Wednesday, July 14, 2010
Nearly classical logic
Here's something interesting: The rule set consisting in Felscher's D minus rules D11 and D12 -- which at one point we thought corresponded to classical logic, but then we found out does not because though LEM is valid, Peirce's formula is not, and so which we've since been calling "Nearly Classical Logic" even though we're not sure it's even a logic, much less that it's close to CL -- if it corresponds to a logic corresponds to one that is strangely sensitive to atomic formulas. The validities that it does have (LEM, WEM, Dummett's formula), remain valid under a number of negation-translations: double-negating the whole formula, DNing each subformula, Kuroda's translation, negation of atomic formulas, DN of atomics. However, the translations which replace $p$ with $p\vee p$ or $p\wedge p$ and Gödel-Gentzen both fail to preserve validity. The problem is that as soon as P makes a defendable attack (such as ?R or ?L or ?), O can stall indefinitely by continuously defending the attack. Despite this oddity, we have been unable to come up with a counterexample to the composition problem. Even simple cases such as LEM->LEM^GG fail to be valid, so it's unlikely that we'll find a more complicated counterexample in that realm. What this sensitivity to atomic formulas corresponds to is at this point still utterly opaque to me.