David Ullrich on Identity
John
"And yes, identity is in _fact_ reflexive. To
refute that statement you need to give an
example of something which is not identical
to itself. The idea that there is something
which is _not_ identical to itself is simply
ludicrous: That's what identity _means_: A
thing is identical to itself and to nothing
else." (news:<mtmumugpksk46sr74...@4ax.com>)
For a contrasting standpoint, see
news:<c37480a7.03090...@posting.google.com>
**************************************************************
David Ullrich asks:
"What's an example of something that's not identical
to itself?" (news:<ai69nus52egl0rqjs...@4ax.com>)
See news:<c37480a7.03090...@posting.google.com>
**************************************************************
David Ullrich dares:
"Exhibit of proof of Ex~(x=x) from
C1-C4 and someone will point out the error."
>> >C1 AxAy[x=y -> Az(z in x <-> z in y)] LL1
>> >C2 AxAy[Az(z in x <-> z in y) -> Az(x in z <-> y in z)] LL2
>> >C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in A)
>> >Classification
>> >C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}]
Weak
>> >Extensionality (news:<0e14nu484feb15mpa...@4ax.com>)
Would someone be kind enough help David out with a proof?
*************************************************************
David Ullrich remonstrates:
"I 'blunder' by saying that equality is reflexive by definition?
Huh. Do you have any idea what the word "definition" means?"
news:<2ogpmuoudtj4l0eqj...@4ax.com>
Homework for David Ullrich:
1) What philosopher said:
"...definitions are available only for transforming
truths, not for founding them."
2) In your own words, explain why (or why not) you think
this is true.
--John
David C. Ullrich
Harris has finally found people who understand him there.
************************
David C. Ullrich
mitch
John wrote:
>
> Homework for David Ullrich:
>
> 1) What philosopher said:
>
> "...definitions are available only for transforming
> truths, not for founding them."
>
I would be interested in this answer...
You should change your terminology from 'identity' to 'scheme of
individuation.' The Luddite mindset still will not understand the
distinction, but the nuance will allow you some freedom of expression.
In the strictest sense of "formal identity" one is identifying duplicated
occurrences of symbol strings that cannot be distinguished from one
another except for spatial locality.
In the logical sense of "identity as a logical symbol of the language" we
are looking at substitutability of distinct symbol strings that are
presumed to have the same referent.
You should have no problems acccepting criticisms where the
contexts/situations are satisfactorily framed so that universal
quantifiers can be taken as parameters over sets.
So, your dispute with "the boyz in the hood" has to do with formal set
theory--class models of that theory, in fact.
Good going John. I know what I had to go through to see past the smoke
screen. I don't know how you figured it out.
Ask questions. I told you once before to look at restricted separation
associated with Goedel's constructible universe. Another place to
look--even more important--is the restricted quantification associated
with language invariance in topological model theory.
:-)
mitch
John
> You should check out http://megafoundation.org/
> Harris has finally found people who understand him there.
What part of "Ax[Qu(x) -> ~(x = x)]" do YOU fail to understand?
(See news:<c37480a7.03090...@posting.google.com>)
John
> You should check out http://megafoundation.org/
> Harris has finally found people who understand him there.
This from a purveyor of store-bought, on-the-shelf knowledge,
whose absolute inability to reason from premises other than
the customary, bought-and-paid-for ones is a common trait
of the unabashedly uptaught.
> >"Exhibit of proof of Ex~(x=x) from
> >C1-C4 and someone will point out the error."
> >
> >>> >C1 AxAy[x=y -> Az(z in x <-> z in y)] LL1
> >>> >C2 AxAy[Az(z in x <-> z in y) -> Az(x in z <-> y in z)] LL2
>
> >>> >C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in A)
> >>> >Classification
>
> >>> >C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}]
> Weak
> >>> >Extensionality (news:<0e14nu484feb15mpa...@4ax.com>)
Here's the proof you couldn't hack.
1) Ax(x in y <-> Et(x in t & ~(x in x)) Instance of (C1)
2) y in y <-> Et(y in t & ~(y in y)) 1, US
3) ~Et(y in t) 2
4) AyAx[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}] C4
5) Az(z in y <-> z in y) -> (Et(y in t) <-> y=y) 4,US
6) Et(y in t) <-> y=y 5
7) ~(y=y) 3,6
8) Ex~(x=x) 7,EG
**********************************************************************
Homework for David Ullrich
What bearing would the non-self-identity of quanta
have on the Leibnizean principle of the Identity of
Indiscernibles?
--John
"And yes, identity is in _fact_ reflexive. To
refute that statement you need to give an
example of something which is not identical
to itself. The idea that there is something
which is _not_ identical to itself is simply
ludicrous: That's what identity _means_: A
thing is identical to itself and to nothing
else."
--David Ullrich
news:<mtmumugpksk46sr74...@4ax.com>
John
> John wrote:
>
> >
> > Homework for David Ullrich:
> >
> > 1) What philosopher said:
> >
> > "...definitions are available only for transforming
> > truths, not for founding them."
> >
>
> I would be interested in this answer...
_Ways of Paradox_, p. 81
> So, your dispute with "the boyz in the hood" has to do with formal set
> theory--class models of that theory, in fact.>
> mitch
In news:<CpG5v3...@cs.cmu.edu>, T. Chow wrote:
> How can *theorems* be *false*? Well, the point here is that
> when one says that a statement about the natural numbers is
> "false" one means that it is false of the *standard* integers,
> i.e., the integers that we normally work with.
Likewise, when David Ullrich says that
statements about individuals such as ExAy~(x=y)
and Ex~(x=x) are false, he has said nothing more
than that these are false of the *standard*
individuals that FOL= deals with. Put back in
the domain those that FOL= excludes, and it is
not ExAy~(x=y) and Ex~(x=x) but AxEy(x=y) and
Ax(x=x) that are false.
It doesn't take a rocket scientist to figure
that one out...
--John
David C. Ullrich
>David C. Ullrich <ull...@math.okstate.edu> wrote in message news:<6vh4mvcro966lua87...@4ax.com>...
>
>> You should check out http://megafoundation.org/
>> Harris has finally found people who understand him there.
>
>This from a purveyor of store-bought, on-the-shelf knowledge,
>whose absolute inability to reason from premises other than
>the customary, bought-and-paid-for ones is a common trait
>of the unabashedly uptaught.
You should _really_ check out http://megafoundation.org/ !
This sort of anyone-who-actually-knows-something-must-be
-too-stupid-to-think-for-himself-the-way-we-geniuses-do is
their motto.
>> >"Exhibit of proof of Ex~(x=x) from
>> >C1-C4 and someone will point out the error."
Just for the record, when I said that you were claiming that
C1-C4 were part of NBG. Later turned out that that was just
something you made up.
************************
David C. Ullrich
John
> On 14 Sep 2003 13:28:55 -0700, john_...@yahoo.com (John) wrote:
>
> >David C. Ullrich <ull...@math.okstate.edu> wrote in message news:<6vh4mvcro966lua87...@4ax.com>...
>
> >> >C1-C4 and someone will point out the error."
>
> >> >>> >C2 AxAy[Az(z in x <-> z in y) -> Az(x in z <-> y in z)] LL2
>
> >> >>> >C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in A)
> >> >>> >Classification
>
> >> >>> >C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}]
> Weak
> >> >>> >Extensionality (news:<0e14nu484feb15mpa...@4ax.com>)
> >
> >Here's the proof you couldn't hack.
> >
> >2) y in y <-> Et(y in t & ~(y in y)) 1, US
> >3) ~Et(y in t) 2
> >4) AyAx[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}] C4
> >5) Az(z in y <-> z in y) -> (Et(y in t) <-> y=y) 4,US
> >6) Et(y in t) <-> y=y 5
> >7) ~(y=y) 3,6
> >
> >**********************************************************************
> Just for the record, when I said that you were claiming that
> C1-C4 were part of NBG. Later turned out that that was just
> something you made up.
"Till their own dreams at length deceive 'em,
And oft repeating, they believe 'em."
--Matthew Prior
Alma [1718], canto III, l. 13
> > Homework for David Ullrich
> >
> > What bearing would the non-self-identity of quanta
> > have on the Leibnizean principle of the Identity of
> > Indiscernibles?
> >
> >--John
>
> David C. Ullrich
Have you forgotten your homework, AGAIN???
John
> On 14 Sep 2003 13:28:55 -0700, john_...@yahoo.com (John) wrote:
>
> >David C. Ullrich <ull...@math.okstate.edu> wrote in message news:<6vh4mvcro966lua87...@4ax.com>...
> >
> >> You should check out http://megafoundation.org/
> >> Harris has finally found people who understand him there.
> >
> >This from a purveyor of store-bought, on-the-shelf knowledge,
> >whose absolute inability to reason from premises other than
> >the customary, bought-and-paid-for ones is a common trait
> >of the unabashedly uptaught.
>
> You should _really_ check out http://megafoundation.org/ !
> This sort of anyone-who-actually-knows-something-must-be
> -too-stupid-to-think-for-himself-the-way-we-geniuses-do is
> their motto.
Against stupidity the very gods
Themselves contend in vain.
--Schiller, _The Maid of Orleans_
Charlie-Boo
> David Ullrich says:
>
> "And yes, identity is in _fact_ reflexive. To
> refute that statement you need to give an
> example of something which is not identical
> to itself. The idea that there is something
> which is _not_ identical to itself is simply
> ludicrous
A variable that returns a random value (common in programming
languages) is often not equal to itself. One that returns the next
sequential number is never equal to itself. Just output the value of
x=x where x is that variable.
But more generally, people from the ancient Greek philosophers who
said things like "Two things equal to the same thing are equal to each
other." to modern day pseudointellectuals miss the real point of
equality.
ANY TWO THINGS ARE EQUAL AT SOME LEVEL OF ABSTRACTION AND ABOVE AND
UNEQUAL AT ALL LOWER LEVELS OF ABSTRACTION.
