N is a Conservative Extension of C
John
theorems of the theory built relative to FOL by adding D1-D8 as
axioms.
D1 "x=y" means "Az(z in x <-> z in y)"
D2 "set x" means "Et(x in t)"
N1 AxAyAz[x=y -> (x in z <-> y in z)] Leibniz 1
N2 AxAyAz[x=y -> (z in x <-> z in y)] Leibniz 2
N3 EyAx[x in y <-> (Et(x in t) & A)] (with y not free in A)
Classification
N4 AxAy[Az(z in x <-> z in y) -> x=y] Extensionality
N5 AaAb[set a & set b -> Ey(set y & Ax(x in y <-> x=a v x=b))] Pairs
N6 Aa(set a -> Ay(Ax(x in y -> x in a) -> set y)) Subsets
N7 Aa[set a -> Ey(set y & Ax(x in y <-> Ew(w in a & x in w)))] Unions
N8 Aa[set a -> Ey(set y & Ax(x in y <-> Aw(w in x -> w in a)))] Power
Set
C1 AxAy[xIy -> Az(z in x <-> z in y)] Leibniz 1
C2 AxAy[Az(x in z <-> y in z) <-> Az(z in x <-> z in y)] Leibniz 2
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) <-> xIy}] Weak
Extensionality
C5 AaAbAy[Ax(x in y <-> (xIa v xIb)) -> yIy] Pairs
C6 AaAy[Ax(x in y -> x in a) -> (aIa -> yIy)] Subsets
C7 AaAy[Ax(x in y <-> Ew(w in a & x in w)) -> (aIa -> yIy)] Unions
C8 AaAy[Ax(x in y <-> Aw(w in x -> w in a)) -> (aIa -> yIy)] Power Set
N1-N8 (hereafter N) is thus a conservative extension of C1-C8
(hereafter C), every theorem of N being provable in C. Proofs to
follow.
" Exhibit a proof of Ex~(x=x) from C1-C4 and someone will point out
the error "
--Dullrich
news:<0e14nu484feb15mpa...@4ax.com>
David C. Ullrich
Um, supposing for the sake of argument that you succeed in showing
that every theorem of N is provable in C. Exactly how does that
show that Ex~(x=x) follows from C1-C4?
>--Dullrich
>news:<0e14nu484feb15mpa...@4ax.com>
John
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
Concerning the above, in news:<0e14nu484feb15mpa...@4ax.com>
you wrote:
> Exhibit of proof of Ex~(x=x) from
> C1-C4 and someone will point out the error.
However, in the post to which you are now 'responding', I use "I" to
symbolize identity-in-C. Therefore, what you shouuld have asked
but didn't, was:
> Um, supposing for the sake of argument that you succeed in showing
> that every theorem of N is provable in C. Exactly how does that
> show that Ex~(xIx) follows from C1-C4?
To which the (obvious!) answer is: it doesn't. To show that N is a
conservative extension of C, it isn't necessary to show that "Ex~(xIx)"
follows from C1-C4.
David C. Ullrich
You've changed your notation, which is wholly admirable. Saying
that I "should" have anticipated the change in notation before it
happened is a little questionable, but never mind that.
>To which the (obvious!) answer is: it doesn't. To show that N is a
>conservative extension of C, it isn't necessary to show that "Ex~(xIx)"
>follows from C1-C4.
I see. Somehow I thought this had to do with that other issue.
It didn't, you just felt like asserting that N is a conservative
extension of C. Good for you.
John
I claim that N is a sub-theory of C, in which "I" symbolizes IMPUDENT,
UPSTART, IN-ULLRICH'S-FACE identity--which is symmetric, transitive
and non-reflexive.
And if as I claim, N *is* a sub-theory of C, everyone but Ullrich
and the Boyz will be granted free use of "x=y", as an abbreviation
for "Az(z in x <-> z in y)".
But every trick Ullrich and the Boyz turn with "=" will cost them.
--John
David C. Ullrich
In case you're curious, you're _really_ beginning to sound like
a raving lunatic, warning us of the Consequences if we don't
shape up.
>--John
John
Hey Homie:
By this do you mean to suggest that I can't *prove* that N
is a conservative extension of C? Or, do you mean to suggest
that it *makes no difference* to you and the Boyz--Herb
Enderton & Robin Chapman & Daryl McCullough & Torkel Franzen
& Bill Taylor & Mike Oliver and Keith Ramsay--whether or
not N is a conservative extension of C?
What do you mean, Homie? What do you mean?
--John
John
My contempt for you and your ilk could not be more profound.
>
> >--John
John
You appear to have trouble with the vernacular, so
let me put this another way.
What you and the Boyz are accustomed to knocking
off for free, you may soon have to pay for.
Didn't Professor Rudin tell you, there's no
Free Lunch?
--John
David C. Ullrich
Whee. The funny thing is that this is what I _thought_
you meant, and it's certainly the meaning I had in mind
when I said you were sounding like a raving lunatic,
but I wasn't _certain_ that it was what you meant.
(Hence my amusement at your conjecturing that I
had trouble with the vernacular - why would you
deduce that from what I said? It doesn't sound
a little off to you, when you hear someone warning
us that there will be a Price to pay, and Soon,
if we don't shape up and agree that things are
not necessarily equal to themselves? heh-heh.)
Maybe you should clarify exactly what we're
going to have to "pay" and why...
>Didn't Professor Rudin tell you, there's no
>Free Lunch?
>
>--John
David C. Ullrich
David C. Ullrich
What I meant was exactly what I said.
>--John
David C. Ullrich
__________________________
"But every trick Ullrich and the Boyz turn with "=" will cost them."
John Correy, warning us of the consequences of assuming the everything
is the same as itself. <70f94e16.02090...@posting.google.com>
John
In news:<70f94e16.02090...@posting.google.com>, I write:
> And if as I claim, N *is* a sub-theory of C, everyone but Ullrich
> and the Boyz will be granted free use of "x=y", as an abbreviation
> for "Az(z in x <-> z in y)".
>
> But every trick Ullrich and the Boyz turn with "=" will cost them.
In news:<0tfanucgrq5nhpsc7...@4ax.com>
you twist the last line into a threat.
>
> In case you're curious, you're _really_ beginning to sound like
> a raving lunatic, warning us of the Consequences if we don't
> shape up.
In news:<5n0cnuc0cav6k1jr7...@4ax.com>, you display
this line out of context and impute to me something you know to
be false.
> __________________________
> "But every trick Ullrich and the Boyz turn with "=" will cost them."
> John Correy, warning us of the consequences of assuming the everything
> is the same as itself. <70f94e16.02090...@posting.google.com>
--John
" Uh, no. I have never intentionally misrepresented anything you've said. "
--David Ullrich
news:<2ogpmuoudtj4l0eqj...@4ax.com>
David C. Ullrich
That's correct.
David C. Ullrich
John
> >
> >However, in the post to which you are now 'responding', I use "I" to
> >symbolize identity-in-C. Therefore, what you shouuld have asked
> >but didn't, was:
> >
> >> Um, supposing for the sake of argument that you succeed in showing
> >> that every theorem of N is provable in C. Exactly how does that
> >> show that Ex~(xIx) follows from C1-C4?
>
> You've changed your notation, which is wholly admirable. Saying
> that I "should" have anticipated the change in notation before it
> happened is a little questionable, but never mind that.
>
> >To which the (obvious!) answer is: it doesn't. To show that N is a
> >conservative extension of C, it isn't necessary to show that "Ex~(xIx)"
> >follows from C1-C4.
>
> I see. Somehow I thought this had to do with that other issue.
> It didn't, you just felt like asserting that N is a conservative
> extension of C. Good for you.
In what follows, "I" symbolizes non-reflexive, in-Ullrich's-face
identity.
0. The fundamental principles of C are Classification, which asserts that
every formula "P(x)" determines a class containing just those elements which
satisfy "P(x)";
C3 EyAx[x in y <-> Et(x in t) & Px] (with y not free in "Px")
and Weak Extensionality, which asserts that equi-membered classes
are identical just in case these are elements.)
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}] Weak
Extensionality
For "identical-with-something" I'll henceforth write "iws"--and for
"identical-with-nothing", "iwn".)
1. From (C3,C4) it follows that there is a class with no elements (T1); a
class which contains a,b--for arbitrary a,b--iff a and b are iws's (T2); a
class which contains the elements of the elements of any class a (T3); and a
class which contains those sub-classes of a that are elements. (Proofs: 1)
T1 EyAx~(x in y)
(There is a null class.)
T2 AaAbEyAx(x in y <-> xIa v xIb)
(For every a,b, and some y, a and b are in y iff these are iws's.)
T3 AaEyAx(x in y <-> Ew(w in a & x in w))
(For every a, there is a class which contains just the elements of
elements of a.)
T4 AaEyAx(x in y <-> (Et(x in t) & Aw(w in x -> w in a)))
(For every a there is a class of just those sub-classes
of a that are iws's.)
2. From (C3,C4) it follows that identity is symmetric, transitive, and
weakly reflexive--and that identicals belong to the same classes and have the
same members. (Proofs: 2).
3. The proof that N is a conservative extension of C will also make frequent use
of the following theorems. (Proofs: 3)
T5 Ax(xIx <-> Ey(xIy))
(Self-identicals are iws's, and conversely.)
T6 Ax(Ey(xIy) <-> Ey(x in y))
(Iws's are elements, and conversely.)
T7 AxAy[Az(z in x <-> z in y) -> (xIx <-> yIy)]
(If x,y are equi-membered, both or neither are elements.)
Jeffrey Ketland
N isn't even an _extension_ of C, since C contains axioms expressed in
notation ("|") not in the language of N. So, N is certainly not a
conservative extension of C. How for example is C5 proved in N?
--- Jeff
John
> David C. Ullrich <ull...@math.okstate.edu> wrote in message news:<5n0cnuc0cav6k1jr7...@4ax.com>...
>
> In news:<70f94e16.02090...@posting.google.com>, I write:
>
> > And if as I claim, N *is* a sub-theory of C, everyone but Ullrich
> > and the Boyz will be granted free use of "x=y", as an abbreviation
> > for "Az(z in x <-> z in y)".
> >
> > But every trick Ullrich and the Boyz turn with "=" will cost them.
>
> In news:<0tfanucgrq5nhpsc7...@4ax.com>
> you twist the last line into a threat.
>
> >
> > In case you're curious, you're _really_ beginning to sound like
> > a raving lunatic, warning us of the Consequences if we don't
> > shape up.
>
> In news:<5n0cnuc0cav6k1jr7...@4ax.com>, you display
> this line out of context and impute to me something you know to
> be false.
>
> > __________________________
> > "But every trick Ullrich and the Boyz turn with "=" will cost them."
> > John Correy, warning us of the consequences of assuming the everything
> > is the same as itself. <70f94e16.02090...@posting.google.com>
>
> --John
Hint: if N is a conservative extension of C, C *proves* Ax(x=x).
>
> " Uh, no. I have never intentionally misrepresented anything you've said. "
> --David Ullrich
> news:<2ogpmuoudtj4l0eqj...@4ax.com>
" Much malice mingled with a little wit. "
--John Dryden (1631 - 1700)
_The Hind and the Panther_ [1687], pt. III, l. 1
John
C includes the definition D1.
D1: "x=y" means "Az(z in x <-> z in y)"
So, C proves N1
N1 AxAyAz[x=y -> (x in z <-> y in z)],
by proving, "AxAyAz[At(t in x <-> t in y) -> (x in z <-> y in z)].
--John
John
>
PROOFS 2:
The following proofs employ the deductive apparatus of FOL and axioms
C1-C4. For ease of reading, C1-C4 precede each proof.
Proofs 2:
C1 AxAy[xIy -> 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) & P(x)] (with y not free in P(x))
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}] Weak
Extensionality
I1: AxAy(xIy <-> yIx)
(Identity is symmetric)
1. Show AxAy(xIy <-> yIx)
2. Show xIy <-> yIx
3. Show xIy -> yIx
4. xIy Assume
5. Show yIx
6. Az(z in x <-> z in y) 4,C1
7. Et(x in t & y in t) 4,6,C4
8. Et(y in t & x in t) 7
9. Az(z in y <-> z in x) 6
10. yIx 8,9,C4:Ca Sh(5)
11. xIy -> yIx 4,5:Ca Sh(3)
12. Proof of yIx -> xIy is similar
13. xIy <-> yIx 3,12:Ca Sh(2)
14. AxAy(xIy <-> yIx) 2:Ca Sh(1)
C1 AxAy[xIy -> 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) & P(x)] (with y not free in P(x))
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}] Weak
Extensionality
I2: AxAyAz((xIy & yIz) -> xIz)
(Identity is transitive.)
1. Show AxAyAz((xIy & yIz) -> xIz)
2. Show (xIy & yIz) -> xIz
3. xIy & yIz Assume
4. Show xIz
5. At(t in x <-> t in y) 3,C1
6. At(t in y <-> t in z) 3,C1
7. At(t in x <-> t in z) 5,6
8. At(x in t <-> y in t) 5,C2
9. Et(y in t & z in t) 3,6,C4
10. y in r & z in r 9,EI
11. x in r <-> y in r 8,UI
12. x in r 10,11
13. x in r & z in r 10,12
14. Er(x in r & z in r) 13,EG
15. xIz 7,14,C4:Can Sh(4)
16. (xIy & yIz) -> xIz 3,4:Can Sh(2)
17. AxAyAz((xIy & yIz) -> xIz) 2:Can Sh(1)
C1 AxAy[xIy -> 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) & P(x)] (with y not free in P(x))
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}] Weak
Extensionality
I3: Ax[Ey(xIy) -> xIx]
(Identity is weakly reflexive)
1. Show Ax[Ey(xIy) -> xIx]
2. Show Ey(xIy) -> xIx
3. Ey(xIy) Assume
4. Show xIx
5. xIy 3,EI
6. yIx 5,I1
7. xIx 5,6,I2:Can Sh(4)
8. Ey(xIy) -> xIx 3,4:Can Sh(2)
9. Ax[Ey(xIy) -> xIx] 2:Can Sh(1)
C1 AxAy[xIy -> 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) & P(x)] (with y not free in P(x))
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}] Weak
Extensionality
I4: AxAyAz(xIy -> (x in z <-> y in z))
(Identicals belong to the same classes.)
1. Show AxAyAz(xIy -> (x in z <-> y in z))
2. Show AxAy(xIy -> Az(x in z <-> y in z))
3. Show xIy -> Az(x in z <-> y in z)
4. xIy Assume
5. Show Az(x in z <-> y in z)
6. Az(z in x <-> z in y) 4,C1
7. Az(x in z <-> y in z) 6,C2:Can Sh(5)
8. xIy -> Az(x in z <-> y in z) 4,5:Can Sh(3)
9. AxAy(xIy -> Az(x in z <-> y in z)) 3:Can Sh(2)
10.AxAyAz(xIy -> (x in z <-> y in z)) 2:Can Sh(1)
C1 AxAy[xIy -> 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) & P(x)] (with y not free in P(x))
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}] Weak
Extensionality
I5: AxAyAz(xIy -> (z in x <-> z in y))
(Identicals have the same members.)
1. Show AxAyAz(xIy -> (z in x <-> z in y))
2. Show AxAy(xIy -> Az(z in x <-> z in y))
3. AxAy(xIy -> Az(z in x <-> z in y)) C1:Can Sh(2)
4. AxAyAz(xIy -> (z in x <-> z in y)) 2:Can Sh(1)
Jeffrey Ketland
That's OK (and might show that C is an extension of N), but the question
concerns the other way round, whether N is an extension of C, as your
original proposal requires. The proposal is that N is a conservative
extension C. If so, N has to be an extension of C. That means:
(N extends C) every theorem of C is a theorem of N
In particular, you need to show that N proves C1, C2, C3, etc. This would
involve giving a definition of the primitive predicate "x | y".
--- Jeff
David C. Ullrich
Who do you think you are, telling _him_ what the word "extension"
means?
>In particular, you need to show that N proves C1, C2, C3, etc. This would
>involve giving a definition of the primitive predicate "x | y".
>
>--- Jeff
>
David C. Ullrich
Jeffrey Ketland
Yes, I know what you mean ...
More seriously, John seems a bit confused about what an extension is. In
fact, I think he intends his idea the other way round: namely, that C is a
conservative extension of N.
--- Jeff
David C. Ullrich
<ket...@ketland.fsnet.co.uk> wrote:
>David C. Ullrich wrote in message ...
>>On Thu, 5 Sep 2002 13:46:07 +0100, "Jeffrey Ketland"
>><ket...@ketland.fsnet.co.uk> wrote:
>>
[...]
>>>
>>>That's OK (and might show that C is an extension of N), but the question
>>>concerns the other way round, whether N is an extension of C, as your
>>>original proposal requires. The proposal is that N is a conservative
>>>extension C. If so, N has to be an extension of C. That means:
>>>
>>>(N extends C) every theorem of C is a theorem of N
>>
>>Who do you think you are, telling _him_ what the word "extension"
>>means?
>
>Yes, I know what you mean ...
>More seriously, John seems a bit confused about what an extension is. In
>fact, I think he intends his idea the other way round: namely, that C is a
>conservative extension of N.
What's your WARRANT for the assertion that N extends C if every
theorem of C is a theorem of N? All you Boyz can do is parrot what
you've read in books. Just another usenet marauder who's never
heard of non-euclidean geometry.
[Insert some quote explaining that definitions are arbitrary, from
which it follows that N extends C if every theorem of N is a theorem
of C, and anyone who thinks otherwise is in favor of lynching.]
John
Jeff, I'd appreciate your help in matters of terminology. Here--as you
noted, and thanks for correcting the error--I should have said that C
is an extension of N.
What I don't understand about your comment is this. If I show that every
axiom of N is a theorem of C, haven't I shown that C *contains* N, in
the sense that N proves nothing that C doesn't also prove?
--John
John
> >>
> >> N isn't even an _extension_ of C, since C contains axioms expressed in
> >> notation ("|") not in the language of N. So, N is certainly not a
> >> conservative extension of C. How for example is C5 proved in N?
> >>
> >> --- Jeff
> >
> >C includes the definition D1.
> >
> >D1: "x=y" means "Az(z in x <-> z in y)"
> >
> >So, C proves N1
> >
> >N1 AxAyAz[x=y -> (x in z <-> y in z)],
> >
> >by proving, "AxAyAz[At(t in x <-> t in y) -> (x in z <-> y in z)].
>
> That's OK (and might show that C is an extension of N), but the question
> concerns the other way round, whether N is an extension of C, as your
> original proposal requires. The proposal is that N is a conservative
> extension C. If so, N has to be an extension of C. That means:
>
> (N extends C) every theorem of C is a theorem of N
>
> In particular, you need to show that N proves C1, C2, C3, etc. This would
> involve giving a definition of the primitive predicate "x | y".
>
> --- Jeff
Jeff,
I don't understand how "conservative extension" is used. In
News:<9akoq...@drn.newsguy.com>, Darryl McCullough wrote:
>...the Godel-Von Neumann-Bernays theory is ... a conservative
> extension of ZFC (that is, you can't prove any new facts about
> sets).
It was on the basis of my understanding of this that I wanted
to call N a conservative extension of C. For if every
axiom of N is (modulo D1) a theorem of C, then you can't
prove any facts about sets with N that you can't prove
with C.
D1 "x=y" means "Az(x in x <-> z in y)"
Where did I go wrong?
--John
Jeffrey Ketland
Yes. Strictly, for an extension, you need to show that all theorems of N are
theorems of C (as deductive theories, N is a subset of C). But showing this
for just axioms of N is sufficient. If each axiom of N is provable in C,
then each theorem of N is provable in C too (we're assuming the theories use
the same inference rules).
--- Jeff
Jeffrey Ketland
That's right. NBG has an ontology of sets and classes. It has extra axioms,
over those of ZFC, for the existence of classes, but it turns out that these
axioms for classes yield no new theorems about _sets_.
