Hi Brent,
This is perhaps a slightly more advanced answer relatively to the
current thread, so don't be astonished if you don't get the end, I
should recall the notion of "theory" before. My current conversation
with Stathis is based directly on the "multiverse (Kripke) semantics",
but I still should explain the connection with theories, that is axioms
and deduction rules and later he soundness and completeness notions.
But then perhaps you have read Chellas, or Smullyan's FU (Forvere
Undecided), or Hugues and Creswell ?
Le 08-déc.-05, à 02:31, Brent Meeker a écrit :
Bruno Marchal wrote:
...
What could this mean in a real world example?
Take W as the set of places in Brussels. Take R to be "accessible by
walking in a finite number of foot steps". Then each places at
Brussels is accessible from itself, giving that you can access it
with zero steps, or two steps (forward, backward, ...).
Take W as the set of humans, say that aRb if a can see directly,
without mirror, the back of b. Then a can access all humans except
themselves. R is said to be irreflexive.
Another important "concrete" example, which will help us latter to
study the modal logic of quantum logic. Take the worlds to be the
vector of an Hilbert Space (or of the simpler 3-dimensional euclidian
space). Say that a is accessible to b, i.e. aRb, if the scalar
product of a and b is non null (i.e. a and b are not orthogonal).
These are good illustrative examples, but how do they apply to worlds
that just consist of propositions?
To be sure, worlds consisting in sets of propositions are just tools
for building mathematical multiverses (Kripke frame) useful for proving
"completeness theorem" in (modal) logic.
It happens that with comp (or even much weaker assumptions)
self-referential machines (entities) cannot distinguish "consistent
linguistic worlds" (or their arithmetical counterparts), with
"primitive (substantial?) independent ontologies, and so they are
playing some role in the derivation of the comp "propositional
physics".
Now, to answer your question, the relations of accessibility in those
linguistic worlds must be defined just in a way to make it following
the Kripke semantics. For example, a linguistic world in which "Bp"
belongs should reach only worlds in which "p" belongs, p being some
linguistic proposition.
To be more concrete if p is the proposition "17 is a prime number and
Buffy is a vampire slayer", then a linguistic world in which "B(17 is a
prime number and Buffy is a vampire slayer)" belongs, will reach only
linguistic worlds in which "17 is a prime number and Buffy is a
vampire slayer" belongs.
What is the relation of accessibility in the p,q,r world(s)? Is it
negation?
Err... I guess you are talking about the reflexive and symmetric
multiverse (the proximity spaces) and their antimultiverse which are
the antireflexive but also symmetric (see why?) multiverse (the
orthogonality spaces).
I didn't thought to those as linguistic multiverse. (Where worlds are
literally set of propositions, actually even just set of formulas: they
are just made true at a world by just belonging to that world). With
the proximity/orthogonality (anti)multiverses, the worlds were seen as
vector in euclidian vector spaces, and the relation of accessibility
was that the scalar product is not null (proximity) (or null
(orthogonality)).
The idea of proximity here is that if two OMs are represented by the
vectors alpha and beta respectively, then the scalar product is the
probability amplitude for going from alpha to beta. It is null if alpha
and beta are orthogonal.
Now, with the modal theory of what is sound is the reflexive
multiverse, [which is actually
B(p -> q) -> (Bp -> Bq) (named K)
Bp -> p (named T)
p -> BDp (Named B in the literature but LASE
in this list).
+ the derivation rule of modus ponens and necessitation] you will be
able to construct linguistic multiverse behaving like in the euclidian
vector space or the "hilbert space" standard interpretation.
A lot of modal logics can get "linguistic" interpretations, but not
all, unless you decide to use infinitary logics, I guess ...
Bruno
http://iridia.ulb.ac.be/~marchal/
