What goes wrong if we explain Russell's paradox as resulting from an overly rigid link between property satisfaction and elementhood?

Question

Specifically, consider the following: $\exists c \exists m \forall x (x \notin c ⇒ (x \in m ⟺ P(x))) \land (x \in c \Rightarrow (x \notin m ⟺ P(x)))$

We can call the above a "schema of premises" or "conjecture schema" (the phrase "conjecture schema" is to be thought of as analogous to the phrase "axiom schema").

Given that every value of x is either an element of c or not an element of c, the information provided by c and m should be enough to encode or represent everything about the intuitive conception of P(x) as a mapping from the value x to the truth value of P(x).

An answer -- that has unfortunately been deleted -- provided an example of a contradiction that can be deduced from the above conjecture schema (analogous to "axiom schema"), but the deduction relied upon some premises copied directly from ZFC, without adapting the ideas that motivated the formulation of the ZFC premises to the conjecture schema. Nevertheless, that answer at least provided some specific indications of reasoning, and that answer may have been of value in the process of developing a better answer.

What goes wrong with the conjecture schema? It is a simple enough proposal that something must at least appear to go wrong. Maybe an advantage of having at least two people looking at it is that it will be possible to confirm beyond any doubt that what appears to go wrong actually does go wrong.

Answer 1

That axiom is satisfied by empty m,c and P(x) being a false statement.
Nothing goes wrong other than that axiom is vacuous and accomplishes nothing.

Answer 2

Introduce a defined binary relation $[ \in ]$ as follows:

$h [\in] k \iff \exists t \exists u \left( k = (t,u) \land \left( \left( h \notin t \land h \in u \right) \lor \left( h \in t \land h \notin u \right) \right) \right)$, where (t,u) is simply the ordered pair having t as first coordinate and u as second coordinate.

We have among our premises the following:

$\exists c \exists m \forall x (x \notin c ⇒ (x \in m ⟺ \lnot (x [\in] x))) \land (x \in c \Rightarrow (x \notin m ⟺ \lnot (x [\in] x)))$
In particular, plugging in $x = (c,m)$, we obtain:

$\exists c \exists m ((c,m) \notin c ⇒ ((c,m) \in m ⟺ \lnot ((c,m) [\in] (c,m)))) \land ((c,m) \in c \Rightarrow ((c,m) \notin m ⟺ \lnot ((c,m) [\in] (c,m))))$

Thus, by simple manipulation of connectives in propositional logic, we obtain:

To be seen again: $\exists c \exists m \left[((c,m) \notin c \land (c,m) \in m ) \lor ((c,m) \in c \land (c,m) \notin m) \right] \iff \lnot ((c,m) [\in] (c,m))$

However, if $h = (c,m)$ and $k = (c,m)$, then we can simply assign $t=c \land u = m$.

Thus, we have -- directly from our definition of $[ \in ]$ -- the following conclusion:

$(c,m) [\in] (c,m) \iff \left[((c,m) \notin c \land (c,m) \in m ) \lor ((c,m) \in c \land ((c,m) \notin m) \right]$

However, that conflicts with the statement that has the label "To be seen again."

Thus, we have a contradiction and the original conjecture schema or schema of premises is logically inconsistent, rather than vacuous.