>It was on the basis of my understanding of this that I wanted
>to call N a conservative extension of C. For if every
>axiom of N is (modulo D1) a theorem of C, then you can't
>prove any facts about sets with N that you can't prove
>with C.
If every axiom of N is a theorem of C, then C is an extension of N. It
doesn't follow that N is a conservative extension of C. It doesn't even
follow that N is an extension of C, which is the wrong way round.
--- Jeff
Chris Menzel
> Jeff,
> I don't understand how "conservative extension" is used. In
> News:<9akoq...@drn.newsguy.com>, Darryl McCullough wrote:
>
> >...the Godel-Von Neumann-Bernays theory is ... a conservative
> > extension of ZFC (that is, you can't prove any new facts about
> > sets).
>
> It was on the basis of my understanding of this that I wanted
> to call N a conservative extension of C. For if every
> axiom of N is (modulo D1) a theorem of C, then you can't
> prove any facts about sets with N that you can't prove
> with C.
>
> D1 "x=y" means "Az(x in x <-> z in y)"
>
> Where did I go wrong?
You're focusing on the "conservative" part of the definition. An
*extension* of a theory T includes T. Thus, a conservative extension of
T (loosely) will prove everything T does but won't prove anything new
about the things T is about.
-chris
Chris Menzel
> In what follows, "I" symbolizes non-reflexive, in-Ullrich's-face
> identity.
>
> 0. The fundamental principles of C are Classification, which asserts that
> every formula "P(x)" determines a class containing just those elements which
> satisfy "P(x)";
>
> C3 EyAx[x in y <-> Et(x in t) & Px] (with y not free in "Px")
>
> and Weak Extensionality, which asserts that equi-membered classes
> are identical just in case these are elements.)
>
> C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}] Weak
> Extensionality
John,
A couple of questions.
1. How do you answer the charge that C is a VNBG-style theory of classes
that simply replaces (reflexive) identity with the relation "identical
with some set", and hence that you really believe (at least implicitly)
that identity is reflexive after all?
2. In my idiolect, "is identical with" means "is the very same thing as".
Either you share this intuition or you don't. If you don't, then we
simply differ about the meaning of "is identical with" and we can
happily continue doing our respective things, simply being careful to
use different terms (i.e., terms that are not the very same! ;-) to
express the two concepts. I'm happy enough granting you "identical"
here for the sake of clarity (though let's write it "identical*" to
avoid confusion) if I can have "the very same". (Of course, the
question remains about which concept "is identical with" expresses for
the bulk of English speakers, but never mind that for now.) But now at
least we understand each other. You think that there are compelling
reasons for extending basic FOL with axioms for "is identical* with"
rather than "is the very same thing as". I, on the other hand, do not
find those reasons compelling. Is this a fair representation of your
position?
3. If, on the other hand, you do share the semantic intuition above, it
follows that you must think that there is something that is not the very
same thing as itself, since you think there are things that are not
identical with themselves. So do you in fact think there is something
that is not the very same thing as itself? If so, what would be an
example of such a thing?
Regards,
-chris
John
For ease of reading, above the proof of each theorem I have placed
the extra-logical premises on which this proof depends. Each of these
premises is either an axiom, or a theorem whose proof has been
previously displayed.
After each set of proofs, I will list the theorems already
proved and the axioms on which these proofs depend. I will
also provide message id's for prior proof sets.
Proofs 3:
T5 Ax(xIx <-> Ey(xIy))
(Self-identicals are iws's, and conversely.)
I3: Ax[Ey(xIy) -> xIx] Premise
1. Show Ax(xIx <-> Ey(xIy))
2. Ax[Ey(xIy) -> xIx] I3
3. Show Ax(xIx -> Ey(xIy))
4. Show xIx -> Ey(xIy)
5. xIx Assume
6. Show Ey(xIy)
7. Ey(xIy) 6,EG:Ca Sh(6)
8. xIx -> Ey(xIy) 5,6:Ca Sh(4)
9. Ax(xIx <-> Ey(xIy)) 2,3:Ca Sh(1)
T6 Ax(Ey(x in y) <-> Ey(xIy))
(Elements are iws's, and conversely.)
I3: Ax[Ey(xIy) -> xIx] Premise
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}] Premise
1. Show Ax(Ey(x in y) <-> Ey(xIy))
2. Show Ey(x in y) <-> Ey(xIy)
3. Ey(xIy) Assume
4. Show Ey(x in y)
5. xIx 3,I3
6. Az(z in x <-> z in x)
7. Ey(x in y) 5,6,C4:Ca Sh(4)
8. Ey(x in y) Assume
9. Show Ey(xIy)
10. Az(z in x <-> z in x)
11. xIx 8,10,C4
12. Ey(xIy) 11,EG:Ca Sh(9)
13. Ey(x in y) <-> Ey(xIy) 3,4,8,9:Ca Sh(2)
14. Ax(Ey(x in y) <-> Ey(xIy)) 2:Ca Sh(1)
T7 AxAy[Az(z in x <-> z in y) -> (xIx <-> yIy)]
(If x,y are equimembered, then both or neither are
self-identical.)
C2 AxAy[Az(z in x <-> z in y) -> Az(x in z <-> y in z)] Premise
T5 Ax(xIx <-> Ey(xIy)) Premise
1. Show AxAy[Az(z in x <-> z in y) -> (xIx <-> yIy)]
2. Show Az(z in x <-> z in y) -> (xIx <-> yIy)
3. Az(z in x <-> z in y) Assume
4. Show xIx <-> yIy
5. Az(x in z <-> y in z) 3,C2
6. Ez(x in z) <-> Ez(y in z) 5
7. xIx <-> yIy 6,T5:Ca Sh(4)
8. Az(z in x <-> z in y) -> (xIx <-> yIy) 3,4:Ca Sh(2)
9. AxAy[Az(z in x <-> z in y) -> (xIx <-> yIy)] Ca Sh(1)
***************************************************************
C1 AxAy[xIy -> 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) & P(x)] (with y not free in P(x))
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}] Weak
Extensionality
I1: AxAy(xIy <-> yIx)
I2: AxAyAz((xIy & yIz) -> xIz)
I3: Ax[Ey(xIy) -> xIx]
I4: AxAyAz(xIy -> (x in z <-> y in z))
I5: AxAyAz(xIy -> (z in x <-> z in y))
T5 Ax(xIx <-> Ey(xIy))
T6 Ax(Ey(x in y) <-> Ey(xIy))
T7 AxAy[Az(z in x <-> z in y) -> (xIx <-> yIy)]
For proofs of I1-I5, see
news:<70f94e16.02090...@posting.google.com>
Acknowledgement: In question here is not a proof that N
is a conservative extension of C, as I erroneously had
stated, but a proof that C is an extension of N.
*Definition*: A system S' is an *extension* of a system S iff
every theorem of S is a theorem of S'. N.B. If S' is an
extension of S, then every model of S' is a model of S.
Source: _MetaLogic_ (Geoffrey Hunter, UC Press, 1973, p. 177)
John
> For proofs of I1-I5, see
> news:<70f94e16.02090...@posting.google.com>
>
> Acknowledgement: In question here is not a proof that N
> is a conservative extension of C, as I erroneously had
> stated, but a proof that C is an extension of N.
>
> *Definition*: A system S' is an *extension* of a system S iff
> every theorem of S is a theorem of S'. N.B. If S' is an
> extension of S, then every model of S' is a model of S.
>
> Source: _MetaLogic_ (Geoffrey Hunter, UC Press, 1973, p. 177)
For ease of reading, above the proof of each theorem I have placed
the extra-logical premises on which this proof depends. Each of these
premises is either an axiom, or a theorem whose proof has been
previously displayed.
After each set of proofs, I will list the theorems already
proved and the axioms on which these proofs depend. I will
also provide message id's for prior proof sets.
Proofs 4
T8 EyAx~(x in y)
(There is a null class.)
T9 AaAbEyAx(x in y <-> xIa v xIb) Pairs
(For every a,b and some y, a and b are in y iff a,b are iws's.)
T10 AaEyAx(x in y <-> Ew(w in a & x in w))
(For every a, there is a class which contains just the
elements of elements of a.)
T11 AaEyAx(x in y <-> (Et(x in t) & Aw(w in x -> w in a)))
(For every, a there is a class of just those sub-classes
of a that are elements of some class.)
*********************************************************
T8 EyAx~(x in y) Null Class
C3 EyAx[x in y <-> Et(x in t) & P(x)] Premise
1. Show EyAx~(x in y)
2. EyAx[x in y <-> Et(x in t) & P(x)] C3
3. EyAx[x in y <-> (Et(x in t) & ~Et(x in t))] 2
4. Ax[x in y <-> (Et(x in t) & ~Et(x in t))] 3,EI
5. Ex(x in y) <-> Ex[Et(x in t) & ~Et(x in t)] 4
6. ~Ex[Et(x in t) & ~Et(x in t)]
7. ~Ex(x in y) 5,6
8. Ax~(x in y) 7
9. EyAx~(x in y) 8:Can Sh(1)
T9 AaAbEyAx(x in y <-> xIa v xIb) Pairs
C3 EyAx[x in y <-> Et(x in t) & P(x)] Premise
T6 Ax(Ey(x in y) <-> Ey(xIy)) Premise
1. Show AaAbEyAx(x in y <-> (xIa v xIb))
2. Show EyAx(x in y <-> xIa v xIb))
3. ~EyAx(x in y <-> (xIa v xIb)) Assume
4. EyAx[x in y <-> Et(x in t) & P(x)] C3
5. EyAx[x in y <-> (Et(x in t) & (xIa v xIb))] 4
6. AyEx~(x in y <-> (xIa v xIb)) 3
7. Ax[x in r <-> (Et(x in t) & (xIa v xIb))] 5,EI
8. Ex~(x in r <-> (xIa v xIb)) 6,UI
9. ~(x in r <-> xIa v xIb)) 8,EI
10. x in r <-> (Et(x in t) & (xIa v xIb)) 7,UI
11. x in r -> (xIa v xIb) 10
12. (x in r & ~(xIa v xIb)) v ((xIa v xIb) & ~(x in r)) 9
13. (xIa v xIb) & ~(x in r) 11,12
14. xIa v xIb 13
15. Et(xIt) 14
16. Et(x in t) 15,T6
17. Et(xIt) & (xIa v xIb) 16,14
18. x in r 10,17
19. x in r & ~(x in r) 13,18
20. EyAx(x in y <-> xIa v xIb)) 3,19:Can Sh(2)
21. AaAbEyAx(x in y <-> (xIa v xIb)) 2:Can Sh(1)
T10 AaEyAx(x in y <-> Ew(w in a & x in w)) Unions
C3 EyAx[x in y <-> Et(x in t) & P(x)] Premise
1. Show AaEyAx(x in y <-> Ew(w in a & x in w))
2. Show EyAx(x in y <-> Ew(w in a & x in w))
3. ~EyAx(x in y <-> Ew(w in a & x in w)) Assume
4. EyAx[x in y <-> Et(x in t) & P(x)] C3
5. EyAx[x in y <-> (Et(x in t) & Ew(w in a & x in w))] 4
6. AyEx~(x in y <-> Ew(w in a & x in w)) 3
7. Ax[x in r <-> (Et(x in t) & Ew(w in a & x in w))] 5,EI
8. Ex~(x in r <-> Ew(w in a & x in w)) 6,UI
9. ~(b in r <-> Ew(w in a & b in w)) 8,EI
10. b in r <-> (Et(b in t) & Ew(w in a & b in w)) 7,UI
11. (b in r & ~Ew(w in a & b in w)) v
(Ew(w in a & b in w) & ~(b in r) 9
12. b in r -> Ew(w in a & b in w) 10
13. Ew(w in a & b in w) & ~(b in r) 11,12
14. Et(b in t) 13
15. b in r 13,14,10
16. ~(b in r) 13
17. EyAx(x in y <-> Ew(w in a & x in w)) 3,16:Can Sh(2)
18. AaEyAx(x in y <-> Ew(w in a & x in w)) 2:Can Sh(1)
T11 AaEyAx(x in y <-> (Et(x in t) & Aw(w in x -> w in a))) Power Classes
C3 EyAx[x in y <-> Et(x in t) & P(x)] Premise
1. Show AaEyAx(x in y <-> (Et(x in t) & Aw(w in x -> w in a)))
2. Show EyAx(x in y <-> (Et(x in t) & Aw(w in x -> w in a)))
3. EyAx[x in y <-> Et(x in t) & P(x)] C3
4. EyAx(x in y <-> (Et(x in t) & Aw(w in x -> w in a))) 3:Can Sh(2)
5. AaEyAx(x in y <-> (Et(x in t) & Aw(w in x -> w in a))) 2: Can Sh(1)
*******************************************************************
Axioms:
C1 AxAy[xIy -> 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) & P(x)] (with y not free in P(x))
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}] Weak
Extensionality
Proofs 2, news:<70f94e16.02090...@posting.google.com>
I1 AxAy(xIy <-> yIx)
I2 AxAyAz((xIy & yIz) -> xIz)
I3 Ax[Ey(xIy) -> xIx]
I4 AxAyAz(xIy -> (x in z <-> y in z))
I5 AxAyAz(xIy -> (z in x <-> z in y))
Proofs 3, news:<70f94e16.0209...@posting.google.com>
John
> On 4 Sep 2002 18:04:58 -0700, John <john_...@excite.ca> said:
> > In what follows, "I" symbolizes non-reflexive, in-Ullrich's-face
> > identity.
> >
> > 0. The fundamental principles of C are Classification, which asserts that
> > every formula "P(x)" determines a class containing just those elements which
> > satisfy "P(x)";
> >
> > C3 EyAx[x in y <-> Et(x in t) & Px] (with y not free in "Px")
> >
> > and Weak Extensionality, which asserts that equi-membered classes
> > are identical just in case these are elements.)
> >
> > C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}] Weak
> > Extensionality
> > ...
>
> John,
>
> A couple of questions.
>
> 1. How do you answer the charge that C is a VNBG-style theory of classes
> that simply replaces (reflexive) identity with the relation "identical
> with some set", and hence that you really believe (at least implicitly)
> that identity is reflexive after all?
I think identity is reflexive for all the entities in the domain of
Z and its extensions. But I think the entities that these theories
do not admit are entities for which identity is not reflexive.
Incidentally, the proof that N is an extension of C employs the
translation you suggest above:
Def: "xIy" means "x=y & set x & set y"
Thus, to your question my answer would be, "No, I don't think
identity is reflexive, because I think that the relation symbolized
by "I" *is* identity, and that in set theory when we say "x=y", what
we really are saying is "Az(z in x <-> z in y)". To your question, the
opposing answer will be: identity is the relation signified by "=",
and "xIy" is just an economical way of saying "x=y & set x & set y".
Even so, I do think that there are reasons for selecting "I" as
primitive--and for introducing "=" by definition. However, before
I go into those reasons, I need to be able to make
a case that the relation symbolized by "I" has some formal bona
fides. Maybe showing that N and C are each extensions of the other,
will accomplish this.
>
> 2. In my idiolect, "is identical with" means "is the very same thing as".
> Either you share this intuition or you don't.
Actually, when I'm not thinking about logic or semantics, I rarely
have occasion to use "is identical with" in the way that logicians
and semanticists use it. Things often crop up that are *almost*
identical, but when I talk of things being identical I almost
always mean different instances of the same kind. Two people I
know have identical cars, identical haircuts, identical characters,
identical vices and virtues, etc.
For me, the closest paraphrase for "x =/I y" is "x and y are one
and the same." Of course, "x and x are one and the same" is
predictably weird. As for, "this is the very same thing as that,"
it's hard for me to imagine a context in which I would say this.
This doesn't mean there isn't such a context, it just means
I can't think of one right now. (Instead I would probably say,
"These are the same thing.")
> If you don't, then we
> simply differ about the meaning of "is identical with" and we can
> happily continue doing our respective things, simply being careful to
> use different terms (i.e., terms that are not the very same! ;-) to
> express the two concepts. I'm happy enough granting you "identical"
> here for the sake of clarity (though let's write it "identical*" to
> avoid confusion) if I can have "the very same". (Of course, the
> question remains about which concept "is identical with" expresses for
> the bulk of English speakers, but never mind that for now.) But now at
> least we understand each other. You think that there are compelling
> reasons for extending basic FOL with axioms for "is identical* with"
> rather than "is the very same thing as". I, on the other hand, do not
> find those reasons compelling. Is this a fair representation of your
> position?
Yes, although I think there are considerations germane to logic--
and to semantics as well--that remain to be explored.
>
> 3. If, on the other hand, you do share the semantic intuition above, it
> follows that you must think that there is something that is not the very
> same thing as itself, since you think there are things that are not
> identical with themselves. So do you in fact think there is something
> that is not the very same thing as itself? If so, what would be an
> example of such a thing?
Although there is no such thing is phlogiston, we do not go around
saying "Phlogiston isn't phlogiston". Nor, on the other hand,
are we reluctant to assert that there is no such thing is Phlogiston,
although this (arguably) commits us to "Phlogiston isn't Phlogiston."
To put this in another way, if pointing to the people in
question, I say, "John isn't the King of France, Henry isn't the King
of France, Joe isn't the King of France", etc., etc., I am asked,
"How do you know Henry isn't the King of France?", I might say,
"Because nobody is the King of France." But if nobody is the King of
France, how about the King of France? Is *he* the King of France?
(The following space is provided for FF to emote:
If he is the King of France, isn't somebody the King of France--
and not nobody? I guess what I am trying to say is that
informal usage does not provide a clear-cut answer to the
problem at hand.
Regards,
John
>
> Regards,
>
> -chris
John
Having slept (or, as it were, unslept) on your questions, I think I
see more clearly where these questions lead.
> 1. How do you answer the charge that C is a VNBG-style theory of classes
> that simply replaces (reflexive) identity with the relation "identical
> with some set", and hence that you really believe (at least implicitly)
> that identity is reflexive after all?
Suppose that identity (hereafter =i) is, and has always been,
coextensive (in set theory), with two conditions: having the
same members *and* being an element. From the vantage
point of Z and its extensions, whose domain includes only
entities that meet both these conditions, it would be rational
to conclude--from the fact that everything in Z domains
bears =i to itself--that everything in any other domain
bears =i to itself. Indeed, it is this conceptualization
of the constitutive elements of (class-theoretic) being
which leads VNBGers to suppose that classes which are not
elements bear =i to themselves, just as classes which are
elements do. Nevertheless, it is equally rational (or so
it seems to me) to construe elementhood as necessary for
identity, and so to conclude that classes that are not
elements do not bear =i to anything, including themselves.
In deciding which of these conceptualizations 'gets things
right', definitions of =i are a hindrance rather than
a help. For by defining =i as symmetric, transitive and
reflexive; or by defining it as symmetric and transitive,
and reflexive for elements and irreflexive for non-elements,
one imposes, as a fetter on logical/set-theoretic discourse,
what it is incumbent upon such discourse to decide.
> 3. If, on the other hand, you do share the semantic intuition above, it
> follows that you must think that there is something that is not the very
> same thing as itself, since you think there are things that are not
> identical with themselves. So do you in fact think there is something
> that is not the very same thing as itself? If so, what would be an
> example of such a thing?
On the one hand, we are not comfortable in saying that phlogiston
is not phlogiston. On the other hand, neither are we comfortable
in saying both that nothing is phlogiston and that phlogiston is
phlogiston. There are a number of ways to avoid being gored by
either of these horns: (1) Don't count "phlogiston" as a term
(classical logic), (2) Keep "phlogiston" around, but paraphrase
wffs in which "phlogiston" occurs by wffs in which "phlogiston"
does not occur, and only count the latter as belonging to your
language (Russell), (3) Allow that "phlogiston" has a sense but
no reference, so that assertions about phlogiston are neither true
nor false (Frege), (4) Don't worry about sense. Just count
assertions about Phlogiston as neither true nor false (Strawson).
