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Re: [ontolog-forum] Lattices of Lattices (was Apologies and Recommendati

To: ontolog-forum@xxxxxxxxxxxxxxxx
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
Date: Fri, 23 Dec 2011 10:20:28 -0500
Message-id: <4EF49C3C.2010003@xxxxxxxxxxx>
Michael,    (01)

I'm getting ready to leave now, but I just wanted to mention a few
points about the advantage of using lattices instead of hierarchies.    (02)

I recognize the need for talking about signatures (or vocabularies),
but instead of talking about hierarchies of theories with a given
signature, I recommend talking about lattices for a given signature.    (03)

First, let V be the total vocabulary of all the ontologies that you'll
ever want to consider.  Then there is a Lindenbaum lattice of all the
theories that use V.  Very few theories would use the full vocabulary,
and the overwhelming majority would use some subset v of V.    (04)

But the set of all subsets of V happens to form a lattice of sets.
For each v in that lattice, there is also a Lindenbaum lattice (LL)
of all theories that use v as their vocabulary.    (05)

For each v, the LL of theories for v can be embedded in the big LL
for the full vocabulary V.    (06)

  1. If v is the empty set, its LL has only two nodes:  the top, which
     contains all tautologies that can be formed with only the logical
     vocabulary, and the bottom with all contradictions that can be
     formed with only the logical vocabulary.  For Common Logic, top
     contains sentences such as (and), (not (or)), (if (and) (and))...
     and bottom contains all those sentences and their negations.    (07)

  2. The LL for the big vocabulary V has infinitely many more
     tautologies at the top and contradictory sentences at the bottom.
     Every theory in between also inherits all those tautologies that
     can be formed with the big vocabulary.    (08)

  3. Therefore the LL for a proper subset v of V is not identical
     to a sublattice of the big LL, but it is isomorphic to a unique
     sublattice in the big LL.  The only difference is that each
     theory in the sublattice of the big LL acquires a lot of
     irrelevant tautologies that can be safely ignored.    (09)

This observation explains why I objected to calling the big LL
"coarse".  It includes every little LL that could ever be formed
from any subset of V.    (010)

Furthermore, the lattice of subsets of V shows how all the little
LLs are related, which ones can be embedded in others, which ones
are disjoint (except for top and bottom), overlapping, etc.    (011)

It also shows why you need to consider the big LL:  When you want to
relate theories in different hierarchies, you need to go to a larger
LL -- at least to the LL for a vocabulary that is a superset of the
vocabulary of both.  But when you are talking about many modules
in different sublattices, it's convenient to consider the big LL.    (012)

Note that in this discussion, I did not use any hairy formalism
with Greek letters, etc.  But I didn't need to because all the
work has been done for me.  I am just talking about lattice
operations for which all the definitions and theorems were
stated and proved long, long ago.    (013)

As I said, the best mathematicians are lazy.  I suggest that
you and your colleagues stop being so industrious.    (014)

John    (015)

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