On 14 February 2014 07:49, Bruno Marchal <[email protected]> wrote:
> Liz, and others,
>
>
> On 13 Feb 2014, at 10:04, LizR wrote:
>
> On 13 February 2014 21:38, Bruno Marchal <[email protected]> wrote:
>
>> If I reported that there was a flying pig, wouldn't comp just explain,
>> "That's the way arithmetic looks from inside."?
>>
>> Why? No. Not at all.
>> You must (using G & Co.) looks at the way arithmetic looks from inside,
>> and if you find the flying pig, then yes, you can say that comp explains
>> the flying pig, but if you see a white rabbit instead (in the arithmetic),
>> you can say that comp does not explain the flying pig, and might be false
>> in case you don't find the white rabbit in nature.
>>
>
> This sounds like my sort of science!
>
>
> It is the scientist sort of science, yes.
>
>
>
> "One pill makes you larger... And one pill makes you small..."
>
>
> Unfortunately, in the *realist* fairy tales, sometimes you don't have the
> magic wand, nor magic pills, and you have to empty an ocean with a tea
> spoon, if not a sieve, and be confronted with ten thousand Monsters if not
> grin without cat!
> We say that "reality" kick back, and it it is an euphemism. And that
> explains probably why science asks for some amount of works. Even tedious
> one!
>
> Well, just to prepare you for 8, well, no, 7 exercises.
>
> And more official definitions.
>
>
> Definition: A Kripke multiverse (W, R) is a non empty set W, with a binary
> relation R.
>
> That's all you need to know, about a Kripke multiverse.
>
> Conventionally we use the greek letter for its element, alpha, beta, ...
> and we call them world.
>
> OK?
>
> Exercise: give examples of the simplest Kripke multiverses possible.
>
W = { alpha } R = alphaRalpha (R is true or false for accessibility of
alpha from itself - although I'm not sure how a universe can't be
accessible from itself, to be honest).
>
> Solution:
>
> W cannot be empty (by the definition above), so the simplest one is
> probably the singleton {alpha}, with the empty binary relation---alpha does
> not access to any world, not even itself.
>
> Perhaps the next simplest one is {alpha}, with alpha R alpha. Alpha access
> to itself.
>
> OK?
>
Oh, OK, I assumed alpa R alpha was "equivalently simple" whether R was true
of false. (Why is inaccessible considered simpler? In a way I'd consider
self-accessible simpler!)
>
> But Kripke multiverse can be illuminated. By this I mean that we can
> associate a truth value (t, f, or 1, 0 ...) to all propositional letters,
> and this for each world. In english: we just illuminate the multiverse by
> telling the truth value of the atomic propositions (p, q, r, ...) in each
> world. (keep in mind that the goal is to find counterexample in modal
> reasoning).
>
OK
>
> We stipulate also that all worlds obeys classical propositional logic
> (CPL).
>
> In particular, if A is true at alpha, and if B is true at alpha, then (A &
> B) is true at alpha. For example. You cannot have a world with A false, but
> (A & B) true. Etc. OK?
>
OK
>
> Let us consider only one propositional letters. As a matter of fact,
> although the possible truth value of []p does not depend on p, it is
> independent of the value of q, so, to start, we can play with only one
> propositional letter, as most formula have only one propositional letters
> occurring in them.
>
> But how to decide the truth of a modal formula, with occurrence of []p and
> <>p?
>
> All you have to keep in mind is that (Kripke semantics):
>
> []p is true in alpha <=> p is true in all the worlds accessible from alpha
>
> (dually, you have already seen this)
>
> <>p is true in alpha <==> there is at least one world accessible from
> alpha where p is true.
>
So is []p true in the simplest multiverse with nowhere accessible? (It
looks like both []p and ~[]p are true!) <>p is false because there isn't an
accessible world.
>
> OK?
>
> It is really the same semantics as Leibniz, but relativized on each
> relatively accessible worlds.
>
> This determines the truth value of []p and <>p in the illuminated
> multiverse. To see if []p is true in a world, just look at those world
> accessed by alpha, and see if p is true at them. If alpha access ten
> worlds, you have to look at those ten worlds. If alpha access 0 worlds, you
> have zero verification to do, making the truth of []p vacuously true (or
> use []p = ~<>~p).
>
> OK that answers the question. Of course []p = ~<>~p so <>~p says there is
a world accessible from alpha where p is false. With no worlds accessible
that is false, so ~<>~p is true.
>
> OK?
>
OK
>
> Now, ask any question if anything remains unclear up to here, before
> trying the 7) exercises.
>
> But here is the exercise (we work with only one propositional letter)
>
> First illuminate the two simplest multiverses above, that is {alpha} with
> no accessibility relation, and {alpha} with alpha R alpha. That should not
> be long, given that we restrict ourself to only one propositional variable
> p, and have only one world.
>
Well, we get { p=t } and { p=f } regardless of the accessibility relations.
(If that's how you write it)
>
> Which of those propositions are true of false in alpha, in the illuminated
> simplest multiverses.
> And which one are law (meaning true in all worlds, but true for all
> valuation of p, that is valid with A = p, but also with A = ~p)
>
> 1) []A -> A
>
This is true in alpha R alpha (because it's just a Leibniz type world). []p
is "vacuously true" in "alpha" (the disconnected multiverse) - as you said
above - so []p -> p is false, because []p is true regardless of p.
> 2) []A -> [][]A
>
This is true in alpha R alpha, and in alpha I guess it's true too, because
vacuously true implies vacuously true?
> 3) <>A -> []<>A
>
true in alpha R alpha again, because there's only one world to consider so
<>A is equivalent to []A in this case (isn't it?)
not true in alpha because []<>A is vacuously true regardless of <>A - I
think
> 4) []A -> <>A
>
Well I think this is true for reason given above.
> 5)A -> []<>A
>
True in alpha R alpha. In alpha not true because []<>A is always true and A
isn't
> 6) <>A -> ~[]<>A
>
False in alpha R alpha, surely? With one world, <>A -> []<>A (above)
Not true in alpha because ~{}<>A is vacuously false regardless of <>A
> 7) []([]A -> A) -> []A
>
True in alpha R alpha I think. And vacuously true in alpha because both
sides of the rightmost -> have to be true. Not sure if that means it's
implied though...
> 8) []([](A -> []A) -> A) -> A
>
Not true in alpha because to the left of rightmost -> is vacuously true
regarldess of A. Don't know about alpha R alpha because my head exploded...
>
> Let me solve one case, to illustrate. Let us look at []A -> <>A, in the
> illuminated multiverse {alpha}, with empty R, and with p true in alpha.
> Well, what about []A?
> Alpha is a cul-de-sac world, so we have seen that []A must be true (if not
> <>~A has to be true), so []A is true and in particular []p is true
> (whatever the value of p is). What about <>A ? well this say that there is
> some beta accessible from alpha in which A is true. But alpha is a
> cul-de-sac world, so there is no such world, and so <>A is false, whatever
> A is (notably p or ~p). So []A is always true, and <>A is always false, so
> []A -> <>A is always false (by CPL), whatever illumination is chosen (p or
> ~p true at alpha).
>
> OK?
>
OK
>
> It is just CPL used in the world alpha, with the value of "[]A" determined
> by the Kripke semantics.
>
> Can you see the truth value of the 7 other formula in those simplest
> multiverses. I think it is a good training, and a good way to demystify the
> difficulties. Tell me what. You can do one formula at a time, in 8 posts.
>
I did them all (or more likely I didn't) before I got to this
point..........see above..........
>
> Take your time, take it easy, and have fun, if possible!
>
> 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.