(5) Admit "phlogiston" as a singular term but assign to "Phlogiston"
something that does not breed ("Chosen Object Theory")
To steal a page from Bencivenga (see his "Free Logic" _Cambridge
Dictionary of Philosophy_; FF and myself have posted this article
on sci.logic), (1-5) "preserve the structure of classical
quantification theory and make adjustments at the level of
application. The sixth
way does not: (6) Admit "phlogiston" as a singular term and "modify
both the proof theory and the semantics of first-order logic."
(Bencivenga, loc. cit)
My proposal for "phlogiston" modifies the proof theory and
semantics of first-order logic by enlarging its ontology.
The enlarged ontology requires identity to be taken as weakly
reflexive rather than reflexive, the entities that are added
not bearing =i to themselves. In this proposal, the classical
inference rules are preseved; they are not preserved by (5).
Each of these proposals has its advantages and disadvantages.
I may be wrong, but I don't believe that such a question as
"Would you say that phlogiston is or is not self-identical?"
has any bearing at all on which (if any) of these proposals
is correct.
Regards,
John
John
For ease of reading, above the proof of each theorem I have placed
the extra-logical premises on which this proof depends. Each of these
premises is either an axiom, or a theorem whose proof has been
previously displayed.
After each set of proofs, I will list the theorems already
proved and the axioms on which these proofs depend. I will
also provide message id's for prior proof sets.
Proofs 5
The proof of T15 depends upon an axiom that has not figured
in previous proofs:
C5 AaAbAy[Ax(x in y <-> (xIa v xIb)) -> yIy] Axiom of Pairs
T12 Ax(xIx <-> set x)
(All and only self-identicals are sets.)
T13 Ax[Ay~(y in x) -> xIx]
(Every null class is self-identical.)
*********************************************************************
T12 Ax(xIx <-> set x)
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}] Premise
1. Show Ax(xIx <-> set x)
2. AxAz(z in x <-> z in x)
3. Ax(Az(z in x <-> z in x) -> (Et(x in t) <-> xIx)) C4
4. Az(Az(z in x <-> z in x) -> (xIx <-> Et(x in t)) 3
5. AxAz(z in x <-> z in x) -> Ax(xIx <-> Et(x in t)) 4
6. Ax(xIx <-> Et(x in t)) 2,5
7. set x <-> Et(x in t) D2
8. Ax(xIx <-> set x) 7:Ca Sh(1)
T13 Ax[Ay~(y in x) -> xIx]
C3 EyAx[x in y <-> Et(x in t) & P(x)] Premise
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}] Premise
C5 AaAbAy[Ax(x in y <-> (xIa v xIb)) -> yIy] Premise
I1 AxAy(xIy <-> yIx) Premise
1. Show Ay[Ax~(x in y) -> yIy]
2. Show Ax~(x in y) -> yIy
3. Ax~(x in y) Assume
4. Show yIy
5. EyAx[x in y <-> Et(x in t) & ~(x in x)] Instance of C3
6. Ax[x in r <-> (Et(x in t) & ~(x in x))] 5,EI
7. r in r <-> (Et(r in t) & ~(r in r)) 6,UI
8. ~Et(r in t) 7
9. ~Et(rIt) 8,C4
10. AaAbAy[Ax(x in y <-> (xIa v xIb)) -> yIy] C5
11. Ay[Ax(x in y <-> (xIr v xIr)) -> yIy] 10,UI
12. Ay[Ax(x in y <-> xIr) -> yIy] 11
13. Show Ax(x in y <-> xIr)
14. Show x in y <-> xIr
15. ~(x in y) 3,UI
16. x in y -> xIr 15, ~p -> (p -> q)
17. At~(rIt) 9
18. ~(rIx) 17,UI
19. ~(xIr) 18,I1
20. xIr -> x in y 19, ~p -> (p -> q)
21. x in y <-> xIr 16,20:Ca Sh(14)
22. Ax(x in y <-> xIr) 14: Ca Sh(13)
23. Ax(x in y <-> xIr) -> yIy 12,UI
24. yIy 13,23:Ca Sh(4)
25. Ax~(x in y) -> yIy 3,4: Ca Sh(2)
26. Ay(Ax~(x in y) -> yIy) 2: Ca Sh(1)
**************************************************************
Axioms:
C1 AxAy[xIy -> 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) & P(x)] (with y not free in P(x))
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}] Weak
Extensionality
C5 AaAbAy[Ax(x in y <-> (xIa v xIb)) -> yIy] Axiom of Pairs
Proofs 2, news:<70f94e16.02090...@posting.google.com>
I1 AxAy(xIy <-> yIx)
I2 AxAyAz((xIy & yIz) -> xIz)
I3 Ax[Ey(xIy) -> xIx]
I4 AxAyAz(xIy -> (x in z <-> y in z))
I5 AxAyAz(xIy -> (z in x <-> z in y))
Proofs 3, news:<70f94e16.0209...@posting.google.com>
T5 Ax(xIx <-> Ey(xIy))
T6 Ax(Ey(x in y) <-> Ey(xIy))
T7 AxAy[Az(z in x <-> z in y) -> (xIx <-> yIy)]
Proofs 4, news:<70f94e16.02090...@posting.google.com>
T8 EyAx~(x in y)
T9 AaAbEyAx(x in y <-> xIa v xIb) Pairs
T10 AaEyAx(x in y <-> Ew(w in a & x in w))
John
I omitted something crucial here, that free logicians
modify the proof theory by placing conditions on the classical
inference rules, and providing an appropriate semantics for the
resulting proof theory; while I keep the classical inference rules,
expand the ontology by admitting entities that the classical ontology
doesn't recognize, and weakening the 'reflexive law of equality' in
order to accomdodate those entities.
--John
John
Proofs 6
First I prove T14.
T14 AxAy(xIy -> (Az(z in x <-> z in y) & (xIx & yIy)))
Then I show that I show that if "x=y" is defined as in D1, N1-N4 are
theorems of C.
D1: "x=y" means "Az(z in x <-> z in y)"
Beneath each theorem are the extra-logical premises on which its proof
depends. Each premise is an axiom or previously proved theorem of C.
Following each proof set are the theorems already proved, the axioms
these depend, and message id's for earlier proof sets.
N1 AxAyAz[x = y -> (z in x <-> z in y)] LL 1
N2 AxAyAz[x = y -> (x in z <-> y in z)] LL 2
N3 EyAx[x in y <-> (Et(x in t) & P(x))] with y not free in P(x)
Classification
N4 AxAy[Az(z in x <-> z in y) -> x = y] Extensionality
*************************************************************
T14 AxAy(xIy -> (Az(z in x <-> z in y) & (xIx & yIy)))
C1 AxAy[xIy -> Az(z in x <-> z in y)]
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}]
T12 Ax(xIx <-> set x)
D2 "set x" means "Et(x in t)"
1. Show AxAy(xIy -> (Az(z in x <-> z in y) & (xIx & yIy)))
2. Show xIy -> (Az(z in x <-> z in y) & (xIx & yIy))
3. xIy Assume
4. Show Az(z in x <-> z in y) & (xIx & yIy))
5. Az(z in x <-> z in y) 3,C1
6. Et(x in t & y in t) 3,5,C4
7. xIx & yIy 6,D2,T12
N1 AxAyAz[x=y -> (z in x <-> z in y)] LL1
D1 "x=y" means "Az(z in x <-> z in y)"
C2 AxAy[Az(z in x <-> z in y) -> Az(x in z <-> y in z)]
1. Show AxAyAz[x=y -> (z in x <-> z in y)]
2. Show x=y -> (z in x <-> z in y)
3. x=y Assume
4. Show z in x <-> z in y
5. Az(z in x <-> z in y) 3,D1
6. z in x <-> z in y 5,UI:Ca Sh(4)
7. x=y -> (z in x <-> z in y) 3,4:Ca Sh(2)
8. AxAyAz[x=y -> (z in x <-> z in y)] Ca Sh(1)
N2 AxAyAz[x=y -> (x in z <-> y in z)] LL2
D1: "x=y" means "Az(z in x <-> z in y)"
C2 AxAy[Az(z in x <-> z in y) -> Az(x in z <-> y in z)]
1. Show AxAyAz[x=y -> (x in z <-> y in z)]
2. Show x=y -> (x in z <-> y in z)
3. x=y Assume
4. Show x in z <-> y in z
5. Az(z in x <-> z in y) 3,D1
6. AxAy[Az(z in x <-> z in y) -> Az(x in z <-> y in z)] C2
7. Az(z in x <-> z in y) -> Az(x in z <-> y in z) 6,UI
8. Az(x in z <-> y in z) 5,7
9. x in z <-> y in z 8,UI:Ca Sh(4)
10. x=y -> (x in z <-> y in z) 3,4:Ca Sh(2)
11. AxAyAz[x=y -> (x in z <-> y in z)] 2:Ca Sh(1)
N3 EyAx[x in y <-> (Et(x in t) & P(x))] Classification
C3 EyAx[x in y <-> (Et(x in t) & P(x))]
1. Show EyAx[x in y <-> (Et(x in t) & P(x))]
2. EyAx[x in y <-> (Et(x in t) & P(x))] C3:Ca Sh(1)
N4 AxAy[Az(z in x <-> z in y) -> x=y] Extensionality
D1: "x=y" means "Az(z in x <-> z in y)"
1. Show AxAy[Az(z in x <-> z in y) -> x=y]
2. Show Az(z in x <-> z in y) -> x=y
3. Az(z in x <-> z in y) Assume
4. Show x=y
5. x=y 3,D1:Ca Sh(4)
6. Az(z in x <-> z in y) -> x=y 3,4:Ca Sh(2)
7. AxAy[Az(z in x <-> z in y) -> x=y] 2:Ca Sh(1)
*************************************************************
Axioms:
C1 AxAy[xIy -> 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) & P(x))] (with y not free in P(x))
Classification
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}] Weak
Extensionality
C5 AaAbAy[Ax(x in y <-> (xIa v xIb)) -> yIy] Axiom of Pairs
D1: "x=y" means "Az(z in x <-> z in y)"
Proofs 2, news:<70f94e16.02090...@posting.google.com>
I1 AxAy(xIy <-> yIx)
I2 AxAyAz((xIy & yIz) -> xIz)
I3 Ax[Ey(xIy) -> xIx]
I4 AxAyAz(xIy -> (x in z <-> y in z))
I5 AxAyAz(xIy -> (z in x <-> z in y))
Proofs 3, news:<70f94e16.0209...@posting.google.com>
T5 Ax(xIx <-> Ey(xIy))
T6 Ax(Ey(x in y) <-> Ey(xIy))
T7 AxAy[Az(z in x <-> z in y) -> (xIx <-> yIy)]
Proofs 4, news:<70f94e16.02090...@posting.google.com>
T8 EyAx~(x in y)
T9 AaAbEyAx(x in y <-> xIa v xIb) Pairs
T10 AaEyAx(x in y <-> Ew(w in a & x in w))
T11 AaEyAx(x in y <-> (Et(x in t) & Aw(w in x -> w in a)))
Proofs 5, news:<70f94e16.02090...@posting.google.com>
T12 Ax(xIx <-> set x)
John
N5 AaAb[set a & set b -> Ey(set y & Ax(x in y <-> x=a v x=b))] Axiom of Pairs
Beneath N5 are the extra-logical premises upon which its proof
in C depends. Following the proof are the axioms of N proved
so far in C, the axioms, definition and theorems of C upon
which these proofs depend, and message id's for proof sets.
***************************************************************
N5 AaAb[set a & set b -> Ey(set y & Ax(x in y <-> x=a v x=b))]
C1 AxAy[xIy -> Az(z in x <-> z in y)] Premise
C5 AaAbAy(Ax[x in y <-> (xIa v xIb)] -> yIy) Premise
T7 AxAy[Az(z in x <-> z in y) -> (xIx <-> yIy)] Premise
T9 AaAbEyAx(x in y <-> xIa v xIb) Premise
T12 Ax(xIx <-> set x) Premise
T13 Ax[Ay~(y in x) -> xIx] Premise
T14 AxAy(xIy -> (Az(z in x <-> z in y) & (xIx & yIy))) Premise
D1: "x=y" means "Az(x in z <-> y in z)" Definition
1. Show AaAb[set a & set b -> Ey(set y & Ax(x in y <-> (x=a v x=b)))]
2. Show set a & set b -> Ey(set y & Ax(x in y <-> (x=a v x=b)))
3. set a & set b Assume
4. Show Ey(set y & Ax(x in y <-> (x=a v x=b)))
5. AaAbAy(Ax[x in y <-> (xIa v xIb)] -> yIy) C5
6. EyAx(x in y <-> (xIa v xIb)) T9
7. Ax(x in r <-> (xIa v xIb)) 6,EI,UI
8. Show Ex(x in r) -> Ey(set y & Ax(x in y <-> (x=a v x=b)))
9. Ex(x in r) Assume
10. Show Ey(set y & Ax(x in y <-> (x=a v x=b)))
11. Show set r & Ax(x in r <-> (x=a v x=b))
12. Ax[x in r <-> (xIa v xIb)] -> rIr 11,UI
13. rIr 6,11
14. set r 13,T12
15. Ax(x in r -> (xIa v xIb)) 7
16. xIa -> Az(z in x <-> z in a) C1
17. xIb -> Ax(z in x <-> z in b) C1
18. Ax(x in r -> [Az(z in x <-> z in a) v
Ax(z in x <-> z in b)] 15,16,17
19. Ax(x in r -> [x=a v x=b]) 18,D1
20. Ax[(xIa v xIb) -> x in r] 7
21. Ax[(Az((z in x <-> z in a) & (set x & set a)) v
Az((z in x <-> z in b) & (set x & set b))) -> x in r] 20,T14,T12
22. Ax[((x=a & set x & set a) v
(x=b & set x & set b)) -> x in r] 21,D1
23. Ax[((x=a & set x) v (x=b & set x)) -> x in r] 22,3
24. Ax[((Az(z in x <-> z in a) & set x) v
(Az(z in x <-> z in b) & set x)) -> x in r] 23,D1
25. Ax[(Az(z in x <-> z in a) v
Az(z in x <-> z in b)) -> x in r] 24,3,T12,T7
26. Ax((x=a v x=b) -> x in r) 25,D1
27. set r & Ax(x in r <-> (x=a v x=b)) 14,15,26:Ca Sh(11)
28. Ey(set y & Ax(x in y <-> x=a v x=b)) 9,10:Ca Sh(8)
29. Show Ex(x in r)
30. ~Ex(x in r) Assume
31. Ex(x in r) <-> Ex(xIa v xIb) 7
32. ~Ex(xIa v xIb) 30,31
33. Ax~(xIa v xIb) 32
34. ~(bIa v bIb) 33,UI
35. ~(bIa) & ~(bIb) 34
36. ~(set b) 35,T12
37. set b & ~(set b) 3,36
38. Ex(x in r) 30,37:Ca Sh(29)
39. Ey(set y & Ax(x in y <-> (x=a v x=b))) 29,8: Ca Sh(4)
40. set a & set b ->
Ey(set y & Ax(x in y <-> (x=a v x=b))) 3,4:Ca Sh(2)
41. AaAb[set a & set b ->
Ey(set y & Ax(x in y <-> (x=a v x=b)))] 2:Ca Sh(1)
*************************************************************
N1 AxAyAz[x = y -> (z in x <-> z in y)] LL 1
N2 AxAyAz[x = y -> (x in z <-> y in z)] LL 2
N3 EyAx[x in y <-> (Et(x in t) & P(x))] with y not free in P(x)
Classification
N4 AxAy[Az(z in x <-> z in y) -> x = y] Extensionality
N5 AaAb[set a & set b -> Ey(set y & Ax(x in y <-> x = a v x = b))] Axiom of
Pairs
C1 AxAy[xIy -> 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) & P(x))] (with y not free in P(x))
Classification
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}] Weak
Extensionality
C5 AaAbAy(Ax[x in y <-> (xIa v xIb)] -> yIy) (Axiom of Pairs)
D1: "x=y" means "Az(x in z <-> y in z)"
D2: "set x" means "Et(x in t)"
Proofs 2, news:<70f94e16.02090...@posting.google.com>
I1 AxAy(xIy <-> yIx)
I2 AxAyAz((xIy & yIz) -> xIz)
I3 Ax[Ey(xIy) -> xIx]
I4 AxAyAz(xIy -> (x in z <-> y in z))
I5 AxAyAz(xIy -> (z in x <-> z in y))
Proofs 3, news:<70f94e16.0209...@posting.google.com>
T5 Ax(xIx <-> Ey(xIy))
T6 Ax(Ey(x in y) <-> Ey(xIy))
T7 AxAy[Az(z in x <-> z in y) -> (xIx <-> yIy)]
Proofs 4, news:<70f94e16.02090...@posting.google.com>
T8 EyAx~(x in y)
T9 AaAbEyAx(x in y <-> xIa v xIb) Pairs
T10 AaEyAx(x in y <-> Ew(w in a & x in w))
T11 AaEyAx(x in y <-> (Et(x in t) & Aw(w in x -> w in a)))
Proofs 5, news:<70f94e16.02090...@posting.google.com>
T12 Ax(xIx <-> set x)
T13 Ax[Ay~(y in x) -> xIx]
Proofs 6, news:<70f94e16.02090...@posting.google.com>
T14 AxAy(xIy -> (Az(z in x <-> z in y) & (xIx & yIy)))
John
Beneath each theorem are the extra-logical premises on which its proof
depends. Each premise is an axiom or previously proved theorem of C.
Following each proof set are the theorems already proved, the axioms
on which these depend, and message id's for earlier proof sets.
