It wasn't intended as a criticism. I was simply looking to see where you were going.

Here:

So, either we know that something is true or false or we cannot say anything about its truthness or falseness. — Alkis Piskas

You suggest three truth-values - "true", "false" or "cannot say". My bolding. All I was wondering is what variation you might choose. I'm aware of two choices. Intuitionist logic, such that statements are not true until proven, and paraconsistent logic, rejecting ex contradictione quodlibet.

3. p∧¬Kp→◊K(p∧¬Kp)

The logic is straightforward and results in a contradiction. — Michael

It doesn't make any difference expressed in notation. 3 does not follow from 1 and 2.

1 says that p is possible to know (ie there exists a circumstance in which p is known)
2 says that it is the case that p is not known
3. then claims that it is possible to know p and not know p, but it doesn't follow since it could still be possible to know p, just not in the particular circumstance where one knows that one does not know p.

Saying that it is possible to know p doesn't rule out circumstances where it becomes impossible to know p. It says nothing of the contingency of knowing p, only that there exists a set of circumstances where it could be the case.

@Michael just left some of the proof out. From p→♢Kp it does follow* that p→Kp; that is, if it is possible to know anything that is true, then every true proposition is known.

*Well, it follows if one holds to classic logic. As @Alkis Piskas pointed out, and as is explained in the SEP article, there are alternatives.

Thanks.

Here's where I'm having trouble (gone to the SEP)

If this existential claim is true, then so is an instance of it:
(1)p∧¬Kp.

Now consider the instance of KP substituting line 1 for the variable p

in KP:
(2)(p∧¬Kp)→◊K(p∧¬Kp)

Since p is a proposition (a factual claim about the way the world is) the problem seems trivially solved by saying that some proposition exists for which it is not possible to know the truth. Namely p∧¬Kp.

It doesn't make any difference expressed in notation. 3 does not follow from 1 and 2. — Isaac

Maybe if I make it clearer you can see:

1. p→◊Kp (knowability principle)
2. q ≔ p∧¬Kp (define q as something that is true but not known to be true)
3. q→◊Kq (apply the knowability principle to q)
4. p∧¬Kp→◊K(p∧¬Kp) (substitute in the definition of q)

the problem seems trivially solved by saying that some proposition exists for which it is not possible to know the truth. — Isaac

Then that's a denial of the knowability principle. The problem is that if you insist on the knowability principle then the only other way to avoid a contradiction is to deny the non-omniscience principle (i.e. to accept that every true proposition is known to be true).

You suggest three truth-values - "true", "false" or "cannot say". My bolding. All I was wondering is what variation you might choose. I'm aware of two choices. Intuitionist logic, such that statements are not true until proven, and paraconsistent logic, rejecting ex contradictione quodlibet. — Banno

I see, OK, but I'm not familiar with either intuitionist or paraconsistent logic. I never use and never need to use such terms. 1) They render a discussion to a literary one, 2) They require special knowledge from all the persons involved in the discussion, which might not be available, 3) They might be confusing and/or irrelevant to the subject that is discussed and, most importantly, 4) They do not really add anything that is of essence or importance.

A clear statement/argument talks and can stand by itself, however you call or categorize it.

However, I do not claim omniscience. Instead, I would argue that truth implies knowledge. This is the conclusion of the argument, after all: for all p, if p is true, then it is known that p is true. The reason that the (NonO) statement is false is because p is true implies p is known, so there cannot be any p for which p is true and p is unknown. The reason that p is true implies p is known is because p cannot be true without knowing the meaningful proposition represented by p. Again, this results from the equivocation over the meaning of p and the truth of p. — Luke

I think you're the only one guilty of equivocation here. In the context of the argument, Kp means "it is known that the statement p is true". It does not mean "the statement p is known of" or "the meaning of statement p is known".

I'm not following your line of thinking. We know that from nonO we can derive (1), and from that we reach the absurdity of knowing p while knowing we don't know p - that's (3). The question becomes, which of the assumptions do we dump?

p→♢Kp — Banno

False, a given observational dataset is compatible with multiple hypotheses. There's no way of knowing which one is the true hypothesis even when one hasta be true (re the scientific method).

It's taken as true by various philosophical notions, explicitly or more often implicitly. Those notions that do so must explain how they deal with Fitch.

