Liz, Others,
I was waiting for you to answer the last questions to proceed. Any
problem?
I give the correction of the last exercise.
On 14 Feb 2014, at 19:18, Bruno Marchal wrote:
<snip>
On 13 Feb 2014, at 22:23, LizR wrote:
On 14 February 2014 07:49, Bruno Marchal <marc...@ulb.ac.be> wrote:
Liz, and others,
On 13 Feb 2014, at 10:04, LizR wrote:
Well, we get { p=t } and { p=f } regardless of the accessibility
relations. (If that's how you write it)
Well ... OK.
More precisely we get
1) {alpha}, with R = { } with p=t in alpha
2) {alpha}, with R = { } with p=f in alpha
and
3) {alpha}, with R = {(alpha R alpha) } with p=t in alpha
4) {alpha}, with R = {(alpha R alpha) } with p=f in alpha
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)
Very good. It is a law there.
. []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.
OK. It is not a law, but it might still be true in some
circumstances. Give me which among 1), 2), 3), 4), above.
Any problem with this?
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?
Exact.
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?)
Well seen!
not true in alpha because []<>A is vacuously true regardless of <>A
- I think
Not correct. You jump to hastily.
in your language the answer is:
true in alpha, because []<>A is vacuously true, so that <>A -> []<>A
is vacuously true too (as "p -> q" is false only if q is false and p
is true). The type of <>A -> []<>A is really f -> t, which is as
much tautological than f -> A, and A -> f, for any A.
Are you OK with this?
4) []A -> <>A
Well I think this is true for reason given above.
You begin to try to go to quickly. I have some doubt that []A -> <>A
can be a law in a cul-de-sac world, like poor alpha, with R = { }.
OK?
5)A -> []<>A
True in alpha R alpha.
OK.
In alpha not true because []<>A is always true and A isn't
Not a law. OK. Again there are case where it is true, like when A is
true.
6) <>A -> ~[]<>A
False in alpha R alpha, surely? With one world, <>A -> []<>A (above)
Correct.
Not true in alpha because ~{}<>A is vacuously false regardless of <>A
Unfortunately as much as ~{}<>A is vacuously false regardless of
<>A, as you say, we are interested in
<>A -> ~[]<>A and in poor alpha (case 1) 2)) <>A is *also vacuously
false, so that we are in the f -> f, case, which is, vacuously or
not, always true.
Are you OK with this?
Keep in mind that in CPL both
f -> A
and
A -> t
are always tautologies. They are true in all worlds, whatever A is.
7) []([]A -> A) -> []A
True in alpha R alpha I think.
A law? True in both 3), 4) ?
And what about alpha (case 1 and 2)?
Let us look in alpha R alpha (case 3 and 4 above):
in W = {alpha}, with alpha R alpha, and with V(p) = t (V = the
valuation or illumination):
We have p is true in alpha, and p is true in all worlds accessed to
alpha. OK?
So, []p is true, and A -> []p is true, whatever A is, so []([]p -> p) -
> []p is true.
What if A = ~p, in []([]A -> A) -> []A? (that is really case 4)
In that case the right hand side is []A = [] ~p, and is false. In the
left hand side, A is false, but []A is false too, so "[]A -> A" is
true (f -> f is true), and thus, it is also true in all worlds
accessed from alpha, and thus we have that []([]A -> A) is true, and
so []([]A -> A) -> []A is of the type t -> f, and so is false, and so
[]([]A -> A) -> []A is NOT a law.
Same reasoning in the "4" case, with p exchanged with ~p.
Conclusion: []([]A -> A) -> []A is NOT a law in the little reflexive
(alpha R alpha) multiverse.
OK?
And vacuously true in alpha because both sides of the rightmost ->
have to be true.
Correct. It is law. True in 1) 2).
Not sure if that means it's implied though...
I am not sure what you are asking here.
8) []([](A -> []A) -> A) -> A
Not true in alpha because to the left of rightmost -> is vacuously
true regarldess of A.
OK. Not a law.
Precisely, if A is false, []([](A -> []A) -> A) is still vacuously
true, so we get T -> f, which is false.
Don't know about alpha R alpha because my head exploded...
Take a break. I said "one a at a time" !
OK. I do it.
Consider []([](A -> []A) -> A) -> A with A true (for example A = p in
case p is true, or equivalently A = ~p with p false).
In that case we have something like # -> t, but that is always true,
so []([](A -> []A) -> A) -> A is true.
So, the less easy case, consider []([](A -> []A) -> A) -> A with A
false.
So we have <something> -> f, and this can be true only if <something>
is false.
What about the inner A -> []A ? A is false, so A->[]A is true (f -> #
is always true).
But if A -> []A is true in "alpha R alpha", then A -> []A is true in
all worlds accessible from alpha, and so [](A->[]A) is true too. We
get something like [](t -> f) -> A; with A = f. (I have substituted []
(A-> []A) by t, in the initial formula. But t -> f is false, and thus
[](t->f) is false too. OK? So the initial Grz formula is of the type f
-> f, and thus it is true! So []([](A -> []A) -> A) -> A is true, with
A false and A true, and it is a law for the 3 and 4 case.
OK?
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..........
Yeah, I should have said "one at a time" earlier :)
Very good work.
I give you more time for Grzegorczyk []([](A -> []A) -> A) -> A
And just one supplementary exercise. What can be said about
[](A -> B) -> ([]A -> []B), in those simple multiworlds structure?
This formula has two (meta) variables A and B.
Is that a law?
I let you think a bit more on this before providing the solution. Tell
me if you are OK with the correction.
You solved correctly 6 cases on 8. I think that your head go near
explosion, because you still panic in front of "too much symbols", but
I hope having shown here that it is just a matter at looking what
happens, beginning by the sub-formula, in each world, for each
valuation. You must keep in mind the truth table of the implication
too. The tautologies f -> # and # -> t are very handy here.
OK?
About [](A -> B) -> ([]A -> []B), let me ask you a more precise
exercise.
Convince yourself that this formula is true in all worlds, of all
Kripke multiverses, with any illumination.
Hint: you might try a reductio ad absurdum. try to build a multiverse
in which that law would be violated.
Don't hesitate also to test your memory on exercises that you have
already been able to solve, just to be sure you memorize correctly the
definition.
Hope you enjoy this a little bit and that it is not too much
technical. We are at the half of the modal prerequisites for AUDA,
except that the real thing must still be done, but it is not modal
logic per se, it is the provability logic. I will not give all details
(as that would be very long and highly technical, but i will give
enough so that you get the idea). Then I will be able to explain
Solovay, and how to extract the 8 hypostases, including physics (and
why) from there. (That was the question!)
We might need to come back also on UDA step seven, if only to
understand in which sense AUDA "translated" it in arithmetic, and to
add some light on it, as it seems we have to do (given some recent
posts).
Bon courage!
Bruno
http://iridia.ulb.ac.be/~marchal/
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