Re: "separate entity" and individuation in mathematics
From: george (greeneg_at_cs.unc.edu)
Date: 03/22/05
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Date: 22 Mar 2005 10:27:16 -0800
Jim Spriggs wrote:
> george wrote:
> >
> > Jim Spriggs wrote:
> > > Let M_1 and M_2 be two such strings and let phi be an S4
> > > formula, if
> > >
> > > S4 |- M_1 phi <--> M_2 phi
> >
> > Is there supposed to be some connective, like maybe |=,
> > between the M_'s and the phi's in the above?
>
> No.
>
> > Just plain
> > old separation-by-a-space doesn't seem to be conveying much.
>
> M_1 is something like []-<> so M_1 phi in this case is []-<>phi.
I'm sorry; I was reading too fast.
It would've been clear the first time if I'd been
paying attention. When I see an "M" before a "phi",
I think a "model", not a "modal". It was just an unfortunate
coincidence of initials.
> It's
> just a formula of modal logic (S4 specifically) which begins with a
> string of "not"s, "possibly"s and "necessarily"s. Sorry if my
notation
> was confusing but
>
> S4 |- M_1phi <--> M_2phi
>
> looks worse to me!
Right.
>
> > >
> > > for all phi, then M1 and M_2 are equivalent, but if S4 doesn't
prove
> > >
> > > M_1 phi <--> M_2 phi
> > >
> > > for some phi, they are non-equivalent or distinct. Writing - for
> > not,
> > > <> for possibly, and [] for necessarily, the distinct modalities
of
> > S4
> > > are:
> > >
> > > empty, [], []<>[], []<>, <>[], <>[]<>, <>,
> > > -, -[], -[]<>[], -[]<>, -<>[], -<>[]<>, -<>.
> >
> > Mitch's original treatment did this with only 2 operators,
> > complement and closure; presumably this is because [] rewrites
> > to -<>-, and <> is rewritable as -[]-, right?
>
> Right.
> > empty, - ,
> > [] , -[],
> > <> , -<>,
> > []<>, -[]<>,
> > <>[], -<>[],
> > []<>[], -[]<>[],
> > <>[]<>, -<>[]<> .
> >
> > are rewritable as
> > empty , - ,
> > -<>- , <>- ,
> > <> , -<> ,
> > -<>-<> , <>-<>,
> > <>-<>-, -<>-<>-,
> > -<>-<>-<>-, <>-<>-<>-,
> > <>-<>-<>, -<>-<>-<> .
> >
> > > The "14 sets" problem was solved by Kuratowski, Fundamenta
> > Mathematica,
> > > vol 3 (I think).
...
> > the hard part is just to prove that
> > -<>-<>-<>- can't be extended with another <> on either end (i.e.,
that
> > both of these extensions rewrite to two of the above, but I
personally
> > have no idea which two).
>
> Putting another <> on the right hand end of -<>-<>-<>- gives
>
> -<>-<>-<>-<> which is []<>[]<>
>
> if []<>[]<> then []<> (This is a so-called "reduction thesis" in
S4)
Is this thesis one of the axioms of S4?
If so, since Mitch was doing this from topology,
the question arises, what does the analogous/topological proof
of this rewrite look like? Which topology-axioms does it invoke?
> if []<> then []<>[]<> (An S4 theorem)
Topologically, that would be something like
"the interior of the closure is a subset of
the interior of the closure of the interior of the closure".
And we already proved the converse, so
>
> []<> iff -<>-<>
>
> so -<>-<>-<>-<> iff -<>-<>.
In other words, "the interior of the closure", viewed
as ONE operator, is idempotent. You have given the S4
proof, but my question is, does the topology proof look
"exactly" analogous? Or does it look different, but just
serendipitously give the analogous result?
>
> Putting another <> on the left hand end of -<>-<>-<>- gives
>
> <>-<>-<>-<>- which is <>[]<>[]
>
> if <>[]<>[] then <>[] (This is a so-called "reduction thesis" in
S4)
OK, it's a thesis, but is it also an axiom?
> if <>[] then <>[]<>[] (An S4 theorem)
Again, the theorem provides only 1 direction of
the equivalence. There is a corresponding topological theorem,
though, right?
> <>[] iff <>-<>-
>
> so <>-<>-<>-<>- iff <>-<>-.
The next question is, could somebody map these to
the propositional connectives? I wouldn't say that
any mapping is obvious since we are talking about complementing
and closing ONE set, while the binary boolean functions take
TWO operators, but it's worth a shot. By way of filling in
the gap between 14 and 16, I am willing to suggest the following:
on the topology side, there must be both an empty set and a universal
set in the domain. Those could correspond to all-false and all-true.
On the topology side, each operator takes 1 set as an argument and
yields 1 set as a value, so you would have to define a new/3rd operator
that always mapped its argument to the universal set (if we're using
closure as the 2nd operator) or the empty set (if we're using interior
as the 2nd operator). That would get you to 16 classes.
On the S4 side, this 3rd operator would be a mode that always
took every statement to true, or to false. If we call this new
mode F (for always False, this is intended to go with the
closure/possiblity dialect), then it would be governed by
an axiom like -F phi (for all phi). I would hope that S4 would then
derive -<>F phi (for all phi) as a theorem.
This doesn't have any value as an actual mode (any more than you would
want to think of the constant true or the constant false as a function
of 2 variables, as opposed to as a constant); the goal is to just
include
it as a theoretical possibility in order to make mitch's structural
similarity actually look similar.
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