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/
Received on Wed Jul 13 2005 - 12:30:06 PDT