Re: a possible paradox

From: Bruno Marchal <marchal.domain.name.hidden>
Date: Fri, 31 Oct 2003 14:54:22 +0100

At 17:03 30/10/03 +0100, "scerir" <scerir.domain.name.hidden> wrote:
>Principles of World Theory say, more or less, that:
>- a proposition (whatever) is *necessary* iff it is true in all worlds;
>- a proposition (whatever) is *possible* iff there is some world in which
> it is true;
>- there is only one *actual* world;
>- there are propositions which are true at the *actual* world;
>- there are propositions which are not true at the *actual* world, but
> they are true at some *non-actual* *possible* world.
>
>It is not much. But, in any case, we must start from these points :-)

Very nice. Except perhaps that it is the principle of the
Old World Theory, implicit in Aristotle
and Leibniz, where all the worlds are accessible from each other.
It is formalised by the modal logic S5.
Abbreviating "necessary p" by box p, or []p, and "possible p" by <>p
(which itself can be seen as an abbreviation of "not box not p", i.e. -[] - p),
its main axiom are []p -> p, []p->[][]p, and <>p -> []<>p.
Search S5, in the archive, I have given the precise axioms and
inference rules, at some time.
Kripke has relativized such sort of modal logic by saying that []p is true
in a world A if p is true in all world accessible from A. Then for each
choice of an accessibility relation, you get a different modal logic.
And modal logic is the best tool for being precise on the invariant
truth (like laws) in a multiverse, or in contextual frame, etc. Also,
modal logics can be used to simulate in the classical settings,
non classical logics, like intutionistic logic, quantum logic.
I agree with we should, well perhaps not start from that, but
invoking them when we disagree about the validity of an argument
in *apparently* fuzzy context.
I work mainly with the modal logic G and G* (see the archive).
This can help for giving axiomatic notion of self-identities.

Bruno
Received on Fri Oct 31 2003 - 08:50:08 PST