  Lorenzen dialogues provide a two-player game formalism that can characterize
a variety of logics: each set $S$ of rules for such a game determines a set
$\mathcal{D}(S)$ of formulas for which one of the players (the so-called
Proponent) has a winning strategy, and the set $\mathcal{D}(S)$ can coincide
with various logics, such as intuitionistic, classical, modal, connexive, and
relevance logics. But the standard sets of rules employed for these games are
often logically opaque and can involve subtle interactions among each other.
Moreover, $\mathcal{D}(S)$ can vary in unexpected ways with $S$; small changes
in $S$, even logically well-motivated ones, can make $\mathcal{D}(S)$ logically
unusual. We pose the problem of providing a structure theory that could explain
how $\mathcal{D}(S)$ varies with $S$, and in particular, when $\mathcal{D}(S)$
is closed under modus ponens (and thus constitutes at least a minimal kind of
logic).
