Hi Tom,
Le 02-mars-06, à 01:06, [EMAIL PROTECTED] wrote:
>
> Bruno,
>
> In this context, what are you taking to be the truth value of the empty
> set?
I have probably been unclear. Truth values apply only to propositions.
> In other words, how can you say that {Empty set} & p = {Empty set} ?
> I thought that you were taking "&" to operate on propositions, not
> sets.
> Doesn't {Empty set} & p mean "saying nothing" in conjunction with "the
> truth value of p"?
OK. My fault. When I describe some point of view, or hypostase, by an
expression like Bp, or Bp & p, it is a shortcut with some implicit
things (which I think I have already explained when I was talking on
the Theaetetical variant, a long time ago). I should have put them in
quotes.
So when I described briefly the "terrestrial intellect" by Bp, or
better "Bp", it means that I am considering some entity, which normally
has developed some discourse including self-referential statements.
This means I have some language containing a set of elementary
propositions {p, q, r, ...}, logical connectives {->, &, v, <->, ~} + a
unary modal box "B" which is supposed to represent the provability (or
believability, or assertability} of the entity. So Bp means literally
"the entity will assert p". It does not mean that the entity can assert
"Bp" because the entity could a priori lack such an introspective
quality. If the entity obeys classical logic and if it is
self-referentially correct then, in the case the machine does never
assert some p, it will never assert Bp, and if it asserts Bp, soon or
later it will assert p, etc. Now, for most interesting entity, such a
"B" will give rise to some modal logic. This is the one which I call,
inspired by Plotinus, the terrestrial intellect. All interview of
machine are done first at the level of that terrestrial intellect. If
the entity is loebian the modal logic describing that "B" is given by
the logic G, which can be considered as the logic of correct
self-reference of any entity capable of asserting enough elementary
theorems of arithmetic (and in that case the letter p, q, r ... are
supposed to represent arithmetical propositions, and "Bp" represent the
arithmetical godel provability "Beweisbar" as applied to some number
coding description of the arithmetical proposition p).
So, when I describe the 8 hypostases: the occurrence of "Bp" alone is a
symbolic pointer to the communicability power of the entity, together
with the modal logic obeyed by such a "B", together with possible
semantics, etc.
What about "Bp & p"? Well, it happens that the loebian entities, which
"B" obeys to G, will assert "Bp -> p" only when they will actually
assert p. That is B(Bp->p) -> Bp" is true about those entities, and
actually even provable. So "Bp -> p" cannot be a theorem of G (that
would mean that "Bp->p" is always provable, whatever p represents). But
"Bp -> p" is a fundamental assumption of any knowledge theory. So the
intellect "Bp" cannot represent a notion of "knower". This was already
seen by Godel in 1933(*).
But "Bp -> p" is still obviously true for any *correct* (loebian or
not) machine (or entity). For the lobian entity this fact is proved by
G* for example. G* knows the truth about the entity. So we can still
define a new box, sometimes written Cp by Smullyan, by Cp = Bp & p (for
each p). This Cp obeys another logic. It is indeed trivial that Cp ->
p, given that (Bp & p) -> p is a tautology.
The hypostase denoted by "Bp & p" will denote the logic of Cp. Etc.
Now, let us going back to the Vimalakirti machine. She says nothing. Bp
is false for any p, and thus ~Bp is true for any p. So the intellect
hypostase, which represent the set of provable (believable, assertable,
...) statements will be empty.
But, and here I am answering your question, the same is true for the
"Cp" = "Bp & p" hypostase. Indeed if "Cp" is different form the empty
set, it means that there is some p that the entity knows. But "to know
p" has been defined by "Bp & p". But there is no proposition p such
that the entity (here Vimalakirti) asserts p, so, a fortiori, there is
no p such that Vim knows p. And the same reasoning goes through for the
"Bp & Dp & p": if there is no p such that Bp is true, then a fortiori
there is no p such that "Bp & Dp & p" can be true (Bp is always false
for Vimalakirti).
Hope I have not been to long. Please tell me if I have been a little
more clear, or ask me if a sentence is not clear (it is hard for me to
explain without knowing people's background and my philosophy is that
all question can and should be asked).
Tomorrow I give the solution for the "divine" hypostases. Those
contains all the propositions which are true *ABOUT* the Vimalakirti
machine. I let you guess ...
