On Sun, Mar 22, 2015 at 6:19 PM, Bruno Marchal <[email protected]> wrote:
>
> On 21 Mar 2015, at 21:38, Telmo Menezes wrote:
>
>
>>
>>
>>>
>>> It is the p in []p & p, which makes "machine's knowledge" not definable
>>> in term of number and machine. S4Grz formalizable at a level, what the
>>> machine cannot formalize about herself (but can bet on, ...).
>>>
>>> Thanks to incompleteness, the Theaetetus' definition makes sense, and
>>> distinguish the knower from the rational believer for the machine.
>>>
>>> Don't hesitate to ask precision. I am very literal here: the knower is
>>> defined by the true believer. It is a modest definition of knowledge, and
>>> it is not similar with "I know for sure that", which needs some amount of
>>> consistency (like <>t, or <><>t, or <><><> t, etc.).
>>>
>>
>> What's the difference between <>t and <><>t and so on?
>>
>>
>> In the Kripke semantics, <>p means that you can, by starting form the
>> world you are in, access to another world where t is true. It means that
>> you are not in a cul-de-sac world. It means that you are "alive", you can
>> access some world. But <>t -> <>[]f. So you despite being alive, you might
>> be dead in that next world. That next world might be a cul-de-sac world,
>> where <>t is false, and thus []f is true. Now, if you are in a world where
>> <><>t is true, then you can access a world in which <>t is true, so that
>> you are still alive.
>>
>> <>t = I can access some world (t is true in all worlds), but that world
>> might be a cul-de-sac world.
>> <><>t = I can access some world where <>t is true, so from there I can
>> access some other world.
>>
>> Put differently:
>>
>> <>t I am alive (but can die at the next instant)
>> <><>t, I am alive and I can access to an instant where I am still alive.
>>
>
> Ok, but it's not obvious to me how temporality (a sequence of instants) is
> introduced here.
>
>
> Here, it is "introduced" simply through the Kripke semantics of the modal
> logic involved, G in this case.
>
> Keep in mind that we are in arithmetic. <>t abbreviates ~beweisbar '~(
> 2+2=4').
> here '~( 2+2=4' denote some number, denoting some falsity. It is
> equivalent with the arithmetical
> consistent('2+2=4'), which is true, but not provable by the system itself.
>
> By Gödel's completeness theorem, which applies still on most correct
> machine if they follow some recommendation in their way of talking,
> consistent('p') is equivalent (from outside) with "there is a model which
> satisfy my beliefs and p".
>
> So, literally, <>p means if you add p to my beliefs, it will not leads to
> a contradiction (semantically: a cul-de-sac world)
>
> So, there is a sort of interpretation of <>p in "there is some reality in
> which p is true", and, as t (true) is true in all worlds, you can interpret
> <>t by "there is a world" (or there is a reality, or there is some truth,
> or there is some god, or ...).
>
Ok got it.
>
>
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>>>
>>> [0]p = []p, and obeys to G, and fully described by G* (at the
>>> propositional level).
>>> [1]p = []p & p, and obeys to S4Grz,
>>> [2]p = []p & <>t obeys and define the logic Z
>>> [3]p = []p & <>t & p
>>>
>>
>> I definitely don't understand [2]p and [3]p.
>>
>>
>> []p & <>t is a weakening of []p & p. Instead of asking p being true, we
>> ask only for p being consistent
>> (([]p & <>t) -> ([]p & <>p)).
>>
>> If you can prove for all p that []p -> p (like with [1]), then you can
>> prove for all p []p -> <>t or []p & <>p.
>>
>> So the logic of provability-and-consistency is weaker than the logic of
>> provability-and-truth.
>>
>> Provability and consistency avoids the probability one for the false,
>> which would exist if we take []p as provability. The passage []p =====> []p
>> & <>p, approximate the main things for a probability one.
>>
>
> What does "provability one" mean?
>
>
> Come on Telmo, it takes me a lot of concentration to write "probability"
> instead of "provability", with those damn "b" and "v" which are so much
> together.
>
It's a bit funny that you find this to be a problem and meanwhile write
long emails where half of them are square brackets, arrows and other weird
notations :)
> And in our case the distinction is of importance, and can be made quite
> clear in arithmetic:
>
> Provability is beweisbar , what is translated in the modal logic of
> provability by the box [], usually []p.
> Probability is beweisbar-and-consistent, the []p & ~[]~p, or (equivalent)
> []p & <>t.
