:52, 21 July 2021 (UTC)
thread_title:  The knowability thesis