Two logics for singular cardinals

Saharon Shelah introduced a logic called L^1_kappa which is interesting for kappa such that kappa=beth_kappa. This logic has many properties of the first order logic, including Interpolation, Undefinability of Well Order (in a strong form) and a Lindstrom type characterisation. The downpart is that it does not have an a priori syntax, the syntax is defined using certain games. In our joint work with Jouko Väänänen, we make a formal comparison of this logic, using Chu transforms, with the well-known and simply define chain logic. We show that when kappa as above is in addition singular with countable cofinality, the chain logic has all the desired properties, and resolves a question from Shelah’s paper.