On Some Infinitary Logics

We define a new class of infinitary logics ℒ^1_κ,α generalizing Shelah's logic 𝕃^1_κ defined in . If κ=ℶ_κ and α <κ is infinite then our logic coincides with 𝕃^1_κ. We study the relation between these logics for different parameters κ and α. 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 𝕃^1_κ in terms of the ϕ-submodel relation.