Hi Stathis,
> Hi Bruno,
>
> I replied to the first part of your post earlier, but it took a bit
> more time to digest the rest. For what it is worth, I have included my
> "thinking out loud" below.
Thanks for replying, and thanks for authorizing me to comment online.
>
>
>> Mmmmmhhhhhh.... I know this could look like jargon. Let me give
>> "easy" exercises for anybody following this list.
>>
>> Let me define a Multiverse (called also "frame" by Kripke) as any
>> non-empty set W together with an accessibility relation R defined on
>> the set. Elements of that set are called "world", by definition, and
>> I follow the convention to denote worlds by greek letters (or their
>> english transcription: alpha, beta, gamma, delta, eta, epsilon, iota,
>> kappa, omega, nu, theta, etc.). R is called the accessibility
>> relation.
>
> Is W the entire multiverse, containing everything past, present and
> future, or is it a snapshot of the multiverse at a particular time
> point? If it contains everything, are the elements then snapshots of
> particular worlds at particular times?
>
> Does R operate on W mapping its elements to another set, or does it
> indicate that a particular element of W is related to another element?
> What exactly does "accessibility" mean in reality? Does it mean a
> world is accessible from my present situation if it is possible that I
> will experience that world in my subjective future? What is and isn't
> accessible in this sense is subject to debate, rather than being
> self-evident.
I totally agree with you. The point of modal logic and their multiverse
semantics does not consist in closing that debate, but just in giving
us a tool to manage the subtleties inherent with the subject. As you
correct yourself below, W is any (non-empty) set, and R is any relation
mapping some element of W to element(s) of W. One element goes to zero,
one, or more elements. W cannot be empty, by definition, but R can be.
I am really acting like mathematicians are used to act. I provide a
very general notion of multiverse: any set with any binary relation
will do. Only later we will reason to find out which multiverses are
more interesting than others, for example with respect to the measure
problem, or the 1-3 distinction problem, or the comp hyp, etc.
At this point anybody should be able to begin a list of all
multiverses, or at least the finite one ...
The list will be like (with set notation):
{a} with R = empty
{a} with R = {(a,a)}
{a,b} with R = empty
{a,b} with R = {(a,a)}
{a, b} with R = {(b,b)}
{a, b} with R = {(a,a),(b,b)}
{a, b} with R = {(a,a), (a,b), (b,b)}
{a, b} with R = {(a,a) (a, b)}
{a, b} with R = {(a,b)(b,b}}
etc.
I use set notations because I cannot make drawings here, but please
"translate" all those description into little drawings, it is much more
readable. To say R = {(a,a) (a,b)} is the same as saying that aRa and
aRb, that is (in multiverse terms): the world a can reach the world a,
and the world a can reach the world b. In a drawing the world are
points and the relation are arrows from a to b when aRb, etc. OK?
The very nature of the worlds a, b, c is left unspecified, and the
relation of accessibility R is entirely defined by its graph (its set
of inputs/outputs).
The multiverse {a} and the multiverse {b} really denote the same
multiverse if nothing special is said about a and b. "Snobbish
algebraists" would say they are isomorphic, and that could be wise at
some point, but that would be too much at this stage.
>
>
>> So the simplest example of multiverse is given by the set {alpha} +
>> the empty relation (so just one dead end!). Another example is the
>> set of natural numbers with the divisibility relation ( n R m iff n
>> divide m iff there is a k such that n * k = m).
>
> OK, I think this means R indicates one element in W is related to
> another element in W.
Yes indeed.
>
>> Let me define a notion of illuminated multiverse (called "model" by
>> the modal logicians). It is just a Kripke multiverse where we
>> associate to each world a value 1 or 0 to each sentence letter. The
>> Kripke multiverse is "illuminated" when a truth value (1 or 0) is
>> assigned to each proposition, in each world.
>> Remember that in (propositional) logic we have sentence letter p, q,
>> r, etc.
>> We also say that p is true in alpha for p has value 1 in alpha (in
>> some illuminated multiverse).
>
> To clarify: each world alpha, beta etc. in W contains many
> propositions p, q, r etc., and W is called "illuminated" when we know
> for each proposition whether it is true (1) or false (0).
Yes, except I prefer to say that a proposition is true at or in a
world, instead of saying the proposition is in the world.
> This confuses me a bit, because I think of physical worlds as
> containing things, not propositions.
Just what I was saying! Except that I am not limiting myself to
"physical worlds".
> I suppose you could redefine a world as containing only propositions.
There is a sense to do that, once the multiverse is illuminated. There
is a sense to identify a world with the set of propositions which are
true at that world.
Now beware a little bit: there is also a sense to identify a
proposition with a set of worlds: the set of worlds at which the
proposition is true. Modal logicians do that all the time. Yes, you can
conclude that (in an illuminated multiverse) a world is a set of of
set of worlds. Thanks to God, we will not do such identification ...
until the lobian machine will show us some multiverse where such
identification will accelerate our reasoning ... (no need to worry
before).
