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).

💡 Research Summary

The paper investigates the relationship between sets of rules governing Lorenzen dialogue games and the logical properties of the corresponding sets of formulas for which the Proponent (P) has a winning strategy. Each rule set S determines a set 𝔇(S) of formulas that are “S‑valid”. While many familiar logics (intuitionistic, classical, modal, connexive, relevance) can be captured by appropriate choices of S, the authors observe that the mapping S ↦ 𝔇(S) is highly sensitive: minor, even well‑motivated, modifications of the rule set can produce unexpected logical behaviour.

The formalism distinguishes particle rules, which prescribe how attacks and defenses are made based on the main connective of a formula, from structural rules, which constrain the order and reuse of attacks and defenses. The standard particle rules (Table 1) are kept fixed throughout; the paper varies structural rules. Three canonical structural rule sets are introduced:

• D – the basic set containing rules such as “P may assert an atomic formula only after O has asserted it” (D10) and several constraints on attack‑response sequencing (D11‑D13).
• E – D plus an additional rule (E) requiring Opponent to respond immediately to the most recent P‑statement.
• CL – obtained from E by removing D11 and D12, thereby yielding a rule set that characterises classical propositional logic.

The authors recall Felscher’s theorems establishing that for all formulas ϕ, intuitionistic validity ↔ 𝔇(D)‑validity ↔ 𝔇(E)‑validity, and that classical validity ↔ 𝔇(CL)‑validity.

The central research question is the Composition Problem: given a structural rule set S, is the set 𝔇(S) closed under modus ponens? Formally, does ϕ∈𝔇(S) and (ϕ→ψ)∈𝔇(S) imply ψ∈𝔇(S)? Closure under modus ponens is taken as a minimal requirement for 𝔇(S) to constitute a genuine logic.

Two formulations are presented:

1. Composition Problem (specific) – decide closure for a particular S.
2. Uniform Composition Problem (general) – identify criteria on S (perhaps within a restricted class of rule sets) that guarantee modus‑ponens closure for any S satisfying those criteria.

The authors argue that while for D and CL the composition problem is trivially solved by the existing correspondence theorems, a uniform solution is far from obvious because the notion of a “dialogue rule” is itself under‑specified. They discuss two philosophical perspectives on dialogue games:

• Rational interaction – where rules should support a rational conversational pattern. Here, closure under modus ponens is part of the rationality requirement.
• Calculus view – where dialogue games are treated as a proof system; closure under modus ponens corresponds to admissibility of the inference rule.

To explore the landscape beyond the known maximal rule sets, the paper reports three experimental case studies:

• Logic N – a newly defined logic obtained by tweaking structural rules. Although N sits between intuitionistic and classical logics, the authors exhibit formulas ϕ and ϕ→ψ that are N‑valid while ψ fails to be N‑valid, thus violating the composition property.
• Intermediate logic LQ – an attempt to capture the “weak excluded middle” (¬ϕ→(ϕ∨¬ϕ)). The proposed rule set, motivated by the structure of LQ, also fails the composition test, illustrating that even well‑motivated intermediate logics may not be obtainable by simple rule modifications.
• Stable logic – a logic where double negation elimination holds for atomic propositions (¬¬p→p). The authors construct a rule set intended to model this property, but again find that modus‑ponens closure is not guaranteed without additional constraints.

These experiments demonstrate that the space of rule sets is populated by many “non‑maximal” configurations that do not yield closed logics, underscoring the need for a systematic structure theory.

Finally, the paper advocates for the development of independent rule sets—rule collections that are not merely derived by adding or removing constraints from D, E, or CL but are conceived as autonomous building blocks. Such independent sets could serve as the basis for a uniform composition criterion, potentially leading to a full-fledged structure theory for Lorenzen dialogue games.

In conclusion, the work frames the composition problem as a central obstacle to a comprehensive theory of dialogue‑based logics. It provides initial negative results, proposes a uniform formulation, and outlines a research agenda: (i) formalise what counts as a dialogue rule, (ii) identify meta‑properties of rule sets that ensure modus‑ponens closure, and (iii) map the landscape of rule sets to known logics and novel non‑logics. The paper thus lays the groundwork for a systematic exploration of how the syntactic design of dialogue games determines their logical strength.