Does 1+1 equal 2? At a low, physical level, they are different
strings of characters. ("1+1" will not equal "2" in string
maniplulation processes in most programming languages.) But at a
higher, Mathematical level of abstraction, they are.
Does 1 equal 1? At a simple Mathematical level, they are. But at a
more exact level, they are not, becaue one of them is the 2nd word in
the question and the other is the 4th word. Or look at them under a
microscope and you will detect faint differences in the displays.
At a very high level of abstraction, they are both things.
From a distance, can you tell the difference between a rock and an
apple? But up close you can.
Like everything, equality is context sensitive.
Charlie Volkstorf
Cambridge, MA
David C. Ullrich
>john_...@yahoo.com (John) wrote
>> David Ullrich says:
>>
>> "And yes, identity is in _fact_ reflexive. To
>> refute that statement you need to give an
>> example of something which is not identical
>> to itself. The idea that there is something
>> which is _not_ identical to itself is simply
>> ludicrous
>
>A variable that returns a random value (common in programming
>languages) is often not equal to itself.
The value of the variable is always equal to itself. For a while
the variable has the value 3. And 3 = 3. A little later the variable
has the value 17. And 17 = 17.
The value at one time is not equal to the value at another time.
So what? They're not equal, because they're different things.
>One that returns the next
>sequential number is never equal to itself. Just output the value of
>x=x where x is that variable.
>
>But more generally, people from the ancient Greek philosophers who
>said things like "Two things equal to the same thing are equal to each
>other." to modern day pseudointellectuals miss the real point of
>equality.
>
>ANY TWO THINGS ARE EQUAL AT SOME LEVEL OF ABSTRACTION AND ABOVE AND
>UNEQUAL AT ALL LOWER LEVELS OF ABSTRACTION.
>
>Does 1+1 equal 2?
Yes.
>At a low, physical level, they are different
>strings of characters.
No, neither 1 + 1 nor 2 is a string of characters.
> ("1+1" will not equal "2" in string
>maniplulation processes in most programming languages.)
One hopes not. So what? If 1 and "1" were the same thing you'd
have a point.
>But at a
>higher, Mathematical level of abstraction, they are.
>
>Does 1 equal 1? At a simple Mathematical level, they are. But at a
>more exact level, they are not, becaue one of them is the 2nd word in
>the question and the other is the 4th word. Or look at them under a
>microscope and you will detect faint differences in the displays.
>
>At a very high level of abstraction, they are both things.
>
>From a distance, can you tell the difference between a rock and an
>apple? But up close you can.
>
>Like everything, equality is context sensitive.
Whether two _things_ are equal has nothing to do with
context. Whether the objects denoted by two symbols
are equal does have a lot to do with context. This
says nothing about whether things are equal to
themselves - in a context where two symbols denote
things that are not equal they denote two _different_
things.
Duh.
>Charlie Volkstorf
>Cambridge, MA
************************
David C. Ullrich
humanist
ch...@aol.com says...
Yes I agree.
I'm not equal to myself (so says my scale after a big meal). IMHO the
flaw is to ponder on the identify of the things themselves. IMHO, we
should rather ponder on the identity of their properties. Which is the
same thing you're saying: identity seems to be an artifact of
intellectual resolution. Zooming in (and out) of levels of precision
and abstraction reveals or dissolves differences. Equivalence is just
as context-sensitive as equality but is not as misleading.
Charlie-Boo
>
> >john_...@yahoo.com (John) wrote
> >> David Ullrich says:
> >>
> >> "And yes, identity is in _fact_ reflexive. To
> >> refute that statement you need to give an
> >> example of something which is not identical
> >> to itself. The idea that there is something
> >> which is _not_ identical to itself is simply
> >> ludicrous
> >
> >A variable that returns a random value (common in programming
> >languages) is often not equal to itself.
>
> The value of the variable is always equal to itself. For a while
> the variable has the value 3. And 3 = 3. A little later the variable
> has the value 17. And 17 = 17.
Right. Only when you use the word (concept) "itself" is there
equality, but then you are not referring to two things.
Call this variable $R (the name used in my favorite programming
language.) Then we agree that $R is not equal to $R. But you
maintain that 3 is equal to 3. However, I pointed out the fact that 3
is not actually equal to 3 because the former is the 8th word in this
sentence and the latter is the 14th word in this sentence. So at some
level they are still not equal. Now if we say that 3 is equal to
itself, we are not referring to two things. We are referring to one
thing and then referring to "itself".
Whenever there is a distinction, and there are in fact two things,
then whether they are equal or not depends on the level of abstraction
at which we are dealing, whether it makes that same distinction.
Since abstracting is the removal of distinctions, at some level, and
above, they will be equal.
In other words, it is not a question of whether two things are equal
or not. Rather, it is a question of, at what level of abstraction,
and above, they are equal? f(one thing,something)=the lowest level of
abstraction at which they are equal.
> They're not equal, because they're different things.
What's the difference between "not equal" and "different things"?
That is, when are these not the same?
> >Does 1+1 equal 2?
>
> Yes.
>
> >At a low, physical level, they are different strings of characters.
>
> No, neither 1 + 1 nor 2 is a string of characters.
How else can we communicate other than by an exchange of strings of
characters (i.e. elements of an agreed upon recursively enumerable
set)?
> Whether two _things_ are equal has nothing to do with
> context. Whether the objects denoted by two symbols
> are equal does have a lot to do with context.
Are not objects things?
David C. Ullrich
>David C. Ullrich <ull...@math.okstate.edu> wrote
>> On 15 Sep 2003 11:40:16 -0700, ch...@aol.com (Charlie-Boo) wrote:
>>
>> >john_...@yahoo.com (John) wrote
>> >> David Ullrich says:
>> >>
>> >> "And yes, identity is in _fact_ reflexive. To
>> >> refute that statement you need to give an
>> >> example of something which is not identical
>> >> to itself. The idea that there is something
>> >> which is _not_ identical to itself is simply
>> >> ludicrous
>> >
>> >A variable that returns a random value (common in programming
>> >languages) is often not equal to itself.
>>
>> The value of the variable is always equal to itself. For a while
>> the variable has the value 3. And 3 = 3. A little later the variable
>> has the value 17. And 17 = 17.
>
>Right. Only when you use the word (concept) "itself" is there
>equality, but then you are not referring to two things.
Uh, that's correct. That's why the word "itself" appears prominently
in the statement that you seemed to think you were refuting.
>Call this variable $R (the name used in my favorite programming
>language.) Then we agree that $R is not equal to $R.
No, we don't agree to that.
>But you
>maintain that 3 is equal to 3.
You don't realize how funny this sounds, saying that I "maintain"
that 3 equals 3?
>However, I pointed out the fact that 3
>is not actually equal to 3 because the former is the 8th word in this
>sentence and the latter is the 14th word in this sentence.
Yes, you "pointed this out". That doesn't make it true. In fact
3 is not a word.
>So at some
>level they are still not equal. Now if we say that 3 is equal to
>itself, we are not referring to two things. We are referring to one
>thing and then referring to "itself".
>
>Whenever there is a distinction, and there are in fact two things,
>then whether they are equal or not depends on the level of abstraction
>at which we are dealing, whether it makes that same distinction.
>Since abstracting is the removal of distinctions, at some level, and
>above, they will be equal.
>
>In other words, it is not a question of whether two things are equal
>or not. Rather, it is a question of, at what level of abstraction,
>and above, they are equal? f(one thing,something)=the lowest level of
>abstraction at which they are equal.
>
>> They're not equal, because they're different things.
>
>What's the difference between "not equal" and "different things"?
There is none. That's what "equal" means.
>That is, when are these not the same?
>
>> >Does 1+1 equal 2?
>>
>> Yes.
>>
>> >At a low, physical level, they are different strings of characters.
>>
>> No, neither 1 + 1 nor 2 is a string of characters.
>
>How else can we communicate other than by an exchange of strings of
>characters (i.e. elements of an agreed upon recursively enumerable
>set)?
We can't. The fact that we need character strings to talk about
things does not imply that things _are_ character strings.
>> Whether two _things_ are equal has nothing to do with
>> context. Whether the objects denoted by two symbols
>> are equal does have a lot to do with context.
>
>Are not objects things?
Yes...
>Charlie Volkstorf
>Cambridge, MA
>
>> ************************
>>
>> David C. Ullrich
************************
David C. Ullrich
Undeniable
Because I'm stupid, can anyone tell me what's the point of this discussion?
mitch
Undeniable wrote:
> Because I'm stupid, can anyone tell me what's the point of this discussion?
Don't worry about responses to my reply here. My main detractor is quite expert with slander.
In point set topology, the singleton {x} is often shortened to x. Consequently, there is an ambiguity of
notation in mathematics associated with reference to objects.
This presents no problem for mathematics because of how invariance is characterized in algebraic topology.
However, set theory--and its metaphysics of paradox--is a particular place where logic and mathematics
share interests. Unfortunately, the people who develop an understanding of set-theoretic identity within
the topological framework find their reasonable intuitions continually subject to "appeal-to-ridicule" in
logic communities.
If you have any interest in pursuing this matter sufficiently to understand what I mean by *reasonable
intuition* here is an excerpt from a citeseer abstract that I found with a quick Google search on "'David
Lewis' mereology":
"Just as mereotopology can be seen as an extension
of mereology through the addition of some topological
primitive such as connection or interior part, so also set
theory can itself be seen as an extension of mereology
through the addition of the primitive set theoretic notion
of singleton. David Lewis (1991) has shown how, with
the help of this one single notion, all the standard axioms
of set theory can be derived within a mereological
framework. The theory of sets and the 3 theory of
mereological sums (or fusions ) of singletons are, it turns
out, formally indistinguishable."
David Lewis is not a mathematician. He is recently deceased and was a tenured professor in philosophy at
Princeton University. The 1991 publication being referenced here is entitled "Parts of Classes." The
advocates of this mereological perspective cite Husserl as originator. Husserl was a student of
Weierstrass along with Cantor. In his third Logical Investigation, he discusses a theory of parts and
wholes and--being aware of Cantor's work--actually distinguishes between "parts" and "pieces." In any
case, this is part of the reason why the revision is coming from philosophy departments even though it
seems to be a mathematics/logic issue.
