For the Helsinki Logic Seminar, I gave (virtually) the lecture

**One Puzzling Logic, Two Approximations… and a Bonus.**

**Abstract:** The puzzling logic (called for a singular strong limit cardinal) I will speak about was introduced by Saharon Shelah in 2012. The logic has many properties that make it very well adapted to model theory, despite being stronger than. However, it also lacks a good syntactic definition.

With Väänänen, we introduced the first approximation (called ,) as a variant of with a transparent syntax and many of the strong properties of Shelah’s logic.

The second approximation (called Chain Logic), while not new (it is due to Karp), has been revisited recently by Dzamonja and Väänänen) also in relation to Shelah’s and the Interpolation property.

I will provide a description of these three logics, with emphasis on their relevance to model theory.

As a bonus, I will make a connection between these logics and axiomatizing correctly an arbitrary AEC. This last part is joint work with Shelah.