> For example, the world I find myself in now contains the propositions
> p="I am typing on my computer"=1; q="there are unicorns in China"=0;
> r="all bachelors are unmarried"=1; and so on, thus accounting for
> every possible true and false fact about the world. Is this what is
> meant?
Yes. One of the reason we tend to shift from "p is true at world a" is
the drawing of the illuminated multiverse: to say that p is true at or
in a, we draw the world a itself by a circle and we put p in it.
Only latter, when I will explain the notion of maximal consistent sets
of formula, we will meet a situation where worlds, in some multiverse,
will be concrete set of formulas; but no worry before we meet them.
Those are the erzats worlds that David lewis did criticize, although he
seems to have change its mind since. Will come back on this...
>
>> Now Kripke semantics can be given in a very simple way, by just
>> asking that,
>>
>> 1) each world obeys to classical logic (that is: if 1 is assigned to
>> p in the world alpha, and if 1 is assigned to q in alpha, then 1 is
>> assigned to (p & q) in alpha, etc. The "etc" is just a pointer to the
>> usual truth table of classical propositional logic. I have already
>> explain this on this list but I can do it again if asked). In
>> particular each classical tautologies are true in all worlds, whatver
>> the illumination chosen (whatever the truth value of the sentence
>> letter are in each world: like (p -> p) or (p v ~p), etc.
>
> What do you mean by "whatever the illumination chosen"? If you define
> "illumination" as above, it seems to mean only one thing, so where is
> the choice?
For the sake of simplicity let us say that we have in our propositional
language only the three sentence letters {p, q, r}. It could be the
three proposition you mentionned before:
p = "I am typing on my computer"
q = "there are unicorns in China"
r = "all bachelors are unmarried"
Let us take now the simplest of all multiverse: the set {a}, and R is
empty.
Then it is exactly like classical (non modal) logic: we have 8 possible
illuminations
p q r
1 1 1
1 1 0
1 0 1
1 0 0
0 1 1
0 1 0
0 0 1
0 0 0
giving just 8 illuminated multiverse (of *that* kind of multiverse).
Each line is one illumination. In a multiverse with two words, you
would have 8 times more illumination, given the remaining 8 choices for
the illumination in the second world.
The last one of those (one element) illuminated multiverse,
corresponding to the last line of the tableau above, is such that you
are not typing on your computer, they are no unicorns in China, and
some bachelor are married.
No contradiction, because p, q, r ... in propositional logic are not
suppose to be analysed further. We are interested in propositions true
in all worlds of all multiverses *of a certain kind". In this example
it should be clear that all classical tautologies are true in all world
(there is only one). verify for (p -> p), (p V ~p), (p -> (q -> p),
etc.
Now is Bp -> p respected in {a}+R empty, that is: is "Bp -> p" true in
all world whatever the illumination is?
No. Look at the fifth line, where p = 0, q = 1, and r = 1. Bp is true
in a (because R is empty, and so a is a dead end where we have already
see that B<anything> is true, and D<anything> is false. In particular
Bp is true, but given that p is false, Bp -> p is false.
If R was not empty and such that aRa, then the Kripke semantics forces
p to be true in a. (I recall that Bp is true in a if p is true at all b
such that aRb). In that case Bp -> p is true in a (whatever the value
of p is).
>
>
>> 2) Kripke says that Bp (also written box p, []p, etc.) is true in the
>> world alpha if p is true in all worlds beta accessible from alpha.
>> From this it follows that Dp (defined as an abbreviation of ~B~p)
>> will be true in some world alpha if there is some world beta,
>> accessible from alpha, and with p true in the world beta.
>
> I am familiar with the idea that p is necessarily true if p is true in
> all possible worlds. How does this relate to the above?
p -> p is true in all worlds, (I mean: of the fixed simplest
multiverse {a} with R empty).
B(p -> p) also
Bp -> p is true only in the world where p is true, given that Bp is
always true (given that a is a dead end). So Bp -> p is not respected
by the class of one-element-multiverse with R empty.
etc.
> And a point on terminology: you say "true in all worlds beta", but I
> thought beta refers to only one particular world... I suppose you are
> using beta here as a variable rather than a constant.
Yes. Just to avoid the greek letter xi, zeta, ...
>
>> Now I will say that a formula A of modal logic is valid in a
>> illuminated multiverse (W, R, V) if A is true in all the worlds of
>> that illuminated multiverse.
>
> V means "illuminated"?
You get the idea. V denotes some illumination. An attribution of truth
value (0 or 1) to any sentence letter {p, q, r, ...} for each world in
W.
