Re: "separate entity" and individuation in mathematics

From: Jim Spriggs (jim.sprigs_at_ANTISPAMbtinternet.com.invalid)
Date: 03/17/05 

Date: Thu, 17 Mar 2005 19:35:35 +0000 (UTC)

george wrote:
>
> Jim Spriggs wrote:
> > > > Ooh, I have! D'you know it is the very same problem as
> > > >
> > > > How many distinct modalities are there in S4?
> > >
> > > This sounds interesting. What's a modality, may I ask?
> >
> > A string of "not"s, "possibly"s and "necessarily"s, including the
> empty
> > string. 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. 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!

> >
> > 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.

>
> > In topology take [] to correspond to interior, <> to closure, and -
> to
> > complement to get the 14 sets.
>
> This means that mitch's version could eliminate interior
> by describing it as the complement of the closure of the complement
> (rewrite [] as -<>-). Mitch is looking for structure in this
> so it maybe it would help to try to do these paired.
> In any case, the reason why there are so few of these is that
> topologically,
> closure and interior are both idempotent ( <> == <><> and [][] == [] ),
> and complement is an involution ( - - == empty/identity).
>
> empty, - ,
> [] , -[],
> <> , -<>,
> []<>, -[]<>,
> <>[], -<>[],
> []<>[], -[]<>[],
> <>[]<>, -<>[]<> .
>
> are rewritable as
> empty , - ,
> -<>- , <>- ,
> <> , -<> ,
> -<>-<> , <>-<>,
> <>-<>-, -<>-<>-,
> -<>-<>-<>-, <>-<>-<>-,
> <>-<>-<>, -<>-<>-<> .
>
> > The "14 sets" problem was solved by Kuratowski, Fundamenta
> Mathematica,
> > vol 3 (I think).
>
> Only one of these 14 cases is hard to prove; I mean, you trivially
> prove
> at the beginning that -- == empty and <><> == <>.
> That means all the "shortest" representatives of the equivalence
> classes have to alternate. So 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)

if []<> then []<>[]<> (An S4 theorem)

[]<> iff -<>-<>

so -<>-<>-<>-<> iff -<>-<>.

Putting another <> on the left hand end of -<>-<>-<>- gives

<>-<>-<>-<>- which is <>[]<>[]

if <>[]<>[] then <>[] (This is a so-called "reduction thesis" in S4)

if <>[] then <>[]<>[] (An S4 theorem)

<>[] iff <>-<>-

so <>-<>-<>-<>- iff <>-<>-.

[Unless someone tells you otherwise:-]

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