So, what is going on here is fairly straightforward. John falls into that group of people who understand
the identity predicate within that topological framework. He is trying to defend his ideas in the place
where everyone else thinks the expertise lies.
lol
Have a good day. Hope that helps a little.
:-)
mitch
John
Because you're stupid, why would anyone want to bother?
John
> Undeniable wrote:
>
> > Because I'm stupid, can anyone tell me what's the point of
> > this discussion?
>
> So, what is going on here is fairly
> straightforward. John falls into that group of people who
> understand the identity predicate within that topological
> framework. He is trying to defend his ideas in the place
> where everyone else thinks the expertise lies.
>
I haven't the faintest idea what you are talking about.
--John
Undeniable
Because, under certain situations, like the one you are involved,
stupidity suffices in providing the answer.
John wrote:
"True, the provability of Ex~(x=x) will vary from one system to
the next. Nevertheless, there is more to truth than
'truth-in-a-model';
and just as the truth of Ex~(x is red) depends on how things stand
with individuals and redness so the truth of Ex~(x=x) depends on
how things stand with individuals and identity."
Because I'm stupid, can you offer an example where Ex~(x=x) is true?
G. Frege
> >
> > So, what is going on here is fairly
> > straightforward. John falls into that group of people who
> > understand the identity predicate within that topological
> > framework. He is trying to defend his ideas in the place
> > where everyone else thinks the expertise lies.
> >
>
> I haven't the faintest idea what you are talking about.
>
Well, *I* would translate...:
"[...] John falls into that group of people who understand
_identity_ in a very special way. [...]"
Not meant as an insult, just as a clarification (if so).
F.
mitch
Of that, I have little doubt. But that does not mean that
I am incorrect, here.
However, I am certainly mistaken if you are a different John--different from the one who, in
<c37480a7.03090...@posting.google.com>
explained haeccity to Immortalist with:
"The most straightforward definition I have
found is from Garth Kemerling's Dictionary of
Philosophical Terms and names
(http://www.philosophypages.com/dy/index.htm):
"haecceity {Lat. haecceitas}
Thisness; the property that uniquely distinguishes
each individual thing from others of its kind.
Introduced by Duns Scotus as a name for
the individuating essence of any particular, the term
has been used more recently in connection with the view
that rigidly designated individuals can exist in each
of many possible worlds."
Thus, (1) is a theorem of all standard set theories:
AyAx(y=x <-> y in {x}).
Accordingly, granted that membership in {x} is both
necessary and sufficient for identity-with-x, {x}
is (arguably) a haecceity of x: if you belong to
{x} you are identical with x, and if you do not
belong to {x} you are not. The passages you cite
are by people whose concern is with what Teller calls
the 'metaphysically robust' notion of *haecceity*:"
And, the quote above follows what was written in the original post for that thread. That is, in the post
<c37480a7.03090...@posting.google.com>
someone named John wrote:
"To this objection one, who (like myself) takes
identity to be nonreflexive, has a non-standard
response--to wit, that the usual axiom of
extensionality (which makes identity reflexive) is
not indispensable: it can be supplanted by C4.
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}]
For from (C4) it follows that sets are self-
identical, but that proper classes (like quanta?)
are not."
Now, if you are the John who wrote both of these quotes, then you might be quite interested in some of the things that David Lewis has to say about singletons, and about what the mereological and mereotopological advocates are saying.
If I am wrong, and you are not that individual, I apologize for having tread where I should not have.
By the way, no one bothered to critique your proof. You made a mediate assumption between the steps
2) y in y <-> Et(y in t & ~(y in y))
3) ~Et(y in t)
That is, you assumed ~(y in y). At no point in your proof did you discharge this assumption.
:-)
mitch
John
I'll provide not one but THREE examples. The first
depends on a (very simple) proof that DAVID ULLRICH COULDN'T HACK!
A) Take any 'pure' first order logic, that is, any FOL with no
singular terms other than variables (just to keep things simple).
B) To your FOL add this formation rule;
If a and b are variables, "a=b" is an atomic formula.
and this axiom scheme:
Let be C and D be wff's which differ only in that a occurs free
in C where b occurs free in D. If a=b, then C <-> D.
C) Call the resulting logic FOL+.
In FOL+, identity is symmetric and transitive and identicals are
indiscernible, but neither Ax(x=x) nor ~Ax(x=x) is a thesis. So,
FOL+ is a subtheory of FOL= (because every thesis of FOL+ is
also a thesis of FOL=), but not conversely (because Ax(x=x)
is a thesis of FOL= but not of FOL+).
Example 1: Set Theory
Add to FOL+ the NBG axiom scheme, C3, and Correy's principle of
extensionality, C4:
C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in A) Classification
C4 AyAx[Az(z in y <-> z in x) -> {(set y & set x) <-> y=x}]
(Equi-membered classes are identical iff these are sets.)
From C3 it follows that there is a class of non-self-membered classes.
1) EyAx(x in y <-> Et(x in t) & ~(x in x)) C3
Hence
2) Ax(x in r <-> Et(x in t) & ~(x in x)) 1,EI
and
3) r in r <-> (Et(r in t) & ~(r in r)) 2,UI
so that
4) ~Et(r in t) 3
and
5) ~(set r) 4
and
6) ~(r=r) 5,C4
so that
7) Ex~(x=x) 6,EG
To be continued.
--John
John
> John wrote:
>
> > mitch <mit...@rcnNOSPAM.com> wrote in message news:<3F68E0C9...@rcnNOSPAM.com>...
> > > Undeniable wrote:
> > >
> > > > Because I'm stupid, can anyone tell me what's the point of
> > > > this discussion?
>
> > >
> > > So, what is going on here is fairly
> > > straightforward. John falls into that group of people who
> > > understand the identity predicate within that topological
> > > framework. He is trying to defend his ideas in the place
> > > where everyone else thinks the expertise lies.
> > >
> >
> > I haven't the faintest idea what you are talking about.
> >
> > --John
>
> Of that, I have little doubt. But that does not mean that I am
> incorrect, here.
>
> However, I am certainly mistaken if you are a different John--
> different from the one who, in
>
> <c37480a7.03090...@posting.google.com>
That is, news:<c37480a7.03090...@posting.google.com>
That is, news:<c37480a7.03090...@posting.google.com>
>
> someone named John wrote:
>
>
> "To this objection one, who (like myself) takes
> identity to be nonreflexive, has a non-standard
> response--to wit, that the usual axiom of
> extensionality (which makes identity reflexive) is
> not indispensable: it can be supplanted by C4.
>
> C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}]
>
> For from (C4) it follows that sets are self-
> identical, but that proper classes (like quanta?)
> are not."
>
>
> Now, if you are the John who wrote both of these quotes,
> then you might be>quite interested in some of the things
> that David Lewis has to say about singletons, and about what
> the mereological and mereotopological advocates are saying.
I am that John, but I am not interested in what David Lewis
has to say about singletons, nor am I interested in what David
Lewis has to say about what the mereological and mereotopological
advocates are saying.
>
> If I am wrong, and you are not that individual, I apologize
> for having tread where I should not have.
Apology accepted.
>
> By the way, no one bothered to critique your proof. You made
> a mediate assumption between the steps
>
> 2) y in y <-> Et(y in t & ~(y in y))
> 3) ~Et(y in t)
>
> That is, you assumed ~(y in y). At no point in your proof
> did you discharge this assumption.
See news:<c37480a7.03091...@posting.google.com>.
>
> :-)
>
> mitch
Undeniable
Good. What's your thoughts on this:
I believe Aristotle defined identity as (x --> x) and not as (x = x).
Don't you loose the transitive property if you use the former
definition?
Then your proof breaks down.
What about that?
As far as I'm conserned, identity means x --> x and not x = x, as the
latter definition leads to paradoxes. Anyway, I'm trying to call
Aristotle on his mobile phone but it's out of cell reach...:)
William Elliot
> David Ullrich says:
>
> "And yes, identity is in _fact_ reflexive. To
> refute that statement you need to give an
> example of something which is not identical
> to itself.
>
You never meet a person who was 'just beside himself'
or 'not feeling himself'?
> The idea that there is something
> which is _not_ identical to itself is simply
> ludicrous: That's what identity _means_: A
> thing is identical to itself and to nothing
> else." (news:<mtmumugpksk46sr74...@4ax.com>)
>
Once upon a modern neotime, in secrete of course, a politician cloned
himself. This went unnoticed for a few decades until..., but just a
nanosec, that's a digression. The morrow of the story is political clones
can be literately beside themselves.
Witt
Hi John,
Your very interesting proof leads to some questions.
1. What is the domain of your variables ? I assume you admit
sets and non-sets. Are physical objects and proper classes and sets
values of the variable x ?
2. Do you claim: Et(x in t & y in t) <-> (set y & set x), or,
Et(x in t)& Et(y in t) .<->. set x & set y. ?
C4 AyAx[Az(z in y <-> z in x) -> {(set y & set x) <-> y=x}]
(Equi-membered classes are identical iff these are sets.)
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}] Weak
Extensionality (news:<0e14nu484feb15mpa...@4ax.com>)
3. Is C3 <-> EyAx[x in y <->. Et(x in t) & set y & A] (with y not free
in A) ?
4. It seems your C4 entails x=x <-> set x. That is, x=x fails for
empirical values (non sets). Where does this leave us with respect to
Russell's descriptions?
5. How do you distinguish: E!x, x=x, set x, Ey(x in y), Ez(z in x) ?
Thanks in advance,
Witt
John
> > >
> > > "True, the provability of Ex~(x=x) will vary from one system to
> > > the next. Nevertheless, there is more to truth than
> > > 'truth-in-a-model';
> > > and just as the truth of Ex~(x is red) depends on how things stand
> > > with individuals and redness so the truth of Ex~(x=x) depends on
> > > how things stand with individuals and identity."
