I took a unit in predicate calculus at Sydney Uni, and I didn't find it difficult. I didn't find it that interesting either. My point is that, however difficult it might be to do, if the argument cannot be expressed informally then it has nothing interesting to say; any interest it might have could only be found within the hermetically sealed formal game.

Ha. Rather, modal arguments are far more difficult to present informally. They are just difficult arguments.

I took a unit in predicate calculus at Sydney Uni, — Janus

Bully for you. So rather than set the task for poor @Michael, have a go at it yourself. You have the background, and doing so will give you a much better understanding of how it works.

They are just difficult — Banno

They were conceived in sin.

All true statements are knowable. (1)
All unknown true statements are knowable. (2, from 1)
.......................................................................... (3)
All unknown true statements are known. (4)

Nice try!

This is a good example of how informality introduces problems. I don't see the "knowable/known" distinction in the formal version. There's just "K" and ♢K, which makes it clear that the move from "knowable" to "Known" is modal.

IS this better...

(1) All true statements might be known
(2) All unknown true statements might be known (1, sub)
.........................(3)
All unknown true statements are known (4)

??

Using the SEP proof...

(KP) All true statements might be known
(NonO) There are unknown truths

(1) There is a truth that is not known (instantiation from NonO)
(2) If there is a truth that is not known, then it might be known that there is a truth that is not known
....(sub (1) into KP)
(3) It might be known that there is a truth that is not known

I'll stop there. that should be enough.

Now, is this a reasonable English representation... and if not, can you do better?

And, for @Janus, do you really think this scholasticism clearer than the more formal presentation?

(edit: fixed 3. )

makes it clear that the move from "knowable" to "Known" is modal. — Banno

Yes but modality is obscure. Give us the Venn diagram.

(1) All true statements might be known — Banno

Or maybe

(1) Some judgements of true statements are knowledge.

??

For me, this shows that the formal version is clearer.

Hocus pocus.

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

In fact the opposite: Kp → ◊Kp. — Michael

But, according to the independent argument, starting with the assumption K(p ∧ ¬Kp) leads to the conclusion ¬◊K(p ∧ ¬Kp). That is, if the conjunction is known, then the conjunction is not knowable.

Just a thought.

At the beginning of the discussion @Agent Smith made reference to Meno's paradox, and I think there could be an interesting parallel to Fitch's. An ambiguity is noted wrt Meno's paradox:

Suppose Tom wants to go to the party, but he doesn't know what time it begins. Furthermore, he doesn't even know anyone who does know. So he asks Bill, who doesn't know when the party begins, but he does know that Mary knows. So Bill tells Tom that Mary knows when the party begins. Now Tom knows something, too—that Mary knows when the party begins.

So Tom knows what Mary knows (he knows that she knows when the party begins). Now consider the following argument:

Tom knows what Mary knows.
What Mary knows is that the party begins at 9 pm.
What Mary knows = that the party begins at 9 pm.
Therefore, Tom knows that the party begins at 9 pm.

What is wrong with this argument? It commits the fallacy of equivocation.

In (A), “what Mary knows” means what question she can answer. But in (B) and (C), “what Mary knows” means the information she can provide in answer to that question. — Meno's paradox ambiguity

I wonder whether the same/similar type of ambiguity applies to your Riemann hypothesis examples. You can say (about unknown truths) which set of statements are truth apt, but not which statements are true. In other words, you can know of unknown truths. but you cannot know them (or which of them) to be true.

Not keen on judgement... not modal enough!

(1) Some affirmations of any true statement are justified.

(2) Some future affirmations of any true statement not previously affirmed justifiably are justified.

(3) ...................................................

(4) Some previous affirmations of any true statement not previously affirmed justifiably are justified.


Modalities excised (or easily so). Missing line exposed.

(1) There is a truth that is not known (instantiation from NonO)
(2) If there is a truth that is not known, then it might be known that there is a truth that is not known
....(sub (2) into KP)
(3) It might be known that there is a truth that is not known — Banno

What about

(1) There are truths that are not known (instantiation from NonO)
(2) If there are truths that are not known, then it might be known that there
are truths that are not known
....(sub (2) into KP)
(3) It might be known that there are truths that are not known

It seems obvious that there are truths that are not known; for example someone cited the example that the Earth is (roughly) spherical, which at one time was not known. There must be many truths about other planets or yet to be discovered flora and fauna which are not known.

I'm not seeing how the (apparent) fact that there are unknown truths proves either that there are or are not unknowable truths. And I'm also not seeing how there being knowable (in the sense of becoming, obviously not presently, known) unknown truths proves that all truths are known. There must be some (formal) sleight of hand going on, it seems to me.

There is an unknown truth $\neq$ p & ~Kp for the simple reason that ~Kp means we can't assert p for to assert p implies Kp.

1. p & ~Kp (assume for reductio)
2. p $\to$ Kp (premise)
3. p (1 Simp)
4. Kp (2, 3 MP)
5. ~Kp (1 Simp)
6. Kp & ~Kp (4, 5 Conj)
7. ~(p & ~Kp) (1 - 6 reductio)
8. ~p v ~~Kp (7 DeM)
9. ~p v Kp (8 DN)

Either p is false Or we know p (is true).

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

I'll have a go. It might not be correct (or helpful) but maybe others can chime in to correct and clarify.

