On 3/28/2013 2:02 PM, Zuhair wrote: > See: http://zaljohar.tripod.com/sets.txt > > > Below is the full quote from the above link. > > ********************************************************************************* > What Are Classes! > > This account supplies THE final answer as to what classes are, > and what is class membership relation, those are defined in > a rigorous system with highly appealing well understood primitive > notions that are fairly natural and easy to grasp. It is aimed to be > the most convincing answer to this question. The formulations are > carried out in first order logic with Identity, Part-hood and Naming > binary relations. Identity theory axioms are assumed and they are > part of the background logical language of this theory. The > mereological > axioms are those of GEM (Generalized Extensional Mereology), they are > the standard ones. The two axioms of naming are very trivial. > The definitions of classes and their membership are coined with the > utmost care to require the least possible assumptions so they don't > require grounds of Atomic Mereology or unique naming or the alike.., > so they can work under more general situations. Also utmost care was > taken to ensure that those definitions are nearer to the reality of > the > issue and not just a technical fix. I simply think that what is given > here > do supply the TRUE and FINAL answer to what classes are and to > what is their membership! > > The General approach is due to David Lewis. Slight modifications are > adopted here to assure more general and nearer to truth grounds. > > Language: FOL(=,P,name) > > Axioms: ID axioms + > > 1.Reflexive: x P x > > 2.Transitive: x P y & y P z -> x P z > > 3.Antisymmetric: x P y & y P x -> x=y > > Define: x O y iff Exist z. z P y & z P x > > 4.Supplementation: ~y P x -> Exist z. z P y & ~ z O x > > 5.Composition: if phi is a formula, then ((Exist k. phi) -> > Exist x (for all y. y O x iff Exist z. phi(z) & y O z)) is an axiom. >
Is the premise here supposed to be
(Exist k. phi(k))
> Definition: x is a collection of phi-ers iff > for all y. y O x iff Exist z. phi(z) & y O z >
So, 'is a collection of phi-ers' is a predicate for each monadic formula phi(x) with x free?
> 6.Naming: n name of y & n name of x -> y=x > > Definition: n is a name iff Exist x. n name of x >
Do you need an axiom stating that every x of the domain is named? (This is actually a presuppostion of your stated logic)
> 7.Discreteness: n,m are names & ~n=m -> ~n O m > / > > > Definitions of "Class" and "Class membership": > > Define: x E y iff Class(y) & Exist n. n P y & n name of x. > > 1. Class(x) iff x is a collection of names. > > 2. Class(x) iff x is a collection of names Or x never overlap with a > name. > > when x never overlaps with a name then it is to be called an inert > object. >
There is something wrong here. Although you can state 1 and 2 together, the logical form makes what is expressed in 2 meaningless.
Eliminating 'Class(x)' between the two statements yields the form,
A <-> (A \/ B)
You cannot have the right side true with the left side false.
> Definition: x is inert iff ~Exist n. n is a name & x O n > > 3. Class(x) iff > x is a sum of an inert object and (an inert object or a collection of > names) > > Sum defined as: > > Sum(x,y) = z iff for all q. q O z iff q O x Or q O y. >
One of these statements of 'Class(x)' needs to be a definition and the others theorems.
Sum(x,y) takes objects as its arguments. But, 3 suggests that x be an object and y be a formula. What I think you are trying to combine is
where I have used 'Nameclass(z)' because of the apparent ambiguity in your definitions.
> 1 is incompatible with the empty class.
Perhaps with the empty class of names. But, it characterizes membership as a relation of objects to names. Although you have presented no axiom as such, the metaphysics of first-order logic without qualifications presumes that every object has a denotation. If xEy and x is empty (in relation to what one usually intends) then it is reasonable that x has a name and your definition of xEy applies.
Although I am unfamiliar with 'general extensional mereology', the subject usually denies empty classes. So, you seem to be reading this into your definition. But, the definition makes no constraint on 'x'.
Some mereologists, such as Peter Simons, work in free logic. If you read the first paragraph of
you will understand what I am saying about the metaphysics of first-order logic. In the history, "naming" had been generally taken to be "extra-logical". The sense of "denotes" in this explanation is that any particular name is irrelevant, but that the definiteness of a name is retained with respect to how a singular term is understood as standing in representation of an object.
