Tuesday, July 5, 2011

Summary of the trip

Tomorrow I leave Vienna and head back to Amsterdam, and thus I thought it would be useful to put a brief recap of the important results of the last month:
  • Jesse and I finished up our extended paper on N, "A Curious Dialogical Logic and Its Composition Problem", and sent it off.
  • Significant new features have been added to the dialogues website, including the long-awaited ability to compute strategies interactively and the possibility of selecting arbitrary rulesets (from a pre-defined set of rules).
  • Our goal with Chris was to find out general conditions under which it can be proved that E is redundant; we have not gotten as far as we like, but the results of my various pokings and proddings on the subject are contained in "Some Remarks on the E rule in Dialogical Logic".
This research visit was funded by a grant from the European Science Foundation EUROCORES Short-Term Visit scheme within the framework of the ESF EUROCORES Programme entitled 'Modelling Intelligent Interaction', which we gratefully acknowledge.

Friday, July 1, 2011

Would you look at that!

OK, now this result does surprise me. I think I just found a ruleset that is not closed under modus ponens! D10+D12+D13+E:

  • validates ~~φ->φ.
  • validates (~~φ->φ)->(φv~&phi).
  • does not validate φv~φ.

And here are my proofs to double check my work, because this is very, very strange:

In fact, the same also holds of D10+D12+D13, without E:

probably surprising to no one other than me

And I wouldn't say it's exactly surprising to me, since that seems to imply I expected the opposite, whereas I didn't have any expectations at all.

I spent a lot of yesterday testing a number of other formulas under D10+D12+D13 and D10+D12+D13+E, and I think I have another conjecture about how E works, namely, while adding E may increase the number of validities (e.g., going from N to CL), it will never decrease the number; I have not found a single formula which is valid under the non-E ruleset but invalid when E is added. There is probably an easy, straightforward explanation of this/proof that this always happens, but I haven't spend any time thinking about it yet.

Even if there is, though, it's still something interesting to point out, because it is not a general phenomenon (i.e., that increasing the number of rules in your ruleset will never decrease the set of validities), since D10+D12+D13 validates all four version of DeMorgan's, but this is not the case if you add D11, since one of the versions is not intuitionistically acceptable.