*Throughout I'm only interested in the **standard** semantics for second-order logic, and all structures/languages are relational for simplicity.*


If defined naively, second-order logic without equality is equivalent to second-order logic, since $x=y$ is equivalent to $\forall A(x\in A\leftrightarrow y\in A)$. However, there is a natural detrivialization of this. Say that a *quasi-isomorphism* between two structures $\mathfrak{A},\mathfrak{B}$ in the same language $\Sigma$ is a binary relation $I\subseteq\mathfrak{A}\times\mathfrak{B}$ such that:


* $dom(I)=\mathfrak{A}$ and $ran(I)=\mathfrak{B}$.
* $I$ preserves-and-reflects all basic relations: for each $n$-ary $R\in\Sigma$, each $a\_1,...,a\_n\in \mathfrak{A}$, and each $b\_1,...,b\_n\in\mathfrak{B}$ with $a\_iIb\_i$ for all $1\le i\le n$, we have $$(a\_1,...,a\_n)\in R^\mathfrak{A}\quad\iff\quad (b\_1,...,b\_n)\in R^\mathfrak{B}.$$


Write $\mathfrak{A}\cong^q\mathfrak{B}$ if there is a quasi-isomorphism between $\mathfrak{A}$ and $\mathfrak{B}$. Quasi-isomorphisms arise as a natural tool for showing non-expressibility in equality-free $\mathsf{FOL}$ or even $\mathcal{L}\_{\infty,\infty}$. This motivates the following "strongly-without-equality" fragment of second-order logic:



> 
> Let $\mathsf{SOL}\_\*$ be the set of second-order formulas such that for every pair of structures $\mathfrak{A},\mathfrak{B}$, every quasi-isomorphism $I:\mathfrak{A}\cong^q\mathfrak{B}$, and every pair of appropriate-arity tuples of elements $a\_1,...,a\_n\in\mathfrak{A}$ and $b\_1,...,b\_n\in\mathfrak{B}$ with $a\_iIb\_i$ for all $1\le i\le n$, we have $$\mathfrak{A}\models\varphi(a\_1,...,a\_n)\quad\iff\quad\mathfrak{B}\models\varphi(b\_1,...,b\_n).$$
> 
> 
> 


In some notes on abstract model theory I'm writing I plan to include a few challenge exercises about $\mathsf{SOL}\_\*$ *(e.g. "give a not-too-terrible syntax for it")*, but I have a strong suspicion that it's already present in the literature. So my question is: what's a good source on this logic?



Source on equality-free second-order logic (nontrivially construed)
It currently appears that this logic is not already treated in the literature.


*(I'm posting this answer to move this question off the unanswered queue, but if someone does find a source on it of course please add it!)*


