So there are unknown truths? — Luke

In reality, yes. However, Fitch's paradox shows that the knowability principle entails that there are no unknown truths. That's why Fitch's paradox shows that the knowability principle is false.

Not according to Fitch's argument. — Luke

Technically speaking Fitch's argument shows that the knowability principle entails that all truths are known. This conclusion is then a reductio ad absurdum to disprove the knowability principle, given that there are unknown truths.

Technically speaking Fitch's argument shows that the knowability principle entails that all truths are known — Michael

Known by everyone always, or known only by someone at some time? I take it all truths are known implies that no truths are knowable (because they are known)? But if they are known only by someone at some time, would that imply they can be knowable by others, in order to save KP?

So, either we know that something is true or false or we cannot say anything about its truthness or falseness.
— Alkis Piskas
So you are going with the rejection of classical logic ... — Banno

BTW, I just realized that my above statement was wrong. And you had the opportunity to easily refute it, if you had paid attention to a detail instead of wondering about what is the type of logic that this statement belongs to. The detail is the word "something". Because one might simply ask: "An apple is 'something'. Can we say that an apple is true or false?" Of course not. It makes no sense. Only a statement or an assertion or a report and that sort of things can be true or false. So my statement was clearly wrong.

Well, another mistake I did was stating that "I stick to simple logic". One might well ask "What is simple logic?", "Simple in comparison with/to what?"[/i], "Simple in what way?", "Why, is there a complicated logic?" and so on. You shouldn't miss that either. I like to have strong "opponents"! :smile:

In philosophical discussions we must pay attention to these things. I'm careless sometimes, too.

The truth is in the detail! :smile:

This has nothing to do with Fitch's paradox. — Michael

It does have a bearing, but you are not interested, which is fine.

By everyone always, or by someone at some time? — Luke

By someone at some time.

I take it all truths are known implies that no truths are knowable (because they are known)? — Luke

In fact the opposite: Kp → ◊Kp.

But if they are known only by someone at some time, would that imply they can be knowable by others, in order to save KP? — Luke

No, because if you address the formal logic of the argument you will see that it entails a contradiction:

a. p → ◊Kp (knowability principle)
b. p ∧ ¬Kp (some proposition that is true but not known to be true)
c. b → ◊Kb (apply the knowability principle to b)
d. p ∧ ¬Kp → ◊K(p ∧ ¬Kp) (substitute in the terms of b)

However, K(p ∧ ¬Kp) is a contradiction, and so isn't possible, as shown below:

e. K(p ∧ ¬Kp) (assumption)
f. K(p ∧ q) ⊢ Kp ∧ Kq (knowing a conjunction entails knowing each of the conjuncts)
g. Kp ∧ K¬Kp (from e and f)
h. Kp ⊢ p (knowledge entails truth)
i. Kp ∧ ¬Kp (from g and i)

i is a contradiction. We cannot know that p is true and not know that p is true. Therefore d is false. Therefore either a (the knowability principle) or b (there is some unknown truth) is false.

There are paradoxes that are not self-referential. — Banno

This is true, but Fitch's paradox is self-referential. Actually, after looking at it more, including SEP, I'm not sure it is. It seems more like a tautology, or at least a trivial statement, a language game. Calling a particular statement a truth means the same thing as saying it is true. If I know something is true, it isn't unknown.

This is true, but Fitch's paradox is self-referential. Actually, after looking at it more, including SEP, I'm not sure it is. It seems more like a tautology, or at least a trivial statement, a language game. Calling a particular statement a truth means the same thing as saying it is true. If I know something is true, it isn't unknown. — T Clark

Where's the language game here?

1. p → ◊Kp (knowability principle)
2. q ≔ the Riemann hypothesis is correct
3. r ≔ the Riemann hypothesis is not correct
4. q ∨ r (law of excluded middle)
5. ¬Kq ∧ ¬Kr (whether or not the Riemann hypothesis is correct is not known)

6. (q ∧ ¬Kq) ∨ (r ∧ ¬Kr) (from 4 and 5)
7. q ∧ ¬Kq → ◊K(q ∧ ¬Kq) (from 1)
8. r ∧ ¬Kr → ◊K(r ∧ ¬Kr) (from 1)
9. ◊K(q ∧ ¬Kq) ∨ ◊K(r ∧ ¬Kr) (from 6, 7, and 8)

10. K(s∧ ¬Ks) (assumption)
11. K(s ∧ t) ⊢ Ks ∧ Kt (knowing a conjunction entails knowing each of the conjuncts)
12. Ks ∧ K¬Ks (from 10 and 11)
13. Kt ⊢ t (knowledge entails truth)
14. Ks ∧ ¬Ks (from 12 and 13)

14 is a contradiction, therefore 10 isn't possible, therefore 9 is false, therefore either 1 or 5 is false.

Sorry. Not good with logical symbology.

See here for an explanation in ordinary language.

7. q ∧ ¬Kq → ◊K(q ∧ ¬Kq) (from 1)
8. r ∧ ¬Kr → ◊K(r ∧ ¬Kr) (from 1)
9. ◊K(q ∧ ¬Kq) ∨ ◊K(r ∧ ¬Kr) (from 6, 7, and 8) — Michael

This is where I think the mathematical formalism is missing something important: the time variable. Knowledge is not static. You are talking of a process of discovery, of the possibility of solving the Riemann conjecture one day. Note that if and when this happens, our knowledge about it will evolve. What we knew not at time t1 will become known at time t2. Which you could write: Kt1(r)<>Kt2(r)

So by adding the time variable, there is no reflexivity anymore. You don't end up knowing what you know not, but knowing now what you knew not back then.

