Hi Liz,
Logician have a large notion of "world". A world is a element of a
set, called the set of worlds, or multiverse.
Statisticians do the same, with the notion of population, which is
also just a set. In fact classical logic and classical statistics have
a sufficiently large common base so that George Boole deemed them both
under the label "the laws of thought".
Propositional logician have variable propositional letter, supposed to
assign to true or false in each worlds. I will assume the letter p, q,
r.
They have some grammar for the formula. I hope you can solve the
following exercise:
Which among the next symbolic expressions is the one being a well
formed formula:
((p -> q) -> ((p& (p V r)) -> q))
))(p-)##à89-< a -> q)
OK?
(to be sure the irst one might contain a typo, but I assure you there
are no typo in the second one (and there is no cat walking on the
keyboard).
***
Then a set of worlds get alive when each proposition (p, q, r), in
each world get some truth value, t, or f. I will say that the
mutiverse is illuminated.
And we can decide to put f and t is the propositional symbol for the
boolean constant true and false.
(meaning that "p -> f" is a proposition, or well formed formula).
In modal logic it is often simpler to use only the connector "->" and
that if possible if you have the constant f.
For example you can define ~p as an abbreviation for (p -> f), as you
should see by doing a truth table. OK?
(Can you define "&", "V", with "->" and "f" in the same way? This is
not an exercise, just a question!).
Each world, once "illuminated" (that is once each proposition letter
has a value f or t) inherits of the semantics of classical proposition
logic.
This means that if p and q are true in some world alpha, then (p & q)
is true in that world alpha, etc.
in particular all tautologies, or propositional laws, is true in all
illuminated multiverse, and this for all illuminations (that for all
possible assignment of truth value to the world).
OK?
Question: If the multiverse is the set {a, b}, how many illuminated
multiverses can we get?
Answer: there is three letters p, q, r, leading to eight valuations
possible in a, and the same in b, making a total of 64 valuations, if
I am not too much distracted. I go quick. This is just to test if you
get the precise meanings.
Of course with the infinite alphabet {p, q, r, p1, q1, r1, p2, ... }
we already have a continuum of multiverses.
Well, that was Leibniz sort of multiverse, with all worlds quite
independent of each other.
With Kripke, we introduce a binary relation R on the set of world.
That's all. We read alpha R beta, as beta is accessible from alpha.
OK. Time for the main recall:
We add then new unary connector "[]", and define <> by ~[]~
In Leibniz semantics, []A is true (absolutely) means that A is true in
all worlds.
In Kripke semantics []A is true in a world alpha means that A is true
in all worlds accessible from alpha.
And the only one exercise:
prove that "[]A -> A" is true in all worlds of a multiverse, for all
illumination possible (choice of valuation for the letter)
iff the relation is reflexive (that is: all world can access
themselves).
Hint: this should be easy. Any difficulty here is due to my probable
unclarity, or my excess of verbosity, or a lack of familiarity with
math of your part. I suggest you might search for counterexample.
And yes, this is truly two exercises, because to prove an iff, you
have to prove two if. You must prove:
1) if a multiverse is reflexive, then, whatever the illumination is,
each world satisfy []A -> A (for all formula A).
2) If, whatever the illumination is, each world satisfy []A -> A (for
all formula A), then the multiverse is reflexive.
"whatever the illumination" is important: for example in the simple
multiverse with one world: {alpha}, and the empty accessibility
relation (so that alpha does not access to itself, ~ (alpha R alpha),
and with p valuated to 1 in alpha, you have that []p is true, p is
true, so []p -> p is true in alpha, yet the mutiverse is not reflexive.
OK?
Please, ask any question to clarify. Note in passing the beauty: a
modal formula, made into a law, impose some structure on a Kripke
multiverse, and inversely, an accessibility structure on a multiverse
impose a modal law.
And now a free subject of reflexion :) (to prepare the sequel)
If reflexivity in Kripke multiverse characterizes []A -> A
Which relations can characterize the following formula?
The Leibnizian one:
[]A -> [][]A
[]A -> <>A
p -> []<>A
<>A -> []<>A
[](A->B) -> ([]A -> []B)
And what about (more hard) the non Leibnizian one, which will play
some role (as scheme of some machines discourses)
<>A -> ~[]<>A (related to Gödel)
[]([]A -> A) -> []A (related to Löb)
[]([](p -> []p) -> p) -> p (related to Grzegorczyk, the Grz of S4Grz).
Bruno
On 29 Jan 2014, at 11:23, Bruno Marchal wrote:
On 29 Jan 2014, at 01:05, LizR wrote:
On 29 January 2014 08:29, Bruno Marchal <[email protected]> wrote:
Hi Liz, Others,
"Good morning Professor Marchal!"
