**LQ**of weak excluded middle: is it true that if, in Felscher's ruleset E for intuitionistic logic, one changes the rule

Proponent may assert an atomtoponly if Opponent has already assertedp

Proponent may assert an atombut leaves all other rules unchanged? If one relaxes the restriction on Proponent's use of atoms as described, then the law of weak excluded middle ¬ponly if Opponent has already either assertedpor asserted ¬p

*q*∨¬¬

*q*becomes valid, while the usual law of the excluded middle

*q*∨¬

*q*remains invalid. This naturally suggests, in light of the soundness and completeness of the ruleset E for intuitionistic logic, that the modified ruleset characterizes the logic

**LQ**of weak excluded middle, a fascinating superintuitionistic logic. My interest in

**LQ**comes from constructive mathematics, but in any case, the problem is now purely a dialogical one. I'm not invested one way or the other in this result, but I would like to try to show that this simple change in one of Felscher's rules gives rise to an interesting known logic. Of course, a simple counterexample would be nice, too, but so far I've not found any. The problem here is unlike the case of

**N**, which (for the moment) has no known characterization or meaning outside the context of dialogues.

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ReplyDeleteI should add that there is a paper on applying dialogue games to logics like

ReplyDeleteLQ, and other intermediate logics, by Christian Fermüller: “Parallel dialogue games and hypersequents for intermediate logics”. His methods go far beyond the modest approach toLQthat I'm pursuing.