>
>> And I will say that a formula A of modal logic is respected by a
>> multiverse (W,R) if A is valid
>> in all illuminated multiverse (W, R, V). Or equivalently: A is
>> respected in (W,R) if A is true in all worlds in W and this for all
>> "illuminations" V, i.e. for all assignment of truth value of the
>> sentence letters in all worlds.
>>
>> Last definition: a multiverse (W,R) is said to be reflexive if the
>> relation R is reflexive (that is: if for all world in W we have xRx,
>> i.e. if each world is accessible to itself by the relation R.
>
>
> What could this mean in a real world example?
Take W as the set of places in Brussels. Take R to be "accessible by
walking in a finite number of foot steps". Then each places at Brussels
is accessible from itself, giving that you can access it with zero
steps, or two steps (forward, backward, ...).
Take W as the set of humans, say that aRb if a can see directly,
without mirror, the back of b. Then a can access all humans except
themselves. R is said to be irreflexive.
Another important "concrete" example, which will help us latter to
study the modal logic of quantum logic. Take the worlds to be the
vector of an Hilbert Space (or of the simpler 3-dimensional euclidian
space). Say that a is accessible to b, i.e. aRb, if the scalar product
of a and b is non null (i.e. a and b are not orthogonal). Each a can
reach itself, so again this "multiverse" is reflexive. Can you see that
is also symmetrical?
Take its "anti-multiverse", that is the mutiverse (W,R') where aR'b iff
not aRb. This means that aR'b iff a is orthogonal to b. Then (W, R') is
irreflexive (no vector is orthogonal to itself) but still symmetrical
(if aR'b then bR'a).
Such "multiverse" will help us to find out a modal logic where B
captures the quantum probability 1 (the sure bets).
>
>> The easy exercise is the following: show that if the multiverse is
>> reflexive then the multiverse respects the formula Bp -> p.
>
> Bp is true in alpha if p is true in all worlds accessible from alpha.
> If the multiverse is reflexive, then for Bp to be true, p must be true
> in alpha (which is one of the worlds accessible from alpha). Since
> this applies to any world alpha in W, the formula is respected in
> (W,R).
That's exact.
>
>> Slightly less easy: show that the reverse is true: show that if a
>> multiverse respects Bp -> p, then the multiverse is reflexive.
>
> Yes, I have to think about this longer.
All right. It is less easy. I tend to do it by a reductio ad absurdum.
I suppose the multiverse respects Bp -> p, and I suppose it to be non
reflexive, then it is enough to find one precise valuation V making a
contradiction with Kripke semantics.
>
>> I would like to know if that exercise *seems* difficult. For those
>> who cannot do it, it just means there is a need to refresh some
>> "naive set theory" knowledge, and I will think about a book who can
>> help.
>>
>> Don't hesitate to answer out of line if you prefer.
>>
>> Sorry to annoy you with that modal stuff, but we are at a point I
>> could no more comment the posts without making nuances which will
>> resemble jargon if you don't invest a little bit in modal logic.
>> This, by the way, could provide us with a language capable of make
>> clearer many other "everything-like philosophy clearer.
>
> I have two general difficulties with the above. One is that I am
> always trying (even without wanting to) to relate the mathematical
> logic to real world examples, and it gets confusing.
It is always a good idea to find examples. "Real world" examples can be
confusing, because "real worlds" are generally very complex.
> The other problem is that it is not clear to me where the theorems/
> axioms/ definitions, such as what is a "valid" and what is a
> "respected" formula , come from.
I will probably explain it all along. They really come from the goal of
studying deduction and independencies among modal formulas.
> Should I accept them as self-evident? My understanding is that there
> are several possible modal logics, so how does one choose between
> them?
That is *the* question! Many classical logicians just hate modal logic
because there are so many of them (instead of the secure cocoon of
"one" (propositional) classical logic).
But here I have a sort of incredible answer, which actually appears
also in the book by Boolos. You know that what I really want to explain
here is how to derive physics from computer science, and what I am
doing is "just" to interview the lobian machine. This will give a
unique modal logic: the logic of self-reference G. And this modal logic
just reflect a mathematical phenomenon (the incompleteness of sound
machines or theories). Then all the other modal logics will appear when
the machine, taking its own incompleteness into account, will introduce
by itself the modal nuances between proof and truth (to begin with, and
this will explain G*), and the modal nuances between 1-person and
3-person, knowledge and probabilities, and eventually between comp and
not-comp.
the comp + 1-person + truth nuances, motivated by the UDA, and applied
to the self-reference logic will give the searched quantum logic.
Poetically (to breath a little), the physical laws will appear to be
the necessary invariant structure living on the border of the machine's
ignorance. And all (honest, sound) machine can discover that, soon or
later (in platonia, but remember with comp we *are* already in
platonia).
Bruno
http://iridia.ulb.ac.be/~marchal/
Received on Wed Dec 07 2005 - 12:31:45 PST