In the general semantic of Leibniz, we have a non empty set of
worlds W, and some valuation of the propositional variables (p, q,
r, ...) at each world.
And we should be convinced than all formula, with A, B, C, put for
any formula, of the type
[](A->B) -> ([]A -> []B)
[]A -> A
[]A -> [][]A
<>A -> []<>A
A -> []<>A
are all laws, in the sense that they are all true in all worlds in
all Leibnizian multiverse. OK?
Yes.
Good.
Most are "obvious" (once familiarized with the idea 'course). Take
[]A -> A. Let us prove by contradiction, to change a bit. Imagine
there is world with []A -> A is false. That means that in that
world we have []A and ~A. But []A means that A is true in all
world, so in that world we would have [A and ~A. Contradiction (all
worlds obeys classical CPL).
Test yourself by justifying in different ways the other
propositions, again and again.
But now, all that was semantic, and logicians are interested in
theories. They want axioms and deduction rules.
So, the question is: is there a theory capturing all the laws, true
in all worlds in all Leibnizian multiverse?
Answer: YES.
Ah? Which one.
S5.
S5?
Yes, S5. The fifth system of Lewis. Who did modal logical purely
deductively, and S5 was his fifth attempt in trying to formalize a
notion of deducibility.
The axioms of S5 are (added to some axiomatization of CPL, like the
one I gave you sometimes ago):
[](A->B) -> ([]A -> []B)
[]A -> A
[]A -> [][]A
<>A -> []<>A
The rules of S5 are:
The modus ponens rule, like CPL axiomatization.
The necessitation rule: derive []A from A.
? derive []A from A ???
Yes. That rule is valid in the Kripke multiverse. It looks rather
strong, and we will have to relinquish it for the "true" hypostases,
the one with a starred name, like G*, Z*, X*, ...).
It really means that if A is satisfied in all worlds in a
multiverse, then []A is also satisfied in all worlds of that
multiverse. It does not mean that if A is true in some world then
[]A is true in that world. We will come back on this, but it is
preferable to focus first on the semantics, and come back to
deducibility later.
Just don't confuse the rule "derive []A from A", and the formula A -
> []A.
I think that you have already refuted that formula (A -> []A) in the
Leibniz multiverse. Can you use this to refute it in some Kripke
multiverse.
Can you, more generally, find an accessibility relation that we
could add on a Leibniz multiverse to make it into a Kripke multiverse?
I recall you basic relation properties.
R is reflexive iff x R x for all x.
R is symmetrical iff x R y -> y R x, for all x and y
R is transitive iff x R y & y R z entails x R z, for all x, y, z
R is irreflexive if ~(x R x) for all x
etc.
(Here, the x, y, z = the apha, beta, gamma, ... worlds in the
multiverse). OK?
It can be proved that S5 can prove all the laws satisfied by all
worlds in the Leibnizian multiverse.
Those axioms are independent. For example you cannot prove <>A ->
[]<>A from the other axioms using those rules. But how could we
prove that? This was rather well known by few modal logicians.
There is a curious article by Herman Weyl, "the ghost of modality",
were Herman Weyl illustrate both that he is a great genius, and a
great idiot (with all my very deep and sincere respect).
He said that our minds "crawl up our worldlines" didn't he? Thereby
giving lots of people the wrong idea about how a block universe
works.
Ah? I read his book on GR. It is a bit old but still pleasant. Not
sure that "our minds crawl up our worldlines" is wrong for block
universe. Maybe you can elaborate a little bit.
Modal logic has been very badly seen by many mathematicians and
logicians. In the field of logic, modal logicians were considered
as freak, somehow. Important philosopher, like Quine were also
quite opposed to modal logic.
So it was very gentle from Herman Weyl to attempt to give modal
logic some serious considerations.
He tried to provide a semantic of modal logic with intuitionist
logic, but concluded that it fails, then with quantum logic, idem,
then with provability logic (sic), but it fails. It fails because
each time some axiom of S5 failed!
This shows he was biased by the Aristotelian Leibnizian
metaphysics. In fact he was discovering, before everybody, that
there are many modal logics, and that indeed they provide classical
view on many non standard logics. In fact, somehow, it is the first
apparition of the hypostases in math (to be short).
I really love that little visionary paper (if only I could put my
hand on it).
It comes up a lot if you google - I think you have to belong to
various academic groups to read it...maybe you would be able to?
I should, but it is more easy from my office than here. I will see.
But if S5 is characterized by the Leibnizian multiverse. What will
characterize the other modal logics?
Well, there has been many other semantics, but a beautiful and
important step was brought by Kripke.
It is almost like the passage from the ASSA to the RSSA! The
passage from absolute to relative. The passage from Newton to
Einstein.
