For a cardinal of the form κ = בκ, Shelah’s logic L 1κ has a characterisation as the maximal logic above \({ \cup _{\lambda < \kappa }}{L_{\lambda ,\omega }}\) satisfying a strengthening of the undefinability of well-order. Karp’s chain logic [20] L cκ, κ is known to satisfy the undefinability of well-order and interpolation. We prove that if κ is singular of countable cofinality, Karp’s chain logic [20] is above L 1κ . Moreover, we show that if κ is a strong limit of singular cardinals of countable cofinality, the chain logic \(L_{ < \kappa , < \kappa }^c{ \cup _{\lambda < \kappa }}L_{\lambda ,\lambda }^c\) is a maximal logic with chain models to satisfy a version of the undefinability of well-order.We then show that the chain logic gives a partial solution to Problem 1.4 from Shelah’s [28], which asked whether for κ singular of countable cofinality there was a logic strictly between \({L_{{\kappa ^ + },\omega }}\) and \({L_{{\kappa ^ + },{\kappa ^ + }}}\) having interpolation. We show that modulo accepting as the upper bound a model class of Lκ, κ, Karp’s chain logic satisfies the required properties. In addition, we show that this chain logic is not κ-compact, a question that we have asked on various occasions. We contribute to further development of chain logic by proving the Union Lemma and identifying the chain-independent fragment of the logic, showing that it still has considerable expressive power.In conclusion, we have shown that the simply defined chain logic emulates the logic L 1κ in satisfying interpolation, undefinability of well-order and maximality with respect to it, and the Union Lemma. In addition it has a completeness theorem.