If "All statements are false" is false, then either 1) all statements are true, or else 2)some statements are true and some statements are false. Of course, if the statement is false, then that rules out implication #1.

Then the falsity of "All statements are false" would just mean that some statements are true, and some statements are false.

So, for "All statements are false" to be false, doesn't mean that it, itself, can't be false. It only means that there are at least some true statements.

Michael Ossipoff

But I think this is wrong. S is logically & semantically equivalent to (S') "All statements, but this statement, are false and this statement is false". But S' is illogical since its part "this statement is false" can't be given a truth value. — Pippen

But I remind you that no one's saying that that's so, because people are saying that the statement isn't true. Yes, if the statement is true, then it's false. But if it's false, that doesn't make it true. It just means that some startements are true.

Michael Ossipoff

A.Typo. Here's what I meant to say:

Then the falsity of "All statements are false" would just mean that some statements are true, and some statements are false. — Michael Ossipoff

Michael Ossipoff

But if (S) "All statements are false" is illogical - like I want to prove - then it can't be false (and therefore it's negation be true) like the majority wants us to believe. And (S) "All statements are false" seems to have the exact same meaning like (S') "All statements, but this statement, are false and this statement is false", just that the last one explicitly shows what's implicit in the first one. If S' is an illogical statement - which it is - then since S' has the same content than S, S must also be illogical.

I hope it makes my point clearer.

In formal logic directly self referential statements like the one you showed do not exist and can not be defined. So your problem is a problem of naive logic or just playing with words and is not exact at all.

We know other self referential sentences like:
A: A is false
B: If B is true then 1+1=1
C: C is true
D: All statements are false

A and B leads to contradiction directly.
C and D do not, however.

I think it is a matter of opinion whether we say the truth value of C and D can be defined or not. Again: in formal logic these sentences do not exist hecne the problem has no meaning.

My personal opinion is that if there is a truth value (true or false) for which a statement do not imply contradiction then we can potentially assign that truth value to the statement, so D is false.
(If we have 3 truth values: T,F and X then T or X is true imo. your logic can be different, but that has nothing to do with being right or wrong)

The statement:

"All statements are false" is false...

means:

Some statements are true.

That's completely uncontroversial and unproblematic.

Saying that that statement is true would be meaningless, self-contradictory, without truth-value.

Michael Ossipoff

But I say that "All statements are false" has no truth value and therefore can't be false.

And my proof is simple: We know that (S') "All statements, but this one, are false and this statement is false" has no truth value because of the paradoxical sencond half sentence. But wouldn't we all agree that S' means the same like "All statements are false"? But then "All statements are false" must have the same fate: no truth value.

And that would be huge, e.g. truth skepticism "All statements are false" would be non-refutable instead of plain false.

S = All statements are false
S' = All statements, but S, are false AND S is false

So you're saying S can't be false because S', the equivalent statement, can't be false because of the "S is false" part.

But there's a contradiction in your claim:

All statements, but S, are false is literally saying All statements, but S, are false AND S is true ("but S") and then you contradict this claim by saying, in the latter part, S is false.

But I say that "All statements are false" has no truth value and therefore can't be false. — Pippen

Are you sure it doesn't have truth-value?

That means you're saying that it can't be true or false. But of course it obviously can be false, as we've discussed.

You seem to be saying that it doesn't have truth value because it can't be true.

The statement is false-if-true, but not true-if-false.

It can be false without any problem..

Michael Ossipoff

So you're saying S can't be false because S', the equivalent statement, can't be false because of the "S is false" part. — TheMadFool

Yes, but maybe this proof makes everything clearer:.

If "All statements are false" would have a truth value and therefore would be a statement, it'd follow by universal instantiation + introducing conjunction: "All statements are false and this statement is false". But that deduction has no truth value since it's a conjunction containing "this statement is false" which has no truth value which poisons the whole conjunction eventually. Therefore "All statements are false" cannot be a statement with a truth value as well.

Again: This is HUGE, basically all philosophers agree that truth skepticism (All statements are false) is refuteable, but that seems just false, because it's not even a statement. Is nobody here with real university logic knowledge?

Actually, if you look at a truth-table for three-valued logic then a conjunction of a false statement ("all statements but this statement are false") and a statement that is neither true nor false ("this statement is false") is itself false.

