  For a cardinal of the form $\kappa=\beth_\kappa$, Shelah's logic $L^1_\kappa$
has a characterisation as the maximal logic above $\bigcup_{\lambda<\kappa}
L_{\lambda, \omega}$ satisfying Strong Undefinability of Well Order (SUDWO).
SUDWO is a strengthening of the Undefinability of Well Order (UDWO). We prove
that if $\kappa$ is singular of countable cofinality, Karp's chain logic
\cite{Karpintroduceschain} is above $L^1_\kappa$, while it is already known
that it satisfies UDWO and Interpolation. Moreover, we show that in these
circumstances, the chain logic is -- in a sense -- maximal among logics with
chain models to satisfy UDWO. We then show that the chain logic gives a partial
solution to Problem 1.4. from Shelah's \cite{Sh797}, which asked whether for
$\kappa$ 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 $L_{\kappa,
\kappa}$, Karp's chain logic satisfies the required properties. In addition, we
show that this chain logic is not $\kappa$-compact, a question that we have
asked on various occasions. We contribue to the 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 $L^1_\kappa$ in satisfying Interpolation, undefinability of well-order
and maximality with respect to it, and the Union Lemma. In addition it has a
Completeness Theorem.