Proof 8
N6 Aa(set a -> (Ay(Ax(x in y -> x in a)) -> set y)) Axiom of Subsets
N7 Aa[set a -> Ey(set y & Ax(x in y <-> Ew(w in a & x in w)))] Axiom of Unions
N8 Aa[set a -> Ey(set y & Ax(x in y <-> Aw(w in x -> w in a)))] Axiom of Power Sets
N9 Ex(set x & Ay~(y in x)) Empty Set Axiom
*******************************************************************************
N6 Aa(set a -> (Ay(Ax(x in y -> x in a)) -> set y))
C6 AaAy(Ax(x in y -> x in a) -> (aIa -> yIy)) Premise
T12 Ax(xIx <-> set x) Premise
1. Show Aa(set a -> (Ay(Ax(x in y -> x in a)) -> set y))
2. AaAy(Ax(x in y -> x in a) -> (aIa -> yIy)) C6
3. AaAy(aIa -> (Ax(x in y -> x in a) -> yIy)) 2
4. AaAy(set a -> (Ax(x in y -> x in a) -> set y)) 3,T12
5. Aa(set a -> (Ay(Ax(x in y -> x in a)) -> set y)) 4:Ca Sh(1)
N7 Aa[set a -> Ey(set y & Ax(x in y <-> Ew(w in a & x in w)))]
C7 AaAy[Ax(x in y <-> Ew(w in a & x in w)) -> (aIa -> yIy)] Premise
T12 Ax(xIx <-> set x) Premise
T10 AaEyAx(x in y <-> Ew(w in a & x in w)) Premise
1. Show Aa[set a -> Ey(set y & Ax(x in y <-> Ew(w in a & x in w)))]
2. Show set a -> Ey(set y & Ax(x in y <-> Ew(w in a & x in w)))
3. set a Assume
4. Show Ey(set y & Ax(x in y <-> Ew(w in a & x in w)))
5. AaAy[Ax(x in y <-> Ew(w in a & x in w)) -> (aIa -> yIy)] C7
6. AaAy[aIa -> (Ax(x in y <-> Ew(w in a & x in w)) -> yIy)] 5
7. AaAy[set a -> (Ax(x in y <-> Ew(w in a & x in w)) -> set y)] 6,T12
8. AaEyAx(x in y <-> Ew(w in a & x in w)) T10
9. Ax(x in r <-> Ew(w in a & x in w)) 8,UI,EI
10. set a -> (Ax(x in r <-> Ew(w in a & x in w)) -> set r) 7, UI
11. Ax(x in r <-> Ew(w in a & x in w)) -> set r 3,10
12. set r 9,11
13. set r & Ax(x in r <-> Ew(w in a & x in w)) 9,12
14. Ey(set y & Ax(x in y <-> Ew(w in a & x in w))) 13,EG:Ca Sh(4)
15. set a -> Ey(set y & Ax(x in y <->
Ew(w in a & x in w))) 3,4:Ca Sh(2)
16. Aa[set a -> Ey(set y & Ax(x in y <->
Ew(w in a & x in w)))] 2:Ca Sh(1)
N8 Aa[set a -> Ey(set y & Ax(x in y <-> Aw(w in x -> w in a)))]
C6 AaAy(Ax(x in y -> x in a) -> (aIa -> yIy)) Premise
C8 AaAy[Ax(x in y <-> Aw(w in x -> w in a)) -> (aIa -> yIy)] Premise
T11 AaEyAx(x in y <-> (Et(x in t) & Aw(w in x -> w in a))) Premise
T12 Ax(xIx <-> set x) Premise
1. Show Aa[set a -> Ey(set y & Ax(x in y <-> Aw(w in x -> w in a)))]
2. Show set a -> Ey(set y & Ax(x in y <-> Aw(w in x -> w in a)))
3. set a Assume
4. Show Ey(set y & Ax(x in y <-> Aw(w in x -> w in a)))
5. AaAy[Ax(x in y <-> Aw(w in x -> w in a)) -> (aIa -> yIy)] C8
6. AaEyAx(x in y <-> (xIx & Aw(w in x -> w in a))) T11
7. Ax(x in r <-> (xIx & Aw(w in x -> w in a))) 6,UI,EI
8. Show set r & Ax(x in r <-> Aw(w in x -> w in a))
9. Ax(x in r <-> Aw(w in x -> w in a)) -> (aIa -> rIr) 5,UI
10. Ax(x in r -> (xIx & Aw(w in x -> w in a))) 7
11. Ax(x in r -> Aw(w in x -> w in a)) -> (aIa -> rIr) 9
12. Ax(x in r -> Aw(w in x -> w in a)) 10
13. aIa -> rIr 11,12
14. set a -> set r 13,T12
15. set r 3,14
16. Show Ax(x in r <-> Aw(w in x -> w in a))
17. Show x in r <-> Aw(w in x -> w in a)
18. x in r -> Aw(w in x -> w in a) 12,UI
19. Show Aw(w in x -> w in a) -> xIx
20. Aw(w in x -> w in a) Assume
21. Show xIx
22. AaAy(Aw(w in y -> w in a) -> (aIa -> yIy)) C6
23. Aw(w in x -> w in a) -> (aIa -> xIx) 22,UI
24. aIa -> xIx 20,23
25. set a -> xIx 24,T12
26. xIx 3,25:Ca Sh(21)
27. Aw(w in x -> w in a) -> xIx 20,21:Ca Sh(19)
28. Ax[(xIx & Aw(w in x -> w in a)) -> x in r] 7
29. (xIx & Aw(w in x -> w in a)) -> x in r 28,UI
30. Aw(w in x -> w in a)) -> x in r 29,19
31. x in r <-> Aw(w in x -> w in a) 30,18:Ca Sh(17)
32. Ax(x in r <-> Aw(w in x -> w in a)) 17:Ca Sh(16)
33. set r & Ax(x in r <-> Aw(w in x -> w in a)) 15,16:Ca Sh(8)
34. Ey(set y & Ax(x in y <-> Aw(w in x -> w in a))) 8:Ca Sh(4)
35. set a -> Ey(set y &
Ax(x in y <-> Aw(w in x -> w in a))) 3,4:Ca Sh(2)
36. As(set a -> Ey(set y &
Ax(x in y <-> Aw(w in x -> w in a)))) 2:Ca Sh(1)
N9 Ex(set x & Ay~(y in x))
T8 EyAx~(x in y) Premise
T12 Ax(xIx <-> set x) Premise
T13 Ax[Ay~(y in x) -> xIx] Premise
1. Show Ex(set x & Ay~(y in x))
2. Show Ex(xIx & Ay~(y in x))
3. EyAx~(x in y) T8
4. Ax~(x in t) 3,EI
5. Ay[Ax~(x in y) -> yIy] T13
6. Ax~(x in t) -> tIt 5,UI
7. tIt 4,6
8. tIt & Ay~(y in t) 7,4
9. Ex(xIx & Ay~(y in x)) 8,EG:Ca Sh(2)
10. Ex(set x & Ay~(y in x)) 9,T12:Ca Sh(1)
***************************************************************
C1 AxAy[xIy -> Az(z in x <-> z in y)] LL1
C2 AxAy[Az(x in z <-> y in z) -> Az(z in x <-> z in y)] LL2
C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in A)
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}] Weak Extensionality
C5 AaAbAy(Ax[x in y <-> (xIa v xIb)] -> yIy) Axiom of Pairs
C6 AaAy(Ax(x in y -> x in a) -> (aIa -> yIy)) Axiom of Subsets
C7 AaAy[Ax(x in y <-> Ew(w in a & x in w)) -> (aIa -> yIy)] Axiom of Unions
C8 AaAy[Ax(x in y <-> Aw(w in x -> w in a)) -> (aIa -> yIy)]Axiom of Powersets
D1: "set x" means "Et(x in t)"
D2: "x=y" means "Az(x in z <-> y in z)"
*************************************************************
C1 AxAy[xIy -> 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) & P(x))] (with y not free in P(x))
Classification
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}] Weak
Extensionality
C5 AaAbAy(Ax[x in y <-> (xIa v xIb)] -> yIy) Axiom of Pairs
C6 AaAy(Ax(x in y -> x in a) -> (aIa -> yIy)) Axiom of Subsets
C7 AaAy[Ax(x in y <-> Ew(w in a & x in w)) -> (aIa -> yIy)] Axiom of Unions
C8 AaAy[Ax(x in y <-> Aw(w in x -> w in a)) -> (aIa -> yIy)] Axiom of Power Sets
Proofs 7, news:<70f94e16.02090...@posting.google.com>
N5 AaAb[set a & set b -> Ey(set y & Ax(x in y <-> x=a v x=b)) Axiom of Pairs
Proofs 8 (see above)
N6 Aa(set a -> (Ay(Ax(x in y -> x in a)) -> set y)) Axiom of Subsets
N7 Aa[set a -> Ey(set y & Ax(x in y <-> Ew(w in a & x in w)))] Axiom of Unions
N8 Aa[set a -> Ey(set y & Ax(x in y <-> Aw(w in x -> w in a)))] Axiom of Power Sets
N9 Ex(set x & Ay~(y in x)) Empty Set Axiom
John
Proofs 9
In Proof Sets 2-8 (Proofs 2 - Proofs 8), I showed that C1-C8|-N1-N9,
provided that "=" is added to the language of C by means of D1. C is
thus an extension of N, for every axiom of N is a theorem of C.
D1 "x=y" means "Az(z in x <-> z in y)"
In this proof set, I show that N1-N9|-C1-C8, provided that "I"
and "set" are added to the language of N by means of D2 and D3.
D2 "xIy" means "x=y & set x & set y"
D3 "set x" means "Et(x in t)"
From the formal equivalence of N and C (N1-N9|-C1-C8 and
C1-C8|-N1-N9), it follows that every model of C is a
model of N, and conversely.
The differences between C and N are thus not model-theoretic in
nature.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
C Axioms
C1 AxAyAz[xIy -> (z in x <-> z in y)] LL1
C2 AxAyAz[Az(z in x <-> z in y) <-> ((x in z <-> y in z))] LL2
C3 EyAx[x in y <-> Et(x in t) & P(x)] (with y not free in P(x))
Classification
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}] Weak
Extensionality
C5 AaAbAy(Ax[x in y <-> (xIa v xIb)] -> yIy) Axiom of Pairs
C6 AaAy(Ax[x in y -> x in a] -> (aIa -> yIy)) Axiom of Subsets
C7 AaAy[Ax(x in y <-> Ew(w in a & x in w)) -> (aIa -> yIy)] Axiom of Unions
C8 Aa(Ay[Ax(x in y <-> Aw(w in x -> w in a))) -> (aIa -> yIy]) Axiom of Power
Sets
NBG Axioms
N1 AxAyAz[x=y -> (x in z <-> y in z)] LL 1
N2 AxAyAz[x=y -> (z in x <-> z in y)] LL 2
N3 EyAx[x in y <-> (Et(x in t) & P(x))] (with y not free in (x))
Classification
N4 AxAy[Az(z in x <-> z in y) -> x = y] Extensionality
N5 AaAb[set a & set b -> Ey(set y & Ax(x in y <-> x=a v x=b))] Axiom of Pairs
N6 Aa[set a -> Ay(Ax(x in y -> x in a) -> set y)] Axiom of Subsets
N7 Aa[set a -> Ey(set y & Ax(x in y <-> Ew(w in a & x in w)))] Axiom of
Unions
N8 Aa[set a -> Ey(set y & Ax(x in y <-> Aw(w in x -> w in a)))] Axiom of
Power Sets
N9 Ex(set x & Ay~(y in x)) Empty Set Axiom
D2 "xIy" means "x=y & set x & set y"
D3 "set x" means "Et(x in t)"
C1 AxAyAz[xIy -> (z in x <-> z in y)]
N1 AxAyAz[x=y -> (z in x <-> z in y)] Premise
D2 "xIy" means "x=y & set x & set y"
1. Show AxAyAz[xIy -> (z in x <-> z in y)]
2. Show xIy -> (z in x <-> z in y)
3. Show (x=y & set x & set y) -> (z in x <-> z in y)
4. x=y & set x & set y Assume
5. Show (z in x <-> z in y)
6. z in x <-> z in y 4,N1:Ca Sh(5)
7. (x=y & set x & set y) -> Az(z in x <-> z in y) 4,5:Ca Sh(3)
8. xIy -> Az(z in x <-> z in y) 3, D2:Ca Sh(2)
9. AxAyAz[xIy -> (z in x <-> z in y)] 2:Ca Sh(1)
C2 AxAyAz[Az(z in x <-> z in y) -> ((x in z <-> y in z))]
N2 AxAyAz[x=y -> (z in x <-> z in y)] Premise
N4 AxAy[Az(z in x <-> z in y) -> x = y] Premise
1. Show AxAy[Az(z in x <-> z in y) -> Az(x in z <-> y in z)]
2. Show Az(z in x <-> z in y) -> Az(x in z <-> y in z)
3. Az(z in x <-> z in y) Assume
4. Show Az(x in z <-> y in z)
5. x=y 3,N4
6. Az(z in x <-> z in y) 5,N2:Ca Sh(4)
7. Az(z in x <-> z in y) -> Az(x in z <-> y in z) 3,4:Ca Sh(2)
8. AxAy[Az(z in x <-> z in y) -> Az(x in z <-> y in z)] 2:Ca Sh(1)
C3 EyAx[x in y <-> Et(x in t) & P(x)]
N3 EyAx[x in y <-> Et(x in t) & P(x)] Premise
1. Show EyAx[x in y <-> Et(x in t) & P(x)]
2. EyAx[x in y <-> Et(x in t) & P(x)] N3:Ca Sh(1)
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}]
D2 "xIy" means "x=y & set x & set y"
D3 "set x" means "Et(x in t)"
N4 AxAy[Az(z in x <-> z in y) -> x = y] Premise
N5 AaAb[set a & set b -> Ey(set y
& Ax(x in y <-> x=a v x=b))] Premise
1. Show AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}]
2. Show Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}
3. Az(z in x <-> z in y) Assume
4. Show Et(x in t & y in t) <-> xIy
5. Show Et(x in t & y in t) -> xIy
6. Et(x in t & y in t) Assume
7. Show xIy
8. Show x=y & set x & set y
9. set x & set y 6,D3
10. x=y 3,N4
11. x=y & set x & set y D2,9,10:Ca Sh(7)
12. Et(x in t & y in t) -> xIy 6,7:Ca Sh(5)
13. Show xIy -> Et(x in t & y in t)
14. xIy Assume
15. Show Et(x in t & y in t)
16. set x & set y & x=y 14,D2
17. AaAb[(set a & set b) ->
Et(set t & As(s in t <-> s=a v s=b))] N5
18. (set x & set y) ->
Et(set t & As(s in t <-> s=x v s=y)) 17,UI
19. Et(set t & As(s in t <-> s=x v s = y)) 16,18
20. set t & As(s in t <-> s=x v s=y) 19,EI
21. As(s in t <-> s=x v s=y) 20
22. x in t <-> x=x v x=y 21,UI
23. x in t 16,22
24. y in t <-> y=x v y=y 21,UI
25. y in t 24,16
26. x in t & y in t 23,25
27. Et(x in t & y in t) 26,EG:Ca Sh(15)
28. xIy -> Et(x in t & y in t) 14,15:Ca Sh(13)
29. Et(x in t & y in t) <-> xIy 5,13:Ca Sh(4)
30. Az(z in x <-> z in y) ->
{Et(x in t & y in t) <-> xIy} 3,4:Ca Sh(2)
31. AxAy[Az(z in x <-> z in y) ->
{Et(x in t & y in t) <-> xIy}] 2:Ca Sh(1)
C5 AaAbAy(Ax[x in y <-> (xIa v xIb)] -> yIy)
N4 AxAy[Az(z in x <-> z in y) -> x = y] Premise
N5 AaAb[set a & set b -> Ey(set y &
Ax(x in y <-> x=a v x=b))] Premise
N6 Aa[set a -> Ay(Ax(x in y -> x in a) -> set y)] Premise
N9 Ex(set x & Ay~(y in x)) Premise
D2 "xIy" means "x=y & set x & set y"
D3 "set x" means "Et(x in t)"
1. Show AaAbAy(Ax[x in y <-> (xIa v xIb)] -> yIy)
2. Show Ax[x in y <-> (xIa v xIb)] -> yIy
3. Ax[x in y <-> (xIa v xIb)] Assume
4. Show yIy
5. Show set y
6. AaAb[set a & set b -> Ey(set y & Ax(x in y <-> x=a v x=b))] N5
Case 1: a,b are sets. Then by 6 we have a set s such that for every
x: x in s <-> x=a v x=b. I claim y is included in s. Thus, suppose
that t in y. Then by (2), tIa v tIb. Hence, by D2 t=a v t=b.
Hence, t in s. So y is included in s.
Case 2: a is a set and b is not. Then by 6 we have a set r such
that for every x: x in r <-> x=a. I claim y is included in r.
Suppose t in y. Then, by (2) tIa v tIb. Hence by D2,
set a v set b. But ~(set b). So tIa. By D2, therefore,
t=a. Hence t in r. So y is included in r.
Case 3: b is a set and a is not. Then, by 6 we have a set q
such that for every x: x in q <-> x=b. By the same reasoning
as in Case 2, y is included in q.
Case 4: Neither a nor b is a set. Then, by 2 and D2, y has
no members. On the other hand, by N9 and N4 every class
with no members is a set.
But (1-4) exhaust all possibilities. In (1-3) y is a sub-class
of some set and so a set, and in (4) y is a set. So y is a set.
Moreover, by N4 y=y. So by D2, yIy.
Hence, modulo D2 and D3: N4,N5,N6,N9|-C5
C6 AaAy(Ax[x in y -> x in a] -> (aIa -> yIy))
N4 AxAy[Az(z in x <-> z in y) -> x = y] Premise
N6 Aa[set a -> Ay(Ax(x in y -> x in a) -> set y)] Premise
D2 "xIy" means "x=y & set x & set y"
1. Show AaAy(Ax[x in y -> x in a] -> (aIa -> yIy))
2. Show Ax[x in y -> x in a] -> (aIa -> yIy)
3. Ax[x in y -> x in a] Assume
4. Show aIa -> yIy
5. aIa Assume
6. Show yIy
7. Aa[set a -> Ay(Ax(x in y -> x in a) -> set y)] N6
8. set a -> Ay(Ax(x in y -> x in a) -> set y) 7,UI
9. a=a & set(a) 5,D2
10. Ay(Ax(x in y -> x in a) -> set y 7,8
11. Ax(x in y -> x in a) -> set y 10,UI
12. set y 3,11
13. y=y N4
14. yIy 12,13,D2:Ca Sh(6)
15. aIa -> yIy 5,6:Ca Sh(4)
16. Ax[x in y -> x in a] -> (aIa -> yIy) 3,4:Ca Sh(2)
17. AaAy(Ax[x in y -> x in a] -> (aIa -> yIy)) 2:Ca Sh(1)
C7 AaAy[Ax(x in y <-> Ew(w in a & x in w)) -> (aIa -> yIy)]
N2 AxAyAz[x=y -> (z in x <-> z in y)] Premise
N4 AxAy[Az(z in x <-> z in y) -> x = y] Premise
N7 Aa[set a -> Ey(set y &
Ax(x in y <-> Ew(w in a & x in w)))] Premise
D2 "xIy" means "x=y & set x & set y"
D3 "set x" means "Et(x in t)"
1. Show AaAy[Ax(x in y <-> Ew(w in a & x in w)) -> (aIa -> yIy)]
2. Show Ax(x in y <-> Ew(w in a & x in w)) -> (aIa -> yIy)
3. Ax(x in y <-> Ew(w in a & x in w)) Assume
4. Show aIa -> yIy
5. aIa Assume
6. Show yIy
7. Aa[set a -> Ey(set y & Ax(x in y <-> Ew(w in a & x in w)))] N7
8. set a -> Ey(set y & Ax(x in y <-> Ew(w in a & x in w))) 5,UI
9. Ey(set y & Ax(x in y <-> Ew(w in a & x in w))) 5,8,D2
10. set m & Ax(x in m <-> Ew(w in a & x in w)) 9,EI
11. Ax(x in m <-> Ew(w in a & x in w)) 10
12. Ax(x in m <-> x in y) 3,11
13. set m 10
14. set y 12,13,N4,D3,N2
15. yIy 14,N4,D2:Ca Sh(6)
16. aIa -> yIy 5,6:Ca Sh(4)
17. Ax(x in y <-> Ew(w in a & x in w)) -> (aIa -> yIy) 3,4:Ca Sh(2)
18. AaAy[Ax(x in y <-> Ew(w in a & x in w)) ->
(aIa -> yIy)] 2:Ca Sh(1)
C8 AaAy[Ax(x in y <-> Aw(w in x -> w in a)) -> (aIa -> yIy)]
N8 Aa[set a -> Ey(set y &
Ax(x in y <-> Aw(w in x -> w in a)))] Premise
N2 AxAyAz[x=y -> (z in x <-> z in y)] Premise
N4 AxAy[Az(z in x <-> z in y) -> x = y] Premise
D2 "xIy" means "x=y & set x & set y"
D3 "set x" means "Et(x in t)"
1. Show AaAy[Ax(x in y <-> Aw(w in x -> w in a)) -> (aIa -> yIy)]
2. Show Ax(x in y <-> Aw(w in x -> w in a)) -> (aIa -> yIy)
3. Ax(x in y <-> Aw(w in x -> w in a)) Assume
4. Show aIa -> yIy
5. aIa Assume
6. Show yIy
7. Aa[set a -> Ey(set y & Ax(x in y <->
Aw(w in x -> w in a)))] N8
8. set a -> Ey(set y & Ax(x in y <->
Aw(w in x -> w in a))) 7,UI
9. Ey(set y & Ax(x in y <-> Aw(w in x -> w in a))) D2,5,8
10. set m & Ax(x in m <-> Aw(w in x -> w in a)) 9,EI
11. Ax(x in m <-> Aw(w in x -> w in a)) 10
12. Ax(x in m <-> x in y) 3,11
13. set m 10
14. set y 12,13,N4,D3,N2
15. yIy 14,N4,D2:Ca Sh(6)
16. aIa -> yIy 5,6:Ca Sh(4)
17. Ax(x in y <-> Aw(w in x -> w in a)) ->
(aIa -> yIy) 3,4:Ca Sh(2)
18. AaAy[Ax(x in y <-> Aw(w in x -> w in a)) ->
(aIa -> yIy)] 2:Ca Sh(1)
*****************************************************************
Proofs 7, news:<70f94e16.02090...@posting.google.com>
Proofs 8, news:<70f94e16.02090...@posting.google.com>
N6 Aa(set a -> (Ay(Ax(x in y -> x in a)) -> set y)) Axiom of Subsets
N7 Aa[set a -> Ey(set y & Ax(x in y <-> Ew(w in a & x in w)))] Axiom of Unions
N8 Aa[set a -> Ey(set y & Ax(x in y <-> Aw(w in x -> w in a)))] Axiom of Power Sets
N9 Ex(set x & Ay~(y in x)) Empty Set Axiom
Proofs 9 (see above)
C1 AxAyAz[xIy -> (z in x <-> z in y)] LL1
C2 AxAyAz[Az(z in x <-> z in y) <-> ((x in z <-> y in z))] LL2
C3 EyAx[x in y <-> Et(x in t) & P(x)] (with y not free in P(x))
Classification
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}] Weak
Extensionality
C5 AaAbAy(Ax[x in y <-> (xIa v xIb)] -> yIy) Axiom of Pairs
C6 AaAy(Ax[x in y -> x in a] -> (aIa -> yIy)) Axiom of Subsets
C7 AaAy[Ax(x in y <-> Ew(w in a & x in w)) -> (aIa -> yIy)] Axiom of Unions
C8 Aa(Ay[Ax(x in y <-> Aw(w in x -> w in a))) -> (aIa -> yIy]) Axiom of Power
Sets
John
> john_...@excite.ca (John) wrote in message news:<70f94e16.02091...@posting.google.com>...
