Friday, July 1, 2011

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.


  1. I'm not sure this is all that helpful, but one extreme case of your conjecture seems manageable: the skeletal ruleset has fewer validities than the skeletal+E ruleset.

    Nice spot about D11. Perhaps that rule is a counterexample to the general claim by virtue of it applying to both players, whereas E applies only to O.

    This motivates me to add a couple new experimental structural rules: D11-only-for-O and D11-only-for-P.

  2. I'm not sure D11 is a counterexample, because I'm not making any general claim yet. So far I'm only willing to make the claim about E, that the set of S-valid formulas is monotonically increasing when S is extended to S+E, for arbitrary S. In fact, I would not be surprised at all if this was something special to E, rather than something unusual in D11.