Quoting "John F Sowa" <sowa@xxxxxxxxxxx>: (01)
> Dear Matthew, Avril, William, and Doug,
>
> MW
>> I do not see set theory and mereology as alternatives that you choose
>> one from for your ontology, rather I see them as being appropriate
>> in different circumstances. One of the tests I use to determine which
>> is appropriate is whether I am or could be interested in the weight
>> of the collection. Sets are abstract and so do not have a weight.
>> A mereological sum on the other hand does.
>
> That's a very good, one-paragraph summary of the difference. Formally,
> I would emphasize that set theory has two operators (subset and isIn),
> but mereology has only one operator (partOf).
>
> The partOf operator need not change the ontological category: a car
> and a mereological sum of cars are both physical. That's also the
> reason why mereology is better suited to plurals in NLs. A single
> person, say Bob, can be an animate agent of an action, and a plural
> such as Bob and Sue can be the agents of the same verb.
>
> But the isIn or elementOf operator always treats the second operand
> as abstract. Even if the first operand is a set, the second belongs
> to the category 'set of sets'. That formalism does not have a simple
> mapping to natural languages, in which plurals do not change category. (02)
Thanks for the summary for my part also. I have only one thing to add.
If we consider only sets with rank 1, that is, no inner sets, then
mereology and set theory become identical. Parts and subsets work
identically: you can talk about the set {a,b,c} as well as about the
aggregate abc. But, if only rank 1 sets are needed, then mereology is
simpler because there are no brackets. It is also indifferent if you
talk about a mereological sum or about a set theoretic sum. You can
add whatever interpretation, weight or anything else, with both notions. (03)
To be more accurate, mereology and rank 1 set theory become identical,
if the empty set is discluded from set theory. This would entail that
the intersection of {a} and {b} does not exist, whereas in
ZF(U)/KP(U)/NBG the intersection of {a} and {b} is {}. The disclusion
of the least element seems to be the greatest contribution that
mereology gives to science. After all, Boolean algebra without the
least element is isomorphic to mereology, and Boolean algebras were
invented before mereology. (04)
-Arvil (05)
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