> > >
Variables range over classes. Classes that are elements are sets,
and classes that are not are proper classes. I'm uncertain whether
physical objects have any role to play in set theory. But I do
agree with G. Frege that proper classes can be taken as *models*
of quanta--not proper classes as they are treated by Ullrich and
Chapman, but proper classes à la (C3,C4). (I'm sending you Teller's
article, which would seem to bear such a construction out. See
especially his footnote 6.)
>
> 2. Do you claim: Et(x in t & y in t) <-> (set y & set x), or,
> Et(x in t)& Et(y in t) .<->. set x & set y. ?
(i) is an instance of C3:
(i) EyAx(x in y <-> (Set x & x=w v x x=z)).
But (ii) follows from (i,C4) (proof supplied upon request):
(ii) EyAx(x in y <-> (x=w v z=z))
On the other hand, from (ii,C4) and "Ax(Set x <-> x=x)" it
both follows that Et(x in t & y in t) <-> (set y & set x),
and that Et(x in t)& Et(y in t) .<->. set x & set y.
>
> C4 AyAx[Az(z in y <-> z in x) -> {(set y & set x) <-> y=x}]
> (Equi-membered classes are identical iff these are sets.)
>
> C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}] Weak
> Extensionality (news:<0e14nu484feb15mpa...@4ax.com>)
>
> 3. Is C3 <-> EyAx[x in y <->. Et(x in t) & set y & A] (with y not free
> in A) ?
No. It does not follow that y is a set. To prove that y is a set,
it must be established that every class with the same members as y
is identical to y. In other words, it must be established that
Leibniz's principle of the identity of indiscernibles holds for
every such class and y. (To establish this, it suffices to
establish that this principle holds for y itself.) If Leibniz's
principle does not hold, then y (along with every class equi-membered
with y) is a proper class.
>
> 4. It seems your C4 entails x=x <-> set x. That is, x=x fails for
> empirical values (non sets). Where does this leave us with respect to
> Russell's descriptions?
The answer to this question is (I think) provided by the following
critique of Quine by Alberto Cortes:
"My final comment about variables and names deals with a comment
of Quine's: he has claimed that in a logically austere language,
names are quite superfluous:
Chief among the omitted frills is the *name*. This . . . is a
mere convenience and strictly redundant, for the following
reason. Think of 'a' as a name, and think of 'Fa' as equivalent
to '(Ex(a = x & Fx)'. We see from this consideration that 'a'
need never occur except in the context 'a='. But we can as
well render 'a=' always as a simple predicate 'A'. thus
abandoning the name 'a'. 'Fa' gives way thus to '(Ex)(Ax & Fx)',
where the predicate 'A' is *true solely of the object a*.
[Cortes's emphasis].
It may be objected that this paraphrase deprives us of an
assurance of uniqueness that the name has afforded. It is
understood that the name applies to only one object, whereas
the preciate 'A' supposes no such condition. However, we
lose nothing by this, since we can always stipulate by further
sentences, when we wish, that 'A' is true of one and only
one thing. (_Philosophy of Logic_ [Englewood Cliffs, New
Jersey: Prentice Hall, 1970]: p. 25)
According to Quine, then, one can always eliminate a name by
concocting an appropriate predicate which is true only of the
object designated by that name.
But how is one to be assured that the predicate one has concocted
does not apply simultaneously to more than one object? One
cannot merely stipulate that it be so. One has to presuppose
that if two objects are numerically distinct they must not
possess all properties in common. And this is nothing less
than PII(L) [PII(L): "No two substances are completely sim-
ilar, or differ *solo numero". -JC] Therefore, it is evident
that in order to eliminate proper names, *à la Quine*, one has
to presuppose PII(L). Needless to say, we shall not admit
such a presupposition in a critical analysis of PII(L).
Therefore, from the point of view of this paper, proper names,
or their logical equivalents (e.g., prounouns), are logically
fundamental ingredients of both ordinary and formal languages."
("Leibniz's Principle of the Identity of Indiscernibles: A
False Principle," _Philosophy of Science_, Vol.43 Dec., 1976,
p. 496)
The eliminability of proper names à la Quine--which Cortes
opposes on the grounds that PII(L) fails to hold for quanta--
marches in lockstep with Russell's Theory of Descriptions.
For Cortes's critique of Quine applies to Russell's
Theory of Descriptions as well.
>
> 5. How do you distinguish: E!x, x=x, set x, Ey(x in y), Ez(z in x) ?
The first three are equivalent. "Ez(z in x)" and "~Ez(z in x)" are
both compatible with the first three.
>
> Thanks in advance,
>
> Witt
My Thanks to You,
--John Correy
PS I am pleased to have been dismissed as a "crass formalist"
by the likes of Robin Chapman and David Ullrich. Not
uncoincidentally, these are PROFESSORS OF MATHEMATICS!
George Greene
: In point set topology, the singleton {x} is often shortened to
: x. Consequently, there is an ambiguity of notation in
: mathematics associated with reference to objects.
There is a deeper problem here.
If 1 branch can treat x and {x} as though they
were indiscernible, and the other CAN'T, then the
question must arise: can they BOTH be RIGHT?
Or is one of them just BETTER than the other?
And is the other one therefore WRONG?
:: This presents no problem for mathematics because of how
: invariance is characterized in algebraic topology. However, set
: theory--and its metaphysics of paradox--is a particular place
: where logic and mathematics share interests. Unfortunately, the
: people who develop an understanding of set-theoretic identity
: within the topological framework find their reasonable
: intuitions continually subject to "appeal-to-ridicule" in logic
: communities.
That is over-simplified to the point of being bullshit.
What ACTUALLY happens is that people who claim to have
an "understanding of set-theoretic identity" get laughed
out of court for the simple reason that there is NO SUCH THING
as set-theoretic identity. In set theory, identity (a=b) is re-
defined as Ax[xea<->xeb]. Everywhere you see an = sign in set
theory, you can simply REPLACE a=b with this macro. In other
words, since set-theoretic identity DOES NOT EXIST, there is
simply NOTHING to have ANY kind of understanding OF. THAT
is why you get ridiculed.
: "Just as mereotopology can be seen as an extension
: of mereology through the addition of some topological
: primitive such as connection or interior part, so also set
: theory can itself be seen as an extension of mereology
: through the addition of the primitive set theoretic notion
: of singleton. David Lewis (1991) has shown how, with
: the help of this one single notion, all the standard axioms
: of set theory can be derived within a mereological
: framework.
But that is simply an alternative axiomatization of set theory.
Why don't you just lay ZFC on one side and mereology + singletons
on the other and argue, directly, that one axiomatization is
more appealing than the other?
: The theory of sets and the 3 theory of
: mereological sums (or fusions ) of singletons are, it turns
: out, formally indistinguishable."
"The 3 theory of mereological fusions of singletons" is obviously
just one of many mereological theories. You could try to say
something about why the mereological framework is richer, given
that it also includes all these other mereological theories
that are 2- theories and 4- theories, instead of just 3- theories.
You could further allege that they are richer because they can do
more than just fusions. But we're not holding our breath until
you get around to that. Especially not in light of the
fact that set-theoretical characterizations of mereology are every
bit as easy to construct as mereological characterizations of set
theory. You are not going to find it easy to support any claim
of superior merit as a foundation.
: So, what is going on here is fairly straightforward. John falls
: into that group of people who understand the identity predicate
: within that topological framework.
Set theory has a rather simple and straightforward conception
of identity. If the other one is not equally simple then he
is not going to have a lot of hope of defending it --
: He is trying to defend his
: ideas in the place where everyone else thinks the expertise
: lies.
not here OR ANYwhere.
--
---
"It's difficult ... you need to be united to have any
strength, but internal issues have to be addressed."
--- E. Ray Lewis, on liberalism in America
Witt
Sets are then existent classes.
> I'm uncertain whether
> physical objects have any role to play in set theory.
Perhaps we can define physical objects as the class of properties that it has,
or the class of those classes that it is a member of.
> But I do
> agree with G. Frege that proper classes can be taken as *models*
> of quanta--not proper classes as they are treated by Ullrich and
> Chapman, but proper classes à la (C3,C4). (I'm sending you Teller's
> article, which would seem to bear such a construction out. See
> especially his footnote 6.)
I hope I can follow it.
>
> >
> > 2. Do you claim: Et(x in t & y in t) <-> (set y & set x), or,
> > Et(x in t)& Et(y in t) .<->. set x & set y. ?
>
> (i) is an instance of C3:
>
> (i) EyAx(x in y <-> (Set x & x=w v x x=z)).
>
> But (ii) follows from (i,C4) (proof supplied upon request):
>
> (ii) EyAx(x in y <-> (x=w v z=z))
I don't see how you use C3 here.
>
> On the other hand, from (ii,C4) and "Ax(Set x <-> x=x)" it
> both follows that Et(x in t & y in t) <-> (set y & set x),
> and that Et(x in t)& Et(y in t) .<->. set x & set y.
>
> >
> > C4 AyAx[Az(z in y <-> z in x) -> {(set y & set x) <-> y=x}]
> > (Equi-membered classes are identical iff these are sets.)
> >
> > C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}] Weak
> > Extensionality (news:<0e14nu484feb15mpa...@4ax.com>)
> >
> > 3. Is C3 <-> EyAx[x in y <->. Et(x in t) & set y & A] (with y not free
> > in A) ?
>
> No. It does not follow that y is a set. To prove that y is a set,
> it must be established that every class with the same members as y
> is identical to y. In other words, it must be established that
> Leibniz's principle of the identity of indiscernibles holds for
> every such class and y. (To establish this, it suffices to
> establish that this principle holds for y itself.) If Leibniz's
> principle does not hold, then y (along with every class equi-membered
> with y) is a proper class.
I thought: (x in y)->. E!x & E!y.
That is, (x in y) is false unless x and y are existent classes (sets).
