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Topic: Background Theory
Replies: 4   Last Post: Dec 8, 2012 6:13 AM

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Posts: 1,968
Registered: 12/4/12
Re: Background Theory
Posted: Dec 7, 2012 11:21 AM
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On 12/7/2012 8:46 AM, Zuhair wrote:

> One might wonder if it is easier to see matters in the opposite way
> round, i.e. interpret the above theory in set theory? the answer is
> yes it can be done but it is not the easier direction, nor does it
> have the same natural flavor of the above,
> it is just a technical formal piece of work having no natural
> motivation. Thus I can say with confidence that the case is that Set
> Theory is conceptually reducible to Representation Mereology and not
> the converse!

I have no doubt that you are correct. In another post
in your thread I summarized the work of Lesniewski which
uses the part relation to characterize classes. His
method was specifically designed to circumvent the
grammatical form that leads to Russell's paradox.

Moreover, the historical record of the philosophy sides
with you. Aristotle, Leibniz, Kant, and Russell all
have had positions asserting how parts are prior to

The issue becomes, however, whether the theory informs
differently. I turned to the investigation of parts
without any knowledge of mereology because I believe
that the ontological interpretation of the theory of
identity is inappropriate for foundational purposes.
Find any distinction in the literature between the
application of logic to a linguistic analysis and
the linguistic synthesis of a theory.

That is why my proper part relation has a the
character of a self-defining predicate. There must
be a first asserted truth, and, it must be an
assertion that constructs the language.

The second sentence has parallel syntax to the
first. It took me a long time to understand
why I should accept that situation. In Liebniz'
logical papers, one finds the remark that individuals
should be identified with a mark. The membership
relation is precisely that relation which marks an
individuated context. It does so in relation to
a plural context.

The priority of the part relation as it applies
to set theory is that the fundamental relation
between objects of a domain should reflect the
object type of the domain. To the extent that
sets are

"collections taken as an object"

the fundamental relation should relate collections
to collections. To the extent that a set is

"determined by its elements"

requires that the individuated context be situated
relative to the defining syntax of the fundamental

One of the primary focuses of my investigation
was to understand the principle of identity of
indiscernibles. If you read Leibniz, you will
find that what is taught about the identity of
indiscernibles is not entirely representative
of what Leibniz said.

He attributes his motivation to a position held
by Thomas Aquinas and explains that the typical
formulation is a consequence of the phrase

"an individual is the lowest species"

This is best interpreted mathematically by
Cantor's nested closed set theorem with vanishing
set diameter.

The following assumption:

Assumption of Aquinian individuation:
AxAy((Az(ycz -> xez) /\ Ez(xez /\ -ycz)) ->
Az((xez /\ -ycz) -> (Ew(xew /\ wcy) \/ Aw(zcw -> ycw)))))

can be used to define membership

AxAy(xey <-> (Az(ycz -> xez) /\
Az((xez /\ -ycz) -> (Ew(xew /\ wcy) \/ Aw(zcw -> ycw))))))

relative to a prior proper part relation "c". In both
sentences, the complexity arises from capturing the sense
of a nested sequence of sets.

To be quite honest, here, I still need to consider
these expressions further. These universals are
difficult because both the assertion and its
negation must be considered.

So, let me suggest that while you continue
to refine your ideas, you also develop some of
the claims you have made concerning
the representations of arithmetic and such.

Do not take these remarks wrong. I see that you
are reading other authors and developing your
ideas. So, please continue. I would like to
see more.

Inform us.

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