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)

The error's in the failure to handle tense correctly. 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 on 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 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. Any attempt to make the statement cover all times fails to restore the paradox because the claim has become: "the box contains a cat is an eternally unknown truth", and this is never true if the content of the box will at any time become known. I have yet to find a valid paradox anywhere - I'm confident that they all contain an error. Djvyd (talk) 16:23, 10 May 2018 (UTC)