Le 17-juil.-05, à 11:07, Stathis Papaioannou wrote :

Bruno,

There's a lot to digest in this post.


Take your time. No problem.



I should clarify that in my original post I had in mind two different usages of the word "death". One is what happens to you in destructive teleportation: you vanish at one set of spacetime coordinates, then reappear in almost exactly the same material configuration at a different set of spacetime coordinates.


OK.



Ordinary moment to moment life is a special case of this process where the difference between the "before" and "after" coordinates is infinitesimal, and therefore there is no subjective discontinuity between one moment and the next.


The first person point of view cannot be aware of "the time" between annihilation and reconstitution. (Step 4 of the UD Argument (UDA)).



I would call what happens when you vanish "provisional death". Provisional death becomes "real death" if (contra QTI) there is no successor OM (or no next moment, or no reachable world): if the teleporter breaks down and loses the information obtained in a destructive scan before it can be sent, or if you are killed in an accident in ordinary life. It is interesting to note that memory loss is effectively the same as real death, or real death with a backup that is not up to date (eg. the original is killed a few minutes after undergoing non-destructive teleportation) if the memory loss is incomplete. Real death and memory loss cause a cul-de-sac in a stream of consciousness, whereas provisional death does not.


OK. Now we can never be sure that there will be a next observer moment, and this makes the provisional death a possible absolute death. I thought this was your justification that we die at each instant/observer-moment. Each accessibility arrow bifurcates, and there is always one leading to a dead-end, so that we (can) die at each instant/observer-moment.




If you can convince yourself that you undergo provisional death all the time, and real death when you experience memory loss, then it may be possible to convince yourself that death is no big deal. However, our evolved minds would fight very hard against such a conclusion.


It is counterintuitive indeed.





In this post I will try to make clearer my argument with Lee by using a minimal amount of modal logic (and so it's good "revision" ;)

Then I will explain how Stathis seems to have (re)discovered, in its "DEATH" thread, what I call sometime "The Smallest Theory of Life and Death", or "Near Death Logic", or just C. I have never abandon C, but the interview of the Lobian machine will give C again, but through some of its most notable extensions which are G and G*.

To prevent falling in the 1004-fallacy, I will use (at least temporarily) the words "state", "world", "situation", "observer-moment", "OM", etc. as synonymous. I will use "world" (if you don't mind), and I will designate individual world by w, w1, w2, w3, w4, etc.

Like Stathis (and Kripke!), I will accept that some world can have *successor* world (successor OMs in Stathis terminology). More generally we suppose a relation of accessibility among worlds (that's Kripke's idea how to enrich Leibniz).

These words - successor, accessibility, reachability - are figures of speech, right? What is important is the relationship between the worlds, not that someone or something "reaches" physically from one world to the next.


I am not sure I understand what you mean by "figure of speech". All theories build their concept from "figure of speech" (being the punctual mass in Newton or the strings in String theory, or perhaps just the real numbers, etc.).



I will be interested in the discourse which are true at each world, and I will assume that classical logic holds at each world. p, q, r, ... denotes propositions. And a "semantics" is given when it is said which one of p, q, r ... are true or false in each world.



I suppose you know some classical logic:
(p & q) is true if both p and q is true, else it is false
(p v q) is true if at least one among p, q is true, else it is false
(~p) is true if and only if p is false
(p -> q) is true if p is false or q is true
(to be sure this last one is tricky. "->" has nothing to do with causality: the following is a tautology (((p & q) -> r) -> ((p -> r) v (q -> r))) although it is false with "->" interpreted as "causality", (wet & cold) -> ice would imply ((wet -> ice) or (cold -> ice)). Someday I will show you that the material implication "->" (as Bertrand Russell called it) is arguably the "IF ... THEN ..." of the mathematician working in Platonia.

That last one always got me: a false proposition can imply any proposition. All the rest seem like a formalisation of what most people intuitively understand by the term "logic", but not that one. Why the difference?


This is important. It was the object of the thread "just a question".
Suppose that you are in a room with only men inside. The statement "all women in the room are 42 km high" will be trivially correct. It really means that if x is a woman in the room she is 42 km high. it is true because the premiss "x is a woman in the room" is false. Nobody can build a contradiction from it. Of course, in the same condition you can say "all woman in the room are not 42 km high". It is again true. From this you can conclude that "all women in the room are both 42 km high and not 42 km high", from this you can conclude that "if x is a woman in the room then false":

(x is a woman in the room) -> f

but (p -> f) is the same as (~p) as you can verify with a two-line truth table, so you can conclude ~(x is a woman in the room), from which you can conclude that there is no woman in the room (which we knew of course).

The idea that false implies anything explain why in math an error in a paper jeopardize in principle the whole paper.

The idea is natural in some common expression, which I know better in french than in english. People says things like this "if you are good at joke then I am Napoleon", meaning "you are not good at joke". Or if "if bin Laden is a gentleman then you can put Paris is mach box".

In any case you can just remember that "p->q" is the same by definition as ((~p) v q). For example: "if 0 is bigger than 2 then 0 is bigger than 1", is the same as (~(0 is bigger than 1) or (0 is bigger than 1)) which gives (t v f) which gives t.

OK?