Don't you also need: x=y ->. E!x & E!y & (Fx<->Fy),
and E!x -> x=x.
Why not define: x=y defined E!x & E!y & AF(Fx<->Fy).
And, E!x defined EF(Fx).
Are your first order axioms:
1. Fx -> E!x
2. E!x -> x=x
3. x=y ->. E!x & E!y & (Fx<->Fy).
His 'Equals a' predicate can't be expressed without the term a.
But, names are superflous anyway. Pure logic, deals without names.
I can't see how we can eliminate variables though.
e.g. Ay(AxFx -> Fy), Ay(Fy -> ExFx).
> But how is one to be assured that the predicate one has concocted
> does not apply simultaneously to more than one object? One
> cannot merely stipulate that it be so.
In virtue of AxE!x, ie. Ax(x=x), we are assured that only a satisfies
(a=). Everything exists and is unique for him.
> One has to presuppose
> that if two objects are numerically distinct they must not
> possess all properties in common. And this is nothing less
> than PII(L) [PII(L): "No two substances are completely sim-
> ilar, or differ *solo numero". -JC] Therefore, it is evident
> that in order to eliminate proper names, *à la Quine*, one has
> to presuppose PII(L). Needless to say, we shall not admit
> such a presupposition in a critical analysis of PII(L).
> Therefore, from the point of view of this paper, proper names,
> or their logical equivalents (e.g., prounouns), are logically
> fundamental ingredients of both ordinary and formal languages."
> ("Leibniz's Principle of the Identity of Indiscernibles: A
> False Principle," _Philosophy of Science_, Vol.43 Dec., 1976,
> p. 496)
>
> The eliminability of proper names à la Quine--which Cortes
> opposes on the grounds that PII(L) fails to hold for quanta--
> marches in lockstep with Russell's Theory of Descriptions.
> For Cortes's critique of Quine applies to Russell's
> Theory of Descriptions as well.
I don't think we can eliminate descriptions, do you?
IMHO, I think your free set theory can be produced as an extension
of classical logic, i.e. Description theory.
EF(~E!(ixFx)), and EF(~((ixFx)=(ixFx))) .. instead of your
Ex(~E!x) and Ex(~(x=x)).
And classical logic is included in your free set theory:
AxE!x -> Ax(x=x), i.e. by restricting your free classes to sets.
>
> >
> > 5. How do you distinguish: E!x, x=x, set x, Ey(x in y), Ez(z in x) ?
>
> The first three are equivalent. "Ez(z in x)" and "~Ez(z in x)" are
> both compatible with the first three.
I think E!x <-> Ey(x in y) too.
(set x) ->. ~Ez(z in x) <-> x={}.
Witt
John
> I hope I can follow it.
Your Yahoo e-mail address is not functioning. Let me know if you get another one.
--John
John
> john_...@yahoo.com (John) wrote in message news:<c37480a7.03092...@posting.google.com>...
> > oori...@yahoo.com (Witt) wrote in message news:<196b22e7.03092...@posting.google.com>...
> > > john_...@yahoo.com (John) wrote in message news:<c37480a7.03091...@posting.google.com>...
> > > > undenia...@yahoo.com (Undeniable) wrote in message news:<1f366ae.03091...@posting.google.com>...
> > > > > john_...@yahoo.com (John) wrote in message news:<c37480a7.03091...@posting.google.com>...
> > > > > > undenia...@yahoo.com (Undeniable) wrote in message news:<1f366ae.03091...@posting.google.com>...
> > > > > > > john_...@yahoo.com (John) wrote in message news:<c37480a7.03091...@posting.google.com>...
> > > > A) Take any 'pure' first order logic, that is, any FOL with no
The logic of existence/non-existence and the logic of sets/proper
classes is (arguably) the same logic: a logic in which identity
is not reflexive. From this, however, it does not follow that
sets *are* existent classes, but that "x=x" holds for sets
and existents while "~(x=x)" holds for proper classes and
non-existents. Indeed, granted that "x=x" holds for
sets and classical particles, it doesn't follow either
that sets *are* classical particles.
>
> > I'm uncertain whether
> > physical objects have any role to play in set theory.
>
> Perhaps we can define physical objects as the class of properties
> that it has, or the class of those classes that it is a member of.
>
> > But I do
> > agree with G. Frege that proper classes can be taken as *models*
> > of quanta--not proper classes as they are treated by Ullrich and
> > Chapman, but proper classes à la (C3,C4). (I'm sending you Teller's
> > article, which would seem to bear such a construction out. See
> > especially his footnote 6.)
>
> I hope I can follow it.
>
> >
> > >
> > > 2. Do you claim: Et(x in t & y in t) <-> (set y & set x), or,
> > > Et(x in t)& Et(y in t) .<->. set x & set y. ?
> >
> > (i) is an instance of C3:
> >
> > (i) EyAx(x in y <-> (Set x & x=w v x x=z)).
> >
> > But (ii) follows from (i,C4) (proof supplied upon request):
> >
> > (ii) EyAx(x in y <-> (x=w v z=z))
>
> I don't see how you use C3 here.
Here's a proof:
C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in A) Classification
C4 AyAx[Az(z in y <-> z in x) -> {(set y & set x) <-> y=x}]
(Equi-membered classes are identical iff these are sets.)
1. Show EyAx(x in y <-> (x=w v x x=z))
2. EyAx(x in y <-> (Set x & x=w v x x=z)) Instance of C3
3. x in y <-> (Set x & x=w v x x=z) 2,EI,UI
4. x in y -> (x=w v x x=z) 3
5. (x=w v x=z) -> x=x "=" is weakly reflexive
6. x=x -> Set x C4
7. (x=w v x=z) -> Set x 5,6
8. (Set x & (x=w v x x=z)) -> x in y 3
9. (x=w v x x=z) -> x in y 7,8
10. x in y <-> (x=w v x x=z) 4,9
11. EyAx(x in y <-> (x=w v x x=z)) 10,UG,EG
>
> >
> > On the other hand, from (ii,C4) and "Ax(Set x <-> x=x)" it
> > both follows that Et(x in t & y in t) <-> (set y & set x),
> > and that Et(x in t)& Et(y in t) .<->. set x & set y.
> >
> > >
> > > C4 AyAx[Az(z in y <-> z in x) -> {(set y & set x) <-> y=x}]
> > > (Equi-membered classes are identical iff these are sets.)
> > >
> > > C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}] Weak
> > > Extensionality (news:<0e14nu484feb15mpa...@4ax.com>)
> > >
> > > 3. Is C3 <-> EyAx[x in y <->. Et(x in t) & set y & A] (with y not free
> > > in A) ?
> >
> > No. It does not follow that y is a set. To prove that y is a set,
> > it must be established that every class with the same members as y
> > is identical to y. In other words, it must be established that
> > Leibniz's principle of the identity of indiscernibles holds for
> > every such class and y. (To establish this, it suffices to
> > establish that this principle holds for y itself.) If Leibniz's
> > principle does not hold, then y (along with every class equi-membered
> > with y) is a proper class.
>
> I thought: (x in y)->. E!x & E!y.
> That is, (x in y) is false unless x and y are existent classes (sets).
--From x in y it follows only that x is a set. It does not also follow
that y is a set. To show that y is a set, it's necessary to show that
it is a member of some class.
--"x=x" holds for sets and "~(x=x)" for proper classes. From this I
don't conclude, however, that proper classes are non-existents, just as
I wouldn't conclude, from the fact that "~(x=x)" holds for quanta, that
quanta are non-existents. What I would conclude instead is that
if identity is non-reflexive, non-existents and proper classes
and quanta all turn out to be identical-with-nothings, granted
certain assumptions about non-existents, classes, and quanta.
The relevant assumption about existence is that "x exists" means
"Ey(x=y)"; the relevant assumption about classes is that these
are governed by C3,C4; and the relevant assumption about quanta
is that these lack haecceities.
>
> Don't you also need: x=y ->. E!x & E!y & (Fx<->Fy),
> and E!x -> x=x.
Yes. But these are all theorems, as I show in the first or second
of the following postings:
news:<70f94e16.02090...@posting.google.com>
news:<70f94e16.0209...@posting.google.com>
news:<70f94e16.02090...@posting.google.com>
news:<70f94e16.02090...@posting.google.com>
news:<70f94e16.02090...@posting.google.com>
news:<70f94e16.02090...@posting.google.com>
news:<70f94e16.02090...@posting.google.com>
news:<70f94e16.02091...@posting.google.com>
>
> Why not define: x=y defined E!x & E!y & AF(Fx<->Fy).
> And, E!x defined EF(Fx).
Other than the last (which is second order), these all follow
from the axioms, so there is no need to define them.
>
> Are your first order axioms:
> 1. Fx -> E!x
> 2. E!x -> x=x
> 3. x=y ->. E!x & E!y & (Fx<->Fy).
No, but these follow from the axioms.
There are already a lot of proposals about how to do this:
a Google search on "logic without variables" turned up 120 hits.
This is not to say that I think that this is necessary or that
I know how it is to be done...
In a logic with non-reflexvie identity, the key definitions
of Russell's Theory of Descriptions can be introduced as axioms.
>
> EF(~E!(ixFx)), and EF(~((ixFx)=(ixFx))) .. instead of your
> Ex(~E!x) and Ex(~(x=x)).
>
> And classical logic is included in your free set theory:
> AxE!x -> Ax(x=x), i.e. by restricting your free classes to sets.
I don't understand this.
>
> >
> > >
> > > 5. How do you distinguish: E!x, x=x, set x, Ey(x in y), Ez(z in x) ?
> >
> > The first three are equivalent. "Ez(z in x)" and "~Ez(z in x)" are
> > both compatible with the first three.
>
> I think E!x <-> Ey(x in y) too.
> (set x) ->. ~Ez(z in x) <-> x={}.
