On Some Infinitary Logics

We define a new class of infinitary logics ${\mathcal{L}}_{\kappa ,\alpha }^1$ generalizing Shelah’s logic ${\mathbb{L}}_{\kappa }^1$ defined in \cite{MR2869022}. If $\kappa ={\mathrm{\beth }}_{\kappa }$ and $\alpha <\kappa$ is infinite then our logic coincides with ${\mathbb{L}}_{\kappa }^1$. 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 Lindström’s Theorem for ${\mathbb{L}}_{\kappa }^1$ in terms of the $\phi$-submodel relation.