>
> The difference is that with Provability, we can access cul-de-sac world,
> in which you have []f. (= ~<>t).
> For proVability, we need to add the "by-default hypothesis" that we will
> not die during the experience.
> When you bet on a coin: the events
>
Ok, so even if you prove you have to bet on consistency, right?
But are you saying then then that cul-de-sac worlds are not consistent?
>
> {H} or {T}, {H, T} have probability one, because semantically we abstract
> from the asteroïds which can smash the playing table, the dice and the
> betters.
>
> By forcing <>t on each state (like Theaetetus did with the truth) we
> abstract from the cul-de-sac world, by imposing the existence of some
> accessible world.
>
Why are cul-de-sac worlds not accessible? I never explored these ideas very
thoroughly (I suspect they have been discussed to death here before my
time). You propose that comp entails quantum immortality, correct?
> Logically, at first sight, it looks like a weakening of Theaetetus' idea
> of restricting []p on the truth, but arithmetically, or mechanicalistically
> (if I can say), that is by incompleteness, it appears that replacing "p" by
> <>p, is an strengthening, yet dual of some sort of []p.
>
> You might need to revise a little bit the Kripke semantics of modal logic,
> and the arithmetical logic of (machine) self-reference. It works also for
> machine + oracles.
>
Ok.
>
>
>
>
>
>> It models what the guy in Helsinki can be sure if, like drinking a cup of
>> coffee (in the protocol where he get a cup in both W and M). It abstract
>> (locally) from the cul-de-sac world. It use the bet on <>t implicit in the
>> yes-doctor.
>>
>
> I don't know about the cup of coffee protocol either, could you explain it?
>
>
>
> Imagine that you are in Helsinki, and you will be duplicated (cut and
> copied) and sent (by waves) to Washington and Moscow, where you are
> reconstituted-reincarnated, OK?
>
> But this time, we add in the protocol that both in Moscow, *and* in
> Washington, they will offer a cup of coffee to the reconstituted person.
>
> The question (that, btw, I have asked many times to J. Clark without any
> answer) is: what is your expectation, when you are still in Helsinki (of
> course) of drinking a cup of coffee after the duplication?
>
> Of course, we, including the guy in Helsinki, *assumes* computationalism.
> We assume the correctness of the substitution level chose, and we agree
> with the default hypotheses (no asteroïds). We already agree that the
> expectation of drinking a cup of coffee is the same (modulo comp and the
> default hypothesis) after a simple digital teleportation than with using a
> plane (assuming it does not crash and the usual default hypotheses).
>
> So, what do you think?
>
Ok, I think the expectation is the same (p_coffee = 1).
>
> What the Löbian Universal Machine thinks, is that provability, knowledge,
> observable, sensible, obeys different modal logics, and that most of those
> modalities splits into a justifiable part and a non justifiable part.
>
> Then we can test the theory by comparing the machine internally defined
> notion of observable ([]p & <>p + p sigma_1) with the empirically
> observable. Curiously, at first sight, is that the quantum only appears on
> the non justifiable part(*), but that is normal: physics is first person
> plural, and that is (arguably) confirmed: QM (without collapse) makes the
> splitting of observers contagious to their colleagues. If you look at the
> schroedinger cat, the story differentiates into a story where you see the
> dead cat, and a story where you see the cat alive, but the day after, when
> your colleagues (in the respective stories) ask you how was the cat, they
> will split too at the occasion. To be sure, they would have already
> splitted/differentiated, by the natural gossip of the sufficiently hot
> environment.
>
> It *does* look like the quantum aspect of reality is explained by the way
> the sigma_1 arithmetical truth can look to itself, in the observable mode.
>
> Bruno
>
> (*) (Z1*, X1*, not X1 nor Z1. To be sure a quantization and a quantum
> logic also appear on S4Grz1 which does NOT split. The outer God does not
> add anything to what the inner God already knew. And, now, the inner god
> has already a foot in matter! This shocked my intuition, as it makes matter
> still more on the first person side than I thought, but it fits better with
> Plato, Plotinus, East and West Mystics, and even with the salvia reports
> and some other reports you can find on Erowid or on Salvia webpages.
>
Ok, most of what you say here makes sense to me intuitively, but there are
some technical aspects that I have to study by myself, I think (I study
some modal logic when I have time, but the progress is slow).
Thanks!
Telmo.
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> Thanks!
> Telmo.
>
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> http://iridia.ulb.ac.be/~marchal/
>
>
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