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.



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