Why are my Math LaTex commands causing the characters to get squished up together? Help!

Which argument are you referring to? — InPitzotl

Fitch's paradox is about true propositions.
— TheMadFool

I know. I'm extending it to false propositions as well. Sue me. — Olivier5

If anyone is claiming there is an incorrect step, then I'd like to know where it is here:

Axiom schemata:

(a) Kq -> q

(b) K(q & r) -> (Kq & Kr)

(c) q -> LKq [this is the axiom presumably to be rejected]

Inference rule:

(d) from |- ~q infer |- ~Lq

Proof:

1. K(p & ~ Kp) assume toward RAA

2. Kp & K~Kp from 1 and (b)

3. ~Kp from 2 and (a)

4. ~K(p & ~Kp) from 1 by RAA

5. ~LK(p & ~Kp) from 4 and (d)

6. p & ~Kp assume toward RAA

7. LK(p & ~Kp) from 6 and (c)

8. ~(p & ~Kp) from 6 by RAA

9. p -> Kp from 8 [note that non-intuitionistic logic is used here]

Charitably, you take a false proposition p. You extend that by building a true proposition ¬p from it. You then use ¬p as you would any true proposition. I'm not sure your interpretation is charitable.

Charitably, you take a false proposition p. You extend that by building a true proposition ¬p from it. I'm not sure your interpretation is charibable — InPitzotl

Notice that in your example: q = ~p , you used q (~p is true) and not p (p is false). So, you didn't expand the scope of Fitch's paradox to include falsehoods.

@InPitzotl @Olivier5

How can we embiggen the scope of Fitch's paradox to cover falsehoods?

An attempt was made by saying $\neg p \rightarrow K \neg p$ but

1. Isn't actually doesn't cut it because $\neg p$ is true.

and

2. Knowing also implies truth. We cannot know a falsehood.

Notice that in your example: q = ~p , you used q (~p is true) and not p (p is false). — TheMadFool

p: <- the false proposition.
q: <- the true proposition.
q = ~p <- makes true proposition q out of false proposition p.

Does your proof need a true proposition? Use q.

What's the problem?

p→Kp — TheMadFool

p is true here, right?

Let's change labels from p to q. q→Kq. Now q must be true (because we changed labels), right?

So let's take the case where q = ~p, where p is false. q is still true, right? What did q have to be? True? Okay, well it is true. What did p have to be to extend to falsehoods? False? Okay, well, it's false.

Now, we can talk about p's that are false. And when you do your proof, you use q's for where you used to use p's. What's wrong?

We can do the same thing in reverse. Just take p, where p is true, and that's your typical application. To do falsehoods, take p where p is false. But we can't do the typical, we have to convert that to a true proposition. No problem; add a complement; if p is false, ~p. But the proof only works when p is a true proposition. Okay, no problem; relabel p's in the proof as q's, and say q=~p. Now we have a false p, and a proof that uses the fact that q is true. What's wrong?

I'm not inclined to go through the earlier posts. But, from what I do see, I don't know exactly what people are claiming about truth in this context.

'p' is a meta-variable ranging over sentences in the object language.

The proof is syntactically correct, as far as I can tell.

Whether any given sentence is true or false in any given semantics for a modal language with two modal primitives, I don't see what that has to do with the correctness of the proof.

And, especially, I don't know what "cover falsehoods' is supposed to mean. Whether 'p' stands for a sentence that is true or is false in any given model, the proof is correct as far as I can tell, thus the conclusion:

It is not the case that for every sentence p we have p -> LKp.

Notice that in your example: q = ~p , you used q (~p is true) and not p (p is false).
— TheMadFool
p: <- the false proposition.
q: <- the true proposition.
q = ~p <- makes true proposition q out of false proposition p.

Does your proof need a true proposition? Use q.

What's the problem?

p→Kp
— TheMadFool
p is true here, right?

Let's change labels from p to q. q→Kq. Now q must be true (because we changed labels), right?

So let's take the case where q = ~p, where p is false. q is still true, right? What did q have to be? True? Okay, well it is true. What did p have to be to extend to falsehoods? False? Okay, well, it's false.

Now, we can talk about p's that are false. And when you do your proof, you use q's for where you used to use p's. What's wrong?

We can do the same thing in reverse. Just take p, where p is true, and that's your typical application. To do falsehoods, take p where p is false. But we can't do the typical, we have to convert that to a true proposition. No problem; add a complement; if p is false, ~p. But the proof only works when p is a true proposition. Okay, no problem; relabel p's in the proof as q's, and say q=~p. Now we have a false p, and a proof that uses the fact that q is true. What's wrong? — InPitzotl

It seems we've been talking past each other.