>
> Witt
>
> >
> > >
> > > Thanks in advance,
> > >
> > > Witt
> >
> > My Thanks to You,
> > --John Correy
> >
> > PS I am pleased to have been dismissed as a "crass formalist"
> > by the likes of Robin Chapman and David Ullrich. Not
> > uncoincidentally, these are PROFESSORS OF MATHEMATICS!
--John
IMSHUR...@spammotel.com
of contradiction? If something can be what it is not, there is a
problem.
Pat Harrington
false. It is not false that it is false, as it IS false. When you
boil it down to the Boolean concepts, what you say is true regarding
"true" but false regarding "false" because "false" is truly false.
I.e., the only legitimate adverb that could modify "false" such that
it retains its identity, is "truly".
It is logically required that false is truly false, otherwise it
would lose its identity and meaning. I believe this is the only
exception to the general rule but it is the one that allows us to
logically negate anything and the engine behind the power of
discernment.
Cheers,
Pat
IMSHUR...@spammotel.com wrote in message news:<b496f240.03092...@posting.google.com>...
Russell Blackadar
>
> What about the identity of the concept "False"?
*Statements* are true or false. Is 'the concept "False"'
a statement? (Rhetorical question.)
It is true that it is
> false.
It is the *concept* "False", yes. But don't confuse a concept
with its (nonexistent) truth value.
Pat Harrington
You mean non-physical existence. Things can exist and not be
physical. "False" exists as a concept and resides in a place where
thinking beings can access it and use it (or not?). The rules of
logic must exist or we couldn't use them. But they don't have
physical existence, for if they did, they wouldn't be of as much use.
Dreams exist. They even show more form than some concepts. And they
exist over time, which exists. But they aren't physical. They can
still frighten some people to death or teach us new ways to look at
life. Again, it's a good thing they aren't physical, as they wouldn't
be as potentially useful.
Cheers,
Pat
Russell Blackadar
>
> Russell Blackadar <rus...@mdli.com> wrote in message news:<3F744B88...@mdli.com>...
> > Pat Harrington wrote:
> > >
> > > What about the identity of the concept "False"?
> >
> > *Statements* are true or false. Is 'the concept "False"'
> > a statement? (Rhetorical question.)
> >
> > It is true that it is
> > > false.
> >
> > It is the *concept* "False", yes. But don't confuse a concept
> > with its (nonexistent) truth value.
By "nonexistent" here, I meant "undefined". Sorry my
wording seems to have misled you a bit.
>
> You mean non-physical existence. Things can exist and not be
> physical. "False" exists as a concept and resides in a place where
> thinking beings can access it and use it (or not?).
I don't have a problem with that; Platonism is fine and
dandy by me. I was only objecting to your careless usage
of words. It is possible to make false statements *about*
the concept "False", but the concept *itself* is not false
(or true) any more than the number 42 is false (or true).
Once you take this rather basic distinction properly into
consideration, it seems to me there's not much useful content
left in your original posting.
John
> of contradiction? If something can be what it is not, there is a
> problem.
Is there anything contradictory about there being no such thing
as Vulcan (~Ex(Vulcan = x))? If not, how can any *consequence* of
such an assertion give rise to a contradiction?
--John
Witt
Yes, I agree with that, if we restrict our vaiables "x" to be classes
in the classical sense, G(x:Fx) <-> Ey(Ax(x in y <-> Fx) & Gy)), then,
your axioms are true.
That is, your set theory is included in classical logic by extending
it in this way.
> From this, however, it does not follow that
> sets *are* existent classes, but that "x=x" holds for sets
> and existents while "~(x=x)" holds for proper classes and
> non-existents.
I don't agree.
Sets are existent classes and proper classes are non-existent classes.
How can there be exceptions?
E!x <-> x=x. (set x) <-> x=x. E!x <-> (set x). Therefore ~E!x <->
~(set x).
I don't yet understand your formulations, eg. what does (x=w v x x=z)
mean?
>
> >
> > >
> > > On the other hand, from (ii,C4) and "Ax(Set x <-> x=x)" it
> > > both follows that Et(x in t & y in t) <-> (set y & set x),
> > > and that Et(x in t)& Et(y in t) .<->. set x & set y.
> > >
> > > >
> > > > C4 AyAx[Az(z in y <-> z in x) -> {(set y & set x) <-> y=x}]
> > > > (Equi-membered classes are identical iff these are sets.)
> > > >
> > > > C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}] Weak
> > > > Extensionality (news:<0e14nu484feb15mpa...@4ax.com>)
> > > >
> > > > 3. Is C3 <-> EyAx[x in y <->. Et(x in t) & set y & A] (with y not free
> > > > in A) ?
> > >
> > > No. It does not follow that y is a set. To prove that y is a set,
> > > it must be established that every class with the same members as y
> > > is identical to y. In other words, it must be established that
> > > Leibniz's principle of the identity of indiscernibles holds for
> > > every such class and y. (To establish this, it suffices to
> > > establish that this principle holds for y itself.) If Leibniz's
> > > principle does not hold, then y (along with every class equi-membered
> > > with y) is a proper class.
> >
> > I thought: (x in y)->. E!x & E!y.
> > That is, (x in y) is false unless x and y are existent classes (sets).
>
> --From x in y it follows only that x is a set.
(I sense a contradiction lurking somewhere nearby)
> It does not also follow
> that y is a set. To show that y is a set, it's necessary to show that
> it is a member of some class.
>
> --"x=x" holds for sets and "~(x=x)" for proper classes. From this I
> don't conclude, however, that proper classes are non-existents,
But, you only admit classes. ~E!x <-> ~(set x). Proper classes are
non-sets.
That is to say, Proper classes are non-existents, in your logic.
I see that your axioms are consistent with this translation:
(from Corry logic to Classical logic)
Classical D1: G(x:Fx) defined Ey(Ax(x in y <-> Fx) & Gy)
1. Gx ---> G(x:Fx)
2. ExGx ---> EF(G(x:Fx))
3. AxGx ---> AF(G(x:Fx))
4. E!x ---> E!(x:Fx)
5. Ex(~E!x) ---> EF~(E!(x:Fx))
6. Ex~(x=x) ---> EF~((x:Fx)=(x:Fx))
To translate to Classical logic from Corry logic we need only preface
that all classes exist, ie. that they are sets.
Witt
John
> > > > and classes that are not are proper classes.
> > >
> > > Sets are then existent classes.
> >
> > The logic of existence/non-existence and the logic of sets/proper
> > classes is (arguably) the same logic: a logic in which identity
> > is not reflexive.
>
> in the classical sense, G(x:Fx) <-> Ey(Ax(x in y <-> Fx) & Gy)), then,
> your axioms are true.
>
> That is, your set theory is included in classical logic by extending
> it in this way.
>
> > From this, however, it does not follow that
> > sets *are* existent classes, but that "x=x" holds for sets
> > and existents while "~(x=x)" holds for proper classes and
> > non-existents.
>
> I don't agree.
> Sets are existent classes and proper classes are non-existent classes.
> How can there be exceptions?
> E!x <-> x=x. (set x) <-> x=x. E!x <-> (set x). Therefore ~E!x <->
> ~(set x).
From the fact that all men are mortals, it doesn't follow that
all mortals are men. Similarly, from the fact that all
non-existents are identical-with-nothings, it doesn't follow
that all identical-with-nothings are non-existents.
> > > > > 2. Do you claim: Et(x in t & y in t) <-> (set y & set x), or,
> > > > > Et(x in t)& Et(y in t) .<->. set x & set y. ?
> > > >
> > > > (i) is an instance of C3:
> > > >
> > > > (i) EyAx(x in y <-> (Set x & x=w v x x=z)).
> > > >
> > > > But (ii) follows from (i,C4) (proof supplied upon request):
> > > >
> > > > (ii) EyAx(x in y <-> (x=w v z=z))
> > >
> > > I don't see how you use C3 here.
> >
> > Here's a proof:
> >
> > C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in A) Classification
> > C4 AyAx[Az(z in y <-> z in x) -> {(set y & set x) <-> y=x}]
> > (Equi-membered classes are identical iff these are sets.)
> >
> > 1. Show EyAx(x in y <-> (x=w v x x=z))
> > 2. EyAx(x in y <-> (Set x & x=w v x x=z)) Instance of C3
> > 3. x in y <-> (Set x & x=w v x x=z) 2,EI,UI
> > 4. x in y -> (x=w v x x=z) 3
> > 5. (x=w v x=z) -> x=x "=" is weakly reflexive
> > 6. x=x -> Set x C4
> > 7. (x=w v x=z) -> Set x 5,6
> > 8. (Set x & (x=w v x x=z)) -> x in y 3
> > 9. (x=w v x x=z) -> x in y 7,8
> > 10. x in y <-> (x=w v x x=z) 4,9
> > 11. EyAx(x in y <-> (x=w v x x=z)) 10,UG,EG
>
> I don't yet understand your formulations, eg. what does (x=w v x x=z)
> mean?
"x=w v x=z" means "x=w or x=z"
From C3 it follows that for any w and z there is a class y,
whose members are just those things that are identical with w or
with z:
C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in A)
(i) AwAzEyAx(x in y <-> (Set x & x=w v x x=z))
In the foregoing proof, (2) follows from (i) by two applications of UI.
>
> >
> > >
> > > >
> > > > On the other hand, from (ii,C4) and "Ax(Set x <-> x=x)" it
> > > > both follows that Et(x in t & y in t) <-> (set y & set x),
> > > > and that Et(x in t)& Et(y in t) .<->. set x & set y.
> > > >
> > > > >
> > > > > C4 AyAx[Az(z in y <-> z in x) -> {(set y & set x) <-> y=x}]
> > > > > (Equi-membered classes are identical iff these are sets.)
> > > > >
> > > > > C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}] Weak
> > > > > Extensionality (news:<0e14nu484feb15mpa...@4ax.com>)
> > > > >
> > > > > 3. Is C3 <-> EyAx[x in y <->. Et(x in t) & set y & A] (with y not free
> > > > > in A) ?
