I was a bit sloppy there. When I said A <-> not A I meant that the equivalence is the defining equivalence of A and in order to be a valid definition the equivalence must be true. The liar paradox is there. In propositional calculus. Not A cant be the definition of A.

For me the fact that the negation of a sentence occurs at the right side of the definition of that sentence does not make it a paradox (in naive logic), see my previous post.

Question: do you think that "All statements are true, false or paradoxical." has no truth value?

Why can't I have "is a statement" and "is false" in predicate logic? — Pippen

The short answer is: to avoid crap like this. The predicate logic we use was designed to formalize mathematics. It's supposed to help, not hinder, and there is no reason to think you can do everything in it.

The long answer is: in part Tarski, and in the other part some metalogic I don't know.

Partial explanation, ignoring the metalogic, which someone else would have to speak to: truth is the fundamental primitive in the system. It's already there, so there's no reason to introduce an "… is true" predicate. (This is easy to overlook because modern notation dropped Frege's early assertion stroke and judgment stroke. Start reading "p" as "It is true that p" and you'll see what I mean.)

If you had an "… is true" predicate, you'd want to apply it to propositions. But it's not perfectly clear that propositions are objects. What we've got is a way to say things like "If x is divisible by 5, then x is not prime." Does what's in between the quotation marks look like an object to you?

Keep in mind: you should not be able to get the result you want. Your argument, in brief, is that if A is false and A is equivalent to B which is neither true nor false-- wait, what? If it is equivalent, this won't happen. If they're not equivalent, then you've nothing to say. Pick your poison.

I think what you really have is an "apparent" paradox. That is, an ambiguity. You use the ordinary English word "equivalent" to mean something like, "Another way to say this is …" but then you take it to mean "must have the same truth value."

So in fact S is not false, but illogical!!! — Pippen

S = all statements are false
S' = all statements but S is false AND S is false
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Argument A
You say S can't be true because that would make S false: the contradiction S & ~S. Then you conclude, by RAA, that S is false.

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Argument B. You go on...
S = S'

You say S' can't be false because of the clause ''S is false'' which is the Liar statement and is neither true nor false. So S can't be false too.
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Your argument rests on the assumption that a conjunction of a proposition with a nonproposition can't have a truth value. Am I right?

The problem is such conjunctions, as you've used, are syntactically wrong. Conjunctions, or any logical operator, are restricted to propositions. Since ''S is false'' (the Liar statement) isn't a proposition, S' is not a proposition and so has no truth value. Do you mean this?

What's important is that S is interpreted as ''all statements but S are false''.

In fact if you want to define S without having a truth value then you have to give up the X <-> X & (A or not A) tautology.
Because lets say A <-> X is true.
Based on your logic A <-> X & (A or not A) does not have a truth value. And therefore X and
X & (A or not A) are not equivalent.
Your logic is fatally broken I think. (Unless you specify it even more to get a standard naive multivalued logic)

First of all I disagree with Srap that you can't have a predicate "is true/false" in PL. I think you can have, but sometimes it leads to inconsistencies or violation of formal rules and only in that last case it is forbidden. That's why "This statement is false" is forbidden in PL, because you couldn't assign a truth value to it which is necessary to apply PL.

I disagree with Meta that one can construt the liar paradox in propositional calculus. It's too weak.

I think I found the solution (at least for myself). "All statements are false" can never be equivalent to "All statements are false AND this statement is false" because the last sentence would not be a wff in predicate logic. From "Alle statements are false" one cannot deduce "All statements are false AND this statement is false" because of the same formal reason. Therefore I cannot prove "All statements are false" to be illogical. I could only argue that when you say "All statements are false" you really mean "Alle statements are false AND this statement too" and that's indeed illogical. And of course that's what a truth skeptic would argue. But this is not logic anymore, but how you formalize everyday-language into PL.