>
>
> > IMO, philosophers of mathematics write all too infrequently about logic.
> > Two questions of interest to me: (1) What is logical truth?
>
>
> Very good, concise article:
>
> LOGICAL TRUTH:
>
> (The Oxford Companion to Philosophy, © Oxford University Press 1995 )
>
>
> The expression has various meanings, all connected to the idea of a
> logical system.
>
> Logical systems have always shared two features: they are at least
> partly symbolic, using letters or similar devices, and they assert, or
> preferably prove, results about their symbolic expressions (in the
> modern jargon, the 'formulae' of their 'logical language'), results
> such as: any argument of the form 'No Bs are Cs, some As are Bs, so
> some As are not Cs' is valid; '¬P' is a consequence of '(P -> ¬P)'.
>
> 1. One current meaning of 'logical truth' is 'result in some sound
> logical system' ('sound' is not redundant here: it excludes faulty
> logical systems in which not all the results are true). A true result
> will usually be a proved result, therefore a theorem, for example (as
> above):
>
> '¬P' is a consequence of '(P -> ¬P)'.
>
> 2. Sometimes certain symbolic expressions are themselves described as
> logical truths, for example:
>
>
> If some As are Bs, then some Bs are As.
> ((P -> ¬P) -> ¬P).
>
> Here explanation is needed, since strictly speaking these expressions
> are not truths at all (they do not say anything). What is meant is
> that all their instances are true, where an instance is what you can
> express by uniformly replacing certain schematic or - in a loose sense
> - 'variable' symbols (the letters A and B in the first example, the
> letter P in the second) by syntactically permissible words from an
> adequately rich vocabulary; or, alternatively, that they are true
> under all interpretations, where an interpretation assigns meanings
> uniformly to those same 'variables' from a syntactically limited but
> adequately rich range of meanings. In this usage, truth and falsity do
> not exhaust the field: in between logical truths, all of whose
> instances are true, and logical falsehoods, all of whose instances are
> false, are symbolic expressions such as 'P or not Q', having some true
> and some false instances.
>
> 3. Finally, and perhaps most commonly, 'logical truth' may mean 'truth
> that is true in virtue of some result in a sound logical system'. The
> basic kind of case is a truth that is an instance (or interpretation)
> of a symbolic expression all of whose instances (or interpretations)
> are true, i.e. an instance of a type 2 logical truth, for example:
>
>
> If some men are Greeks, then some Greeks are men.
> If a condition for your believing erroneously that you exist is that
> the belief is not erroneous, then it is not erroneous.
>
> The range of type 3 logical truths is indeterminate, since it depends
> on which sorts of system you are willing to count as logical.
> Propositional logic, predicate logic, and syllogistic are accredited
> systems, but not all philosophers are so happy about, say, modal
> logic, epistemic logic, tense logic, deontic logic, set theory,
> mereology. On the other hand it is disputable whether any boundary
> conditions can rationally be set; certainly none are agreed.
>
> Type 3 logical truths can be defined in other roughly equivalent ways:
> 'true in virtue of its (logical) form', that is, in virtue of being an
> instance of some type 2 logical truth; 'true in virtue of the meanings
> of its logical words', that is, of the words in it that can be
> represented by constants in some logical system; or 'true under all
> reinterpretations of its non-logical words', similarly.
>
> Basic type 3 logical truths are often described as 'logically
> necessary', as if their origin in logic guarantees their necessity.
> Part (only part) of the guarantee comes from using intuitively
> satisfying methods to prove the logical results, the type 1 truths,
> methods which may be semantic, resting on the truth-conditions of the
> system's constants, or logistic, resting on self-commending
> manipulation of ('derivation from') self-commending primitive
> expressions ('axioms').
>
> Other truths can be deduced from the basic logical truths by means of
> definitions; for example, 'A mastax is a pharynx' from 'The pharynx of
> a rotifer is a pharynx' by the definition of 'mastax'. But usually
> these aren't counted as logical truths, though they are counted as
> logically necessary.
>
> There's a warning in all the above: it would be mistake to suppose
> that you can always tell at a glance whether some proposition is a
> type 3 logical truth. You must know your type 1 truths, the theorems
> of sound systems, many of which are far from obvious; you must judge
> whether the systems they belong to deserve to be called logical; you
> must take care over the notions of 'instance' and 'interpretation'
> (for example, 'If she's wrong, she's wrong' will not be an instance of
> the type 2 logical truth 'If P, P', unless the 'shes' refer to the
> same person); and definitions - if the use of them is allowed - are
> often hazy (for example, is water liquid by definition?).
>
> C.A.K.
> Bibliography W. V. Quine, 'Carnap and Logical Truth', in B. H.
> Kazemier and D. Vuysje (eds.), Logic and Language (Dordrecht, 1962);
> repr. in P. A. Schilpp (ed.), The Philosophy of Rudolf Carnap (La
> Salle, Ill., 1963), and in The Ways of Paradox (New York, 1966).
> W. V. Quine, Philosophy of Logic (Englewood Cliffs, NJ, 1970), ch. 4.
> P. F. Strawson, 'Propositions, Concepts, and Logical Truths',
> Philosophical Quarterly (1957); repr. in Logico-linguistic Papers
> (London, 1971).
>
>
> regards
> PH
What is the bearing of Kirwan's article on the following posting of Moses
Klein's concerning the soundness of first-order logic?
> > In standard first-order logic the formula Ex ~R(x) V Ex R(x)
> > (that is, Ax R(x) -> Ex R(x)) is provable, and it holds in every stru-
> > cture. But if one allows structures to have empty universe the formula
> > does not hold. To put it another way, { ~Ex ~R(x), ~Ex R(x) } is incon-
> > sistent, but it has a model, the structure with M = empty, R_M = empty.
> > Apparently one has to do something more than just stipulate M can be
> > empty; losing the Completeness theorem is no little thing!
>
> Ilias, you understate the crisis. If you have an inconsistent theory
> with a model, you violate not completeness, but its converse,
> soundness. (A complete logic is one for which every valid statement
> is provable. A sound logic is one for which every provable statement
> is valid.) Losing completeness, which could reasonably be called
> the Fundamental Theorem of Model Theory, would certainly be no little
> thing; losing soundness would be catastrophic!
>
> This can certainly be repaired. As others have pointed out,
> we can redefine the rules of inference to make this modified
> first-order logic complete and sound. (This is possible
> because the set of validities is still recursively enumerable,
> and any r.e. set is the set of theorems of some formal system.
> There is still one purely semantic issue concerning logic
> allowing the empty model: how to interpret constant symbols?
>
> That brings me to the real objection to Bill Dubuque's statement. He
> was replying to Dave Seaman's assertion that *groups*, rather than
> model structures in general, cannot be empty. Group theory needs
> either a constant symbol for the identity, or an axiom stating the
> existence of an identity. If the latter, then that axiom is false
> in an empty structure. If the former, then the empty structure can't
> even be a model for the language. Any way you slice it, part of
> the definition of group is that there has to be an identity element,
> and as a trivial consequence there has to be an element.
>
> In brief: it's no great pain to redefine first-order logic to
> allow an empty structure, but there's still no way that structure
> is a group, so Dave's statement remains true.
news:<52r9d6$1q...@news.doit.wisc.edu>
Owen Holden
"John" <john_...@excite.ca> wrote in message
news:70f94e16.02091...@posting.google.com...
> paulholba...@freenet.de (Paul Holbach) wrote in message
news:<881c8779.02091...@posting.google.com>...
> > john_...@excite.ca (John) wrote in message
news:<70f94e16.02091...@posting.google.com>...
> >
> >
> > > IMO, philosophers of mathematics write all too infrequently about
logic.
> > > Two questions of interest to me: (1) What is logical truth?
All truth is ' that which can be shown'
Logical truth is that truth which cannot be otherwise.
The existential quality of 'tauutology', is obvious.
Owen
Neil Nelson
> "John" <john_...@excite.ca> wrote in message
> news:70f94e16.02091...@posting.google.com...
>
>>paulholba...@freenet.de (Paul Holbach) wrote in message
>>
> news:<881c8779.02091...@posting.google.com>...
>
>>>john_...@excite.ca (John) wrote in message
>>>
> news:<70f94e16.02091...@posting.google.com>...
>
>>>
>>>>IMO, philosophers of mathematics write all too infrequently about
>>>>
> logic.
>
>>>>Two questions of interest to me: (1) What is logical truth?
>>>>
>
> All truth is ' that which can be shown'
> Logical truth is that truth which cannot be otherwise.
> The existential quality of 'tautology', is obvious.
There are different logics, see Susan Haack's book, that do
not have the same theorems and hence not the same truths, and
hence otherwise and not generally obvious.
It seems that we can easily say that what any given group of
mathematicians and their philosophers can agree is true will
be true for them. That is, a group can be defined by its
agreement to a given philosophy or method. And then, as a
quick aside, to have different groups requires that we can
identify the differences that define those groups. That is,
to know that some person subscribes to some particular
philosophy or group as against another requires that there is
a method by which the distinction is made and apparently
known to those who assert those differences.
But onto a primary and related theme which is that agreement
(or agreement that there is disagreement) requires, if we
are to speak in some sense where our meaning is to be adopted
by others (why speak otherwise), that there is some objective
standard, some universal method of agreement--though not that
we must agree, but that we can agree that we disagree or agree
that we cannot tell. And the identification of agreement
apparently at the moment is by a predictive word sequence.
That is, to determine that another person agrees it might be
sufficient for that other person to say, "I agree". But to be
certain we may want to more clearly know what it is that they
are agreeing to, at which point we could ask questions as to
the detail of their agreement with what we would say ourselves.
If their responses to our questions are the same as what we
would give ourselves, what we expect, then we would have a
higher confidence in their agreement. And then since we only
have words at the moment by which to ask and give replies, we
have a predictive word sequence that determines agreement.
I suppose we could just say two people agree without getting
a confirming word sequence. But it would seem better to say
that of which we have some evidence, some confidence. That
is, you might say A agrees with B, but I could ask how you
know such is the case, and if you did not give your reason
or evidence I would ask how I would choose between you and
someone you would say otherwise. This then requires that you
give an argument or predictive word sequence that obtains my
agreement as above.
Now comes an interesting point which is that if we require
a predictive word sequence to obtain agreement, but perhaps
before continuing some emphasis on what agreement has to do
with the price of tea. To say 'the price of tea' in a
meaningful sense so that the person being spoken to is
willing to buy (or not) your tea requires that they agree with
you on what tea is and what the units of price represent. All
meaning, by whatever method, between two people requires that
those two people agree on their expected common or same
meaning. The meanings for those two people could easily not
be the same, or we could not know they were the same, unless
there was a method by which that can be known. Meaning for
the current context is where two people have the same
predictive word sequence for a given question, which could
be: what does this word or expression mean?
But since we are requiring predictive word sequences to
determine agreement--and what is true is what we agree that
it is--we can ask how it comes about that two people can
have the same predictive word sequences. One way to do this
is to have rules, say those of logic, by which the word
sequences are generated. If we both use the same word
sequence generating rules then we will have the same
results.
But can we assert that there are other ways other than rules
by which agreement on the word sequences can be obtained?
That is, how is it possible that two people could consistently
show an agreement on word sequences without there being some
prior agreement between those two people on their generating
rules? But perhaps the question is: even though two people
might consistently agree without the rules in evidence (if
there are any rules), what evidence or argument can we give by
which we can assert that those two people will agree in the
future? That is, if we do not have the evidence of the
generating rules that the two people are using, what will our
reason be for asserting that those two people will agree in
the future, that they are in agreement in the general case?
The question is not that is it possible for two people to
agree and where we do not know how. The question is how do
we assert that they will continue to agree if we do not know
how? E.g., for me to say that so-and-so subscribes to some
particular logic or philosophy requires, unless I am only
refering to their prior statements, that I am able to give
a method in words, the rules in words, that will predict
so-and-so's future words to some sufficiency of agreement
with that logic or philosophy.
Now that we have in the current context: (1) that agreement
requires a predictive sequence of words/symbols, and (2)
that an assertion for general or future agreement requires
the generating rules to be in words--we have generalized
word maps--, what agreement can be asserted that is not
bounded by Church's Thesis? What predictive sequences are
there by which agreement may be obtained that are not
bounded by a Turing complete language? Since we do not
yet have a good counterexample to Church's Thesis, no
conclusive opposing assertion can be made unless of course
you do not agree with Church's Thesis.
Regards,
Neil Nelson
nucleus
>Owen Holden wrote:
>
>> "John" <john_...@excite.ca> wrote in message
>> news:70f94e16.02091...@posting.google.com...
>>
>>>paulholba...@freenet.de (Paul Holbach) wrote in message
>>>
>> news:<881c8779.02091...@posting.google.com>...
>>
>>>>john_...@excite.ca (John) wrote in message
>>>>
>> news:<70f94e16.02091...@posting.google.com>...
>>
>>>>
>>>>>IMO, philosophers of mathematics write all too infrequently about
>>>>>
>> logic.
>>
>>>>>Two questions of interest to me: (1) What is logical truth?
>>>>>
>>
>> All truth is ' that which can be shown'
Nope.
'That which can be shown' is a matter of interpretations.
First of all, it depends on your brain conditioning.
The things you "see" depend on your brain programming.
If you look at what is going on in the USA this very moment,
it is simply mind boggling. These people "see" something
entirely different from what the other people in the world
"see". They simply have gone mad.
Secondly, truth is not trueness.
It is not a matter of your belief system.
The best interpretation I've heard is:
THAT WHICH IS.
It does not matter what you see.
It still is.
It does not matter what is the state of your "science".
It still is.
Sooner or later you are bound to "see"
that which is.
>> Logical truth is that truth which cannot be otherwise.
Logical truth is but a delusion.
Limitations of logic can not be applied to
issues of Truth.
Truth includes all aspects, logical, illogical,
emotional, or otherwise.
Logic, on the first place, is broken in the most
profound ways.
First of all, it is the most extreme limitation
of scope.
What is "logic"?
Well, it is essentially the issue of "true" versus
"false".
It is a binary delusion of the most profound magnitude.
Out of entire scope of existence, only black and white
are accepted. Then they are further recombined
in order to "derive" some other conclusions,
thus furthering this binary delusion.
Essentially, logic is but a byproduct of what
is known to be fascism.
When, out of entire scope, you only take two extreme
samples, what you have is reduction of a rainbow
of existence to a fascist, black and white view.
Logic is but a byproduct of morality,
invented by the priest, derving from the notions
of "good" and "bad". Interestingly enough,
those "good" and "bad" aren't even the same
in different traditions, geographical regions, etc.
Furthermore, that which was considered "bad"
can be EASILY reclassified as "good".
Take for example homosexualism.
It is one of the condemned things in Bible,
and yet, at this junction, it has been accepted
as some kind of a "norm" and some countries and
states even accept it as a valid marriage.
So...
What is "good"?
What is "bad"?
Anybody can define?
Interestingly enough, eventually, the scientist's
argument will be reduced to morality. Because it is
all based on manipulations of the initial assumptions,
and, since those assumptions can not be proven in principle,
you would have to resort to resolving the core issues
from the standpoint of morality.
>> The existential quality of 'tautology', is obvious.
>There are different logics, see Susan Haack's book, that do
>not have the same theorems and hence not the same truths, and
>hence otherwise and not generally obvious.
Logic can not possibly have truths.
The so called scientific truth
is the outmost delusion.
Otherwise, you have to completely abandon the very notion
of Truth, and that is not possible, even in principle.
>It seems that we can easily say that what any given group of
>mathematicians and their philosophers can agree is true will
>be true for them.
True is not the same as truth.
Now you reduced it to the level of obscene.
> That is, a group can be defined by its
>agreement to a given philosophy or method. And then, as a
>quick aside, to have different groups requires that we can
>identify the differences that define those groups. That is,
>to know that some person subscribes to some particular
>philosophy or group as against another requires that there is
>a method by which the distinction is made and apparently
>known to those who assert those differences.
Now you are lost here and you will never be able
to get out of the deadliest contradictions there are.
You are merely talking about the brain conditioning
and belief systems that are all contradictory to one another.
Good buy.
Seth Russell
> All truth is ' that which can be shown'
> Logical truth is that truth which cannot be otherwise.
> The existential quality of 'tauutology', is obvious.
I don't think so.
Logical truth is that which cannot be otherwise,
*given the assumptions you have built into your system*.
Seth Russell
Logic is Great, survival is better.
http://robustai.net/
Chris Menzel
> All truth is 'that which can be shown'
Either it is true that there is an even number of stars or it is true
that there is an odd number of stars. No one can show which. So truth
is not "that which can be shown".
> Logical truth is that truth which cannot be otherwise.
It cannot be otherwise that arithmetic is incomplete. But that
arithmetic is incomplete isn't a logical truth.
Chris Menzel
David C. Ullrich
wrote:
You must have a good reason for saying this is not a logical truth,
but I can't figure out what it could be.
>Chris Menzel
David C. Ullrich
Seth Russell
> On Wed, 18 Sep 2002 18:56:38 GMT, Owen Holden <oori...@yahoo.com> said:
>
>>All truth is 'that which can be shown'
>
> Either it is true that there is an even number of stars or it is true
> that there is an odd number of stars.
I doubt that is true even in the domain of human science. The problem
being that what is (or is not) a star is not a thing that is universally
agreeable. There are some gass clouds which some will count as stars
and others will count as not stars.
Seth Russell
http://robustai.net/mentography/3laws.jpg
Chris Menzel
Well, of course, there are huge issues here, but a quick and
pretty widely accepted answer here [Translation: this is my answer but
don't want to argue for it! :-] is that logical truths per se are those
that follow from the meanings of some set of logical constants alone --
quantifiers, boolean operators, etc. What counts as a logical constant,
of course, is a matter of not a little contention, but most folks agree
that (logicism notwithstanding) the mathematical concepts involved in
the incompleteness theorem, while of course an intimate part of the
*metatheory* of pure logic, are not themselves purely logical. The
theorem requires a fair chunk of set theory, for example, and only the
hardiest of souls would argue that its axioms are all truths of logic in
the above sense. It is certainly doubtful, for example, that the axiom
of infinity, or even pairing for that matter, follows from the *meaning*
of "is a member of".
Chris Menzel
Neil Nelson
> On Wed, 18 Sep 2002 18:56:38 GMT, Owen Holden <oori...@yahoo.com> said:
>
>>All truth is 'that which can be shown'
>>
>
> Either it is true that there is an even number of stars or it is true
> that there is an odd number of stars. No one can show which. So truth
> is not "that which can be shown".