So "all statements but this statement are false and this statement is false" is false. Therefore "all statements are false" is false.

which poisons the whole conjunction eventually — Pippen

Is nobody here with real university logic knowledge? — Pippen

Which university taught you about logic "poison"?

What I'm saying is that your interpretation of ''all statements are false'' is a contradiction from the get go. So, it can't be the correct interpretation.

The only way to make sense of ''P = all statements are false'' is to interpret it as ''all but P are false.'' and that we know is false because we can find at least one statement, other than P, that's true e.g. ''Trump is the president of USA''.

University logic says you did not prove anything. Your statements are meaningless. You're welcome.
edit: Who are the "most logicians and philosophers" you are referring to?

University logic says you did not prove anything. Your statements are meaningless. — Meta

Why?

1. We assume (A) "All statements are false" has a truth value.

2. Since we can always go from "All x are y" (x may stand for 1,2,3 here) to for instance "All x are y and 2 are y" without changing the truth value, it must hold that A is logically equivalent to (A') "All statements are false and this statement is false", so A <-> A'.

3. Now we prove that A' can't have a truth value because of its part "this statement is false". This is self referential and can't have a truth value (if it's true it would be false and vice versa), so it follows for the whole conjunction if A'.

4. That leads to a contradiction, because it's impossible that A <-> A' (2.) and A' doesn't have a truth value (3.). That means that the assumption, A has a truth value, leads to a contradiction and is false, A has no truth value.

That looks like a legit proofsketch to me.

"Why?"
Because Your arguments can not be formalized. You can't speak about "all statements" formally.

Nvm I like your idea. I just don't see why we should choose your idea of giving truth value instead of the other one.

Because Your arguments can not be formalized. You can't speak about "all statements" formally. — Meta

That's incorrect. Assuming a modicum amount of arithmetic, it's possible to formalize the syntax of first-order logic inside a given mathematical theory (say, primitive recursive arithmetic). This technique is known as the arithmetization of syntax and is due to Gödel. Given one such formalization, we can define a predicate S(x) which is true of all and only the (codes of) sentences of the language. So a sentence such as "all statements" would be regimented as "for every x, if S(x), then ... ". Notice that this allows for self-reference, by employing the Carnap-Gödel diagonalization lemma, which is an element in the original proof of Gödel's incompleteness theorem (though it's not essential for proving the result).

2. Since we can always go from "All x are y" (x may stand for 1,2,3 here) to for instance "All x are y and 2 are y" without changing the truth value, it must hold that A is logically equivalent to (A') "All statements are false and this statement is false", so A <-> A'. — Pippen

As I mentioned in a private message, I'm not entirely sure this step holds. What is the reference of "this" above? If it is "this statement is false" (taking it to have small scope), then your proposed rule would take us from "All statements are false" to ""This statement is false" is false". But the latter one is false, not truth-valueless. So the whole conjunction is false.

I have to correct myself. Can't speak about all statements in a definition of a statement.

edit: But.... If we think about statements more naively and generally (or philosophically) then my statement is correct, because your post is about formal statements and not all statements. I think OP counts every self-referential sentence as a statement. In this sense I was correct in my post.

Using the Godel function we can create a statement that is analogous in some sense to the unary predicate S with the properties you describe. But I don't think we would one formalise 'is false', could we? The Godel function arithmetises syntax, not semantics, and 'false' is concerned with semantics.

IIRC, in some interpretations of Godel, his diagonalised sentence is associated with the purely syntactic notion of 'is provable'.

I am rusty on these issues - It's been a few years since I was involved with them, so I'm happy to have my leaky memory corrected.

You are right that we can't define a truth predicate for the language in question in the language itself---that's Tarski's theorem (though you can define it in a metalanguage, and one can then restrict the quantifier of the op to "All statements of the object language"; admittedly, this would make his argument evaporate for rather trivial reasons). But there are other options, namely to introduce a new predicate, say "Tr" to the language, try to fix its extension by introducing new axioms, say "Tr("Q") <--> Q" for every sentence Q. Of course, you need to be careful if you want both to preserve classical logic and avoid inconsistencies, and things can get complicated rather quickly here, especially considering iterations ("Tr("Tr("Q")")", for instance). But there are some reasonable ways of doing it.

