On Some Infinitary Logics

We define a new class of infinitary logics $\mathscr L^1_{\kappa,\alpha}$ generalizing Shelah's logic $\mathbb L^1_\kappa$ defined in \cite{MR2869022}. If $\kappa=\beth_\kappa$ and $\alpha <\kappa$ is infinite then our logic coincides with $\mathbb L^1_\kappa$. We study the relation between these logics for different parameters $\kappa$ and $\alpha$. We give many examples of classes of structures that can or cannot be defined in these logics. Finally, we give a different version of Lindstr\"{o}m's Theorem for $\mathbb L^1_\kappa$ in terms of the $\phi$-submodel relation.