The composition problem for Lorenzen dialogue games

The problem: given a set S of dialogue rules, show that

If Proponent has a winning strategy for the S-dialogue game commencing with φ and a winning strategy for the S-dialogue game commencing with φ→ψ, then Proponent has a winning strategy for the S-dialogue game commencing with ψ.

The composition problem for a set of dialogue rules S asks whether the set of formulas φ for which Proponent has a winning strategy in the S-dialogue game commencing with φ is closed under modus ponens. From a rule of inference R one can introduce a further parameter into the definition of the composition problem:

Is the set of formulas φ for which Proponent has a winning strategy in the S-dialogue game commencing with φ closed under the rule R?

The original composition problem—which depends on a set S of dialogue rules—is evidently just the specialization of the more general two-parameter composition problem where we have plugged in modus ponens for the rule of inference parameter R. From this more general perspective, here are two further compositions problems:

• modus tollens:

  If Proponent has a winning strategy for the S-dialogue game commencing with φ→ψ; and a winning strategy for the S-dialogue game commencing with ¬ψ, then Proponent has a winning strategy for the S-dialogue game commencing with ¬φ.

• “implication weakening”:

  If Proponent has a winning strategy for the S-dialogue game commencing with ψ then Proponent has a winning strategy for the S-dialogue game commencing with φ→ψ.

These variants of the original composition problem may or may not be more tractable than it. Either kind of solution to the composition problem or its variants—positive or negative—would be valuable. Of course, positive and negative solutions (counterexamples) to the general composition problem abound: for any interesting rule of inference at hand, we can simply concoct ad hoc dialogue rules that ensure that a positive or a negative counterexamples. These counterexamples, though, lack interesting logical content. A challenge then is to formulate some minimally contentful dialogue rules that give the discussion at least some semblance of serious logical investigation. There may be many such minimally contentful sets. The composition problem is valuable because it is, for of all, so basic and gets at the very heart of the claim that (strategies for) dialogues (à la Lorenzen) really do give us another view on validity. The problem is tantalizing because we do know, for at least one or two sets of dialogue rules, that there is a positive solution to it. Felscher's so-called equivalence theorem, showing that Proponent has a winning strategy for the dialogue game commencing with φ iff φ is intuitionistically valid, amounts to a positive (though not entirely straightforward) solution in the case of Felscher's dialogue rules. Moreover, there are cases where we expect a positive solution (I have in mind here cases where we drop one or two of Felscher's rules, or add an “innocent” extension of them), but which—to my knowledge at least—are open.