(p <-> q) is true if (p->q) is true and (q->p) is true. I could have said (p <-> q) is true if p and q have the same truth value. The truth value are true and false, and I will write them t and f. You can see t as a fixed tautology like (p -> p), and f as a fixed contradiction like (p & (~p)), or add t and f in the proposition symbols and stipulate that
f is always false
t is always true

That classical logic holds in the worlds means the "usual things", for example that

- if p holds at w, and if q holds at w, then (p & q) holds at w,
- if p holds at w, then p v q (read p or q) holds at w,
- if p holds at w and p -> q holds at w, then q holds at w.
- t holds in all world
- f does not hold in any world
- etc.

Etc. All "tautologies" will be true in all world (p -> p), (p -> (q -> p)), ((p & q) -> p), etc.
(whatever the truth value of p, q, r, ... in the worlds).
I hope most of you knows the "truth table method" to verify if a proposition is a tautology or not. But I can explain or give reference or you could google.

Remark.
Note that if the excluded middle principle (p v (~p))is a classical tautology, it is not an intuitionist logic, and (much later) we will met this logic. We live the modern time where even the classical (Platonic) logician must aknowledge the importance of the many many many many possible logics. For example in Quantum Logic and in the Relevant Logics, the classical tautology which is "guilty" is the "a fortiori principle": (p -> (q -> p)) One of the main utility of modal logic, imo, is to give a tool to "modelize" non-classical logics in a classical setting. But this we don't need to know now.


KRIPKE:

Now, and this is the important line, with Kripke, some worlds can be reachable from others; and I will say that the modal proposition Bp, also often written []p or \Box p (in LATEX), is true at some world w if and only if p is true in each world which are successor of w.

Is this the same B which is the modal logic operator for necesssity = true in all possible worlds?


Yes and no. B, out of context represents any box. In Aristotle/Leibniz this has indeed the meaning of "true in all possible worlds". With Kripke it means "true in all accessible possible world", and it is supposed that worlds can be linked by some accessibility relation. Kripke relativizes the notion of "necessity" and "possibility" to each world. I think that the passage from Leibniz to Kripke is a good modal mirror of the passage from the ASSA to the RSSA.




I say it again:

KRIPKE IMPORTANT LINE: Bp is true in w if for all world x such that wRx we have that p is true in x.

You can read wRx as the world w reaches the world x, or x is accessible from w.

For example, with a drawing, where the (broken) line represents the oriented accessibility relations (please add an arrow so you see that it is the worlds on the top which are accessible from the world at the bottom:

p          p
  \        /
    \    /
      \/
     Bp


Let us consider that "multiverse" M with only three worlds: w, w0, w1, and with "successor" or "accessibility relation" R given by wRw0, and wRw1. Meaning obviously that w0 and w1 are accessible from w, and that's all.

Now what I was trying to say to Lee was just that if p is true in w0, and if q is true in w1, then, B(p v q) is true in w0.


p          q
  \        /
    \    /
      \/
     B(p v q)


And if the world represents subjective observer moment a-la Bostrom, and if the accessibility relation represents scanning-annihilation followed by reconstitutions, the diagram with w, w0, w1 + wRw0 and wRw1 fits well the situation.


OBJECTION?
Ah! but Lee could have build an objection by saying that in Stathis' theory we die, or can die, at each "instant", or at each teleportation experiment. He told us this in its death thread.

I don't think Lee accepts my idea of death.


I have seen that. But Science is supposed to avoid wishful thinking. The question is "does Lee proposes some alternative? Lee sees clearly the difference between first and third point of view, but it seems he want to identify himself with some fuzzy notion of recent duplicates, and this apparently forces him to abstract from the 1-3 person pov difference.




Stathis was doing Kripke semantics, perhaps like Jourdain was doing prose. He suggests to define a state (world, OM, ..) as being "alive" when it is "transient":

The state/world/OM... x is "alive" when there is a y such that xRy

and a state is "dead" when there is no such accessible world from x. x is terminal, or cul-de-sac, dead-end, etc.

Now in Stathis' theory, we die at each instant and this means that all transient states reach dead-end worlds!

What do you mean by "transient state"? Aren't all states transient?


Certainly not the dead-end states. I took the world "transient" from some of your post.




Now suppose x is alive. This means there is y such that xRy. But the proposition true, t, is true in all world, and thus it is true in y. This means Bf is false in x (by KRIPKE IMPORTANT LINE). It is just false that f is true in all accessible world from x, giving that in y t is true (and xRy). So in any world x which is alive, Bf is false. This means that ~Bf is true (worlds obeys classical logic). and giving that f equivalent with ~t, this means that ~B~t is true in the alive state.
What about ~B~t, or ~Bf,  in a dead-end state?
What about Bf in a dead-end state?

If you specifically define B as above, you could say Bf is false because there are no reachable worlds for it to be true in. However, the same could be said for Bt, which doesn't seem right, since t is by definition true in any possible world. Could I go for a third option, "undefined"?


Nothing is left undefined in (arithmetical) Platonia, except from the point of view of the creatures living *in* that Platonia. The option "undefined" is premature at this stage. Bf and Bt are just "trivially" true in the dead-end worlds. It is just elementary classical logic. I mean it is not a crazy idea by me, it is a fact admitted and used by all mathematicians, even before Boole (Boole did just make this explicit).

It would help me to proceed if you tell me, you Stathis or any reader of the list, if you have understand (or not) that the fact that [Bf, Bt, actually any B<something> hold in the dead-end world] is as "obvious" as the fact that [for ALL numbers x, if x is bigger than 2 then x is bigger than 1.]


Bruno

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


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