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.
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?
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).
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?
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.
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?
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.
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
2) []A -> [][]A
3) <>A -> []<>A
4) []A -> <>A
5)A -> []<>A
6) <>A -> ~[]<>A
7) []([]A -> A) -> []A
8) []([](A -> []A) -> A) -> A
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?
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.
Take your time, take it easy, and have fun, if possible!
Bruno
http://iridia.ulb.ac.be/~marchal/
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