Thursday, November 4, 2010

Breaking: counterexample to a simple proposal for characterizing the logic LQ of excluded middle

A while ago, I proposed a simple dialogical characterization of the logic LQ of weak excluded middle (also known as Jankov's logic). (See my earier post on the subject.) The idea is, simply, to replace the usual Felscher rule D10, which says
P can assert an atom p only if O has already asserted p
P can assert an atom p only if O has either already asserted p or already asserted ¬p.
Chris Fermüller has kindly provided a counterexample to the claim that the usual rules for intutionistic logic, with D10 replaced by this modified version of it, generates LQ: the formula ¬pp is valid in this setting! The idea is that O asserts ¬p, then P can assert p and win the game. That ¬pp is not valid in LQ is clear because the formula is not classically valid, and LQ is an intermediate logic. Back to the drawing board…

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