> > > >
> > > > No. It does not follow that y is a set. To prove that y is a set,
> > > > it must be established that every class with the same members as y
> > > > is identical to y. In other words, it must be established that
> > > > Leibniz's principle of the identity of indiscernibles holds for
> > > > every such class and y. (To establish this, it suffices to
> > > > establish that this principle holds for y itself.) If Leibniz's
> > > > principle does not hold, then y (along with every class equi-membered
> > > > with y) is a proper class.
> > >
> > > I thought: (x in y)->. E!x & E!y.
> > > That is, (x in y) is false unless x and y are existent classes (sets).
> >
> > --From x in y it follows only that x is a set.
>
> (I sense a contradiction lurking somewhere nearby)
From the fact that x is an element of y, why should it follow that for
some z, y is an element of z? If this followed in NBG theories, then
then every such theory would be inconsistent!
>
> > It does not also follow
> > that y is a set. To show that y is a set, it's necessary to show that
> > it is a member of some class.
> >
> > --"x=x" holds for sets and "~(x=x)" for proper classes. From this I
> > don't conclude, however, that proper classes are non-existents,
>
> But, you only admit classes. ~E!x <-> ~(set x). Proper classes are
> non-sets. That is to say, Proper classes are non-existents,
> in your logic.
Once again, from the fact that all non-existents are identical-with-
nothings, it doesn't follow that all identical-with-nothings are
non-existents. A logic in which identical-with-nothings are accessible
to variables has varying applications. IMHO the only thing proper
classes have in common with non-existents is that both satisfy
~(x=x). (Certainly we would not want to identify you-know-who
with a horse's ass, JUST on the grounds that both satisfy x=x.)
> > > I don't think we can eliminate descriptions, do you?
> > > IMHO, I think your free set theory can be produced as an extension
> > > of classical logic, i.e. Description theory.
> >
> > In a logic with non-reflexvie identity, the key definitions
> > of Russell's Theory of Descriptions can be introduced as axioms.
> >
> > >
> > > EF(~E!(ixFx)), and EF(~((ixFx)=(ixFx))) .. instead of your
> > > Ex(~E!x) and Ex(~(x=x)).
> > >
> > > And classical logic is included in your free set theory:
> > > AxE!x -> Ax(x=x), i.e. by restricting your free classes to sets.
> >
> > I don't understand this.
>
> I see that your axioms are consistent with this translation:
> (from Corry logic to Classical logic)
>
> Classical D1: G(x:Fx) defined Ey(Ax(x in y <-> Fx) & Gy)
>
> 1. Gx ---> G(x:Fx)
> 2. ExGx ---> EF(G(x:Fx))
> 3. AxGx ---> AF(G(x:Fx))
> 4. E!x ---> E!(x:Fx)
>
> 5. Ex(~E!x) ---> EF~(E!(x:Fx))
> 6. Ex~(x=x) ---> EF~((x:Fx)=(x:Fx))
>
> To translate to Classical logic from Corry logic we need only preface
> that all classes exist, ie. that they are sets.
>
> Witt
From the fact that all non-existents are identical-with-nothings it
does not follow that all identical-with-nothings are non-existents.
Otherwise, you are correct: to translate from Correy
set theory to ZF set theory (cutting some slack to those
who would rather that set theory not be a chapter of logic),
one need only assume that all classes bear "=" to themselves.
>
--John
Witt
Witt:
Are your first order axioms,
1. Fx -> E!x
2. E!x -> x=x
3. x=y ->. E!x & E!y & (Fx<->Fy).
John:
No, but these follow from the axioms.
From 3. x=y ->. E!x & E!y & (Fx<->Fy), we get,
3a. x=x -> E!x, i.e. Ax(~E!x -> ~(x=x))
(all non-existents are identical-with-nothings)
From 2. E!x -> x=x, we get,
2a. Ax(~(x=x) -> ~E!x)
(all identical-with-nothings are non-existents)
4. Ax(~E!x <-> ~(x=x)), from 2a and 3a.
(all identical-with-nothings are equal to non-existents)
Where am I going wrong?
Witt
Witt
If you agree that we can represent your set theory within classical logic,
such that your variable x is represented by (x:Fx), then,
A. G(x:Fx) <-> Ey(Ax(x e y <-> Fx) & Gy).
B. E!(x:Fx) <-> Ey(x e y <-> Fx)
C. y e (x:Fx) <->. E!(x:Fx) & Fy.
Your (x e y) becomes:
1. (x:Fx) e (x:Gx) <-> Ey(Ax((x e y <-> Fx) & y e (x:Gx)), by A.
2. (x:Fx) e (x:Gx) <-> Ey(Ax((x e y <-> Fx) & Ez(Ax(x e z <-> Gx) & Gy), by C.
3. (x:Fx) e (x:Gx) ->. Ey(Ax((x e y <-> Fx) & Ez(Ax(x e z <-> Gx),
by p & q & r .->. p & q, MP.
4. (x:Fx) e (x:Gx) ->. E!(x:Fx) & E!(x:Gx), by B.
That is, x e y ->. E!x & E!y, is true in your set theory.
Witt
John
> john_...@yahoo.com (John) wrote in message news:<c37480a7.03092...@posting.google.com>...
> > > > > I thought: (x in y)->. E!x & E!y.
> > > > > That is, (x in y) is false unless x and y are existent classes (sets).
> > > >
> > > > --From x in y it follows only that x is a set.
> > >
> > > (I sense a contradiction lurking somewhere nearby)
> >
> > From the fact that x is an element of y, why should it follow that for
> > some z, y is an element of z? If this followed in NBG theories, then
> > then every such theory would be inconsistent!
>
> If you agree that we can represent your set theory within classical logic,
> such that your variable x is represented by (x:Fx)
"x" cannot be 'represented by' {x:Fx}, because variables in class
theories (including mine) range over classes--of which some (but
not all) are sets. But "{x:Fx}" designates a set. Therefore,
"x" cannot be 'represented by' "{x:Fx}".
>
> A. G(x:Fx) <-> Ey(Ax(x e y <-> Fx) & Gy).
> B. E!(x:Fx) <-> Ey(x e y <-> Fx)
> C. y e (x:Fx) <->. E!(x:Fx) & Fy.
>
> Your (x e y) becomes:
No, it doesn't. See above.
>
> 1. (x:Fx) e (x:Gx) <-> Ey(Ax((x e y <-> Fx) & y e (x:Gx)), by A.
> 2. (x:Fx) e (x:Gx) <-> Ey(Ax((x e y <-> Fx) & Ez(Ax(x e z <-> Gx) & Gy), by C.
> 3. (x:Fx) e (x:Gx) ->. Ey(Ax((x e y <-> Fx) & Ez(Ax(x e z <-> Gx),
> by p & q & r .->. p & q, MP.
> 4. (x:Fx) e (x:Gx) ->. E!(x:Fx) & E!(x:Gx), by B.
>
> That is, x e y ->. E!x & E!y, is true in your set theory.
--John
John
I notice you posted on the epsilon operator some time back.
I am wondering whether, where quanta are concerned, an
epsilon operator might be of some use in referring to
buggers that can't be named:
"The epsilon calculus is a logical formalism developed by David
Hilbert in the service of his program in the foundations
of mathematics. The epsilon operator is a term-forming operator
which replaces quantifiers in ordinary predicate logic.
Specifically, in the calculus, a term x A denotes some x
satisfying A(x), if there is one. In Hilbert's Program,
the epsilon terms play the role of ideal elements; the
aim of Hilbert's finitistic consistency proofs is to
give a procedure which removes such terms from a formal
proof. The procedures by which this is to be carried out
are based on Hilbert's epsilon substitution method. The
epsilon calculus, however, has applications in other
contexts as well. The first general application of the
epsilon calculus was in Hilbert's epsilon theorems,
which in turn provide the basis for the first correct
proof of Herbrand's theorem. More recently, variants
of the epsilon operator have been applied in linguistics
and linguistic philosophy to deal with anaphoric pronouns."
(from Jeremy Avigad's piece on the epsilon calculus, at
<http://plato.stanford.edu/entries/epsilon-calculus/>)
--John
"One-a-quanta two-a-quanta three-a-quanta four,
On both sides of the bathroom door."
--A Nony Mous
Witt
"John" <john_...@yahoo.com> wrote in message
news:c37480a7.03092...@posting.google.com...
> oori...@yahoo.com (Witt) wrote in message
news:<196b22e7.03092...@posting.google.com>...
>
> I notice you posted on the epsilon operator some time back.
> I am wondering whether, where quanta are concerned, an
> epsilon operator might be of some use in referring to
> buggers that can't be named:
Yes, you are a funny dude.
It seems to me that all references to 'unlikely objects' are buggers.
I am delighted that you noticed my remarks about indefinite descriptions.
Your opinions are always, more than welcome.
The Boyz of the group notwithstanding.
It is disconcerting that the Boyz always resort to insults and flaming,
in one way or other. Franz is simply more blatent than David,
But, bad manners 'sucks', don't you think so?
My expression of 'indefinite descriptions' is derived from Russell's 'On
Denoting'.
And, I believe that it shows that Hilbert's 'epsilon function' rendition is
in error.
If you are interested, I will persue this point further.
Please keep in mind that, I am not educated like you and most others on this
board are.
I have not attended any logic courses at all. I am self-decieved, (hahaha).
Please show that I am wrong, slowly: so that I can understand .
Witt
Witt
news:<196b22e7.03092...@posting.google.com>...
> > john_...@yahoo.com (John) wrote in message
news:<c37480a7.03092...@posting.google.com>...
>
> > > > > > I thought: (x in y)->. E!x & E!y.
> > > > > > That is, (x in y) is false unless x and y are existent classes
(sets).
> > > > >
> > > > > --From x in y it follows only that x is a set.
> > > >
> > > > (I sense a contradiction lurking somewhere nearby)
> > >
> > > From the fact that x is an element of y, why should it follow that for
> > > some z, y is an element of z? If this followed in NBG theories, then
> > > then every such theory would be inconsistent!
