*S*of dialogue rules, show that

If Proponent has a winning strategy for theThe composition problem for a set of dialogue rulesS-dialogue game commencing with φ and a winning strategy for theS-dialogue game commencing with φ→ψ, then Proponent has a winning strategy for theS-dialogue game commencing with ψ.

*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 theThe original composition problem—which depends on a setS-dialogue game commencing with φ closed under the ruleR?

*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.

*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.

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