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Re: [ontolog-forum] Universal Basic Semantic Structures

To: ontolog-forum@xxxxxxxxxxxxxxxx
From: John F Sowa <sowa@xxxxxxxxxxx>
Date: Sun, 30 Sep 2012 08:43:08 -0400
Message-id: <50683E5C.3060808@xxxxxxxxxxx>
Chris and Matthew,    (01)

> both set theory and many versions of mereology can be considered
> specializations of the theory of Boolean collections.    (02)

> I'm not sure how this addresses the claim you made to which I objected
>  that set theory has two operations (better, relations) and mereology one,
> which I took to imply that set theory has two primitive relations. That's
> just not faithful to the way set theories are almost always developed.    (03)

I agree that textbooks usually take isIn as primitive in developing
set theory.  But the notion of primitive does not generalize well
when you are looking at the way various theories are related.    (04)

Before Cantor, there was a great deal of vacillation about which
operators to use for talking about sets or collections (or Menge).
Cantor wasn't the first to suggest an isIn or elementOf operator,
but he used it -- as you observed -- as the primitive.  Frege
denounced Boole for not recognizing the need for that operator.    (05)

But pedagogically, it helps to start with Boolean algebra and show
how many other theories can be developed as specializations of it.    (06)

George Boole used the same operators (+, x, and -) for propositions,
monadic predicates, and sets.  For any monadic predicate, there is
a set or collection of everything for which the predicate is true.
But the same symbols with the same axioms can be used to relate them.    (07)

(A x B) is 'and' for predicates of any arity (0-arity predicates are
propositions).  For collections, (A x B) is intersection and (A + B) is
union (or mereological sum).  Peirce introduced a less-than-or-equal
symbol for implication of propositions and predicates and for the
subset operator of collections.    (08)

> One of the tests I use to determine [whether set theory or mereology]
> is appropriate is whether I am or could be interested in the weight
> of the collection.    (09)

If you use Boolean algebra, there is a simple solution:    (010)

  1. Note that for any Boolean lattice, you get a sublattice
     of all the nodes that fall under any node P.    (011)

  2. If that node P represents the characteristic predicate for
     your category (say HasWeight), then the Boolean operators on
     that sublattice are guaranteed to preserve the constraints
     specified by the predicate P.    (012)

  3. Therefore, the Boolean operators, when restricted to the HasWeight
     sublattice will guarantee that the results will be a collection
     for which HasWeight is true.    (013)

  4. This solution can be applied to linguistics.  For any verb (or
     preposition or other part of speech) that imposes a selectional
     constraint on its arguments, you can use a monadic predicate for
     that constraint to specify the sublattice for acceptable plurals.    (014)

> ... if you want to model, say, my arm as a set then you can use the
> subset relation to show that the set my arm is "part of" the set my body.
> However, I think not many people would be convinced that my arm is a set.    (015)

With Boolean algebra, you can call the operators any names you like.
Boole started with the symbols +, x, and -.  Then Peirce added the
less-than-or-equal symbol, which you can call implication for
propositions and predicates or subset for collections.  But if you
prefer, you can call it partOf.    (016)

It's the same theory.  The English words you associate with the symbols
can be adapted to the domain of application.  For linguistics, you can
call Boolean algebra a theory of plurals.    (017)

> The bottom line here is that set theory and mereology are not
> interchangeable at any level.    (018)

But they are both specializations of Boolean algebra.    (019)

John    (020)

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