## "All statements are false" is NOT false!?!

• 52
Most logicians and philosophers seems to agree that the statement (S) "All statements are false" is plain false, because if S is true it's a contradiction, so by RAA it follows ~S ("Not all statements are false") and since that statement is not inconsistent it must be true.

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. Since S = S' this result must also hold for S.

So in fact S is not false, but illogical!!! So who's right, me or the rest of the world?
• 1.1k
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
• 1.1k
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.

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
• 1.1k
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
• 52
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.
• 185
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

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)
• 1.1k
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
• 52
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.
• 2.3k

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.

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.
• 1.1k
But I say that "All statements are false" has no truth value and therefore can't be false.

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
• 52
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.

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?
• 6.5k
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

Is nobody here with real university logic knowledge?

Which university taught you about logic "poison"?
• 2.3k
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''.
• 185

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?
• 52
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.
• 185
"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.
• 76
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).
• 76
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'.

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.
• 185
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.
• 1.2k
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.
• 76

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.
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