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A curious dialogical logic and its composition problem

Dialogue games are two-player logic games between a Proponent who puts forward a logical formula A as valid or true and an Opponent who disputes this. An advantage of the dialogical approach is that it is a uniform framework from which different logics can be obtained through only small variations of the basic rules. We introduce the composition problem for dialogue games as the problem of resolving, for a set S of rules for dialogue games, whether the set of S-dialogically valid formulas is closed under modus ponens. Solving the composition problem is fundamental for the dialogical approach to logic; despite its simplicity, it often requires an indirect solution with the help of significant logical machinery such as cut-elimination. We give a set N of dialogue rules that is quite close to a set of rules known to characterize classical propositional logic, and which is evidently well-justified from the dialogical point of view, but whose set of dialogically valid formulas is quite peculiar (and non-trivial). Its peculiarity notwithstanding, the composition problem for N can be solved directly.