Liz, Brent, others,
Just a revision, before you forget the definitions :)
A multiverse (W, R), or frame, is a set W, with a binary relation R.
The elements of the set are called "world" , and denoted often by
greek letter (alpha, beta, gamma, ...). The binary relation is called
accessibility relation, and constitutes the main ingredient in Kripke
semantic.
Now we are doing logic. So we suppose the usual classical
propositional language, with atomic sentence letters p, q, r, ..., and
the symbol "->" and "f".
(The other logical symbols can be defined from "->" and "f", for
example ~p can be defined as an abbreviation of (p -> f), as it clear
when we give the usual classical semantic).
From them, we can build the usual formula like p, q, (p -> q), ((p ->
f) -> (q -> r)), etc.
"A" will denote such arbitrary formula. (It is a metavariable, and it
does not belong to the formal symbols).
A valuation or illumination is a function from {p, q, r, ...} to {1, 0}.
An multiverse becomes illuminated when each world get a valuation. So
in each world, the sentence letters can be said true, or false,
according to the valuation. Let us call V that valuation which
attributes a 1 or a 0 to each propositional letter, in each world of
the multiverse.
So we can denote an illuminated multiverse by (W, R, V).
We suppose that each world obeys classical logic. basically this
means, that:
- The propositional letter are true or false, according to the
valuation in the illuminated multiverse.
- f is false in alpha (any alpha)
- A -> B is true in the world alpha iff A is false in alpha or B is
true in alpha.
Ah, but we do modal logic. So we have one unary connector symbol more:
[]p, and Kripke semantics for modal logic is that:
[]A is true in alpha iff A is true in all beta such that alpha
R beta. That is, such that beta is accessible from alpha.
<>p is defined as an abbreviation of ~[] ~, and you might enjoy
verifying that
<>A is true in alpha iff there exist a world beta, with A true
in beta, and beta accessible from
Last but not least definitions. Now that we know what it means for a
formula to be true in a world, we say that :
An illuminated multiverse (W, R, V) satisfies a formula if that
formula is true in all worlds of the multiverse.
A multiverse (W, R) respect a formula if that formula is true in all
worlds for any of its illumination (W, R, V).
That's all you have to know. Print this or recopy this by hand in the
diary, as this will remains with us, for sometime.
****
Then you have shown (Brent and Liz, at least):
(W, R) respects []A -> A
iff
R is reflexive (that is for all alpha in W, alpha R alpha)
and
(W, R) respects []A -> <>A
iff
R is ideal on W, that means that from any world you can access to some
world (another one or itself). It means that there is no cul-de-sac
worlds.
OK?
I will send one post with all the proofs.
Brent, in some post you tell me you were working on the proof of
(W, R) respects []A -> [][]A
iff
R is transtive (aRb and bRc implies aRc) (writing quickly a b c for
alpha beta gamma, it is also clearer:
R is reflexive iff for all a aRa,
R is symmetrical iff for all a and B, aRb implies bRa.
I will prove the transtitive case soon, unless you ask some delay.
New exercise:
show
(W,R) respects A -> []<>A
iff
R is symmetrical.
This one plays some role in the specific derivation of physics from
arithmetic. This is due to the fact that the logic B, with axioms:
[](A -> B) ->( []A -> []B)
[]A -> A
A -> []<>A,
translates a minimal Quantum logic in modal terms. "[]<>A" quantized
the "truth" of A in some sense. Quantum logic are usually handled by
the algebraical structure of the observable, and quantum proposition
are structured in terms of orthospace, that is a space with a
orthogonality notion (a scalar product). The complementary relation
("not-orthogonal") defines a proximity relation.
In arithmetic we will get a weakening/strengthening, of this as we
will get A->[]<>A, only for the atomic sentences (the arithmetical
interpretation of the letters p in the modal logic), and we will loss
the necessitation rules, losing some quantum tautologies, perhaps, but
not necessarily. It is a strengthening by the axioms corresponding to
the Löb formula, and the arithmetical reality (intensional and
extensional).
Bruno
http://iridia.ulb.ac.be/~marchal/
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