The argument doesn't asserting it, but uses it hypothetically to show that consequence,

OK, so perhaps inadvertently you had hit on one of the possible responses.

Not much point in complaining about he use of specialised language in a thread on logic.

Anyway, have you further thoughts, given the consequence of your proposal? Are you happy to throw out classical logic? I think it a very promising line.

It's taken as true by various philosophical notions, explicitly or more often implicitly. Those notions that do so must explain how they deal with Fitch.

The argument doesn't asserting it, but uses it hypothetically to show that consequence, — Banno

Two categories of propositions then:

1. p's that are provable, belief-apt, and true e.g. Biden is POTUS

2. p's that are unprovable though belief-apt and true e.g. scientific hypotheses.

No. Just a hypothetical. If p then q.

Thanks. That is clearer.

Then that's a denial of the knowability principle. — Michael

and

I think it's trivially true that the knowabilty principle cannot apply to propositions about our own knowledge where knowledge is treated as JTB, since the truth of the proposition is contained within the definition of knowing the proposition. To say that "I know I know" is to say (ignoring the justifications for now) "It is true that it is true" which is nonsense. We cannot know things about our knowledge under JTB (one of the reasons I don't like it) because it is senseless to make a truth claim about another truth claim. Whatever uncertainty we had about the first truth claim is automatically propagated to the second.

So, in the terms of the argument, to say that for all p it is possible to know p is to say that for all p it is possible to have p true and be justified believing p.

If q (our substitution) is "p is true" plus some proposition about my knowledge of p, then "p is true" is already a claim (contained within the knowledge claim). "p is true and I don't know p" can be rendered as "p is true and I lack justification", or "p is true and p is not true", or "p is true yet I do not believe p". Or some combination of the three. All of which are clearly contradictory.

No. Just a hypothetical. If p then q. — Banno

Why? Some propositions are provable (e.g. Fermat's last theorem) and others not (e.g. the theory of relativity).

I think that what you are suggesting is correct, but also that it is taken into account in the structure of the argument.

SO, to give an example of who the argument might address, suppose that someone argues that only things that have been proved true are true.

Then for them, if some statement is true, then it has been proven - that's their definition of "true". And if it is proven, it is justified. So we know it. Hence, if a statement is true in this system, it is known.

And it follows from the argument that everything that is true, is known.

So is our conclusion to be that those who thinks that only things that have been proved true are true is muddled, or that Fitch's paradox is faulty?

So is our conclusion to be that those who thinks that only things that have been proved true are true is muddled, or that Fitch's paradox is faulty? — Banno

Well. In my opinion, the whole field of 'truth', and 'knowledge' is made into a quagmire by the use of a JTB definition of knowledge. To my mind, the 'truth' of a proposition is the extent to which it is actually the case, something we ourselves assess by testing the hypothesis that it is actually the case (note, I'm only saying that this is how we test it's truth, not what it's truth actually means). So our knowledge can only ever be a state of the results from those tests. The truth of "p" doesn't enter into it, the results from our latest tests of assuming p is all we ever have. I can't see a place for a mental state (knowledge) which relies on an external state (the truth of something) to be defined. The mental state (knowing that p) doesn't change dependant on p since p might be completely disconnected from our mental state (teapot orbiting Jupiter).

What is often arrived at by way of compromise is a sense that a claim "I know p" and a claim "John knows p " are two different types of claim, with only the latter assessable by JTB. I don't like that solution (though I grant it's coherent), but then we cannot make the claim made in the knowability proposition that 'We' know anything (by JTB) since 'We' necessarily includes 'I'.

I think it's trivially true that the knowabilty principle cannot apply to propositions about our own knowledge — Isaac

Yes, I was considering the same sort of thing. I think this kind of self-referential knowledge is victim to the same problems as other self-referential knowledge/truth claims like the Liar Paradox. Technically speaking, if all meaningful propositions have a truth value and if "this statement is false" is a meaningful proposition then we have a contradiction. But is it really a problem to say that all meaningful propositions except propositions like "this statement is false" have a truth value? Or is that special pleading?

Perhaps we can say (as me and @Banno discussed in the other thread) that empirical truths are subject to the knowability principle, but that the truth of self-referential knowledge claims, counterfactuals, predictions, mathematics, etc. work differently?

is it really a problem to say that all meaningful propositions, except propositions like "this statement is false", have a truth value? Or is that special pleading? — Michael

Personally, I think statements like this are fine, and I think so on the following ground...