Suppose both of these principles:

All truths are knowable (the knowability principle)
We are non-omniscient; there is an unknown truth (the non-omniscience principle)

Combine these principles:

If one of all of the knowable truths (KP) is that we are non-omniscient or that there is an unknown truth (NonO) - in other words, if it is possible to know that there is an unknown truth - then it follows that an unknown truth is knowable.

However, it can be independently shown that an unknown truth is unknowable.

Given the contradiction that an unknown truth is both knowable and unknowable, one of the starting principles (KP or NonO) must be rejected.

However, if we reject the non-omniscience principle (which says that there is an unknown truth) such that there are no unknown truths, then it follows that not only are all truths knowable, but all truths are in fact known.

On the other hand, if we reject the knowability principle (which says that all truths are knowable) such that not all truths are knowable, then it follows that not only is there an unknown truth, but there is an unknowable truth.

Or, as the archived SEP article puts it:

The paradox of knowability is a logical result suggesting that, necessarily, if all truths are knowable in principle then all truths are in fact known. The contrapositive of the result says, necessarily, if in fact there is an unknown truth, then there is a truth that couldn't possibly be known [i.e. an unknowable truth]. — Archived SEP article

When we say we don't know then we mean, for a proposition p, p v ~p ( p or not p).

An unknown truth: p is true but we don't know p is true = p & ~K(p v ~p).

We know that p is an unknown truth = K(p & ~K(p v ~p)).

No K(p v ~Kp), no paradox.

1.8k
Can you lay out the argument clearly in plain English? — Luke

Good call!

If one of all of the knowable truths (KP) is that we are non-omniscient or that there is an unknown truth (NonO) - in other words, if it is possible to know that there is an unknown truth - then it follows that an unknown truth is knowable.

However, it can be independently shown that an unknown truth is unknowable. — Luke

I don't see how it follows from the fact that we know (if we do know) there are unknown truths that an unknown truth is knowable; the fact that there are unknown truths (if there are) is not itself an unknown truth (if it is known).

I don't see how it follows from the fact that we know (if we do know) there are unknown truths that an unknown truth is knowable — Janus

It isn't that we do know there are unknown truths, it is that it is possible to know there is an unknown truth. If it is possible to know, then it is knowable. These terms are simply synonymous.

A reminder here that this comes from combining the two starting principles, KP and NonO.

the fact that there are unknown truths (if there are) is not itself an unknown truth (if it is known). — Janus

No, but why do you think it should be?

(1) There are truths that are not known (instantiation from NonO) — Janus

That's not an instantiation.

https://en.wikipedia.org/wiki/Existential_instantiation
is the rule being used.


But also
https://en.wikipedia.org/wiki/Universal_instantiation

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

Yes, something like that. Poor logical formalism. The paradox is about the capacity to learn, to know not some truth at time t and then to know it at time t'. This implies that 'the knowledge of x' changes, that it depends on the knower and the time of the knowing, especially if we are talking of a learning process, as Fitch objectively is.

The logical contradiction stems therefore from postulating a change in knowledge in the problem statement, but then ignoring such change in the formalism.

If in the formalism of Fitch you introduce the idea that knowledge changes over time, you may arrive at something that in English means: he now knows what he knew not before. That is an unproblematic statement about learning something new. But erase time from Fitch (or from that bold sentence), and you get: he knows what he knows not, ie a contradiction.

Kp & ~Kp

Kp
1. p is true
AND
2. Someone believes p
AND
3. p is justified

~Kp
1. p is false
AND/OR
2. No one believes p
AND/OR
3. p isn't justified

Is Kp & ~Kp a contradiction? No! :snicker:

It isn't that we do know there are unknown truths, it is that it is possible to know there is an unknown truth. If it is possible to know, then it is knowable. These terms are simply synonymous. — Luke

OK, that seems fine: so it is possible to know there is an unknown truth; that does not mean it is possible to know an unknown truth (which would be a contradiction) but that it is possible to know that there is an unknown truth (which is not a contradiction).

the fact that there are unknown truths (if there are) is not itself an unknown truth (if it is known). — Janus


No, but why do you think it should be? — Luke

I don't think it should be.

That's not an instantiation. — Banno

Right, not an instantiation, but many instantiations? Why should non-omniscience not entail that there be more than one unknown truth?

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

I'm sure you did. And I'm sure you also aware that assumptions can be big traps. :smile:

The principle of charity at work. — Banno

Thank you for your kindness, Banno. :smile:

There is an unknow truth: p & ~Kp

That there's an unknown truth can be known = K(p & ~Kp)

Because K(r & s) $\to$ Kr & Ks, we can say that Kp & K~Kp

Aside: K~Kp = I know that I don't know p (is true). Socratic.

Kp $\to$ p. Ergo, K~Kp $\to$ ~Kp.

Kp & ~Kp (contradiction).

Hence, ~K(p & ~Kp). Meno! It isn't possible to know that there's an unknown truth. Inquiry is ~◇.

Right, not an instantiation, but many instantiations? — Janus

It needs to be singular to substitute in to (2), so as to get (3) right.

It needs to be singular to substitute in to (2), so as to get (3) right. — Banno

If the singular substitutes into (2) as you laid it out, why doesn't the plural substitute into (2) as I laid it out?

It does, but then the conclusion wouldn't be what we want...

Wouldn't it then just be "it might be known that there are truths that are not known" rather than " It might be known that there is a truth that is not known" ? Is there a salient difference?

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

I've done so a couple of times: here and here.