Hi,

`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).`

`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.`

`(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.`

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.`

`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!`

`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?

`This is a little bit tricky and I let you think (I must go now). It is`

`important also for getting a "theory" (set of propositions through in`

`all worlds in some multiverse, where a multiverse is just a set of`

`worlds (OMs) with some specified accessibility relation among worlds`

`(OMs).`

`The main exercise now consists in finding all formulas true in all`

`worlds, whatever is the valuations of p, q, r in the worlds, when the`

`worlds belongs to Papaioannou multiverse, and I recall that a`

`Papaioannou multiverse is characterized by the fact that all transient`

`(alive) state reaches dead-end.`

`For this the first problem is the truth/false status of Bf in a`

`cul-de-sac world.`

`Stathis, dont' hesitate to accuse me of betraying your idea if that is`

`the case,`

Sorry for those holiday exercises,

`Don't hesitate to ask questions, it is cumbersome to explain Kripke`

`semantics without easy drawing abilities (I already regret the`

`combinators ;-)`

I will come back on this asap (or just a little bit later), Bruno http://iridia.ulb.ac.be/~marchal/