If first-order formula is true in every expansion of some structure, is it true in any expansion of any elementarily equivalent structure?

Question

Let $M$ be a $\Sigma$-structure and $p\not\in\Sigma$ be a predicate symbol. Let $T := Th(M)$ in $\Sigma$. Let $\phi$ be a formula in $\Sigma(p)$ with the following property: for all $\Sigma(p)$-structures N,

\begin{align} \text{if }N\mathord{\upharpoonright}\Sigma = M\text{, then }N\vDash\phi. \end{align}

In other words, let every $\Sigma(p)$-expansion of $M$ be a model of $\phi$. If we consider $T$ as a set of $\Sigma(p)$-sentences, I wonder if

\begin{align} T\vDash\phi. \end{align}

In other words, if $p$ "occurs tautologically" in $\phi$ with respect to $M$, does it hold for every elementarily equivalent to $M$ $\Sigma$-structure?

Answer 1

Here is a counterexample. Let $\Sigma = \{\leq\}$, and consider the $\Sigma(p)$-sentence $\varphi$ asserting that if $p$ defines a non-empty set with an upper bound, then the set defined by $p$ has a least upper bound. Then $\varphi$ is true in every $\Sigma(p)$-expansion of $(\mathbb{R},\leq)$, since this is a complete linear order, but it is not true in every $\Sigma(p)$-expansion of every structure elementarily equivalent to $(\mathbb{R},\leq)$. For example, if we interpret $p$ as $(-\infty,\sqrt{2})$ in $(\mathbb{Q},\leq)$, $\varphi$ is false.

On the other hand, if $\varphi$ satisfies your desired conclusion, i.e. if $\varphi$ is true in every $\Sigma(p)$-expansion of every model of $T = \text{Th}(M)$, then it follows immediately from Beth's definability theorem that $\varphi$ is equivalent modulo $T$ to a $\Sigma$-sentence.