# [FOM] New Umbrella?/big picture

W.Taylor at math.canterbury.ac.nz W.Taylor at math.canterbury.ac.nz
Tue Nov 11 00:39:33 EST 2014

```Sorry for the mistakes!!  - b

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Quoting Mitchell Spector <spector at alum.mit.edu>:

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Let's continue in this vein, carrying the line of reasoning to its
natural conclusion. If understanding the type-0 objects requires type-1
relations as a separate sort, it would seem that understanding the
type-1 relations would require a third sort, that of type-2 relations
(predicates on the collection of type-1 relations, rather than
predicates on objects).  Leibniz' principle would then suggest that
there are so many type-2 relations that, given any two distinct type-1
relations, there's a type-2 relation that distinguishes between those
two type-1 relations.
Just a simple-minded note here, to bring to attention a sig-line
I used to see on the newsgroups, that seems slightly relevant.

If one doesn't wish to go to a hierarchy of predicate variables,
and is prepared (largely ignoring 1st/2nd-order distinctions)
to treat objects and properties on the same footing, then
Leibnitz' law has an interesting "dual" form, which might be useful.

IDENTITY OF INDISCERNABLES:    (Leibnitz)

[all x,y]  x = y  <=>  [all P] Px <=> Py

IDENTITY OF INDISCRIMINABLES:      (dual)

[all P,Q]  P = Q  <=>  [all x] Px <=> Qx

i.e. if no property distinguishes two objects, they are the same;
and  if no object discriminates two properties, they are the same.

-- Bill Taylor

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