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



fom
Posts:
1,968
Registered:
12/4/12


Re: Background Theory
Posted:
Dec 7, 2012 11:21 AM


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 individuals.
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 selfdefining 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 relation.
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.



