(Note that the logic systems described in this question *only refer* to logic systems restricted to the language $\in$)


**1. Can $\mathcal{L}\_{\kappa,\kappa}$ express $n$-th order finitary logic?** It is clear that it can express first-order logic ($\Pi\_{<\omega}^0$), but it seems unlikely that $\mathcal{L}\_{\kappa,\kappa}$ can express even $\Pi\_{<\omega}^1$.


By the answer to this question: [Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem)](https://mathoverflow.net/questions/156940/g%C3%B6dels-constructible-universe-in-infinitary-logics-a-possible-approach-to-hod), it is known that $L^{\Pi\_{<\omega}^1}=HOD$. Assuming $\mathcal{L}\_{\kappa,\kappa}$ expresses $\Pi\_{<\omega}^1$, it is then not difficult to show that $HOD\subseteq L^{\mathcal{L}\_{\kappa,\kappa}}\subseteq L(V\_\kappa)$. Thus, unless $L(V\_\kappa)=V$ (which is true iff $L=V$), $V\neq HOD$.


Therefore, the existence of such a $\kappa$ is inconsistent with $V\neq L\land V=HOD$. This is very doubtful.


**2. Can $\mathcal{L}\_{\kappa,\kappa}$ express $\Pi\_n^1$ for $n>0$?** For $n=0$, it is once again easy to see that, yes, this is true ($\Pi\_0^1=\Pi\_0^0$) for every $\kappa$. Other than this, I have no information for this question.



How expressive can $\mathcal{L}_{\kappa,\kappa}$ be?
The answer is no. 


Suppose that $\kappa$ is a Mahlo cardinal. This is $\Pi^1\_1$ expressible in $V\_\kappa$, since it amounts to the assertion that every closed unbounded subset of $\kappa$ contains a regular cardinal. But it cannot be expressible by an assertion of $\mathcal{L}\_{\kappa,\kappa}$, since there is a forcing extension $V[G]$ in which $V\_\kappa$ is preserved, but which destroys the Mahloness of $\kappa$. Since $V\_\kappa$ is preserved, the truth of any $\mathcal{L}\_{\kappa,\kappa}$ assertion is invariant by the forcing, but the Mahloness of $\kappa$ is not.


One can make other similar kinds of examples. For example, "$\kappa$ is regular" is expressible in $V\_\kappa$ by a $\Pi^1\_1$ assertion, but it cannot be provably equivalent to any $\mathcal{L}\_{\kappa,\kappa}$ assertion, since for example with a measurable cardinal, this can be changed by forcing without changing $V\_\kappa$. 


Here is another way to argue, without forcing. Let $\kappa$ be an inaccessible cardinal. The assertion that $\kappa$ is regular is $\Pi^1\_1$ expressible in $V\_\kappa$. If we fix a $\mathcal{L}\_{\kappa,\kappa}$ truth predicate for $V\_\kappa$, we can find many $\gamma<\kappa$ with $V\_\gamma\prec\_{\mathcal{L}\_{\gamma,\gamma}}V\_\kappa$. In particular, there will be such $\gamma$ of cofinality $\omega$ above the size of any particular $\varphi$ in $\mathcal{L}\_{\kappa,\kappa}$. So $V\_\gamma$ and $V\_\kappa$ will agree on the truth of $\varphi$, but not on the assertion that the ordinals are regular.


I believe that one can likely press this further to make an example in ZFC, without any large cardinals.