I'm not sure I understand. What do you mean by "can't speak about all statements in a definition of a statement"? In any case, it seems to me false; in defining the predicate "S", we will presumably use universal quantifiers, e.g. "(x) (Sx <--> ...)", where "..." is the definition of "S". But then, this quantifier will range over all statements (and more besides), so we will be in a sense "speaking about" all statements. If you mean that we can't mention all statements in the definition of a particular statement, that seems false too. We could define a new statement, say Q, such that Q is "All statements imply themselves", which is presumably true.

As for your point regarding taking statements naively, I'm unconvinced. Formal linguistics try to capture what is meant by a "naive" statement, and it's not unreasonable to suppose that in such a theories we quantify over all such statements. There may be other problems lurking in the background, though: if you construe statement broadly enough (naively?), consider a statement about every statement which is not about itself. Is it about itself?

Let me clarify what I want to say.
Let's take the Berry paradox. In the paradox there is a definition D which is talking about every (arithmetical) definition. D is a well formed definition in the natural language. However if we could formally model D in arithmetic that would mean arithmetic is inconsistent. When I call a concept "naive" I think of this. So there is a (naive) definition that can exist only outside of formal logic.

Same goes for sentences. We can't have a statement A where the definition of A mentions A. When you are talking about the S predicate you automatically restrict the universe to well-formed formulas and the sentence in the OP is no such statement. So there are statements outside of logic and outside of the S predicate. This is why the OP's sentence is outside of the scope of logic and his proof is not a formal proof.

We can't define Q formally (only in the informal metalanguage). There isn't such definition as
Q <-> (Q->Q and X)
Q is talking about all statements but Q itself is a statement and not part of the (subject) universe of statements.

D is a well formed definition in the natural language. — Meta

How can we say it is well-formed when natural language is informal, and hence does not have a notion of well-formed statement? Do you just mean it is grammatically correct? If so, that doesn't tell us much as the statement 'The cheese of five is sad' is also grammatically correct.

Same goes for sentences. We can't have a statement A where the definition of A mentions A. — Meta

Depending on what language we are using, this may be possible. For instance, the sentence:

'This sentence contains the first letter of the alphabet'

is meaningful and true, and can be formalised without difficulty.

If we have a first order language L we can define a relation symbol (with one variable) R by saying:
R(x) <-> Phi(x)
Where Phi(x) is a formula of L. R is not part of L so Phi does not contain R.
I call a statement a (naive) well formed definition if there is no such restriction.
In this sense 'This sentence contains the first letter of the alphabet' is not a formal definition.

You can say we can be more general and use higher order logic but higher order logic is also problematic.

edit: And yes i think being grammatically correct and being a well formed statement (or definition or whatever) are very close concepts in the natural language.

There is no need to use higher-order logic. Let T be a first-order theory containing enough arithmetic to capture the primitive recursive functions (you can let T be primitive recursive arithmetic, Robinson's Arithmetic, etc.). Then T has enough resources to code its own syntax: in particular, it has enough resources to express the following notions: <is a (well-formed) term>, <is a well-formed formula>, <is a sentence>, <is an atomic formula>, <is a theorem>, <is Q> (where Q is any particular expression of the language: it can be part of the alphabet, a term, a formula, anything), etc. The basic idea here is that we can use numbers to stand as codes for things, for example people (ID cards), products (bar codes), strings of symbols (word processing), etc. Anyway, the important point is that using any reasonable coding scheme, T can prove the following lemma:

Carnap-Gödel Diagonalization Lemma: Let Q(x) be any formula of the language with one free variable. Then there is a sentence D such that T proves "D <-> Q("D")", where "D" is the number coding D.

Notice that the above implies that, for any property Q, there is a sentence D which says of itself that it has the property Q (notice also that this immediately implies that, if the theory is consistent, truth is not definable, otherwise we would have a version of the liar paradox---again, which is not to say we cannot introduce a predicate Tr by axiomatic stipulation). In particular, if we order the alphabet of the language and construct a predicate F(x) such that a (code of a) sentence satisfies F(x) iff it contains the first letter of the alphabet (a tedious, but entirely routine exercise, once you get the hang of the coding machinery), there will be a sentence D which says of itself that it contains the first letter of the alphabet. So that particular sentence can be given a formal definition.

