# Re: modal logic and possible worlds

From: Tim May <tcmay.domain.name.hidden>
Date: Tue, 13 Aug 2002 11:15:28 -0700

On Monday, August 12, 2002, at 11:18 PM, Wei Dai wrote:

> Tim, I'm afraid I still don't understand you.
>
> On Mon, Aug 12, 2002 at 06:00:26PM -0700, Tim May wrote:
>> 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.
>
> Ok, but what about my point that you can state this by explicit
> quantification over possible worlds rather than using modal operators?
> I.e., "There exist a world accessible from this one where WWIII happens
> before the end of this year." instead of "It is possible that WWIII will
> happen before the end of this year."?

That is indeed saying just the same thing (though the language is
slightly different).

The important part of modal logic is not in the "accessible from this
one" or "it is possible" language.

Rather, the "forking paths" (a la Borges) picture that is described by
posets and lattices.

>
>> There can be no implication from one world to the other. That is, we
>> can't say "A implies B" or "B implies A."
>
> What does that have to do with my question? Anyway A and B are supposed
> to
> be worlds here, not propositions, so of course you can't say "A implies
> B". I don't know what point you're trying to make here.

Worlds _are_ propositions. And the "causal operator" (time) is the same
as implication.

With some important caveats that I can't easily explain without drawing
a picture. In conventional logic, implication is fully-contained or
defined from some event A (or perhaps some combination of events A, B,
C, etc., all causally contributing to a later event).

There are two interesting cases to consider where implication does not

1. Possible worlds. The event A forks down two (or more) possible paths.
A future where war occurs, a future where war does not. A future where
Fermat's Last Theorem is proved to be true. A future where it is not. A
future of heads, a future of tails.

2. Quantum mechanics. Schrodinger's cat.

(It was Einstein and Podolsky's belief that classical logic must apply
that led to their belief that there _must_ be some other cause, some
hidden variable, that makes the outcome follow classical logic. Bohm,
too. But we know from Bell's Theorem and the Kochen-Specker "no-go"
theorems that, basically, these hidden variables are not extant.)

(By the way, the book "Interpreting the Quantum World," by Jeffrey Bub,
has an interesting section on how modal logic applies to QM.)

Bruno is much more of a logician than I am, but the various terms of
logic, lattices, and set theory are analogous (probably a very efficient
category theory metaview, but I don't yet know it).

1 is True
0 is False
lattice infimum or Boolean meet, ^ , is conjunction (AND)
lattice supremum or Boolean join, v , is disjunction (OR)
lattice or Boolean orthocomplement is negation (NOT)

(Understanding this is not essential to my arguments here...I just
wanted to make the point that there are mappings between the languages
of logic, set theory, and lattices. In a deep sense, they are all the
same thing. Definitions do matter, of course, but e-mail is not a great
place to lay out long lists of definitions!)

>
>> 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.
>
> I'm with you so far in this paragraph.
>
>> We cannot say "A or not-A."
>
> Now I'm lost again. Again A is a world not a proposition so what
> would "A
> or not-A" mean even if A and B are comparable?

The two forks in the road are given the same truth value weighting in
this "possible worlds" approach.

We have _assumed_ A in this fork I described, so "not-A" is certainly
not necessarily the other path. In fact, the meaningful interpretation
of "not-A" in the complement sense is "that which precedes A," that is,
the events leading up to A in this world.

I realize this sounds confusing. Draw a picture. Just have three points
in it, arranged in a triangle:

A B

\ /

X

Time is in the upward direction. The points/events/states X, A, B form a
poset. One arrow between any two points. Pre-ordering (reflexive,
transitive) and partial-ordering (reflexive, transitive, antisymmetric).

We cannot, however, say "X implies A" because X has given rise to _both_
A and B.

Besides the possible worlds situation, where we "assume" X could give
rise to either of these events, there is also the distinct possibility
that this will be the only logic we ever know for quantum mechanics. The
situation X gives rise to either the cat being dead or alive at the time
we make the measurement, and it makes no logical sense to sense that
"but there must be some hidden cause because nature works by Boolean or
Aristotelian logic."

Further, this situation describes many cases of incomplete knowledge. As
Lee Smolin and Fotini Markopoulou have pointed out so convincingly,
there are events outside our light cone which will combine in various
ways with other events to produce futures which look exactly like these
posets and lattices. (I described this in some detail in some earlier
posts.)

>