Saturday, December 31, 2011
Tuesday, July 5, 2011
- 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".
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.
Thursday, June 30, 2011
One of the original motivations our work with Chris here in Vienna started with is contexts under which we can show that E is redundant; IL is an obvious case, and hence D10+D11+D13 is as well. D10+D12+D13 was a surprising (to some) countercase. CL is an obvious countercase...or is it?
It appears that the unique redundancy of E in IL noted above is actually closely tied to Felscher's formulation of the rules. If we stray from there (keeping E: "The Opponent must immediately respond to the previous Proponent move") and start formulating rules on our own, can we perhaps come away with sets of rules where the presence or absence of E is irrelevant?
When we talked about this a few days ago, my conjecture was that it's not the immediate-response consequence of E that is so important for CL, but rather the restriction on the number of moves O can make. I propose the following:
- D10: P cannot assert an atom before O does.
- D13: A P-statement can be attacked at most once.
- D14: A P-attack can be defended at most once.
Certainly D10+D13+E implies D10+D13+D14. But D10+D13+D14 does not imply E, since it relaxes the immediate-response constraint.
So one hypothesis for how to go about constructing rules that would allow us to prove the redundancy of E in a particular ruleset is by looking at variants of the bounded rules that Krabbe proposed in his 1985 Synthese article which put constraints on the number of times O can repeat a move.
However, if you are in the context of E, what is it that separates IL from CL? It is the ability of P to go back and essentially "change his mind" about a defense, e.g., in the case of excluded middle. So here, it appears that the important distinction is not how many times O can repeat a move, but how many times P can. The case is not quite the same, though: In IL, there are contexts where P is allowed to repeat a move (same formula/symbolic attack, same stance, same reference), and in CL, it is not that we are allowing P to repeat a move that he's already made, but rather to re-refer to a move of O that he's already referred to. So the cases aren't quite parallel, but it still shows a tantalizing dimension to consider.
Tuesday, June 28, 2011
|Excluded middle||not valid||not valid|
|WEM||not valid||not valid|
|Cond. EM||not valid||not valid|
|Dummett||not valid||not valid|
|Scott||tried to depth 36||tried to depth 20|
|Smetanich||tried to depth 24||valid|
|not CL||X||X||X||excluded middle, Peirce not valid|
|not CL||X||X||X||X||validates Peirce, does not validate excluded middle|
|IL||X||X||X||U, blog post|
|IL||X||X||X||X||U, blog post|