This argument fails when applied to a natural language, because there is no precise definition for <is a well-formed formula>. Only our intuition can tell us what do we consider a WFF. There isn't a fixed set of relation symbols either.

edit: Any anyways we are not talking about real self-reference just some kind of reflection. Let's modify the statement of the Berry paradox:
"The definition with the least Godel number not definable in fewer than 20 words."
This is also paradoxical.
Or "The statement with the least Godel number that does not contain the first letter of the alphabet."
This can be a paradox.

This argument fails when applied to a natural language, because there is no precise definition for <is a well-formed formula>. Only our intuition can tell us what do we consider a WFF. There isn't a fixed set of relation symbols either. — Meta

I don't agree with the premises here, but even granting them that's irrelevant: you originally claimed that the op's argument was not formalizable, since it contained self-referent statements. So the claim was about the expressive powers of formal languages, not natural languages. The Carnap-Gödel diagonal lemma literally disproves this assertion: if T is a nice arithmetic theory, for any one-place predicate of the language, we can form a sentence which says of itself that it has that predicate.

Any anyways we are not talking about real self-reference just some kind of reflection. Let's modify the statement of the Berry paradox:
"The definition with the least Godel number not definable in fewer than 20 words."
This is also paradoxical. — Meta

This just shows that "definable" is not, well, definable in a formal language (even this, strictly speaking, is not true: we can both define definability for a given language in a meta-language with richer expressive resources, or, some times, we can make do with "local" definability---e.g. the constructible sets in ZFC). The problem is with the concept of definability, not with self-reference.

Or "The statement with the least Godel number that does not contain the first letter of the alphabet."
This can be a paradox. — Meta

But it's not, and is perfectly definable in, say, Robinson's Arithmetic.

We use the expression self-reference differently. When I say self-reference I mean (as I have explained earlier) that the definiens can not contain the definiendum. And I am not talking about Godel numbers and codings.

You restrict the universe of statements to a set of formulas of a given language. Even in this case the truth predicate is undefinable so the OP's sentence can't be formalized. Or if you formalize the truth predicate in a metalanguage then when talking about all statements you will only talk about all statements of the object language.

I think OP meant something much more general than statements of a fixed object language L when he was talking about all statements.

So either:
1) "All statements are false" is self-referential (as I mean it), or
2) if we assume Godel numberings and that the sentence talks about its Godel-number then
2a) we don't have a truth predicate defined or
2b) we have a truth predicate but then we have meta statements outside of the quantification range
of "All statements are false".

This is why it can't be formalized.

"But it's not, and is perfectly definable in, say, Robinson's Arithmetic."
But it is because "a" is the first letter of the alphabet and
"The statement with the least Godel number that does not contain the first letter of the alphabet." contains "a".

edit: "the definiens can not contain the definiendum" in an explicit definition.

As I mentioned in a private message, I'm not entirely sure this step holds. What is the reference of "this" above? If it is "this statement is false" (taking it to have small scope), then your proposed rule would take us from "All statements are false" to ""This statement is false" is false". But the latter one is false, not truth-valueless. So the whole conjunction is false. — Nagase

You are right with the reference, but if the part "...this statement is false" is false then it'd be true (and vice versa false if it's true), isn't it? So therefoe it can't have a truth value and so does the conjunction.

Just for reference the proof again:

1. We assume (A) "All statements are false" has a truth value.

2. Since we can always go from "All x are y" (x may stand for 1,2,3 here) to for instance "All x are y and 2 are y" without changing the truth value, it must hold that A is logically equivalent to (A') "All statements are false and this statement is false", so A <-> A'.

3. Now we prove that A' can't have a truth value because of its part "this statement is false". This is self referential and can't have a truth value (if it's true it would be false and vice versa), so it follows for the whole conjunction of A'.

4. That leads to a contradiction, because it's impossible that A <-> A' (2.) and A' doesn't have a truth value (3.). That means that the assumption, A has a truth value, leads to a contradiction and is false, A has no truth value.

Now we prove that A' can't have a truth value because of its part "this statement is false". This is self referential and can't have a truth value (if it's true it would be false and vice versa), so it follows for the whole conjunction of A'. — Pippen

I addressed this mistake here.