Tuesday, September 14, 2010

Returning to assumptions

In an earlier post I discussed how to handle reasoning from assumptions in the dialogic framework. My first proposal was: Reasoning from Assumptions Attempt #1 (Naive): Fix a set of formulas Γ. Γ\vDashφ iff P has a winning strategy for φ when he is allowed to assert any ψ\inΓ, either as an attack or as a defense. I quickly realized that this was not acceptable, since it doesn't work to capture reasoning from implication in CL; take the consequence (p&q)\vDash(p&q): If P asserts the right-hand side, then the material on the left won't help him at all when O attacks (p&q). This lead to: Reasoning from Assumptions (Naive) Attempt #2: Γ\vDashφ iff P has a winning strategy for φ when he is allowed to attack any &psi&\inΓ and to defend with anything in Γ. Now, if we think that any formula is a consequence of its own assumption in N (which may or may not be a reasonable thought), then this approach doesn't work since it fails to make (pVq)\vDash(pVq); if P asserts pVq (the right-hand side), and O attacks it, then if P counterattacks by attacking pVq (the left-hand side), this allows O to defend, and the dialogue will never end. This is only a problem if we think that this consequence should be valid in N. Assuming we do, let's get a bit more creative: Reasoning from Assumptions Attempt #3: Let a(Γ) be the set of atoms which are subformulas of elements of Γ. Then Γ\vDashφ iff P has a winning strategy for φ when he is allowed to assert anything in a(Γ) whenever he wants. This fails because then (pVq)\vDash(p&q). We should also consider: Reasoning from Assumptions #4 (Jesse's proposal): Γ\vDashφ iff there is Γ'\subseteqΓ and \vDash\bigwedge Γ'\rightarrow φ. I dislike this suggestion because it builds in the validity of the deduction theorem, which begs the question against the possibility of reasoning from assumptions in N. I haven't come up with any other candidates yet.

Wednesday, September 8, 2010

News and events

We have had two short papers accepted to the 17th International Conference on Logic for Programming, Artificial Intelligence and Reasoning, Yogyakarta, Indonesia, October 10-15, 2010: Both Jesse and I will be giving talks on our recent work at the Workshop on Dialogues, Inference, and Proof - Logical and Empirical Perspectives, Vienna, Austria, November 26-28, 2010. There are also plans in the work for a follow-up workshop in Tübingen, tentatively scheduled for February 25-27, 2011. When details are finalized, they will be announced here among other places.