Talk:Fitch's paradox of knowability

Recommendations

Every instance of "L" on the main page should be replaced with "M" to bring it in line with the predominant usage in modal logic where "L" is typically the necessity operator and "M" the possibility operator.

Also, I'd recommend that the importance class be intermediate (if there is such a thing) rather than "low".

How exactly is 'being knowable' defined?

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)

Lengthy deleted text

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)

I wrote the text above based on the first of the two linked pages, namely http://plato.stanford.edu/entries/fitch-paradox/, which describes the paradox (translated from formal logic notation) as:

• Suppose knowability: For all propositions p, if p is true then p can be known at some time.
• Suppose non-omniscience: There exist some proposition(s) p such that all p are true but no p is known at any time.
• Instantiate: There exists a particular proposition p such that p is true and p is not known at any time.
• Substitute above statement into knowability: if the proposition that, "There exists a particular proposition p such that p is true and p is not known at any time" is true, then it can be known at some time.
• Modus Ponens: since we are assuming that "There exists a particular proposition p such that p is true and p is not known at any time" is indeed true, then it must indeed be possible for "There exists a particular proposition p such that p is true and p is not known at any time" to be known.
• But it can't be, because as soon as you know it, you know p, and then the part of the conjunction stating that "p is not known at any time" becomes false.

The "surprise party" example describes a single knower, but the later example concerning Everest is intended to apply to the whole range of agents capable of knowing things, which yields Fitch's Paradox as above. -- Hyphz

Of course, the premise, all true statements are knowable, is false. See, for example, Gödel's_completeness_theorem --Ken —Preceding unsigned comment added by 128.83.61.196 (talk) 15:21, 18 September 2008 (UTC)

Not a paradox

This isn't a paradox. The trickery lies in thinking that it is impossible for the statement "p is an unknown truth" to be both true and knowable at the same time. Any normal human would 1st realize "p is an unknown truth" (making it true and known(able)) and then quickly know p (at which point "p is an unknown truth" becomes untrue). OR, if we treat the knowing of "p is an unknown truth" and knowing p as true as simultaneous in their realization, then they are simply restatements of the same thing and the paradox becomes semantics from the beginning.173.164.218.92 (talk) 17:31, 17 September 2010 (UTC)

I initially thought the error was in the failure to handle tense correctly, but I've changed my mind - exploring this does make it easier to identify the correct resolution though, so let's run with it. Suppose there's a box which contains something, but no one knows what it contains. There is an unknown truth about what is in that box. However, when someone looks in the box and sees a cat, it becomes a known truth. Now let's look at the statement about the truth of "the box contains a cat". "The box contains a cat" is an unknown truth. That statement is true up until the time when someone looks in the box, but is no longer true afterwards. So how can that statement be both right and wrong? Well, it isn't: it's a new statement every time you read it because each reading of it has a new tense. Let's say that I looked in the box at midnight when today began, the statement, when read yesterday, had a tense that tied it to the 9th of May 2018, but when it's read today it has a tense tied to it which is the 10th of May 2018, so it's clearly a different statement. Yesterdays version of the statement remains true, because it's now a past tense, and "the box contains a cat" was an unknown truth yesterday. Today, the statement is a new one with today as its tense, and it's false. But let's now try to make the statement cover all times and see what happens. The claim becomes: "the box contains a cat" is eternally an unknown truth. This is never true if the content of the box will at any time become known by any means, so the statement can never become a known truth, and that means that there can be unknown truths that can never be known to be true, so rule C is false.

At the end of the article there's a part that introduces a rule C' to replace C, and this is claimed to restore the paradox: "There is an unknown, but knowable truth, and it is knowable that there is an unknown, but knowable truth." The article needs to show this in an example which ordinary people can follow to see whether the claim stacks up. This matters, because there doesn't appear to be any example of an unresolvable paradox anywhere, and if this one's an exception to the rule, it's worth exploring properly.Djvyd (talk) 16:23, 10 May 2018 (UTC)

This is a paradox: it shows that the obvious formalisation of a widely held thesis runs into contradication. The puzzle with attempted solutions to the paradox is to show that they can be formalised without contradiction. — Charles Stewart (talk) 11:30, 14 May 2018 (UTC)

I agree. You can consider tenses or different types of knowing (known unknown truth, such as I know there's a box with something, but I don't know that something), no matter what, Line 1 runs into a contradiction. I think the resolution lies in understanding knowledge as the totality of interconnected particles in the universe. If knowledge is limited to the electricity in my neurons, then I don't know what's in the box and I can never know. Not until some new electricity (information) arrives. If that happens, i.e. my hands open the box and the light bounces off the cat to my eye, then it's not the case that I did not know: I simply did not gauge my knowledge correctly due to time. The paradox disregards space and time: it looks at actuality, not possibility: if I don't know something now and never learn it even if it was possible to learn, it's the same as being impossible to know it. Or if I don't know it, then either I will never know it, or I've miscalculated/restricted the parameters of my information (past, present, and future).Cornelius (talk) 06:51, 22 July 2021 (UTC)

The knowability thesis

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)