On Monday, August 12, 2002, at 12:07 PM, Wei Dai wrote:
> According to possible world semantics, "it's necessary that P" means
> that
> P is true in all worlds accessible from this one. Different modal logics
> correspond to different restrictions on the accessibility relation.
> Before
> the invention of possible world semantics, people argued about which
> modal
> logic is the correct one, but now philosophers realize that different
> notions of accessibility (and the corresponding notions of modality) are
> useful at different times, so there is no single correct modal logic.
>
> That's my one paragraph summary of possible world semantics. Please
> correct me if I'm wrong, or read these articles if you're not familiar
> with this topic:
>
> http://www.xrefer.com/entry.jsp?xrefid=552831
> http://www.xrefer.com/entry.jsp?xrefid=553229
>
> My questions is, why not just quantify over the possible worlds and
> refer
> to the accessibility relation directly? This way you can talk about
> multiple accessibility relations simultaneously, and you don't have to
> introduce new logical symbols (i.e. the box and the diamond). Is
> modality just a syntactic shorthand now?
Modal logic is a lot more than syntactic shorthand.
Consider this example, phrased in MWI terms.
It is possible that WWIII will happen before the end of this year. In
one possible world, A, many things are one way...burned, melted,
destroyed, etc. In another possible world, B, things are dramatically
different.
There can be no implication from one world to the other. That is, we
can't say "A implies B" or "B implies A."
This branching future is exactly what I was talking about a week or so
ago in terms of "partially ordered sets." If the order relationship is
"occurs before or at the same time as," which is equivalent to "less
than or equal to," A and B cannot be linearly ordered. In fact, since
both A and B are completely different states, neither can be said to be
a predecessor or parent of the other. In fact, A and B are not
comparable. We cannot say "A or not-A."
We have thus left the world of classical logic and are in the world of
non-classical, or intuitionistic, or Heyting logic.
Posets are not just a different syntactic shorthand from
linearly-ordered sets.
Branching worlds, aka possible worlds, aka MWI (when QM is involved) is
a more accurate way of talking about time and successions of events than
is attempting to force time into a strait-jacket of linearly-ordered
sets (chains).
Besides the topos work of Saul Kripke, Vaughan Pratt at Stanford has
written a lot on concurrency, lattices, and posets.
Lee Smolin's book "Three Roads to Quantum Gravity" is very good at
explaining how this relates to cosmology.
--Tim May
(.sig for Everything list background)
Corralitos, CA. Born in 1951. Retired from Intel in 1986.
Current main interest: category and topos theory, math, quantum reality,
cosmology.
Background: physics, Intel, crypto, Cypherpunks
Received on Mon Aug 12 2002 - 18:06:32 PDT