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Re: What are sets? again
Posted:
Dec 4, 2012 5:50 AM


On 12/2/2012 9:38 PM, William Elliot wrote: > On Sun, 2 Dec 2012, Zuhair wrote: >> On Dec 2, 10:44?am, William Elliot <ma...@panix.com> wrote: >>> On Sat, 1 Dec 2012, Zuhair wrote: > >>>>>> The following is an account about what sets are, >>> >>>>>> Language: FOL + P, Rp >>>>>> P stands for "is part of" >>> >>>>> Does P represent "subset of" or "member of"? >>>> Neither. >>> >>>> P represents "is part of" >>>> review mereology to understand that relation informally. >>> What is simple jargon, a brief intuitive description of "is a part of". > >> Just read Varzi's article on Mereology: >> http://plato.stanford.edu/entries/mereology/ > > It's long winded as philosophy usually is. > Basically, "is a part of" is a (partial) order. > "Subset" is the better interpretation that "is member of". > > So I'll take it as "subset" unless you give a useful > interpretation within 300 words or less. > >> The relation "is part of" is well understood philosophically speaking, >> it has natural examples. > > For example? > >> I think Varzi's account on it is nice and interesting really. You can >> also read David Lewis account on it. The discipline of Mereology is >> well established. > > What's the point of mereology? >
Arguably, mereology as an investigation into foundational mathematics is from Lesniewski
Lesniewski wrote several papers criticizing Russell's Principia as basically being incoherent
He did his own investigations along the lines of logical structure of sentences having existential import that are different from both Frege and Russell
Subsequently, he characterized a notion of class that was different from Russell's
At first, he tried to characterize his ideas in traditional logical formats but ultimately began to pursue it using formal syntax
This actually gets dense and I have not really examined it. But, for example he begins a system he calls protothetic with
A1. ((p <> r) <> (q <> p)) <> (r <> q)
A2. (p <> (q <> r)) <> ((p <> q) <> r)
and then lists 79 theorems about logical equivalence.
He then switches notation to quantify over propositional variables
A1. ApAqAr(((p <> r) <> (q <> p)) <> (r <> q))
A2. ApAqAr((p <> (q <> r)) <> ((p <> q) <> r))
So that he can extend the system using variables ranging over truthfunctions
A3. AGAp(AF(G(p,p) <> ((Ar(F(r,r) <> G(p,p)) <> (Ar(F(r,r) <> G(p <> Aq(q),p)))) <> Aq(G(q,p)))
He then lists 422 theorems in order to obtain the three logical axioms of Lukasiewicz grounding the usual theory of deduction based on implication and negation. His primary stated goal at the outset for extending the original system was to obtain
ApAqAF((p <> q) <> (F(p) <> F(q)))
and the equivalent was obtained at step 381
Next, he turns to his system of ontology grounded upon the logical foundation of his protothetic. His only axiom is
A0. AZAz((Z class of z) <> ( ((AY((Y class of Z))) /\ AYAX(((Y class of Z) /\ (X class of Z)) > (Y class of X))) /\ AY((Y class of Z) > (Y class of z)) ))
In his exposition he comments on Russell's paradox:
"... can be strengthened in ontology by means of the easily proved sentence which says that:
AZAz((Z class of z) > (Z class of Z))
(I call this the 'ontological identity sentence'; it should be noticed that the yet stronger thesis
AZ(Z class of Z)
is not provable in ontology  indeed, its negation is provable.) In connection with this sentence, I want to emphasize expressly that in ontology there is always a very good possibility of proving theses having a single component of the type (Z class of Z) or (what is indifferently the same in ontology) (z class of z). This does not, however, lead to a contradiction via the wellknown schema of Principia Mathematica because the definition directives of ontology have been appropriately formulated so that no thesis of the type
AZ((Z class of x) <> (Z class of Z))
can be obtained."
This assertion might best be viewed much like the situation with general relativity. Philosophers who have been looking at Lesniewski's systems have not run into any contradictions such as Russell's (at least, in so far as Peter Simons has reported accurately)
The character of his predicate in ontology allows him to formulate terminological explanations about ontology within the language of ontology.
The axiom of ontology, and the equivalent axioms he discusses in his exposition derive from his analysis of Russell's paradox.
The Lesniewskian notion of class is based upon a part relation and this is the formal mereology associated with his investigations:
A1: If P is a part of object Q, then Q is not a part of object P
A2: If P is a part of object Q, and Q is a part of object R, then P is a part of object R
D1: P is an ingredient of an object Q when and only when, P is the same object as Q or is a part of object Q
D2: P is the class of objects p, when and only when the following conditions are fulfilled:
a) P is an object
b) every p is an ingredient of object P
c) for any Q, if Q is an ingredient of object P, then some ingredient of object Q is an ingredient of some p
These conditions formalize a statement Lesniewski made in his analysis of Russell's paradox:
"I use the expressions "the set of all objects m" and "the class of all objects m" to denote every object P which fulfills the two following conditions:
1) every m is an ingredient of the object P
2) if I is an ingredient of object P, then some ingredient of object I is an ingredient of some m"
Since ingredient is effectively the reflexive subset relation in set theory by Zermelo's 1908 language, you can see why Zuhair chose the language he did to describe atoms. In an atomistic theory, I and m must at least share some atom of P
I expressed this idea in a formal sense long ago only to be flamed (singed by you and firebombed by someone else)
What actually makes mereology work is something that is associated with the constructible universe. It is called almost universality. Of course, that is not how Lesniewski referred to it:
"Lukasiewicz writes in his book as follows: 'we say of objects belonging to a particular class, that they are subordinated to that class'
"It most often happens that a class is not subordinated to itself, as being a collection of elements, it generally possesses different features from each of its elements separately. A collection of men is not a man, a collection of triangles is not a triangle, etc. In some cases, it happens in fact to be otherwise. Let us consider e.g., the conception of a 'full class', i.e., a class to which belong, in general, some individuals. For not all classes are full, some being empty; e.g., the classes: "mountain of pure gold', 'perpetual motion machine', 'square circle', are empty, because there are no individuals which belong to those classes. One can then distinguish among them those classes to which belong some individuals, and form the conception of a 'full class'. Under this conception fall, as individuals, whose classes, e.g., the class of men, the class of triangles, the class of first even number (which contains only one element, the number 2), etc. A collection of all those classes constitutes a new class, namely 'the class of full classes'. So that the class of full classes is also a full class and therefore is subordinated to itself."
In almost universal models of set theory, every subclass of the universe is an element of the universe. Thus, "is an element or is equal to" is the same as "is a proper part or is equal to". So, the satisfaction predicate can be reflexive containment... except for one thing. The identity predicate in the theory can not be based on extensionality. It must be based on firstorder object identity as described by Frege and this cannot arise just because one invokes the ontological position of a "theory of identity". The reason that it must be based on object identity is that reference to the universe can only be made if, as Lesniewski has observed
P is an object
In an almost universal model, every proper part of the universe is an element of a class of which the universe is not an element. Thus every proper part of the universe is distinguished from the universe on the basis of object identity. Since to be a 'full class' the universe can have no other parts, it is unique and may be denoted by a singular term.
I have a very strong suspicion that no one will ever derive a Russellian paradox in Lesniewskian mereology. This is especially true if one considers George Greene's explanation that the paradox arises from grammatical form. As we have seen, Lesniewski specifically devised his mereology to circumvent the grammatical forms he thought would be problematic.

