Wednesday, June 8, 2011

this is what happens when you think too much about dialogues

You start developing gut feelings; and the problem with gut feelings is that they are hard to transfer. It was a gut feeling about how N worked that led me to the characterization of implications in that system; and Jesse didn't believe me until he'd fully worked out a proof. Working with N meant focus on D10 and D13. Today, for reasons I'm not still entirely sure, I found myself back in Felscher's article, looking at D11 and D12, and suddenly I'm not seeing how they aren't redundant w.r.t. each other:
  • D11: Only the latest open attack may be defended.
  • D12: An attack may be answered at most once.
I got Jesse to set up two new rulesets on the dialogues site, D10-D11-D13, and D10-D12-D13, and after some simple playing around with the latter, I'm beginning to develop another feeling in my gut, namely that one or the other of these is sufficient to characterize IL, even though at this point I haven't a clue which. I would LOVE to find a counterexample to this gut feeling, but I've got a feeling there's no systematic way to go about finding one, and thus I must test it via hit-and-miss potential counterexamples. Hmmm. Or maybe I can find a nice axiomatization of IL and test each axiom in both sets. Would that really be enough?


  1. The axiomatiziation approach is promising, but one would need to be careful of getting bit by a lack of uniform substitution. What I have in mind is: even if all the axioms of some suitable axiomatization of IL turn out to be D-minus-D11 valid or D-minus-D12 valid, we might not get all of IL because there could be some instance of one of these that fails to be valid. We also have to show that the composition problem for D-minus-D11/D-minus-D12 is solved positively.

  2. Exactly my worry w.r.t. my comment "Would that really be enough?" It would certainly be a start.