It seems that the essential aspect of this argument could be
retained without regard to what are starts as in the argument
by Seth Russell. E.g., we could say that Chris has selected a
card from a standard 52 card deck and where Jack, Queen, and
King are numbered 11, 12, and 13, respectively, and such that
only Chris knows what card he has selected.
Now we can say that it is true that the number of the card
Chris has is either even or odd. And I can agree with that,
or rather I agree that it is (truely) either one or the other.
As to whether or not there is a truth that I do not know that
Chris knows, the parallel being that the number of stars may
not be known but that number is either even or odd and hence
a truth that is not known, I can agree that there are truths
that I (we) do not know,
But this seems to be a somewhat particular class of truth.
That is, if I do not know a particular truth then I could
not argue from holding that truth. If I did not know if the
card Chris held was even or odd, I could not then say since
the card Chris holds is odd, then .... And I could not
usefully argue in classical logic that since the card Chris
holds is even or odd, then ... because argument of any
useful content does not argue from a (classical) tautology.
Though it might be useful to say that there are truths we
do not know and perhaps we say such things to keep our minds
open, it is quite difficult to argue from a truth you do
not know and hence other than the observation that there may
be such things it is not clear what other useful result of
any kind may be obtained from that position. And, commonly
we want to have a truth of content from which further
content can be derived and not vacuous truth.
Regards,
Neil Nelson
Jesse F. Hughes
> The [first incompleteness] theorem requires a fair chunk of set
> theory, for example, and only the hardiest of souls would argue that
> its axioms are all truths of logic in the above sense. It is
> certainly doubtful, for example, that the axiom of infinity, or even
> pairing for that matter, follows from the *meaning* of "is a member
> of".
Surely, the axiom of infinity is not necessary for the proof of the
incompleteness theorems, although pairing is.
I suppose Ullrich's question was intended thus: Why is the statement,
"The following theory is incomplete," not a logical truth? Granted,
it can't be expressed as a single formula (since the theory includes
the induction scheme), so one might say that it doesn't count as a
logical truth, but that seems a cheap way out.
Note: I'm not expressing an opinion as to whether the incompleteness
theorem is properly called a logical truth or not.
--
Jesse Hughes
"Casting [Demi] Moore as a woman who has come to the New World so that
she can 'worship without fear or persecution' in _The_Scarlet_Letter_
is like casting Bruce Willis as Young Rene Descartes." -Joe Queenan
Seth Russell
> Chris Menzel wrote:
>
>> On Wed, 18 Sep 2002 18:56:38 GMT, Owen Holden <oori...@yahoo.com> said:
>>
>>> All truth is 'that which can be shown'
>>
>> Either it is true that there is an even number of stars or it is true
>> that there is an odd number of stars. No one can show which. So truth
>> is not "that which can be shown".
>
> It seems that the essential aspect of this argument could be
> retained without regard to what are starts as in the argument
> by Seth Russell. E.g., we could say that Chris has selected a
> card from a standard 52 card deck and where Jack, Queen, and
> King are numbered 11, 12, and 13, respectively, and such that
> only Chris knows what card he has selected.
I don't see how that argument applies here. My argument
is that the number of stars depends upon an effective
procedure to count them, and whether *there can be no
dispute* about that procedure - lacking such a procedure,
then the nuber of stars is a subjective choice and are we
are not asking a question that has a binary answer.
It might be easier to understand my argument if the
question was: "How many drops of water are there in
the ocean?" Do you propose that it is a logical
truth that the number of drops is odd or even?
Seth Russell
http://robustai.net/sailor/
Jesse F. Hughes
> cme...@philebus.tamu.edu (Chris Menzel) writes:
>
>> The [first incompleteness] theorem requires a fair chunk of set
>> theory, for example, and only the hardiest of souls would argue that
>> its axioms are all truths of logic in the above sense. It is
>> certainly doubtful, for example, that the axiom of infinity, or even
>> pairing for that matter, follows from the *meaning* of "is a member
>> of".
>
> Surely, the axiom of infinity is not necessary for the proof of the
> incompleteness theorems, although pairing is.
Actually, my comment about pairing is silly, too. All that's needed
is a chunk of number theory itself, of course. I was wrongly
persuaded by Chris's comment that the theorem requires a chunk of set
theory. It does not.
--
"Russell was in my opinion as much a philosopher as a mathematician,
and in my opinion, philosophy counts as a practical industry."
-- James Harris, on the claim Bertrand Russell was a
"pure" mathematician
David C. Ullrich
wrote:
Hmm. It's not clear to me that the incompleteness theorem requires
all that much set theory, nor that saying something is "metalogic"
means that it's not a logical truth. Surely metalogical theorems
are also logical truths - the distinction between logic and metalogic
is really just the distinction between things that are proved _in_
a formal system and things that are proved _about_ a formal system,
which of course is an important distinction, but it doesn't make the
metalogical truths any less logical truths. Or so it seems to me.
The truth that seems to me is a logical truth would be something of
the form "assuming [insert large number of axioms and definitions]
then arithmetic is inconsistent" - I assumed that's what you really
meant, and it seems to me that that statement _is_ a logical truth.
Or so it seems to me.
Neil Nelson
> Neil Nelson wrote:
>>
>> It seems that the essential aspect of this argument could be
>> retained without regard to what are stars as in the argument
>> by Seth Russell. E.g., we could say that Chris has selected a
>> card from a standard 52 card deck and where Jack, Queen, and
>> King are numbered 11, 12, and 13, respectively, and such that
>> only Chris knows what card he has selected.
>
>
> I don't see how that argument applies here. My argument
> is that the number of stars depends upon an effective
> procedure to count them, and whether *there can be no
> dispute* about that procedure - lacking such a procedure,
> then the nuber of stars is a subjective choice and are we
> are not asking a question that has a binary answer.
>
> It might be easier to understand my argument if the
> question was: "How many drops of water are there in
> the ocean?" Do you propose that it is a logical
> truth that the number of drops is odd or even?
I think you make a good point, and perhaps we could see two
issues involved:
(1) whether or not it makes sense to speak of stars or drops
of water and hence anything derived from those ambiguous
terms, and this is a very good point, and
(2) that there is a common assumption that there are truths we
do not know, and hence truths that are at the moment not shown,
and hence truths not shown for a current argument. It is this
second aspect that seems to me to be what Chris was getting at.
And I think this needs to be worked on a bit more.
An argument might be made that all truth is known and shown.
That is though we need to keep our minds open, and hence hold
that it is possible that new truth may be made available to
us and that though it might be new when we obtain that truth
that it was true before we obtained it, the key aspect seems
to be our knowing of the truth. And where knowing implies
some clear evidence for that knowing, that it is shown.
E.g., to say that Chris' card, in my prior example, is either
even or odd, is rather a property of numbers easily shown as
against the particulars of Chris holding a card. That is, we
know that every number is either even or odd and however one
supposes to obtain a number it will be even or odd. There is
nothing unknown about this aspect of numbers. And then we
assume (and know by common assumption and perhaps argued from
frequent physical experience) that certain aspects of physical
reality conform to certain logical and arithmetical properties.
The result being that what we know about the numerical
properties of the numbers of stars, water drops, or a card held
by Chris, is shown precisely to the degree that we know it.
That is, what is known in a commonly acceptable sense, that
for which an acceptable argument may be given, is shown (by
the requirement for an acceptable argument), and that which
is shown (has an acceptable argument) is then known. Truth
and Proof identify the same set.
Neil Nelson
Chris Menzel
Well, that sort of misses the point. Your reply simply exploits the
vagueness of the concept of a star. Focus instead on knowledge of the
past. For any past event E, there is a date E(D) at which it occurred
(or an interval of time over which it occurred). For the large majority
of such events, that date cannot be shown. Nonetheless, it is true, for
any such event E (say the collision of the meteor with earth that led to
the extinction of the dinosaurs), that E occurred at (or over) E(D). So
truth isn't that which can be shown.
Chris Menzel
Neil Nelson
> The truth that seems to me is a logical truth would be something of
> the form "assuming [insert large number of axioms and definitions]
> then arithmetic is inconsistent" - I assumed that's what you really
> meant, and it seems to me that that statement _is_ a logical truth.
Things were going along very well until the association of arthmetic
being inconsistent and logical truth. I am not aware of any argument
for the inconsistency of arithmetic and can rather think of a good
argument for the consistency of any Turing complete language of which
Arithmetic sometimes is held to be, which is that such a language is
comprised only of functions. And I think there are related arguments
by Goedel and Shoenfield. And so it must have been a typo I have
overreacted to.
Perhaps what is meant instead of logical truth is that the statement
would have logical form and then if it were held or asserted would
be held or asserted true. That is, a person if they held that
argument would not then say that they did not hold a truth without
logic (though at the moment it would be illogical to do so).
Neil Nelson
Chris Menzel
> jes...@cs.kun.nl (Jesse F. Hughes) writes:
>
> > cme...@philebus.tamu.edu (Chris Menzel) writes:
> >
> >> The [first incompleteness] theorem requires a fair chunk of set
> >> theory, for example, and only the hardiest of souls would argue that
> >> its axioms are all truths of logic in the above sense. It is
> >> certainly doubtful, for example, that the axiom of infinity, or even
> >> pairing for that matter, follows from the *meaning* of "is a member
> >> of".
> >
> > Surely, the axiom of infinity is not necessary for the proof of the
> > incompleteness theorems, although pairing is.
>
> Actually, my comment about pairing is silly, too. All that's needed
> is a chunk of number theory itself, of course. I was wrongly
> persuaded by Chris's comment that the theorem requires a chunk of set
> theory. It does not.
Hm, well, maybe I'm overgeneralizing from standard presentations of the
theorem; I'm certainly no expert. But typically one lays some
groundwork for the theorem by doing a bunch of recursion theory, which
involves quantification over functions on N, which in turn we usually
cash in terms of set theory. Are you saying this is inessential to the
proof of the theorem?
So, once again, my claim is only that the incompleteness theorem, though
necessarily true, is not one that can be derived from basic logical
principles alone, and hence is not a logical truth. It relies in
addition upon a number of principles from the more specialized abstract
sciences. I'd be happy to be informed about minimal sets of those
principles, but I'll be surprised, to say the least, if we don't find a
number of specialized, nonlogical principles in all of them.
Chris Menzel
Pat
"Neil Nelson" <news_r...@dslextreme.com> wrote in message
news:3D8A6025...@dslextreme.com...
The 3 'R's: aRithemtic, Rhetoric and Religion are all consistent in their
own world. They start becoming inconsistent when you look at them from
outside. Consistency is a viewpoint delimiting the universe around it.
Marvin Minsky used Gödel's incompleteness theorem to point out that logic
itself is inconsistent, as you must be well aware. Take the following simple
construct:
A-->B-->C (Read it: If A then B and if B then C). Transitivity in logic
dictates then that: If A then C.
Step outside the world of logic and to consistent world of language and
interpret the sign "-->" as meaning "next to" and reinterpret the statement
in question. A implies B which implies C. A implies that B is next to it and
B implies that C is next to it. Does A then imply that C is next to it?
Certainly not. I know there is modal logic or something else to that effect
to deal with cases like this and that Implication in formal logic is also a
two-barreled gun. My point is: whenever the subject area becomes
inconsistent you need to go outside of it and redefine. As long as you stick
within, Bob's your uncle. Stray outside, and all the hell gets loose.
Patrick O'Hooligan, the WiseGuy
John
> of "is a member of".
>
> Chris Menzel
Even so, it follows from FOL and (C3,C4) that there's a
collection to which a,b belong if anything does. (See proof
of T9 at news:<70f94e16.02090...@posting.google.com>.)
C3 EyAx[x in y <-> Et(x in t) & P(x)] (with y not free in P(x))
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}] Weak
Extensionality
But you wouldn't want a pairing principle to follow from
the meaning of "in" and the meaning of "identity", now, would you?
I mean, it's ever so much nicer to assemble basic principles AD HOC,
right?
--John
"If giving a list of axioms that implies that there exists something
not identical with itself counted as proving that there actually _is_
something not identical with itself you could just give the axiom
Ex~(x=x), note that this axiom follows from itself, and be done with
it."
--Whatchamacallit
news:<7jb1nucclpj4pvopl...@4ax.com>
David C. Ullrich
<news_r...@dslextreme.com> wrote:
>David C. Ullrich wrote:
>
>
>> The truth that seems to me is a logical truth would be something of
>> the form "assuming [insert large number of axioms and definitions]
>> then arithmetic is inconsistent" - I assumed that's what you really
>> meant, and it seems to me that that statement _is_ a logical truth.
>
>
>Things were going along very well until the association of arthmetic
>being inconsistent and logical truth. I am not aware of any argument
>for the inconsistency of arithmetic
"inconsistent" was a typo for "incomplete". Sorry.
>and can rather think of a good
>argument for the consistency of any Turing complete language of which
>Arithmetic sometimes is held to be, which is that such a language is
>comprised only of functions. And I think there are related arguments
>by Goedel and Shoenfield. And so it must have been a typo I have
>overreacted to.
>
>Perhaps what is meant instead of logical truth is that the statement
>would have logical form and then if it were held or asserted would
>be held or asserted true. That is, a person if they held that
>argument would not then say that they did not hold a truth without
>logic (though at the moment it would be illogical to do so).
>
>Neil Nelson
David C. Ullrich
Neil Nelson
> "Neil Nelson" <news_r...@dslextreme.com> wrote in message
> news:3D8A6025...@dslextreme.com...
>
> The 3 'R's: aRithemtic, Rhetoric and Religion are all consistent in their
> own world. They start becoming inconsistent when you look at them from
> outside. Consistency is a viewpoint delimiting the universe around it.
> Marvin Minsky used Gödel's incompleteness theorem to point out that logic
> itself is inconsistent, as you must be well aware. Take the following simple
> construct:
>
> A-->B-->C (Read it: If A then B and if B then C). Transitivity in logic
> dictates then that: If A then C.
> Step outside the world of logic and to consistent world of language and
> interpret the sign "-->" as meaning "next to" and reinterpret the statement
> in question. A implies B which implies C. A implies that B is next to it and
> B implies that C is next to it. Does A then imply that C is next to it?
> Certainly not. I know there is modal logic or something else to that effect
> to deal with cases like this and that Implication in formal logic is also a
> two-barreled gun. My point is: whenever the subject area becomes
> inconsistent you need to go outside of it and redefine. As long as you stick
> within, Bob's your uncle. Stray outside, and all the hell gets loose.
There is paraconsistent logic, but as to classical logic and the weaker
logics also used for the common undecidability arguments, there seems
to be rather good agreement and arguments that those logics are
consistent. We may need to get the actual reference for Minsky to
see if we need to correct his position. But your position is easily
adjusted.
Obviously I could interpret any string to represent or show an
inconsistency, but typically we do not interpret or choose whatever
we wish when given a language or logic expression. And then this also
gets into the area of models. Typically what one does is find a model
of a given logic, its axioms, or set of logic statements and show
something about that logic via the model. But not just any model that
you may think of that may or may not model the logic will do. You
need to have a model where the consequences of the model are also
consequences of the logic.
And so you have provided a model, interpreting '-->' as next-to that
does not model logic's implication and then saying that such a model
has something to do with the logic. No one knowledgeable would care
to do such a thing. That is, your interpretation for '-->' is not
transitive, as you have noted, and hence not a model of how the
logic would be used and hence useless for making statements about the
logic.
The using of models and interpretations to say things about a logic
has certain rules. If you do not use those rules then we would avoid
your argument.
Neil Nelson
Pat
The reference goes back to 1974 dealing with Minsky's idea about Frame
represenatation of knowledge.
To quote it some more:
"A famous mathematician, warned that his proof would lead to a paradox if he
took one more logical step, replied "Ah, but I shall not take that step." He
was completely serious. A large part of ordinary (or even mathematical)
knowledge resembles that in dangerous professions: when are certain actions
unwise. When are certain approximations safe to use? When do various
measures yield sensible estimates? Which self-referent statements are
permissible if not carried too far? Concepts like "nearness" are to valuable
to give up just because no one can exhibit satisfactory axioms for them. To
summarize:
"Logical" reasoning is not flexible enough to serve as a basis for thinking;
I prefer to think of it as a collection of heuristic methods, effective only
when applied to starkly simplified schematic plans. The Consistency that
Logic demands is not otherwise usually available-and probably not even
desirable-because consistent systems are likely to be too "weak."
see: http://web.media.mit.edu/~minsky/papers/Frames/frames.html
>
> Obviously I could interpret any string to represent or show an
> inconsistency, but typically we do not interpret or choose whatever
> we wish when given a language or logic expression. And then this also
> gets into the area of models. Typically what one does is find a model
> of a given logic, its axioms, or set of logic statements and show
> something about that logic via the model. But not just any model that
> you may think of that may or may not model the logic will do. You
> need to have a model where the consequences of the model are also
> consequences of the logic.
>
> And so you have provided a model, interpreting '-->' as next-to that
> does not model logic's implication and then saying that such a model
> has something to do with the logic. No one knowledgeable would care
> to do such a thing. That is, your interpretation for '-->' is not
> transitive, as you have noted, and hence not a model of how the
> logic would be used and hence useless for making statements about the
> logic.
>
> The using of models and interpretations to say things about a logic
> has certain rules. If you do not use those rules then we would avoid
> your argument.
I get the drift of your argument here. But it seems to me that this approach
is an attempt to divorce logic from the real world. It is the classical way
of saying: lets make a consistent system and call it "Logic". It has got
some consistent rules and regulations. Then you try to apply it in the real
world and notice that it doesn't work consistently there. In this sense any
system I create which is consistent is a logical system no matter how simple
or frivolous it is.
Patrick O'Hooligan, The WiseGuy
Neil Nelson
> The reference goes back to 1974 dealing with Minsky's idea about Frame
> represenatation of knowledge.
> To quote it some more:
> "A famous mathematician, warned that his proof would lead to a paradox if he
> took one more logical step, replied "Ah, but I shall not take that step." He
> was completely serious. A large part of ordinary (or even mathematical)
> knowledge resembles that in dangerous professions: when are certain actions
> unwise. When are certain approximations safe to use? When do various
> measures yield sensible estimates? Which self-referent statements are
> permissible if not carried too far? Concepts like "nearness" are to valuable
> to give up just because no one can exhibit satisfactory axioms for them. To
> summarize:
>
> "Logical" reasoning is not flexible enough to serve as a basis for thinking;
> I prefer to think of it as a collection of heuristic methods, effective only
> when applied to starkly simplified schematic plans. The Consistency that
> Logic demands is not otherwise usually available-and probably not even
> desirable-because consistent systems are likely to be too "weak."
>
> see: http://web.media.mit.edu/~minsky/papers/Frames/frames.html
Minsky like everyone else is entitled his opinion and it is hard
to avoid having an opinion and having it being called an opinion
in areas where opinions abound. But there is an implication early
in Minsky's opinion that has little to do with logic unless the
logic of which Minsky was speaking about was unusual which is not
the assumption we would commonly make from an off-hand statement
of the kind Minsky gives.
The questionable implication is that some famous mathematician doing
logic, and hence we would assume as knowledgeable, has made a statement
acknowledging that his argument would become inconsistent by some
further consequence (a further deduction step). Anyone familiar with
logic, and perhaps Minsky is not sufficiently, would know that any
inconsistent consequence whether the logician chooses to make the
deduction or not identifies an inconsistent set of assumptions in
the usual logic cases. No recognized argument would survive such a
condition and hence the suggested response of the famous mathematician
would either be that the mathematician was quite confused, which is
not likely the case if he was famous or he would be infamous in short
order, or Minsky mis-interpreted the mathematician's remark, and
misinterpreted to fit his own mathematical philosophy, which is most
likely the case.
And I do not think any logician is attempting to give up nearness
or any other useful notion. Saying that we cannot model nearness in
one way does not mean we cannot model it in another.
