Toward a structure theory for Lorenzen dialogue games

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).