Obviously, if the class of names is empty, the existence criterion on the right side fails. But, your definition says nothing about the referent 'x' -- the only constraint is on the relatum 'y'.
> 2 is incompatible with the subclass principle that is : > "Every subclass of x is a part of x". >
You are probably reading something into this as well. I do not really want to say more because the definitions no longer seem to be really definitions any longer.
It appears you were considering alternatives without stating as much.
> 3 does the job but it encourages gross violation of Extensionality > over classes > since having multiple names for an object is the natural expectation! >
It appears to be a bad definition.
> If we assume the subclass principle and use definition 3 then full > Extensionality > over classes is in place and it follows that the empty Class is an > atom.
I do not see atoms defined above. I will assume that it is 'that which is an inert object and has no part'
I will also assume that names have no parts. But, it is unclear whether they constitute atoms. So, I will presume that they do not.
> Although attractive on the face of it (since the empty set is just a > technical fix),
Too much mereology. Forget FOPL+= and go read Leibniz' "Discourse on Metaphysics".
> however it is not that convincing since there is no real > justification > for such atom-hood. >
Depends on how one views "real".
Depends on how one views "principle of sufficient reason".
Depends on how one views "justified true belief".
> If we strengthen the subclass principle into the principle that: > "For all classes X,Y (Y subclass of X iff Y P X)", then only > definition 1 > can survive such a harsh condition, and this would force all names to > be atoms and shuns the existence empty classes altogether!
Once again, that does not seem to be clear from your definitions.
It is an expectation of mereological metaphysics.
> such > a demanding commitment that despite the clear aesthetic gain of > having internally pure classes in the sense that all classes are only > composed of parts that are classes, yet still this is a very > demanding > commitment that do not seem to agree with basic natural expectations > about naming. >
> So a definition of classes that proves Extensionality over them > without > restricting multiple naming per object is what is demanded. >
Multiple names is not the problem. The problem is this:
Given the expectation of multiple names and given the logical equivalence of distinctive properties that might be used to introduce a name, how does one isolate a single name/description pair to serve as a canonical name?
Russell's "On Denoting"
Searle's "Language Acts"
Kripke's "On Naming and Necessity"
Boersema's "Pragmatism and Reference"
> Define: x is an equivalence collection of names iff > there exist y such that x is the collection of all names of y. > > Define: y is a fusion of equivalence collections of names iff > y is a collection of names & for all a,b,c (a P y & a name of b & c > name of b -> c P y) >
I do not want to think too hard here.
You probably want to redefine this along the lines of
'y is a fusion of equivalence collections of names for b'
or replace the universal Ab with the existential Eb
Your definition may work. But, as I said, I do not want to think too hard.
> Define V' as the collection of ALL inert objects. > > 4. Definition: Class(x) iff > x is a sum of V' and (V' or a fusion of equivalence collections of > names) >
I was not going to mention this, but you have not defined 'fusion' in these remarks.
> As far as the concept of class is concerned Extensionality is at the > core of it, > so 4. is the right definition of classes. >
And 4 has what appears to be the same problems as 3 with regard to the definition of Sum(x,y)
Also, can Sum(x,y) take the collection of ALL inert objects as an argument?
> It is nice to see that the *Empty Class* is just the collection of all > inert objects. > > For the sake of completion of this approach, we may say that > Definition 4. > is an Equivalence rendering of Definition 3. Similarly we can > introduce two > further definitions that are Equivalence renderings of Definition 1 > and Definition 2. > But those are rarely applicable in class\set theories. >
I cannot even touch this.
> Now one can easily define a set as a class that is an element of a > class. > An Ur-element is defined as an element of a class that is not a > class. > Or alternatively a non-class object. All kinds of circular membership > can be explained; > paradoxes can be easily understood. Also non definability of some > classes > can be understood. > > This account explains membership and classes in a rigorous manner. > And actually supplies the FINAL answer! > > Somehow those definitions might be helpful in orienting thought about > some > philosophical questions about mathematics founded in set theory. For > example > identity and part-hood are expected natural relations and they can be > reasoned > about as being human independent, but Naming might present some > challenge, > definitely it favors human dependency but still it can be human > independent! > Philosophical debate about the nature of sets would become a debate > about the > nature of naming procedures. >
It needs work Zuhair. You are either presenting 4 different definitions that you believe to be equivalent without proof or you are considering 4 different alternatives. It is very confusing.