> I get the drift of your argument here. But it seems to me that this approach
> is an attempt to divorce logic from the real world. It is the classical way
> of saying: lets make a consistent system and call it "Logic". It has got
> some consistent rules and regulations. Then you try to apply it in the real
> world and notice that it doesn't work consistently there. In this sense any
> system I create which is consistent is a logical system no matter how simple
> or frivolous it is.
I am not attempting to divorce or marry logic to the real world. My
argument was that we could not incorrectly model a logic and then say
the logic was inconsistent. Saying that you can apply logic to some
context and not get consistent or expect results is a rather too common
condition in applying logic. Because we do not get the results we
expect does not require that the logic is useless. If we use a
hammer to scramble our eggs we may end up with broken China in our
omlette. There is no mystery that we can do just about anything wrong.
Would I then say that my hammer was not good for building a house and
instead recommend my usual omlette making fork to drive nails. Logic
and hammers are tools that need to be used properly, and if you do
not want to be smashing your fingers it will take some practice.
Neil Nelson
Pat
I think I better understand your viewpoint now. As long as logic is used as
a tool and not as a sufficient explanatory mechanism of how the real world
works I have no argument. There are many kinds of models i.e. deficient
descriptions of the world that can be used to aid in understanding. These
logical models may claim internal consistency and/or completeness in that
sense. But when we start talking about the relationship of logic to
arithmetic and other phenomena, one can expect some surprises to crop up.
Even the logic that underpins arithmetic cannot claim to represent the world
as it is.
nucleus
>Pat wrote:
>
>
>> The reference goes back to 1974 dealing with Minsky's idea about Frame
>> represenatation of knowledge.
>> To quote it some more:
>> "A famous mathematician, warned that his proof would lead to a paradox if he
>> took one more logical step, replied "Ah, but I shall not take that step." He
>> was completely serious. A large part of ordinary (or even mathematical)
>> knowledge resembles that in dangerous professions: when are certain actions
>> unwise.
The you'd have to define the term "wisdom".
>> When are certain approximations safe to use?
The term "safety" has nothing to do with reality.
Safety is used by the "survivalists",
those people that remain in the middle of the herd.
Those are but cowards.
>> When do various
>> measures yield sensible estimates?
"Sensible estimates" are but a cunning trick,
avoiding the issues entirely.
Again, you'd have to define "sensible"
befoere you can go on.
>> Which self-referent statements are
>> permissible if not carried too far?
Self-referential reasoning is just a brainwashing
procedure.
It refers to itself as evidence of nothing more
than itself.
Anotherwords, a complete delusion.
>> Concepts like "nearness"
Nearness depends on scope.
It all depends on your system of limitations.
Unless you specify the exact "distance",
the term "nearness" is meaningless.
>> are to valuable
>> to give up just because no one can exhibit satisfactory axioms for them.
Axioms are those things that can not be proven
by definition.
They are nothing but the initial assumptions
and those, in their turn, depend on current state
of brainwashing.
You can be led to believe ANYTHING.
Literally so.
>> To
>> summarize:
>> "Logical" reasoning is not flexible enough to serve as a basis for thinking;
Weak argument.
Indeed, logical "reasoning" is something of a delusion.
At the same time, the very term "reasoning" is but a
byproduct of logic, and broken one at that.
>> I prefer to think of it as a collection of heuristic methods,
Heuristics is but another delusion.
It is based in so called notion of "experience".
Now, you "experience" largely those things,
you are programmed with.
Your "experience" changes drastically depending
on a system of belief you have been programmed with.
>> effective only
>> when applied to starkly simplified schematic plans. The Consistency that
>> Logic demands is not otherwise usually available-and probably not even
>> desirable-because consistent systems are likely to be too "weak."
The very idea of consistency is something of profound significance
and consequences.
Basically, consistency has to do with the issues
of stability, which in turn translates into predictability.
Consistency applies to systems.
Unless the system is consistent,
it can not be proven to be stable.
That is ALL there is to it.
In itself, the principle of consistencey
is contradictory to the very notion of intelligence.
Intelligence is INHERENTLY inconsistent.
"In fact", it is that, which goes beyond the very
notion of consistency.
Intelligence is able to operate in the regions
of inconsistency and yet provide the very impetus
to be, able to reconcile those inconsistencies
and yet to "survive".
In the conditions of "consistency",
you won't be able to invent ANYTHING new.
Because those new things will INHERENTLY
lead to inconsistency (with current world view).
In that respect, the term "consistency" is
effectively equivalent with death of intelligence
as such.
>> see: http://web.media.mit.edu/~minsky/papers/Frames/frames.html
>Minsky like everyone else is entitled his opinion and it is hard
>to avoid having an opinion and having it being called an opinion
>in areas where opinions abound.
First of all, do not forget that Marvin Minsky
is but a politician essentially.
He has to maintain his position and remain
something of prominency in order to continue
to "survive", and at MIT, one of the most vicious
"survival" places you can even begin to imagine,
and in the context of what he is doing,
he is BOUND to behave in a certain way.
His last work is but a pile of delusions.
>But there is an implication early
>in Minsky's opinion that has little to do with logic unless the
>logic of which Minsky was speaking about was unusual which is not
>the assumption we would commonly make from an off-hand statement
>of the kind Minsky gives.
He has to be given credit in the respect that
he was one of the first to admit the aspect of
emotional has to be considered in intelligence.
Yes, he tried to tackle the issue of emotional domain,
but he failed so miserably and proven to be such a slick
politician and manipulator, that it will remain with him
until the rest of his life.
>The questionable implication is that some famous mathematician doing
>logic, and hence we would assume as knowledgeable,
This is complete horseshit,
and of the lowest grade.
No matter what the "fame" of that matematician is,
the issues of science are well beyond it.
You can not use the argument that the Ohms law is valid
because Ohm was "famous".
Famous or not, it does not imply there is any knowledge,
unless, of course, you define the scope and domain of that
"knowledge".
>has made a statement
>acknowledging that his argument would become inconsistent by some
>further consequence (a further deduction step).
One more time, the term "consistency"
is but a byproduct of the issue of stability.
Intelligence is INHERENTLY inconsistent.
Consistent are only machines, mechanical, robot like
objects, bound it total slavery of the scheme of things
they were designed to do.
Intelligence is totally freee.
It is not bound by ANY limitations.
It could care less if something remains "consistent"
or not. Just the other way around, the more "inconsistent"
it is, the more intelligent it is.
>Anyone familiar with
>logic, and perhaps Minsky is not sufficiently, would know that any
>inconsistent consequence whether the logician chooses to make the
>deduction or not identifies an inconsistent set of assumptions in
>the usual logic cases. No recognized argument would survive such a
>condition and hence the suggested response of the famous mathematician
>would either be that the mathematician was quite confused, which is
>not likely the case if he was famous or he would be infamous in short
>order, or Minsky mis-interpreted the mathematician's remark, and
>misinterpreted to fit his own mathematical philosophy, which is most
>likely the case.
This is beyond comment.
Otherwise, we would have to repeat the same thing
again and again and again.
Enough.
Neil Nelson
> I think I better understand your viewpoint now. As long as logic is used as
> a tool and not as a sufficient explanatory mechanism of how the real world
> works I have no argument. There are many kinds of models i.e. deficient
> descriptions of the world that can be used to aid in understanding. These
> logical models may claim internal consistency and/or completeness in that
> sense. But when we start talking about the relationship of logic to
> arithmetic and other phenomena, one can expect some surprises to crop up.
> Even the logic that underpins arithmetic cannot claim to represent the world
> as it is.
One of the first things you would do in using logic is to define
the objects and context to which you want to apply a logic. E.g.,
you keep saying 'real world' and that it 'works' in some manner.
I would think that physics is about the real world and that the
logic and arithmetic used in physics, which is just the usual
classical logic and related arithmetic methods are then quite
sufficient in being applied to the real world and how it works.
That is, you keep implying there is some problem with applying logic
to the real world, but my evidence is that it is being applied quite
capably by those who know how to apply it. And then there are others
who do not know how to apply it with the result that they should
learn how to properly apply it.
You use the word 'arithmetic' in an odd way in saying it is a
phenomena as if somehow arithmetic was an artifact of the real
world (as I understand the meaning of 'real world'). There is
a philosophy that does speak this way, but the large majority
of mathematicians regard arithmetic as a language or domain
separate from the real world in the sense of physics or physical
phenomena. If your meaning of 'real world' is mathematical
Platonism, then that might be something to look at. But that
does not seem to be your meaning. You seem to be coming from
a kind of mathematical empiricism viewpoint.
But you are saying that logic and possibly arithmetic
do not represent the real world. My response is that you are
not using logic (arithmetic) properly and such that if it was
used properly you would likely adequately represent the real
world. Many other people are able to make proper
representations, and so with a little work you might be able
to do the same. And perhaps to emphasize this a bit, the logic
and arithmetic commonly used to successfully model the real
world are frequently considered a mundane variety, a kind of
very commonplace kind of logic and arithmetic. There is no
concern among most studied people that the commonplace logic
and arithmetic can do the job. I think that what we need is a
careful argument showing where there might be a problem.
Minsky's argument that nearness was some problem is not a
particularly great problem for a range of common observation
clustering methods known in AI and Statistics. Typically if
we want to know how near on object is to another we compute
Pythagorean theorem for whatever dimensions are being used.
As far as Minsky's position that consistent logics are too
weak, I do not have much excuse for Minsky's position. Perhaps
we just need to recommend a good mathematical logic course to
Minsky where he can get these confused ideas sorted out. There
are the paraconsistent logicians and related logic approaches
that deal with what might be called local inconsistencies, but
we generally realize that there are easy translations that
make such interpretations entirely consistent.
Neil Nelson
Chris Menzel
> It is certainly doubtful, for example, that the axiom
> > of ... pairing for that matter, follows from the *meaning*
> > of "is a member of".
> >
> > Chris Menzel
>
> Even so, it follows from FOL and (C3,C4) that there's a
> collection to which a,b belong if anything does.
> C3 EyAx[x in y <-> Et(x in t) & P(x)] (with y not free in P(x))
>
> C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}] Weak
> Extensionality
>
> But you wouldn't want a pairing principle to follow from the meaning
> of "in" and the meaning of "identity", now, would you?
I couldn't care less if one did, it's simply the case that I don't think
one does. When I talk about the meaning of "is a member of", I do not
mean a highly theoretical axiom stipulated by some sophisticated
theoretician. Your C3 is an extremely strong mathematical principle,
certainly not one that simply drops out of our pre-theoretic
understanding of membership. Indeed, it is significantly massaged to
avoid the paradoxes that arise from axioms that are arguably far better
representations of our naive semantic intuitions.
Perhaps more relevantly, though, I do not think that the set membership
relation is a part of pure logic at all. Rather, it's part of the
subject matter of a more specialized abstract discipline that studies a
certain species of mathematical object, and hence, in my view, is not
sufficiently general to count as part of pure logic. This is, of
course, not a judgment of value, but simply a (far from universal) view
about what counts as pure logic and what doesn't.
> I mean, it's ever so much nicer to assemble basic principles AD HOC,
> right?
You often ask very interesting questions and raise interesting
challenges, John. Your insistence upon framing them with silly
conspiratorial accusations lessens their impact considerably. And it
certainly saps one's motivation to reply.
Chris Menzel
Pat
> Pat wrote:
>
>
> > I think I better understand your viewpoint now. As long as logic is used
as
> > a tool and not as a sufficient explanatory mechanism of how the real
world
> > works I have no argument. There are many kinds of models i.e. deficient
> > descriptions of the world that can be used to aid in understanding.
These
> > logical models may claim internal consistency and/or completeness in
that
> > sense. But when we start talking about the relationship of logic to
> > arithmetic and other phenomena, one can expect some surprises to crop
up.
> > Even the logic that underpins arithmetic cannot claim to represent the
world
> > as it is.
>
>
> One of the first things you would do in using logic is to define
> the objects and context to which you want to apply a logic. E.g.,
> you keep saying 'real world' and that it 'works' in some manner.
> I would think that physics is about the real world and that the
> logic and arithmetic used in physics, which is just the usual
> classical logic and related arithmetic methods are then quite
> sufficient in being applied to the real world and how it works.
To make it clearer what I mean by 'real world'. There may be several layers
of real world depending on the observer's position. If I try to model
something, the object of the modelling is the 'real world'. For example the
formula for the speed of light is a deficient model of the real world
phenomenum Speed of Light. The formula has no sense of the light itself nor
even the speed. It is a mapping and modelling exercise.
> That is, you keep implying there is some problem with applying logic
> to the real world, but my evidence is that it is being applied quite
> capably by those who know how to apply it. And then there are others
> who do not know how to apply it with the result that they should
> learn how to properly apply it.
Of course, you can apply logic on the real world, but it has its limits.
And it is around these limits that the inconcistencies rear their ugly
heads. Some people, as Minsky points out, prefer to ignore these, because
they do not fit the idea of perfect model of reality.
>
> But you are saying that logic and possibly arithmetic
> do not represent the real world. My response is that you are
> not using logic (arithmetic) properly and such that if it was
> used properly you would likely adequately represent the real
> world. Many other people are able to make proper
> representations, and so with a little work you might be able
> to do the same.
I think some people get thoroughly blinded when their semi-religious beliefs
about whatever they are brainwashed with get challenged. They think it is
impossible that if another 100 000 people believe the same that it could be
wrong. This is not an oblique reference to you, Neil, but the hordes of
people who, for example propound their religious convictions. If psychiatry
was properly applied hundreds of thousands of people would end up in mental
asylums for their delusions. Unfortunaly this is not rare in Sciences,
either. You talk about people being able to make 'proper' representations,
and exhort me to follow the lead believing that their path is the right one
and the dissenters have gone astray. A lot you say makes a lot of sense, but
your references to many people do not convince me in any way. You can
adequately represent anything, depending on your purpose, but the point is
the inconcistencies you introduce around the edges. I can do the same with
very little work, indeed.
And perhaps to emphasize this a bit, the logic
> and arithmetic commonly used to successfully model the real
> world are frequently considered a mundane variety, a kind of
> very commonplace kind of logic and arithmetic. There is no
> concern among most studied people that the commonplace logic
> and arithmetic can do the job. I think that what we need is a
> careful argument showing where there might be a problem.
I do agree with you to a certain point. But logic is a construct, an
artifact and it can model the world to a certain extent only. This is the
case with the mathematics and physics also. Newtonian physics has its limits
and clash with Einsteinian at a certain interface. I think we both agree
that they are basically models relating to the "real world" they are
modelling. If you give me the formula for the speed of light you can only
give me a deficient model of the reality it tries to depict. The light is
missing. Where is the speed etc. The logic behind the formula maps the
formula to the reality in a certain way and works OK. Try peeling a potatoe
in your head. Reduce it to a some kind of formula. Base the formula on some
logical axioms. What you have at hand then is a deficient depiction of
reality. It cannot include everything necessary the actual peeling of
potatoe involves. Your representation is inconsistent with the real world in
many respects. It may occassionally work for you and it may prove a useful
tool, but sooner or later you start finding some inconsistencies. To be able
to have a 100% consistent model, you need to replicate the real world in all
of its details. I am not talking about fiddling with logical formulas and
proving mathematical equations here. The model may be internally consistent,
but it has its its interface to reality where things go awry.
> Minsky's argument that nearness was some problem is not a
> particularly great problem for a range of common observation
> clustering methods known in AI and Statistics. Typically if
> we want to know how near on object is to another we compute
> Pythagorean theorem for whatever dimensions are being used.
This just shows that you need a different model to be able to describe
nearness and in order to avoid inconcistencies.
> As far as Minsky's position that consistent logics are too
> weak, I do not have much excuse for Minsky's position. Perhaps
> we just need to recommend a good mathematical logic course to
> Minsky where he can get these confused ideas sorted out. There
> are the paraconsistent logicians and related logic approaches
> that deal with what might be called local inconsistencies, but
> we generally realize that there are easy translations that
> make such interpretations entirely consistent.
Your thoughts about Minsky are interesting. However, I think he saw some
generalities that many practioners of logic fail to see because they are too
deeply involved in the mathematical side and think: mathematics + logic =
reality.
Let me say, that I do not disagree with you argument from your viewpoint.
But I think there is a different viewpoint to logic generally, which may
prove useful to keep in mind when the formal logic reaches its limits.
Patrick O'Hooligan The WiseGuy
Neil Nelson
> Let me say, that I do not disagree with you argument from your viewpoint.
> But I think there is a different viewpoint to logic generally, which may
> prove useful to keep in mind when the formal logic reaches its limits.
You keep saying limits and that these limits have something to do with
how logic and arithmetic are applied that we at times have new theories
about reality as would be seen in the progession of the theories of
physics from Newton to Einstein. And I would say that mathematics and
logic progressed quite a bit in that same period. But in some manner
you are saying that formal logic has inherent limits with regard to
physical theories as exmplified by the physics example.
My understanding is that the change in physical theories has to do
with modifications that allow greater predictive precision with
experimental result. That is, you do the Scientific Method bit and
improve your physical theories. You seem to be saying that at some
point in the advancement of scientific theory that the common
logic and arithmetic are not sufficient for providing the description
or mathematical model required. I do not know of any such case. You
are saying there are limits in the application of logic and arithmetic
but I do not know of any or of an argument why there would be any, and
can rather think of an easy argument why there will not be any.
But we seem to keep missing an essential point: if you do not use
logic and mathematics properly, you will not get what you expect and
could easily conclude that logic and mathematics has limits according
to your experience, but where your application is the limiting problem
and not the tools used in the application. That is, you may have a
physical hypothesis on which you want to run an experiment. If you
do not run the experiment properly and it says your hypothesis is not
good, then you could easily throw away a good hypothesis. What I am
seeing is that you are not running your experiement properly and
holding that the hypothesis that logic and arithmetic may be used
without limitations for the description of physical circumstance is
being unnecessarily thrown out.
No one disagrees that the current models in physics are frequently
good approximations since we expect their continued improvement. But
we do not hold that the approximating aspect is a limitation of logic
and arithmetic. We hold that the approximating aspect derives from
the ability of science to find hypothesis and adequately test them.
Mathematics and logic have little to do, but perhaps more so with AI
methods, with creating scientific hypothesis. That is the job of
guys like Einstein. And testing those hypothesis is frequently
with those who know how to run cyclotrons and can come up with a
good experiment that will decide the hypothesis.
But you only need to show a conclusive example of how logic and
arithmetic does not work. You said
> For example the
> formula for the speed of light is a deficient model of the real world
> phenomenum Speed of Light. The formula has no sense of the light itself nor
> even the speed. It is a mapping and modelling exercise.
and I do not well understand what the point is. If you want to
say or communicate that the speed of light is such and such then you
need a communicated language to do that. Saying that the model in
the communicated language is not the real thing goes without saying.
Saying that the model is enherently deficient since it is only a
model could well go without saying. But to be enherently deficient
would require some idea of what it means to be deficient. If it is
deficient because it is not the real thing then OK. But unless you
want to not communicate at all you will need to tolerate, apparently,
some small deficiency. I do not think Minky's argument is that we
should not communicate since he seems to be hard at it.
Neil Nelson
Pat
> Pat wrote:
>
>
> and I do not well understand what the point is. If you want to
> say or communicate that the speed of light is such and such then you
> need a communicated language to do that. Saying that the model in
> the communicated language is not the real thing goes without saying.
> Saying that the model is enherently deficient since it is only a
> model could well go without saying. But to be enherently deficient
> would require some idea of what it means to be deficient. If it is
> deficient because it is not the real thing then OK. But unless you
> want to not communicate at all you will need to tolerate, apparently,
> some small deficiency. I do not think Minky's argument is that we
> should not communicate since he seems to be hard at it.
>
> Neil Nelson
>
I think this is where I agree with you. But I'd like to restate my point
once more:
Logic, like physics or arithmetics is an artifact, a construct if you like,
or a model. As an artifact, I put logic at par with any other explanation of
reality, not above mathematics or physics or social sciences to that matter.
