Why are models of equality impossible?

Question

Let $\mathcal{L}$ be a first order language. We define an $\mathcal{L}$-structure $\mathcal{M}$ as a tuple consisting of:

1. Universe $M$ (understood implicitly to be that over which the variables of $\mathcal{L}$ will range)
2. An assignment of each of the $k$-ary function symbols $f$ of $\mathcal{L}$ to some $k$-ary function $f^\mathcal{M}$ over $M$
3. An assignment of each of the $k$-ary predicates $P$ of $\mathcal{L}$ to some $k$-ary relation $P^\mathcal{M}$ over $M$. If $\mathcal{L}$ contains the predicate $=$, then $=^\mathcal{M}$ is always the true equality relation over $M$ (i.e. the equivalence relation whose equivalence classes are precisely the singleton subsets of $M$).

We define a weak $\mathcal{L}$-structure $\mathcal{M}$ to be an $\mathcal{L}$ structure without the requirement that $=^\mathcal{M}$ be the true equality relation (i.e. it can be any binary relation over $M$).

My question concerns the following excerpt from Logical Foundations of Proof Complexity by Cook and Nguyen.

Are there sentences $\mathcal{E}$ (axioms for equality) such that a weak structure M satisfies $\mathcal{E}$ iff M is a (proper) structure? It is easy to see that no such set $\mathcal{E}$ of axioms exists, because we can always inflate a point in a weak model to a set of equivalent points.

The book then defines a set of "equality" axioms (which, in line with the previous excerpt, can really only require that $=^\mathcal{M}$ be an equivalence relation):

The first three axioms are the usual equivalence relation axioms, and the axioms EA4 and EA5 require that the functions and relations, respectively, assigned by $\mathcal{M}$ respect the equivalence classes of $=^\mathcal{M}$.

I understand how the quoted claim holds in the case where we $\mathcal{E}$ is this set of axioms EA1-5 which define an equivalence relation. Clearly, any potential list of axioms requiring that $=^\mathcal{M}$ be true equality would need to contain such axioms since true equality is an equivalence relation after all. However, I'm not entirely convinced that there isn't some additional set of axioms we could add to EA1-5 which actually would require that $=^\mathcal{M}$ be true equality.

How does one prove that for any $\mathcal{E}$ and weak structure $\mathcal{M}$, it is not the case that $\mathcal{M}$ satisfies $\mathcal{E}$ iff $=^\mathcal{M}$ is true equality?

Answer 1

The axiom schemaas EA4 and EA5 are telling you that the values of the function symbols and predicate symbols do not distinguish between operands that are $=$-equivalent. So your additional axioms have no way of distinguishing between $=$-equivalent elements in a model. To prove this formally, proceed as the quotation from your book suggests: given a weak model, $\cal M$, pick an element of $x$ in the universe of $\cal M$ and extend the universe of $\cal M$ to include a new element $x'$ and extend the interpretation of the function and predicate symbols to treat $x'$ identically with $x$.

Answer 2

The argument is as you quoted: "we can always inflate a point in a weak model to a set of equivalent points". This doesn't depend on anything about $\mathcal L$ or the chosen axioms. Add a copy $x'$ of a point $x$ to $\mathcal M$ and extend the symbol interpretations so that they behave identically on $x$ and $x'$. Now for any binary relation symbol $R$, we have that $\mathcal M$ satisfies $xRx'$ iff it satisfies $xRx$. But $x\ne x'$ whereas $x=x$, so $R$ does not correspond to true equality.