- 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".
Tuesday, July 5, 2011
Friday, July 1, 2011
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:
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.