A curious dialogical logic and its composition problem

TL;DR

This paper explores the composition problem in dialogue games, a framework for logic where two players dispute formulas, and introduces a set of rules close to classical logic with a solvable composition problem.

Contribution

It introduces the composition problem for dialogue games, providing a solution for a set of rules near classical propositional logic, advancing the understanding of dialogical logic frameworks.

Findings

Set N's dialogically valid formulas are peculiar and non-trivial.

The composition problem for set N can be solved directly.

Dialogue games can be tailored to classical logic with specific rule sets.

Abstract

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.