It's called learning.

There exists a being x and a time t such that x knows at t that proposition p is true: ∃x∃t(Kxtp)

1. p → ◊∃x∃t(Kxtp)
2. p ∧ ¬∃x∃t(Kxtp)
3. ◊∃x∃t(Kxt(p ∧ ¬∃x∃t(Kxtp)))

4. ∃x∃t(Kxtp ∧ Kxt(¬∃x∃t(Kxtp)))

There exists a being x and a time t such that x knows at t that proposition p is true and knows at t that there doesn't exist a being x and a time t such that x knows at t that proposition p is true. This is a contradiction. Therefore 3 is false. Therefore either 1 or 2 is false.

Admittedly this doesn't entail that every true statement is currently known to be true, only that every true statement is known to be true at some point, but that might also be an undesirable conclusion. It's possible that the Riemann hypothesis is never proved nor disproved.

See here for an explanation in ordinary language. — Michael

Thanks.

You should introduce a difference in time, t1 and t2, to account for the progression of knowledge that is assumed in your ◊. E.g.

3. ◊∃x∃t2(Kxt2(p ∧ ¬∃x∃t1(Kxt1p)))

That's not how the rules of inference work.

And yet we don't know which of "the Riemann hypothesis is correct" and "the Riemann hypothesis is not correct" is true, but one of them must be. Therefore not all truths are known. — Michael

The criterion is 'knowable' not 'known'.

I make the rules.

The criterion is 'knowable' not 'known'. — Janus

Fitch's paradox shows that if all truths are knowable then all truths are known. Some truths aren't known, therefore some truths aren't knowable.

This has no bearing on Fitch's paradox. — Michael

That's right.

What's difficult to see is if @Olivier5 has a point or has just not understood the logic of the argument. If he has a point, it remains obscure.

The move to "who is it that dos the knowing" is pretty common in phenomenological discourse, but without setting out explicitly how it is relevant to the argument. Notice that the Kvanvig rendering of the argument does take the knower and the time into account. The argument would then proceed into considering the rigidity of the designation, and all that would involve. If @Oliver5 were to proceed in that direction the conversation might become interesting.

SO, Oliver5, are you proposing that the argument suffers a modal fallacy? Can you set it out explicitly?

Fitch's paradox is self-referential. — T Clark

Self reference in itself is not a problem. This sentence is six word long. This sentence contains thirteen words. No worries. SO saying the paradox involves self reference is neither here nor there.

Otherwise you seem to be making the sam ubiquitous error as others hereabouts.

Fitch's paradox shows that if all truths are knowable then all truths are known. Some truths aren't known, therefore some truths aren't knowable. — Michael

Can you lay out the argument clearly in plain English?

The detail is the word "something — Alkis Piskas

I just assumed your were adopting the convention of restricting that "something" to propositions. And I understood your "simple logic" to be classical logic.

The principle of charity at work.

Ah, I se you have used it. Nice.

Can you lay out the argument clearly in plain English? — Janus

@Michael, Don't - it's a trap!

:wink:

It's not a trap; if it can't be expressed in plain language then it has no bearing on epistemology (or anything else) since it is in plain language that philosophy is practiced and our thinking in general is done.

The argument is expressed clearly in both the Wiki and SEP articles. No more than a basic comprehension of formal logic is needed. It seems to me that those who will not proceed with the formality ought not engage in the discussion. If one cannot understand a bit of basic logic one will not be able to follow the argument if presented informally; and one will be adding the congenital problem of informal argument, the propensity for misunderstanding - the results of which are to be seen in many of the posts here.

Formal logic is nothing more than a formalization of the logical validity that operates, or doesn't, in plain language usage. If a conclusion is reached via formal logic which cannot be translated back into plain language and shown to be valid, then something has gone wrong somewhere, and the problem cannot lie with our everyday language, since that is where the formal language is derived from in the first place.

If formal logic is merely a self-enclosed game with its own rules and practices differing from the rules and practices of plain language, that's fine, but then it cannot be plausibly claimed that it has any entailments outside of its own boundaries.

You can't have it both ways.

If a conclusion is reached via formal logic which cannot be translated back into plain language and shown to be valid, then something has gone wrong somewhere... — Janus

But the conclusion of Fitch's argument can be "translated back" into plain english - and has been, multiple times, in both articles and in this thread. :roll:

Formal logic sets out the grammatical structure of the argument clearly. It is clearer and easier to follow the detail than in an informal argument; that's why we use it.

But the conclusion of Fitch's argument can be "translated back" into plain english - and has been, multiple times, in both articles and in this thread. :roll: — Banno

:roll: So what if the conclusion, but apparently not the argument itself can be translated back into plain English? So what if it is "clearer and easier to follow the detail" in the formal language; the detail should nonetheless be able to be translated into plain language and seen to be valid. If it can't be then it's useless.

, so you find logical notation challenging. So do I. It takes effort to see what is going on. But many arguments are clearer when presented formally.

This is one.

many arguments are clearer when presented formally. — Banno

But far fewer when the formalism is modal.