About a year and a half ago, while I was looking on the Web for papers regarding the Russell paradox, I chanced to find an interesting concept. This concept was contained in what (for want of a better term) was a two-page 'advertisement' for Math 582, "Introduction to Set Theory", taught at the University of Michigan (the 'advertisement' had the title "The Joy of Sets" and can still be found on the Web under the title "The Joy of Sets--University of Michigan"). The author of the 'advertisement' was Karl Liechty, who is an assistant professor there. The concept (which Prof. Liechty called "the MOST IMPORTANT PROPERTY a set S has...") is the following (I quote the sentence it is contained in verbatim):
"For us, the MOST IMPORTANT PROPERTY a set S has is this: if x is an object, then either x $\in$ S or x $\notin$ S, but not both."
This concept can be made purely symbolic as follows:
(x)((x $\in$ S $\lor$ x $\notin$ S) $\land$ $\lnot$(x $\in$ S $\land$ x $\notin$ S)) ( it is interesting to note that the concept, formulated as such, seemingly allows the universal quantifier to be unrestricted--that is, to mean 'for all possible objects'....).
Prof. Liechty uses this concept as follows:
"Russell's paradox provides a non-example of a set. Consider
{S is a set| S $\notin$ S}.
Call this candidate for set-hood T. As you should verify, we have both T $\in$ T and T $\notin$ T. Thus, T does not have the MOST IMPORTANT PROPERTY, and so is not a set."
Can the other "candidates for set-hood" that generate the Curry paradox, The Burali-Forti paradox, Cantor's paradox, etc. be shown to be nonexamples of sets because they, too, do not have the "most important property" (it should be noted that since the candidates for set-hood that generate the paradoxes can be formulated using set-builder notation (i.e. {x|$\phi$(x)}) this notation can be reserved for mere collections (or classes, if you will))? Also if the paradoxical candidates for set-hood can be shown to be non-examples of sets by 'the most important property', can Naive set theory, when restricted to collections deemed sets by 'the most important property', be rendered free of paradox?
No, that doesn't tell us anything relevant about sets.
Since $x\notin S$ is just an abbreviation for $\neg(x\in S)$, we can abbreviate your property $$ \forall x. \; (x\in S \lor \neg(x\in S))\land \neg(x\in S\land \neg(x\in S)) $$ as $$ \forall x. \; (A\lor \neg A)\land \neg(A\land \neg A) $$ where $A$ is the proposition $x\in S$.
And this is a logical tautology -- it is true as a matter of pure logic no matter what the proposition $A$ it, and therefore also independently of what $x\in S$ means or how sets behave in general.
It's just a simple consequence of the fact that we want to reason about sets using ordinary classical logic.
At best, the statement could be taken as a roundabout way to say that $x\in S$ is a logical proposition that obeys the rules that logical propositions do -- in particular whether $x\in S$ is true does not depend on anything else than what $x$ and $S$ are. For example, the truth of $x\in S$ is not allowed to depend on which context we ask the question in, or be different if we ask the same question at different times -- unless the different context assigns different meaning to one or both of the letters $x$ and $S$.
It would be somewhat more meaningful to say that the ONLY property a set $S$ has is the totality of answers to the question "does $x\in S$" for all possible $x$s. That's formalized as the axiom of extensionality, saying that if $\forall x.x\in S\leftrightarrow x\in T$, then $S=T$.