We can only make two kinds of propositions - those about the way the world is, and those about the way the world ought to be. Ignoring the normative for now, and assuming realism, then the world is some way and we determine it to be so by testing the assumption that the world is that way and assessing the result. As such, any idea that 'special pleading' is a fallacy has in it the assumption that the way the world is is simple and contains no special cases. I can see a reason for testing that assumption first, but I can't see a reason why if, on testing that assumption, we find it inadequate, that we shouldn't assume, as our next best assumption, that this is some 'special case'. After all, we've no fundamental reason to assume the world is simple and contains no special cases of otherwise general rules.

I think the same assumption holds even for an idealist. There's no default reason to assume our notions of how we're going to see the world ought contain no special cases of otherwise general rules.

It's not as if the issue hasn't been pretty exhaustively examined. If a special case seems a good solution then, at this late stage, it seems more than a little self-defeatingly stubborn to refuse one.

Perhaps we can say (as me and Banno discussed in the other thread) that empirical truths are subject to the knowability principle, but that the truth of self-referential knowledge claims, counterfactuals, predictions, mathematics, etc. work differently? — Michael

Yes, I think one could almost say that's definitionally true since the dividing out of empirical claims is by finding those to which sense data might apply and to make that delineation one needs to imagine, at least, a way in which one might obtain that sense data (and so 'know' the the proposition).

Claims of the second sort seem to rely more on rule-following and as such encounter the problems Wittgenstein shows about assessing whether a rule is followed, private rules, etc.

Not much point in complaining about he use of specialised language in a thread on logic. — Banno

I wasn't complaining. I just gave you FOUR reasons why I, personally don't use a specialized language. And also because you asked me what kind of logic I'm using, most probably assuming that I would or should know ...

Re "have you further thoughts, given the consequence of your proposal?": What consequence?

Re "Are you happy to throw out classical logic?": I don't know if I have thrown out any kind of logic, classical or other. See, you are still bound to philosophical "lliterature" and generalities.
I asked you to just disprove my statement-position using plausible arguments and/or examples. You still haven't. So I have to assume that you cannot. I'm not surprised ...

Ok. Some of your posts seemed to show a background in philosophy. My bad.

the whole field of 'truth', and 'knowledge' is made into a quagmire by the use of a JTB definition of knowledge. — Isaac

I agree, on Mondays, Wednesdays and Fridays.

While it is interesting to consider the argument in the light of JTB, I don't think that the argument depends on JTB. It has a more general applicability.

I agree, on Mondays, Wednesdays and Fridays. — Banno

Ha! I feel that way about many theories.

It has a more general applicability. — Banno

Your use against idealism seemed apt, certainly. Idealism having it's very own peculiar relationship with the verb 'to know'.

Have you seen Kripke's theory of truth? It denies that all such propositions have a truth value.

Yes, his solution is probably the one I'm most partial to. See here.

I understand that he has antirealist leanings, so I suspect he would follow on and apply the logic from his theory of truth to Fitch's paradox. It would be interesting to see how that would work.

I had thought from his post that was thinking along those lines.

So it would be something like saying that we do know every true proposition, but more propositions keep becoming true as the domain of true propositions expands...

Were'd it go?

Principles:

Knowability: p $\to$ Kp

Non-O: $\exists$p(p $\land$ ~Kp)


Rules:

A. Kp $\to$ p

B. □~p $\to$ ~◇p

C. $\vdash$p $\to \vdash$ □p

D. K(p & q) $\to$ Kp & Kq

p & ~Kp (instantiation of Non-O)

(p & ~Kp) $\to$ K(p & ~Kp) (substitute p & ~Kp in Knowability]

1. K(p & ~Kp) (assume for reductio)
2. Kp $\land$ K~Kp (1, rule D)
3. Kp (Simp 2)
4. K~Kp (simp 2)
5. ~Kp (4, rule A)
6. Kp $\land$ ~Kp (3, 5 Conj)
7. ~K(p $\land$ ~Kp) (1 - 6 reductio)
8. □~K(p $\land$ ~Kp) (7, rule C)
9. ~◇K(p $\land$ ~Kp) (8, rule B)
10. ~$\exists$p(p $\land$ ~Kp) (from 9)
11. $\forall$p(p $\to$ Kp) (from 12)