Is there any state that satisfies for all $x$, $x$ equals $5$ If I have an assertion: for all $x$, $x$ equals $5$
Is there any state that satisfies this assertion?
Does the state $x=5$ satisfy the above?
I believe for every $x$, $x$ cannot be $5$. So no state satisfies this assertion?
 A: What you are calling a state seems to be what it usually called a model.  It is a set of elements and relationships between them that satisfy a set of axioms.  If you have an axiom $\forall x (x=5)$ the model $\{5\}$ satisfies that axiom.  If you have an axiom (your belief) that $\forall x (x \neq 5)$ any model that does not include $5$ satisfies it.
A: Typically, a model for first-order logic specifies a non-empty domain of discourse $D$ (or universe) and an interpretation function $\cal I$ that maps each constant symbol to an element of the domain, each relation symbol to a relation on the domain (e.g., a 2-ary relation symbol is mapped to a subset of $D \times D$), and each function symbol to a function on $D$.
If a model satisfies the sentence $\forall x (x = 5)$, then what must be true?  For every element $e$ of the domain, then $(e,\cal I(5)) \in \cal I(=)$, that is, whatever the interpretation of $5$ in the domain is, every other element of the domain must be the same as it (assuming that $=$ is, in fact, mapped to the actual equivalence relation).  Then the domain must be $\{5\}$.
If you let the $\cal I$ map $=$ to something other than the actual equivalence relation on $D$, that is, if you let it be an arbitrary 2-ary relation, you can also come up with some models, but that's not particularly interesting, since you're just saying $\forall x P(x,5)$ for some relation $P$.
