Re: modal logic KTB (a.k.a. B)

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Le 05-mars-08, à 10:19, <[EMAIL PROTECTED]> a écrit :```
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>
> Bruno Marchal <[EMAIL PROTECTED]> wrote:
>> logic B (KTB) can be used to capture a notion of vagueness, and, by a
>
>> theorem of Goldblatt, it can be used to formalise classicaly a
> minimal
>> form of von Neuman quantum logic in a manner similar to the way the
>> modal logic S4, or S4Grz, capture intuitionistic logic.
>
> The Gödel-McKinsey-Tarski translation from intuitionistic logic to S4
> can be defined in different ways. The most concise one is by saying
> that one has to insert a [] before every subformula.  Can we
> reformulate the translation by Goldblatt in a similar way, e.g., by
> saying that one has to insert []<> before every subformula ?

Hmmm.... I don't think so but I may be wrong. You seem to know some
logic, so perhaps you could try to prove this by yourself (and then let
me know :). Here is the transformation by Goldblatt, going from a
propositional (non modal) language with & and ~ as connectors,  to a
modal language, with &, ~, and the box []  (and the usual abbreviation:
<> = ~[]~, for example).

T(p) = []<>p
T(A & B) = T(A) & T(B)
T(~A) = [] ~T(A)

It is this last line, the case of the negation, which makes me think
that just inserting []<> before the subformula will not work.
(Actually, even if it works for Goldblatt's purpose, it can certainly
not work for the arithmetical quantum logic, because the negation of a
Sigma_1 universal sentence like Gödel's "bew" gives a PI_1 sentences,
usually not provable by the system.

>
>>> Suppose the atomic propositions are what I currently know on a
>>> physical system.
>>
>> This does not make sense.
>
> Really? it made some sense to me...

Of course it makes sense. It makes sense for me too ... in everyday
life. You should have quoted my whole paragraph which was

"This does not make sense. In the way I proceed I will use the
arithmetically derived points of view logics (the arithmetical
hypostases) to derive the logic of observability, knowability,
sensitivity ..."

I was supposing that the UDA argument is already grasped, so that the
only "atomic sentences" which are usable (assuming comp) correspond to
the accessible states of the Universal Dovetailer (aka the Sigma_1
sentences, (with or without oracle).

Your supposition was not making sense, because after UDA the term
"physical system" has none of its everyday meaning. We don't even know
if there is a physical system.
We *know* the smell of coffee and things like that, but this refer to
first person experiences.

>
>> Again. Just remember that I am not supposing any physics at all, nor
>> any "physical world".
>
> My initial question was not referring to your work in particular.

OK.  (of course my answer was referring to it).
You could ask the question explicitly to people who have diiferent
views like those who believes in absolute self sampling, like Hal
Finney, Nick Bostrom ...

> However I would be glad to hear more from your point of view.
>
>>  Did you grasp the UDA's point?
>
> No, but I am interested in and will try to catch up.

Perhaps you could read my 16 may 007 post here:
http://www.nabble.com/Attempt-toward-a-systematic-description-
td10643945.html

Bruno

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

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