> >
> > If you agree that we can represent your set theory within classical
logic,
> > such that your variable x is represented by (x:Fx)
>
> "x" cannot be 'represented by' {x:Fx}, because variables in class
> theories (including mine) range over classes--of which some (but
> not all) are sets. But "{x:Fx}" designates a set. Therefore,
> "x" cannot be 'represented by' "{x:Fx}".
I think you have misunderstood me.
I agree that class theories range over possible classes, some of which do
not exist,
e.g. the Russell class.
The expression {x:Fx} is not a set, rather, if it exists then it does
represent sets,
otherwise it represents proper classes ...non-existent classes.
{x:Fx} defined (iy:Ax(x e y <-> Fx))
If Ax(x e y <-> Fx)) is false then the purported class does not exist.
G{x:Fx} <-> Ey(Ax(x e y <-> Fx) & Gy).
Sets can be defined as those classes for which E!{x:Fx) is true, where,
E!{x:Fx} <-> EyAx(x e y <-> Fx).
i.e. sets are 'existent' classes.
(Sets) ..{x|Fx} defined (iy: Ax(x e y <->. Ez(Ax(x e z <-> Fx) & Fx)) .
This definition has the advantage that all sets exist, including the Russell
set.
And, they behave as the individual variables do in classical logic.
That is, proper classes and ultimate classes (non-existent classes) are each
equal to the null set.
E!{x:Fx} ->. G{x|Fx} <->. Ey(Ax(x e y <-> Fx) & Gy), is a theorem.
And, EyAx(x e y <->. Ez(Ax(x e z <-> Fx) & Fx)), is a theorem.
(Given that the null set exists.)
G{x|Fx} <-> Ey(Ax(x e y <->. EzAx(x e z <-> Fx) & Fx) & Gy).
It seems to me that, You and von Neumann and Quine want to say:
E!(x|Fx) <-> EyAx(x e y <->. Ez(x e z) & Fx).
But this approach denies the universal set and other sets which we cannot
deny.
Zermelos' Aussonderung principle fits in here too.
His: E!(x|Fx) <-> Ey(Ax(x e y <->. x e z & Fx), fails similarily. (imho)
Witt
>
> >
> > A. G(x:Fx) <-> Ey(Ax(x e y <-> Fx) & Gy).
> > B. E!(x:Fx) <-> EyAx(x e y <-> Fx)
John
> "John" <john_...@yahoo.com> wrote in message
> news:c37480a7.03092...@posting.google.com...
> > oori...@yahoo.com (Witt) wrote in message
> news:<196b22e7.03092...@posting.google.com>...
> > > john_...@yahoo.com (John) wrote in message
> news:<c37480a7.03092...@posting.google.com>...
>
> > > > > > > I thought: (x in y)->. E!x & E!y.
> > > > > > > That is, (x in y) is false unless x and y are existent classes
> (sets).
> > > > > >
> > > > > > --From x in y it follows only that x is a set.
> > > > >
> > > > > (I sense a contradiction lurking somewhere nearby)
> > > >
> > > > From the fact that x is an element of y, why should it follow that for
> > > > some z, y is an element of z? If this followed in NBG theories, then
> > > > then every such theory would be inconsistent!
> > >
> > > If you agree that we can represent your set theory within classical
> logic,
> > > such that your variable x is represented by (x:Fx)
> >
> > "x" cannot be 'represented by' {x:Fx}, because variables in class
> > theories (including mine) range over classes--of which some (but
> > not all) are sets. But "{x:Fx}" designates a set. Therefore,
> > "x" cannot be 'represented by' "{x:Fx}".
>
> I think you have misunderstood me.
Many have taken "(a exists)" to *mean* "Ey(a=y)", and "~(a exists)"
to mean "~Ey(a=y)". Now, if you agree with this construal of
(a exists)/~(a exists) (and think there are non-existents), you
are committed to a logic which differs from the standard one,
where variables range over--and singular terms denote--identical-
with-somethings only.
However, granted that non-existents are identical-with-nothings,
IT DOES NOT FOLLOW that identical-with-nothings are non-existents.
Proper classes are a case in point. From C4 it follows that proper
classes are identical-with-nothings.
> > C4 AyAx[Az(z in y <-> z in x) -> {(set y & set x) <-> y=x}]
> > (Equi-membered classes are identical iff these are sets.)
But it does not follow that proper classes are non-existents.
And it will not follow that proper classes are non-existents
until you have fitted out your set theory with a predicate
"exists" and justified WITHHOLDING that predicate from proper classes.
As long as you have made no case for introducing this predicate
and withholding it from proper classes, the proofs that you
display in the latter part of your posting are pointless.
--John
Witt
"John" <john_...@yahoo.com> wrote in message
news:c37480a7.03100...@posting.google.com...
Witt
"Witt" <oori...@yahoo.com> wrote in message
news:ZL8fb.77185$3r1....@news02.bloor.is.net.cable.rogers.com...
Witt
It is an understanding that, we must exercise an expression of identity.
"Witt" <oori...@yahoo.com> wrote in message
news:ffhfb.82344$3r1....@news02.bloor.is.net.cable.rogers.com...
John
> You are wrong and I don't want to discus the matter futher.
You claim two things: (1) that sets are 'existent' classes,
and (2) that in C, x e y ->. E!x & E!y: that is, to use my
notation: x in y -> Ez(x=z) & Ez(y=z).
In support of (1), you might claim that "to exist
means to have a property". Hence (you might conclude) if
classes are properties-in-extension, and sets are elements,
then sets are 'existent' class.
In support of (2) you offer a proof in framed in an extension
of the language of C, which contains (in addition to "E!"),
expressions whose meaning to me is so unclear--I am referring
to your "{x:Fx}"--that I am unable to judge your reasoning.
This being the case, I''ll proceed in the following manner.
First I will prove (in the pertinent fragment of C) that
~AxAy(x in y -> Ez(y=z). I will then ask you to provide
a proof, couched in the language of this fragment,
that AxAy(x in y -> Ez(y=z).
My proof presupposes FOL, three premises and one
definition:
C2 AxAy[x=y -> Az(z in x <-> z in y)]
C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in A)
C4 AyAx[Az(z in y <-> z in x) -> {(set y & set x) <-> y=x}]
(Equi-membered classes are identical iff these are sets.)
D1: "set x" means "Et(x in t)"
To Boyz-proof this proof, I will provide an auxiliary
proof establishing that identity is symmetric and
transitive (in FOL + (C2-C4, D2)) Although elementary
the auxiliary proof is long, so please be patient.
--John
John
> Hello again David,
>
> It is an understanding that, we must exercise an expression of identity.
David Ullrich, I presume.
--John
"This was the unkindest cut of all."
--From Julius Caesar (III, ii, 187)
John
> Evidenly You do not understand what I say, Why is that?
The proof that = is symmetric and transitive employs FOL, C2
and C4, and D1:
C2 AxAy[x=y -> Az(x in z <-> y in z) & Az(z in x <-> z in y)]
C4 AyAx[Az(z in y <-> z in x) -> {(set y & set x) <-> y=x}]
(Equi-membered classes are identical iff these are sets.)
D1: D1: "set x" means "Et(x in t)"
I. Symmetry
Suppose x=y. Then x and y are equi-membered (C2).
Hence y and x are equi-membered. Moreover, since
x=y, x and y are sets (C4). Therefore, since y and
x are sets, and y and x are equi-membered, y=x (C4).
Hence if x=y, then y=x. The same argument shows
that if y=x, then x=y. So x=y <-> y=x. Consequently,
identity is symmetric.
II. Transitivity
Suppose x=y & y=z. Then x and z are sets (C4).
Moreover, since x=y, x and y are equi-membered (C2).
And since y=z, y and z are equi-membered (C2).
Therefore, since x and y are equi-membered and
y and z are equi-membered, x and z are equi-membered.
Therefore, since x and z are equi-membered and sets,
x=z. So, if x=y & y=z, then x=z. Consequently,
identity is transitive.
III. From FOL, the symmetry and transitivity of identity,
and (C2,C3,C4;D1), it follows that some non-empty
class is an identical-with-nothing.
C2 EyAx[x in y <-> Et(x in t) & A] (with y not free in A)[1]
C3 Ew(w=w & ~(w in w))[2]
C4 AyAx[Az(z in y <-> z in x) -> {(set y & set x) <-> y=x}]
(Equi-membered classes are identical iff these are sets.)
D1: "set x" means "Et(x in t)"
1. Show Ey(Ex(x in y) & Ãz~(y=z))
2. EyAx[x in y <-> (Et(x in t) & ~(x in x))] Instance of C2
3. Ew(w=w & ~(w in w)) C3
4. w=w & ~(w in w) 3,EI
5. Ax[x in r <-> (Et(x in t & ~(x in x))] 2,EI
6. w in r <-> (Et(w in t) & ~(w in w)) 5,UI
7. w=w 4
8. Et(w in t) 7,C4
9. w in r 4,8,6
10. r in r <-> (Et(r in t & ~(r in r)) 5,UI
11. ~Et(r in t) 10
12. ~(set r) 11,D1
13. ~(r=r) 12,C4
14. Az~(r=z) From the symmetry & transitivity of =
and ~(r=r) it follows that Az~(r=z):
proof supplied upon request.
15. Ex(x in r) 9,EG
16. Ex(x in r) & Az~(r=z) 14,15
17. Ey(x in y & Az~(y=z)) 16,EG: Cancel "Show" at line 1
**********************************************************
[1] Nothing about this proof changes if NBG classification
is substituted for C2.
[2] C3 is a corollary of the empty set theorem in
news:<70f94e16.02090...@posting.google.com>.
--John
PS From the same assumptions that I employ, please
try to prove: Ay(Ex(x in y) -> Ez(y=z)). If you are
satisfied you can't do this, we can take a look at how
one goes about introducing and eliminating "{x:Fx}".
John
Line 17 should be:
17. Ey(Ex(x in y) & Az~(y=z)) 16,EG: Cancel "Show" at line 1