Le 16-déc.-05, à 16:49, Stathis Papaioannou a écrit :

It may be easy to find logical flaws in the above credo, but I maintain that it is so deeply ingrained in each of us that it would be very difficult to overcome, except perhaps on the intellectual level.

OK but that would not make sense. I don't think any third person (intellectual) belief can put any doubt on a first person conviction. To learn that the sun is not the one going around the earth will never change our conviction that the sun apparently turns around us.

Many difficulties are easier to be approached when we keep up explicitly the 1-3 distinction.

One could imagine other beliefs about personal identity that might have evolved if there were the appropriate selection pressure; for example, identifying as part of a group or swarm organism. The point is, our belief is not scientifically or philosophically "right"; it is just our belief.

All right. I will say more because some people ask me out of line what are G, G*, and G* \ G.

I am thinking to define "machine science" by what machine can prove correctly about themselves, and by machine theology what machine can hope correctly about themselves ; where a machine proves a proposition p correctly when the machine proves p and p is true, and a machine hopes p correctly when the machine hopes p and p is true.

This is a non normative definition of science and theology because such definition does not forces us to revise any scientific or theological prejudices we could have, a priori.

Now let M be a sound machine. By definition a sound machine is a machine which proves only true propositions. Let us write "Bp" the proposition that the machine M proves p. To say that the machine is sound is equivalent as saying that IF the machine proves p THEN p is true. This means the proposition Bp -> p is always true, whatever particular proposition p represents (assuming of course that p is written in some language the machine can understand or at least manipulates formally).

So for each p the proposition Bp -> p is true for M, and this is just equivalent as saying that the machine is sound. Question: is it true that for each p B(Bp -> p) will be true?

Let p b any obviously false sentence like a contradiction (like q & ~q, or better the constant f). If B(Bp -> p) was always true, we would have B(Bf -> f); which means (giving that "Ba" means that the machine proves a) that the machine proves Bf -> f. But Bf -> f is equivalent with ~Bf. (Verify with a thruth table in case of doubt). But ~Bf means that the machine does not prove the false, but this means that the machine is consistent. So if B(Bp -> p) was true for any p, then the machine could prove its own consistency: the machine would prove

Bf -> f, that is ~Bf (consistency; you can write it also Dt). And this is in contradiction with Godel's second incompleteness theorem.

So soundness (Bp -> p) or more particularly consistency (Bf -> f) are typical example of true propositions about the sound machine M, that the machine M is unable to prove. So soundness and consistency, thanks to the second incompleteness theorem of Godel, are example of proposition that such a machine can hope for, and hope correctly, as WE know (giving that we talk by definition about a sound machine).

A natural question is the following: does such a B follow some modal logical laws? The answer is affirmative.

Basically, the theorem of SOLOVAY is just that for such sound machine, the modal logic G formalises completely the science of the machine, and the modal logic G* formalizes completely the theology of the machine.

G has the following axioms and rules:

K B(p -> q) -> (Bp -> Bq)

L B(Bp -> p) -> Bp

Rules : modus ponens (if the machine proves A and if the machine proves A -> B then the machine will prove B) and necessitation (if the machine proves A then the machine will prove BA).

G* has the following axioms and rules:

As axioms: all theorems of G, +

Bp -> p (the machine is sound)

Rules: modus ponens (ONLY!).

In general G (and G*) are thought as being the set of theorems which can be derived from the axioms and rules given, and thus are infinite sets of formulas. What I note G* \ G is the set difference of the (infinite) set G* and G, that is: "pure theology": the set of everything which is true for the machine but that the machine cannot prove. The machine can only hope such proposition, and WE know that such machine can correctly hope those propositions.

I must go. Here are important formulas (and their standard name) that we will meet again and again. I let you think about which sort of "multiverse" (Kripke frame) makes each of those formula automatically true, whatever the illumination (the valuation of the sentence letters) is.

K = B(p->q) -> (Bp -> Bq)

t = Bp -> p,

4 = Bp -> BBp,

B = p -> BDp,

D = Bp -> Dp,

5 = Dp -> BDp,

C = Dp -> ~ BDp,

L = B(Bp->p)->Bp,

Grz = B(B(p -> Bp) -> p) -> p.

Bruno

http://iridia.ulb.ac.be/~marchal/