1. What you're saying is true of course. If p is false then, ~p is true and indeed, $\neg p \rightarrow K \neg p$. If then q = ~p, it follows that $q \rightarrow Kq$. If by Fitch's paradox being extended over falsehoods means this to you, you and @Olivier5 are absolutely right. Negations of falsehoods, truth, are being utilized.

2. To me, if Fitch's paradox includes falsehoods within its scope, it can only be so if falsehoods themselves, not their negations which are true (see 1 above), can be part of Fitch's argument.

What does it mean to say that falsehoods are or are not in the scope of Fitch's paradox? What does being "in the scope" of a paradox exactly mean in this context?

What does it mean to say that falsehoods are or are not in the scope of Fitch's paradox? What does being "in the scope" of a paradox exactly mean in this context? — TonesInDeepFreeze

A very difficult question to answer but let's just stick to the basics: Fitch's paradox is about unknown truths.

Suppose p is a sentence that is an unknown truth; that is, the sentence p is true, but it is not known that p is true. — Wikipedia

That seems a different case than that of an unknown proposition: a known proposition whose truth value is yet unknown. There are many many examples of this kind of proposition.

¬p can be stated as "p is false".

q = ¬p

q→Kq

therefore

¬p→K¬p = all false propositions are known propositions.

This is elementary, really. Reason for which I did not write it down, not wanting to insult people's intelligence. @InPitzotl got it immediately. So make an effort, calm your contrarian demons and for once, TRY and understand these ultra basic logical steps above.

¬p→K¬p = all false propositions are known propositions — Olivier5

You need to work on your logic.

Formally, it's about sentences, regarded in terms of two primitive modal operators, whether true or false, whether known to be true or false or not.

The import is:

It is not the case that for all sentences p, we have p -> LKp.

From that, it does follow that there are true sentences such that it is not possible that they are known to be true.

Moreover, that vitiates certain philosophical views.

Why is

q -> Kq

being stated?

No one believes that as a generalization for all q.

Well, maybe you'll get it one day, if you decide to apply your mind to it.

No one believes that as a generalization for all q. — TonesInDeepFreeze

I do believe that.

Well, maybe you'll get it one day, if you decide to apply your mind to it. — Olivier5

Fitch's paradox is about unknown truths. That's the essence of the argument.

It is not the case that for all sentences p, we have p -> LKp — TonesInDeepFreeze

False that all truths are knowable. For if it is, Fitch is right, all true propositions are known. Gödel's Incompleteness Theorems.

To believe

For all q, we have q -> Kq

is extraordinarily contrarian.

It should not be overlooked that 'Kq' does not stand for 'q is knowable' but rather for 'q is known'.

Contrarian to what, exactly?

Fitch does not claim that all truths are known. That is ridiculous a misunderstanding of him.

What he shows is that

If all truths are knowable then all truths are known. But since not all truths are known, we infer that it is not the case that all truths are knowable.

Clarification

In Fitch's argument, p, whereever it appears, stands for true propositions. (p & ~Kp) = an unknown truth.

Regarding some members who want to say Fitch is talking about falsehoods, yes but only in the sense of their negations which are, well, truths.

Contrarian, I would think, to the preponderance of philosophers and to everyday common sense.

It is an extraordinarily outlandish view that every truth is known.

There is no statement in the expositions I've seen of Fitch that 'p' stands for a true sentence.

One may go back and read the exact expositions to see that.

It is an extraordinarily outlandish view that everything is known. — TonesInDeepFreeze

Everything is not known. But every existing proposition has been proposed by someone or another, by definition of what a proposition is. Thus every proposition in existence will be known at least by he or she who originally made the proposition.

A proposition is a form of knowledge. And there can be no such thing as unknown knowledge, even though there's plenty we don't know.

What we don't know is not neatly set in the form of propositions yet. This work still has to be done.

@Olivier5

The earth is round. True
The earth is flat. False

What's the differenc between:

1. Knowing the earth is round [sounds ok]

and

2. Knowing the earth is flat [believe seems a better fit]

and

3. Not knowing the earth is round [looks, feels right]

and

4. Not knowing the earth is flat [something's off]

?

To cut to the chase, it doesn't look like we can know a falsehood.

'known' in this context doesn't mean 'we know that the sentence itself exists'. 'known' in this context means 'we know that the sentence is true'.

If one conflates 'it is known that the sentence exists' with 'it is known that the sentence is true', then of course the whole discussion falls apart.

Also, in this context, it is not just sentences that have known to have been expressed but sentences in general (in a formal context, that would be all the sentences of the formal language).

"We don't know that the earth is round"

and

"We believe that the earth is flat"?

The differences are so easy to point out that I don't see the sense in asking about it.