What Axiom/Principle of set Theory Is Expressed By This?

[Frylock]

@Liberal

Well, that’s true. Almost every natural language will accomodate the construction, but for logic self-reference more problematic. (For one thing, there’s the problem of circularity.) Godel had to invent his own self-referential language, and there’s nothing wrong with that.

But you can construct a so-called paraconsistent entailment semantics (which is what Graham Priest did ten years ago or so) that will almost resolve the paradox, except that there’s the problem of supervenience (his world is non-normal). One solution offered as recently as a couple years ago (by Beall) is simply to declare that logical laws fail by fiat. Since fiat is arbitrary, it demands a lack of supervenience. And so, problem solved. But that’s like eating air. Not very satisfying.

But now the pendulum is swinging back the other way. Greg Restall proved last year, in Curry’s Revenge: the costs of non-classical solutions to the paradoxes of self-reference (PDF), that non-classical solutions all come up short. You have to reject large disjunctions, the law of distribution, the transitivity of entailment (e.g. A -> B, B -> C means that A would not imply C), or reject the schema itself, which means rejecting the whole logic you’ve invented (thereby leaving only meaningless semantics).

For classical solutions, you need at least referential identity (something to mean “this element”), a law of non-contradiction, a law of identity, a rule of modus ponens, and a self-contained truth predicate (something that says an element is true in your language). Natural language has all these. (And that’s why it’s so easy to read and write the statement in English.) But mathematical languages generally don’t.

It’s really a booger.

Could you tell me what “non-normal” and “supervenience” mean in this context?

-FrL-

[Chronos]

For classical solutions…

Curious choice of words, there. I would call a “solution” something that allowed you to avoid a paradox. You seem to be using the term to mean something which would allow you to create a paradox.

I don’t know if this is a trivial case of one of the “solutions” you mentioned, but one could always just construct a logic in which every statement is proven. It’s not very interesting, but…

[Liberal]

@Frylock

Could you tell me what “non-normal” and “supervenience” mean in this context?

-FrL-

A non-normal world, in this context, is one in which at least one contradiction exists. Kripke came up with the concept in order to model logics weaker than S4 (like S5, for instance). To make a non-normal world normal, you have to introduce a premise. To normalize S5, for example, you have to posit that if something is possible, then it is necessarily possible (the S5 Axiom).

Supervenience is too complex to cover here, but essentially in this context, I mean that if X and Y are two sets of properties, then X supervenes on Y if being a Y-something implies being a X-something. If we may arbitrarily assign a property to an X (that is X-indiscernible), then we may assign a property that is Not-Y. And that introduces a contradiction.

@Chronos

Curious choice of words, there. I would call a “solution” something that allowed you to avoid a paradox. You seem to be using the term to mean something which would allow you to create a paradox.

Well, sure. To solve it, you must be able to state it. What people are looking for is a way to state it so that it can be resolved by a set of rules.

I don’t know if this is a trivial case of one of the “solutions” you mentioned, but one could always just construct a logic in which every statement is proven. It’s not very interesting, but…

Yeah, that’s what rejecting the schema does, which I mentioned.

[Squink]

@Liberal

Supervenience is too complex to cover here, but essentially in this context, I mean that if X and Y are two sets of properties, then X supervenes on Y if being a Y-something implies…

So much for my vision of a Kwik-E-Mart the size of a Walmart.