Kripke will put some structure on the Leibnizian multiverse. He
will relativize the necessities and possibilities.
How?
By introducing a binary relation on the worlds, called
accessibility relation. Then he require this:
[]A is true in a world alpha = A is true in all worlds
*accessible* from alpha.
Exercise: what means <>A here? (cf <>A is defined by ~[]~A).
it isn't the case that in all worlds accessible from alpha, A is
false. Or in at least one world accessible from alpha, A is true.
OK.
So a Kripke multiverse is just a non empty set, with a binary
relation (called accessibility relation). It is a Leibnizian
multiverse, enriched by that accessibility relation. For []A being
true, we don't require it to be true in all worlds, but only in all
worlds accessible from some world (like the "actual world", for
example).
Again a Kripkean law will be a proposition true in all worlds in
all Kripke multiverse.
Now I am a bit tired, so I give you the sequel in 2 exercises, or
subject of meditation.
1) Try to convince yourself that the formula:
[](A->B) -> ([]A -> []B)
is a Kripkean law. It is satisfied in all worlds (meaning also all
valuations of the propositional letters), in all Kripke multiverse.
OK, I think, if the []s refer to the same subset of the multiverse
in each case this reduces to Leibniz case.
I do not understand. You might be a little quick here.
2) try to convince yourself that none of the other formula are laws
in all Kripke multiverse. Try to find little Kripke multiverse
having some world contradicting those laws.
[]p -> p
I take this to mean that the truth of p isn't available if the
world in question isn't accessible from the one under consideration.
I think this is going in the right direction. You mean that []p -> p
is false in the world alpha if
1) p is false in alpha
2) not (alpha R alpha). Calling "R" the relation of accessibility.
OK?
But you must add that p must be true in all world beta that can be
accessed from alpha, so as to guaranty that []p is true in alpha. If
not, []p will be false, and as p is false, we would have []p -> p
(as f -> f). OK?
So, a counterexample kripke multiverse M (putting the valued
propositions in the world) would be
M = {alpha, beta}, R = {(alpha, beta)}
alpha = {~p},
beta = {p}
alpha R beta
In alpha, we have []p and ~p, refuting []p -> p. OK?
Is the following also a counter-example. M = {alpha}, and alpha =
{~p} ?
It would be simpler.
Hint: keep in mind that all unicorns in my living room are bigger
that the mount Everest.
So []p = "(if) p is true in all worlds accessible from some world"
which doesn't imply that p is true in a given world.
You forget the relativization (a typo perhaps): it is
[]p is true in this world alpha = "p is true in all worlds beta
accessible from alpha".
With Leibniz []A is absolute. A will be true in all worlds,
accessible or not.
With Kripke []A is not absolute. []A can be true in some world and
false in some other worlds. For example []A can be true in alpha,
and false in some beta (which needs only to be non accessible from
alpha). OK?
[]A -> [][]A
if A is true in all worlds accessible from a given world, then
wouldn't that imply that it's true that "A is true in all worlds
accessible from a given world"
But it is not an arbitrary given world. It is the world under
consideration.
in all worlds accessible from a given world? It seems like because
we use [] on either side, we are reducing to a multiverse connected
by accessibility, and within that world, "Leibniz still applies".
But I must have that wrong.
Or too much unclear. Keep in mind that in Leibniz "[]A" has some
absolute meaning, not depending on any world. But that is no more
the case with Kripke. []A is true or false in some world, depending
only to the worlds accessible from alpha (and of the truth status of
A in those accessible worlds). OK?
I will have to leave this for now...
Can you find *special* binary relations which would enforce some of
those propositions to be law in those corresponding *special*
Kripke multiverse?
I think we have clarified the meaning of []A in Kripke semantics.
Try to solve this for the proposition []A -> A. You are very close
in your solution above.
Then you might consider the question for all the proposition above.
This asks for a bit of work. I don't want you to be able to solve
this, but to well understand the question, before I provide the
solution. Some kids here really want me not given the solution, but
they love perhaps math without moderation!
Do you see what happens? The geometry of the multiverse determine
the laws satisfied by the worlds.
But we have also that the modal laws satisfied by the worlds
determine a "geometry" on the multiverse.
So if we find the modal laws of the thinking machine, we might
determine the structure of their multiverse.
That's the general idea, but things will get a bit more complex.
Bruno
Advise. Draw potatoes for the worlds, with the valuation inside,
and draw big readable arrow between two worlds when one is
accessible from the other. The binary relation is arbitrary. If
(alpha beta) is in the relation (beta is accessible from alpha), we
don't have necessarily that (beta alpha) is in the relation.
I let you play. (with moderation. No need of brain boiling, and ask
any question if something seems weird, keep in mind I made typo
also ...)
Bruno
http://iridia.ulb.ac.be/~marchal/
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