Bruno
(*)GÖDEL K., 1933, Eine Interpretation des Intuitionistischen
Aussagenkalküls, Ergebnisse eines Mathematischen Kolloquiums, Vol 4,
pp. 39-40, also in FEFERMAN & Al. 1986.
> Going even further back in your statements, how can Bp = {Empty set}
> when Bp corresponds to a truth value?
> The Dharm-Door of Non-Duality states that {Empty set} is neither true
> nor false.
> [If this doesn't show up on the list, could you post it there for me.
> I have been having trouble registering.]
>
> Tom
>
> -----Original Message-----
> From: Bruno Marchal <[EMAIL PROTECTED]>
> To: everything-list@googlegroups.com
> Sent: Wed, 1 Mar 2006 16:42:25 +0100
> Subject: Vimalakirti Machines
>
> Vimalakirti Machines.
>
> Before going back to the lobian hypostases (point of views) and their
> associated possible multiverses (the geometrical structure organizing
> the possible collection of the observer-moments, states, worlds,
> situations, etc.) it could help to study the same hypostases in the
> case of a machine or entity much simpler than a lobian one.
>
> Now there is a machine, reasoner, entity (whatever) which is even
> simpler than the type 1 reasoner of Smullyan. You can consider it as a
> the wisest of all machine, or the dumbest one (your choice). It has
> some relationship with some of Hal Ruhl's intution, I think, and
> actually, even the lobian talk will lead us to special sort of non
> Kripkean world related to the Vimalakirti machine. The machine just say
> nothing. Pure total and eternal silence. It is hard to imagine a
> simpler discourse than this one. I call it "Vimalakirti' in honnor of a
> buddhist who famously said nothing at the right time and place(**).
>
> I recall the 8 hypostases (as I interpret it in the context of the
> interview of some machine or entity):
>
> First there are the four primary hypostases:
>
> p (Truth, the One)
> Bp (the Intellect, which splits into two: the terrestrial one, and
> the divine one, described by G and G* respectively, in the case you
> interview a lobian machine)
> Bp & p (The Soul, which miraculously doesn't split, in the loebian
> case)
>
> Then there are the four secondary hypostases:
>
> Bp & Dp ("Intelligible Matter", which splits in the loebian case)
> Bp & Dp & p ("Sensible Matter", which also splits in the loebian
> case)
>
> Now, in the interview context, Bp means simply: the machine or entity
> will print, or believe, or assert p, if she has not already done so. I
> could write B(p) for the sake of readability. For example B(Alice likes
> puzzles) means that the machine will assert that Alice likes puzzles,
> and B(Bp), = BBp, means the machine will assert Bp, or, given that Bp
> means that the machine will assert p, BBp means that the machine will
> assert that the machine will assert p. Obviously ~Bp means that the
> machine does not asserts p, and B~p means that the machine does assert
> ~p, and ~B~p means that the machine does not assert ~p. Like always I
> will abbreviate ~B~p by Dp.
> Now, given the triviality of the discourse of the Vimalakirti machine
> (she says nothing), the hypostases will be rather simple too.
>
> p (The truth does not change except for some mundane propositions
> concerning perhaps the Vimalakirti machine itself)
> Bp terrestrial: this is the discourse of the machine, it can only be
> the empty set, given that the machine says nothing.
> Bp & p, at the terrestrial level this is again the empty set. OK?
>
> Bp & Dp again empty
> Bp & Dp & p empty again.
>
> So all the terrestrial hypostases are empty!
>
> What can we say about the divine one. I recall that they are defined by
> all the propositions which are true *about* the entity, independently
> of the fact that the entity asserts them or not.
>
> I let you think before giving the answer tomorrow.
>
> Bruno
>
> (**) Googelizing a little bit I realize that the entire teaching of
> Vimalakirti is in english on the net:
> http://www.buddhistinformation.com/vimalakirti_nirdesa_sutra.htm
> See the end of the section 9 for his famous silence. search on the
> dharma-door or on non-duality or on the full title of the 9 section:
> 9. The Dharma-Door of Non-duality
>
>
> http://iridia.ulb.ac.be/~marchal/
>
>
>
>
>
>
http://iridia.ulb.ac.be/~marchal/
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