In this and this questions, relativization occurs with respect to new unary preicates $P$ and $Q$. For example, if the sets $X$ and $Y$ are spectra, say of the $L$-sentence $\varphi$ and the $K$-sentence $\psi$, respectively (where $L$ and $K$ are disjoint relational first order languages). Then the set $$X+Y=\{\mbox{$m+n$ : $m\in X$ and $n\in Y$}\}$$ is the spectrum of the $L\cup K\cup\{P,Q\}$-sentence $$\varphi^{P}\wedge\psi^{Q}\wedge (\forall x\,(Px\Leftrightarrow\neg Qx)),$$ were $P$ and $Q$ are new unary predicates, and $\varphi^{P}$ and $\psi^{Q}$ are the relativizations of $\varphi$ and $\psi$ to $P$ and $Q$, respectively.
Question: Could we simply relativize with respect to $P$ and $\neg P$. That is, $Q$ is simply $\neg P$? The sentence would become $\varphi^{P}\wedge\psi^{\neg P}$. But it seems there may be circumstances when this is not satisfiable, as we have the predicate $P$ in common. Or, where else is the pitfall?