On 6 February 2014 08:25, Bruno Marchal <[email protected]> wrote:
>
> 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?
>
I sure hope so.
>
> (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?
>
p -> f is (~p V f), for which the truth table is indeed the same as ~p
>
> (Can you define "&", "V", with "->" and "f" in the same way? This is not
> an exercise, just a question!).
>
I don't think I can define those *literally* with p, -> and f if that's
what you mean. But that doesn't make sense, because & requires two
arguments, so it would have to be something like ... well, p -> q is (~p V
q) and it's also ~(p & ~q), which contain V and & ... I'm not sure I know
what you mean.
>
> 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?
>
I suppose 4, since we have a world with 2 propositions, and each can be t
or f?
>
> 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.
>
Oh, OK. So a and b are worlds, not ... sorry. I see. So that is 2^3 x 2^3
because a has p,q,r = 3 values, all t or f, as does b. OK now I see what
you meant.
>
> Of course with the infinite alphabet {p, q, r, p1, q1, r1, p2, ... } we
> already have a continuum of multiverses.
>
I can't quite see why it's a continuum. Each world has a countable infinity
of letters, and the number of worlds is therefore 2 ^ countable infinity!
Is that a continuum? My transfinite maths may not be quite up to that one.
>
> 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.
>
This seems like a way of getting subsets from the multiverse, but I'm not
completely sure what accessible means here.
>
> 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)
>
So []A means the proposition A is true in all worlds accessible from ...
somewhere.
Oh dear. I don't seem to be able to get my head around this. Maybe because
I'm not sure what accessible means here...
>
>> 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/
>>
>>
>>
>> --
>> You received this message because you are subscribed to the Google Groups
>> "Everything List" group.
>> To unsubscribe from this group and stop receiving emails from it, send an
>> email to [email protected].
>> To post to this group, send email to [email protected].
>> Visit this group at http://groups.google.com/group/everything-list.
>> For more options, visit https://groups.google.com/groups/opt_out.
>>
>
>
> --
> You received this message because you are subscribed to the Google Groups
> "Everything List" group.
> To unsubscribe from this group and stop receiving emails from it, send an
> email to [email protected].
> To post to this group, send email to [email protected].
> Visit this group at http://groups.google.com/group/everything-list.
> For more options, visit https://groups.google.com/groups/opt_out.
>
>
> http://iridia.ulb.ac.be/~marchal/
>
>
>
>
> --
> You received this message because you are subscribed to the Google Groups
> "Everything List" group.
> To unsubscribe from this group and stop receiving emails from it, send an
> email to [email protected].
> To post to this group, send email to [email protected].
> Visit this group at http://groups.google.com/group/everything-list.
> For more options, visit https://groups.google.com/groups/opt_out.
>
>
> http://iridia.ulb.ac.be/~marchal/
>
>
>
> --
> You received this message because you are subscribed to the Google Groups
> "Everything List" group.
> To unsubscribe from this group and stop receiving emails from it, send an
> email to [email protected].
> To post to this group, send email to [email protected].
> Visit this group at http://groups.google.com/group/everything-list.
> For more options, visit https://groups.google.com/groups/opt_out.
>
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To post to this group, send email to [email protected].
Visit this group at http://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/groups/opt_out.
