On Dec 7, 7:21 pm, fom <fomJ...@nyms.net> wrote: > 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 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 > 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.
I just wanted to note that with this approach a set is a singular entity that represent a plurality of singular entities, while with Lewis's approach a set is a plurality of singular entities that is represented by a singular entity. I see this approach reductive while Lewis's diffusive. However formally speaking they almost mirror each other, but conceptual wise I think the approach given here is more faithful to the general context in which sets are mentioned.