On 06 Mar 2009, at 18:09, Günther Greindl wrote:

> > Hi Bruno, > >>> With COMP it is not so clear. >> >> explicit appeal to self-consistency (= the move from Bp to Bp & Dt; >> the >> "Dt" suppresses the cul-de-sac). With comp, to believe in a next >> instant or in a successor state is already based on an act of faith. > > Please bear in mind that I have not yet studied the AUDA in detail. > How > does Dt suppress cul-de-sac? By Kripke semantics. A Kripke frame is given by a set of "worlds", together with an accessibility relation between those worlds. For a mathematical logician a kripke frame is just a set with a binary relation. By definition a world is just an element of that set, and the accessibility relation is just that binary relation. Those Kripke frames are used to provide a mathematical tools to reason on formal modal logical systems. They provide models of modal theory, that is mathematical structure which satisfy, in a mathematical sense, the theorems of the modal logical system. The idea is that a modal theorem in a modal system should be a formula true in a ll the worlds of some frame. the hope, indeed realized for many theories including G (but not G*), is that there is a binary relation on a type of frame which characterized all and only all the theorem of the modal system. We do logic here, meaning we dispose of a set of propositional variables p, q, r, ... A frame become a model when you assign on each world a function from {p, q, r, ...} to {0, 1} (a valuation). If v() = 1 in world alpha, we say that p is true at world alpha. You make each world obeying classical logic (for exemple if p is true in alpha, and if q is true in alpha, you make (p & q) true in alpha, etc. The key of Kripke semantics is that Bp iis true alpha if and only if p is true in all worlds beta which are accessible (cf the binary relation of the frame) from alpha. Now, what does mean to say that Dp is true in alpha? We have no choice, given that Dp is really an abbreviation of ~B~p, which means that it is false (in alpha) that B~p, which means that it is false (by Kripke key point) that ~p is true in all worlds accessible from alpha, which means (using "false -> false" is a tautology) there is a world, with p true, accessible from alpha. So if Dp, or even just Dt is true in alpha, then there is necessarily a world beta, with p true, or even just t, accessible from alpha. alpha cannot be a culd-de-sac. You can note that in cul-de-sac, Dt is false, so Bf is true. Bf is true because trivially if a world beta is accessible from alpha then f (false) is true in beta. This is trivially true because the proposition "beta is accessible from alpha" is never met, so the condition is always false, and the propositions have the shape f -> f (a tautology). To sum up: the Kripke semantics of Bf is "I am dead" or "I am in a cul- de-sac world". The Kripke semantics of Dt is "I am alive" or "I am in world able to access some other world". World, or moment, or whatever. It is said that Artemov would have interpret jokingly Dt as "I am in country which provides visa". Günther, I will be frank, this is just elementary modal logic, and even advanced modal logic is considered "easy" compared to the provability logic. Solovay theorem made one precise modal logic, G, an incredible tools for simplifying the provability logic field. The modal logic G is to provability logic, what tensor calculus is to general relativity theory. G is just one modal logic among an ocean of possible modal logics. Somehow modal logic is the abstract theory of the multimultiverses. It is just a wonderful result that the formula of Löb, B(Bp->p)->Bp, the only axiom of G, (really), formalizes completely the whole field (at the propositional level). It gives, with the intensional variants, the whole propositional theology of the honest or correct, or sound, universal machine. (Universal machine believing some effective induction principle, they are automatically Löbian). It is an ideal case, of course, in our lives we are far from lobian. But it is what we need, by UDA, to get the correct, assuming comp, big picture, including physics, first and third and first plural physics. Bruno http://iridia.ulb.ac.be/~marchal/ --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---