The terrestrial hypostases correspond to discourses of the Vimalakirty machine, where a discourse is defined by the set of propositions asserted by the machine. Now the machine says nothing, so that each terrestrial hypostases are empty.
The divine hypostases are given by some "omniscient true discourse" on the machine (that is: on its intellect, on its souls, on its intelligible matter and on its sensible matters (see below)).
Let us look at the "divine intellect". Well, the machine says nothing, so it is true that the machine didn't say that 1+1 = 2, all right? So ~B(1+1 = 2) is true. So ~B(1+1 = 2) belongs to the divine intellect of the Vimalakirti machine. OK, but the machine didn't say that 1+1 ≠ 2 either. So ~B(1+1 ≠ 2) is also true, and also belongs to the divine intellect. More generally, given that the machine does not utter any proposition, all proposition saying that the machine does not say <whatever> will be true. So the divine intellect of the Vimalakirti Machine will contains all proposition of the form:
~B(an asteroid will not hurt earth in 2102)
~B(an asteroid will hurt earth in 2102)
~B(1+1 = 4)
~B(1+1 ≠ 4)
~B(PI is rational)
~B(PI is not rational)
This gives an infinite set of true propositions *about* the machine, all beginning by "~B". The modalist will recall that "~B" is equivalent with "D~", and <whatever> is of course the same as <~whatever>, so the divine intellect can be characterized by saying it contains all "possibilities" (the alethic reading of the diamond "D").
This was simple enough, no?
I let you find the divine soul, defined by the logic of Cp with Cp defined from the intellect (Bp) by Bp & p. It is the "true opinion" proposed by Theaetetus. BTW I was saying that Plotinus did use the Theaetetus' trick in its approach to the soul, and I have discovered recently that Emile Brehier confirms this in a footnote in its (first) translation of the V Enneads of Plotinus (1931). Note that the corresponding diamond for the soul can be defined by ~C~.
Remark. In which world could the divine intellect lives? We have there that D<whatever> is always true, and B<whatever> is always false. In the alethic reading: everything is possible, and nothing is necessary. Is it cool? Do we have met such world?
Those who remember our trip in the "Pappaioannou multiverse" does remember that from each world there, you can access to a dead end (cul-de-sac world), and by Kripke semantics, in a dead end everything is necessary but nothing is possible (brrr...). Now it looks like the divine intellect of a Vimalakirti machine/entity lives in an "anti-dead end world", where everything is possible and nothing is necessary. Is it cooler than a dead end? And could such a world exist in a Kripke Multiverse. Certainly not. In a Kripkean Multiverse Dp is true at world alpha if there is a world beta accessible from alpha, and such that p is true in beta. But in the "divine intellect" of a Vimalakirti machine even Df (f = false) is true, so that if such a world was Kripkean, f would be true in some world! Well, that is not the case in any Kripkean multiverse.
Curiously enough Kripke himself will met such "non-normal" anti-cul-de-sac worlds in his study of the system S1 S2 S3 of C.I. Lewis, that is the first modern modal theories; and we will see (soon or later) that the intelligible and sensible matter, both terrestrial and divine, will introduce those "Vimalakirti weirdness".
I hope this helps you to distinguish a discourse made BY a machine/entity from a discourse made ABOUT the machine/entity. This is a key to understand the difference between terrestrial and divine in the mathematical interpretation of Plotinus.
Le 01-mars-06, à 16:42, Bruno Marchal a écrit :
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.
(**) Googelizing a little bit I realize that the entire teaching of Vimalakirti is in english on the net:
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
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