Can $~x~$ be not equal to itself? Can we have something not equal to itself?
$~∃~~x, ~~~~~x ≠ x~~?$
We can have non associative, non commutative, and non transitive.
Does any one have a reference to non identity?
 A: From a mathematical perspective, what you're asking is if there can be some binary relation $=$ such that $x \ne x$, which is definitely possible.
For example, we could easily define a relation $=$ on matrices to be true when their determinants are equal, but not if the matrices themselves are identical, so we could say that
$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$
but
$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \ne \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$
It's entirely dependent on the definition of the relation, and it's entirely possible to define a relation such that there are elements which are not "equal" to themselves, like in eyeballfrog's response. Whether or not such a relation is useful is another question entirely.
A: In the IEEE standard for floating point arithmetic, there is a convention that
NaN == NaN

evaluates to false. What this really means is that the floating point == operation is not strictly speaking an equality test, but it does make it possible for
x != x

to evaluate to true if x happens to have the value NaN. 
A: Not without violating the law of identity.
A: No. $x$ must be equal to $x$. That is a basic identity of math. How else could we be sure of the answer to any problem?