It can be improved to reflect the reality, the state of things, better. But
it will remain a model. It will never be the real thing. Yes, we are forced
to tolerate deficiencies which may seem small because that's how far our
knowledge reaches at any one moment. That's the limit we have reached with
our current state of knowledge. That's where we come across many
inconcistencies, and that's where we need to improve our model be it logic,
nanotechnology or animal husbandry. I get the feeling that many people think
that logic is somehow above everything else, not a human artifact but a
God-given revelation. The Arabic world gave us the words for algebra,
alchemy, alcohol etc., all very useful artefacts, especially the last one.
There is another thing today's science should adapt from today's Arabs,
especially those calling themselves logicians: When you create an artifact,
leave some room for imperfection, because only GOD himself is perfect. God,
I think, could just as well refer to reality we try to model. (Previously I
thought I must be an atheist or an agnostic, but now I notice I have this
religious streak which pops up in very unexpected quarters).
librarian
>Owen Holden wrote:
>
>> "John" <john_...@excite.ca> wrote in message
>> news:70f94e16.02091...@posting.google.com...
>>
>>>paulholba...@freenet.de (Paul Holbach) wrote in message
>>>
>> news:<881c8779.02091...@posting.google.com>...
>>
>>>>john_...@excite.ca (John) wrote in message
>>>>
>> news:<70f94e16.02091...@posting.google.com>...
>>
>>>>
>>>>>IMO, philosophers of mathematics write all too infrequently about
>>>>>
>> logic.
>>
>>>>>Two questions of interest to me: (1) What is logical truth?
>>>>>
>>
>> All truth is ' that which can be shown'
Nope.
'That which can be shown' is a matter of interpretations.
First of all, it depends on your brain conditioning.
The things you "see" depend on your brain programming.
If you look at what is going on in the USA this very moment,
it is simply mind boggling. These people "see" something
entirely different from what the other people in the world
"see". They simply have gone mad.
Secondly, truth is not trueness.
It is not a matter of your belief system.
The best interpretation I've heard is:
THAT WHICH IS.
It does not matter what you see.
It still is.
It does not matter what is the state of your "science".
It still is.
Sooner or later you are bound to "see"
that which is.
>> Logical truth is that truth which cannot be otherwise.
Logical truth is but a delusion.
Limitations of logic can not be applied to
issues of Truth.
Truth includes all aspects, logical, illogical,
emotional, or otherwise.
Logic, on the first place, is broken in the most
profound ways.
First of all, it is the most extreme limitation
of scope.
What is "logic"?
Well, it is essentially the issue of "true" versus
"false".
It is a binary delusion of the most profound magnitude.
Out of entire scope of existence, only black and white
are accepted. Then they are further recombined
in order to "derive" some other conclusions,
thus furthering this binary delusion.
Essentially, logic is but a byproduct of what
is known to be fascism.
When, out of entire scope, you only take two extreme
samples, what you have is reduction of a rainbow
of existence to a fascist, black and white view.
Logic is but a byproduct of morality,
invented by the priest, derving from the notions
of "good" and "bad". Interestingly enough,
those "good" and "bad" aren't even the same
in different traditions, geographical regions, etc.
Furthermore, that which was considered "bad"
can be EASILY reclassified as "good".
Take for example homosexualism.
It is one of the condemned things in Bible,
and yet, at this junction, it has been accepted
as some kind of a "norm" and some countries and
states even accept it as a valid marriage.
So...
What is "good"?
What is "bad"?
Anybody can define?
Interestingly enough, eventually, the scientist's
argument will be reduced to morality. Because it is
all based on manipulations of the initial assumptions,
and, since those assumptions can not be proven in principle,
you would have to resort to resolving the core issues
from the standpoint of morality.
>> The existential quality of 'tautology', is obvious.
>There are different logics, see Susan Haack's book, that do
>not have the same theorems and hence not the same truths, and
>hence otherwise and not generally obvious.
Logic can not possibly have truths.
The so called scientific truth
is the outmost delusion.
Otherwise, you have to completely abandon the very notion
of Truth, and that is not possible, even in principle.
>It seems that we can easily say that what any given group of
>mathematicians and their philosophers can agree is true will
>be true for them.
True is not the same as truth.
Now you reduced it to the level of obscene.
> That is, a group can be defined by its
>agreement to a given philosophy or method. And then, as a
>quick aside, to have different groups requires that we can
>identify the differences that define those groups. That is,
>to know that some person subscribes to some particular
>philosophy or group as against another requires that there is
>a method by which the distinction is made and apparently
>known to those who assert those differences.
Now you are lost here and you will never be able
to get out of the deadliest contradictions there are.
You are merely talking about the brain conditioning
and belief systems that are all contradictory to one another.
Good buy.
>But onto a primary and related theme which is that agreement
>(or agreement that there is disagreement) requires, if we
>are to speak in some sense where our meaning is to be adopted
>by others (why speak otherwise), that there is some objective
>standard, some universal method of agreement--though not that
>we must agree, but that we can agree that we disagree or agree
>that we cannot tell. And the identification of agreement
>apparently at the moment is by a predictive word sequence.
>
>That is, to determine that another person agrees it might be
>sufficient for that other person to say, "I agree". But to be
>certain we may want to more clearly know what it is that they
>are agreeing to, at which point we could ask questions as to
>the detail of their agreement with what we would say ourselves.
>If their responses to our questions are the same as what we
>would give ourselves, what we expect, then we would have a
>higher confidence in their agreement. And then since we only
>have words at the moment by which to ask and give replies, we
>have a predictive word sequence that determines agreement.
>
>I suppose we could just say two people agree without getting
>a confirming word sequence. But it would seem better to say
>that of which we have some evidence, some confidence. That
>is, you might say A agrees with B, but I could ask how you
>know such is the case, and if you did not give your reason
>or evidence I would ask how I would choose between you and
>someone you would say otherwise. This then requires that you
>give an argument or predictive word sequence that obtains my
>agreement as above.
>
>Now comes an interesting point which is that if we require
>a predictive word sequence to obtain agreement, but perhaps
>before continuing some emphasis on what agreement has to do
>with the price of tea. To say 'the price of tea' in a
>meaningful sense so that the person being spoken to is
>willing to buy (or not) your tea requires that they agree with
>you on what tea is and what the units of price represent. All
>meaning, by whatever method, between two people requires that
>those two people agree on their expected common or same
>meaning. The meanings for those two people could easily not
>be the same, or we could not know they were the same, unless
>there was a method by which that can be known. Meaning for
>the current context is where two people have the same
>predictive word sequence for a given question, which could
>be: what does this word or expression mean?
>
>But since we are requiring predictive word sequences to
>determine agreement--and what is true is what we agree that
>it is--we can ask how it comes about that two people can
>have the same predictive word sequences. One way to do this
>is to have rules, say those of logic, by which the word
>sequences are generated. If we both use the same word
>sequence generating rules then we will have the same
>results.
>
>But can we assert that there are other ways other than rules
>by which agreement on the word sequences can be obtained?
>That is, how is it possible that two people could consistently
>show an agreement on word sequences without there being some
>prior agreement between those two people on their generating
>rules? But perhaps the question is: even though two people
>might consistently agree without the rules in evidence (if
>there are any rules), what evidence or argument can we give by
>which we can assert that those two people will agree in the
>future? That is, if we do not have the evidence of the
>generating rules that the two people are using, what will our
>reason be for asserting that those two people will agree in
>the future, that they are in agreement in the general case?
>
>The question is not that is it possible for two people to
>agree and where we do not know how. The question is how do
>we assert that they will continue to agree if we do not know
>how? E.g., for me to say that so-and-so subscribes to some
>particular logic or philosophy requires, unless I am only
>refering to their prior statements, that I am able to give
>a method in words, the rules in words, that will predict
>so-and-so's future words to some sufficiency of agreement
>with that logic or philosophy.
>
>Now that we have in the current context: (1) that agreement
>requires a predictive sequence of words/symbols, and (2)
>that an assertion for general or future agreement requires
>the generating rules to be in words--we have generalized
>word maps--, what agreement can be asserted that is not
>bounded by Church's Thesis? What predictive sequences are
>there by which agreement may be obtained that are not
>bounded by a Turing complete language? Since we do not
>yet have a good counterexample to Church's Thesis, no
>conclusive opposing assertion can be made unless of course
>you do not agree with Church's Thesis.
>
>Regards,
>
>Neil Nelson
>
librarian
>Pat wrote:
>
>
>> The reference goes back to 1974 dealing with Minsky's idea about Frame
>> represenatation of knowledge.
>> To quote it some more:
>> "A famous mathematician, warned that his proof would lead to a paradox if he
>> took one more logical step, replied "Ah, but I shall not take that step." He
>> was completely serious. A large part of ordinary (or even mathematical)
>> knowledge resembles that in dangerous professions: when are certain actions
>> unwise.
The you'd have to define the term "wisdom".
>> When are certain approximations safe to use?
The term "safety" has nothing to do with reality.
Safety is used by the "survivalists",
those people that remain in the middle of the herd.
Those are but cowards.
>> When do various
>> measures yield sensible estimates?
"Sensible estimates" are but a cunning trick,
avoiding the issues entirely.
Again, you'd have to define "sensible"
befoere you can go on.
>> Which self-referent statements are
>> permissible if not carried too far?
Self-referential reasoning is just a brainwashing
procedure.
It refers to itself as evidence of nothing more
than itself.
Anotherwords, a complete delusion.
>> Concepts like "nearness"
Nearness depends on scope.
It all depends on your system of limitations.
Unless you specify the exact "distance",
the term "nearness" is meaningless.
>> are to valuable
>> to give up just because no one can exhibit satisfactory axioms for them.
Axioms are those things that can not be proven
by definition.
They are nothing but the initial assumptions
and those, in their turn, depend on current state
of brainwashing.
You can be led to believe ANYTHING.
Literally so.
>> To
>> summarize:
>> "Logical" reasoning is not flexible enough to serve as a basis for thinking;
Weak argument.
Indeed, logical "reasoning" is something of a delusion.
At the same time, the very term "reasoning" is but a
byproduct of logic, and broken one at that.
>> I prefer to think of it as a collection of heuristic methods,
Heuristics is but another delusion.
It is based in so called notion of "experience".
Now, you "experience" largely those things,
you are programmed with.
Your "experience" changes drastically depending
on a system of belief you have been programmed with.
>> effective only
>> when applied to starkly simplified schematic plans. The Consistency that
>> Logic demands is not otherwise usually available-and probably not even
>> desirable-because consistent systems are likely to be too "weak."
The very idea of consistency is something of profound significance
and consequences.
Basically, consistency has to do with the issues
of stability, which in turn translates into predictability.
Consistency applies to systems.
Unless the system is consistent,
it can not be proven to be stable.
That is ALL there is to it.
In itself, the principle of consistencey
is contradictory to the very notion of intelligence.
Intelligence is INHERENTLY inconsistent.
"In fact", it is that, which goes beyond the very
notion of consistency.
Intelligence is able to operate in the regions
of inconsistency and yet provide the very impetus
to be, able to reconcile those inconsistencies
and yet to "survive".
In the conditions of "consistency",
you won't be able to invent ANYTHING new.
Because those new things will INHERENTLY
lead to inconsistency (with current world view).
In that respect, the term "consistency" is
effectively equivalent with death of intelligence
as such.
>> see: http://web.media.mit.edu/~minsky/papers/Frames/frames.html
>Minsky like everyone else is entitled his opinion and it is hard
>to avoid having an opinion and having it being called an opinion
>in areas where opinions abound.
First of all, do not forget that Marvin Minsky
is but a politician essentially.
He has to maintain his position and remain
something of prominency in order to continue
to "survive", and at MIT, one of the most vicious
"survival" places you can even begin to imagine,
and in the context of what he is doing,
he is BOUND to behave in a certain way.
His last work is but a pile of delusions.
>But there is an implication early
>in Minsky's opinion that has little to do with logic unless the
>logic of which Minsky was speaking about was unusual which is not
>the assumption we would commonly make from an off-hand statement
>of the kind Minsky gives.
He has to be given credit in the respect that
he was one of the first to admit the aspect of
emotional has to be considered in intelligence.
Yes, he tried to tackle the issue of emotional domain,
but he failed so miserably and proven to be such a slick
politician and manipulator, that it will remain with him
until the rest of his life.
>The questionable implication is that some famous mathematician doing
>logic, and hence we would assume as knowledgeable,
This is complete horseshit,
and of the lowest grade.
No matter what the "fame" of that matematician is,
the issues of science are well beyond it.
You can not use the argument that the Ohms law is valid
because Ohm was "famous".
Famous or not, it does not imply there is any knowledge,
unless, of course, you define the scope and domain of that
"knowledge".
>has made a statement
>acknowledging that his argument would become inconsistent by some
>further consequence (a further deduction step).
One more time, the term "consistency"
is but a byproduct of the issue of stability.
Intelligence is INHERENTLY inconsistent.
Consistent are only machines, mechanical, robot like
objects, bound it total slavery of the scheme of things
they were designed to do.
Intelligence is totally freee.
It is not bound by ANY limitations.
It could care less if something remains "consistent"
or not. Just the other way around, the more "inconsistent"
it is, the more intelligent it is.
>Anyone familiar with
>logic, and perhaps Minsky is not sufficiently, would know that any
>inconsistent consequence whether the logician chooses to make the
>deduction or not identifies an inconsistent set of assumptions in
>the usual logic cases. No recognized argument would survive such a
>condition and hence the suggested response of the famous mathematician
>would either be that the mathematician was quite confused, which is
>not likely the case if he was famous or he would be infamous in short
>order, or Minsky mis-interpreted the mathematician's remark, and
>misinterpreted to fit his own mathematical philosophy, which is most
>likely the case.
This is beyond comment.
Otherwise, we would have to repeat the same thing
again and again and again.
Enough.
>And I do not think any logician is attempting to give up nearness
Neil Nelson
> I think this is where I agree with you. But I'd like to restate my point
> once more:
> Logic, like physics or arithmetics is an artifact, a construct if you like,
> or a model. As an artifact, I put logic at par with any other explanation of
> reality, not above mathematics or physics or social sciences to that matter.
> It can be improved to reflect the reality, the state of things, better. But
> it will remain a model. It will never be the real thing. Yes, we are forced
> to tolerate deficiencies which may seem small because that's how far our
> knowledge reaches at any one moment. That's the limit we have reached with
> our current state of knowledge. That's where we come across many
> inconcistencies, and that's where we need to improve our model be it logic,
> nanotechnology or animal husbandry. I get the feeling that many people think
> that logic is somehow above everything else, not a human artifact but a
> God-given revelation. The Arabic world gave us the words for algebra,
> alchemy, alcohol etc., all very useful artefacts, especially the last one.
> There is another thing today's science should adapt from today's Arabs,
> especially those calling themselves logicians: When you create an artifact,
> leave some room for imperfection, because only GOD himself is perfect. God,
> I think, could just as well refer to reality we try to model. (Previously I
> thought I must be an atheist or an agnostic, but now I notice I have this
> religious streak which pops up in very unexpected quarters).
You keep talking about reality but most logicians and mathematicians
are not concerned with reality per se. They could generally care less
if there is any real application for the logic or mathematics they are
thinking about. And further along this line, if you say to a logician
or mathematician, "What does this have to do with reality?" they are
likely to think that you have wandered into the wrong department with
no concept of what mathematics or logic is about. And just so that we
do not get confused with the Platonist reality common in Mathematics,
the reality you seem to be referring to is a physical or empiricist
reality.
Now that we are not in Kansas (reality) anymore, and if you can begin
to understand where you are, the realm of mathematics and logic, then
you may also begin to understand why mathematics in particular and
logic in part identifies certain absolutes that cannot be avoided no
matter what language, model, or method you happen to choose or create.
You say mathematics is on par with physics or social sciences and such
but mathematics is not on par with anything else. Ultimately there is
only mathematics as a language that we can not avoid using when
communicating whatever it may be: physics, social sciences, or anything
that we can possibly communicate and particularly to some degree of
precision. Language is not an artifact in your meaning in that we
need to first have language in order speak about artifacts. You may
think that there artifacts even though you do not speak of them, but
language is a rather fundamental and unique artifact, if we care to
call it that, quite separate from the remainder.
You can choose not to use language but unless those around you feel
there is a good reason why you are not using language you may end up
in an asylum. But usually we use language as a very expected behavior
of our species and once you commit to using language, every sentence,
symbol, word, formula or whatever you can express in language is part
of or contained within the domain of what languages are and can do.
Arithmetic in particular is a kind of universal language for precise
expression. It is precise because it is made of functions over a
discrete domain. It is universal because of Church's Thesis that
observes that every symbolic language of precision has been shown
to be able to be translated to all or some part of arithmetic. From
what anyone has been able to tell for the last 65+ years, there is
only one most expressive language for precision in the sense that
whatever language you choose you could just as well use arithmetic
or any other language that may be translated to arithmetic (or
arithmetic to these other languages).
It is these two points of the necessary use of language and one of
precision, and the universal nature of arithmetic that you are not
appreciating. This is not the corn fields of Kansas, this is all
languages anywhere, anytime, about anything anyone may possibly
think of. It is perhaps the most universal and fundamental aspect of
mathematics and hence of humans as the species that uses language.
(It could easily be that many or not a majority of mathematicians
would hold that Church's Thesis identifies a boundary between,
say, practical or computable math and what many would hold as more
theoretical than computable and I would disagree, but for the
current discussion we are only interested in computable. In, say,
physics we need to be able to compute our prediction values at
some point.)
Now before you can speak about reality with precision you need a
precise language. That is, we are not yet dealing with reality, we
are figuring out what it means to have a precise language so that
if we want to speak about anything else, and one of those things may
be reality, we can do it with some precision.
Now we would seem to agree that some language of precision is
required for these various areas of reality you want to speak about.
What likely needs to be done next is that you understand what
Church's Thesis is, and that should be done from the mathematical
perspective and not from a philosophical perspective. That is,
before worrying about speaking about reality we need to agree on
how to be speaking about anything at all. And then you might
realize how Church's Thesis fundamentally affects how we can speak
about things whatever those things may be. And we are not yet in
Kansas.
When we get the two prior items working then there is another aspect
of Church's Thesis that we can look at when we get to Kansas.
Neil Nelson
John
I do value your opinion, and apologize for not having asked for it
graciously. I guess I had already presumed that you would not say
what you thought--or, worse yet, would not think about this at
all--and then acted in such a way as to bring this situation
about. My mistake: I meant no ill.
--John
Bennett Standeven
No; the reality we try to model is not perfect...
George
Neil, you are weird, ignorant, and you make
me fall asleep. And you are one rotten jerk
with your little bitch friend (Jure).
Here is a big fat stick, shove it up your
greedy Swedish mouths.
Hoo Hoo
nucleus
That is how you talk to these pseudo-scientists,
peddling delusions.
Chris Menzel
<ull...@math.okstate.edu> said:
> Surely metalogical theorems are also logical truths --
In my view, they are not.
> The truth that seems to me is a logical truth would be something of
> the form "assuming [insert large number of axioms and definitions]
> then arithmetic is inconsistent" - I assumed that's what you really
> meant, and it seems to me that that statement _is_ a logical truth.
Well, sure, *that* statement is. But I wouldn't identify that with the
incompleteness.
Chris Menzel
David C. Ullrich
wrote:
>On Thu, 19 Sep 2002 20:16:58 GMT, David C. Ullrich
><ull...@math.okstate.edu> said:
>> Surely metalogical theorems are also logical truths --
>
>In my view, they are not.
Ok, then we differ in what a "logical truth" is.
>> The truth that seems to me is a logical truth would be something of
>> the form "assuming [insert large number of axioms and definitions]
>> then arithmetic is inconsistent" - I assumed that's what you really
>> meant, and it seems to me that that statement _is_ a logical truth.
>
>Well, sure, *that* statement is. But I wouldn't identify that with the
>incompleteness.
Note that "inconsistent" was a typo for "incomplete".
>Chris Menzel
David C. Ullrich
Chris Menzel
> ...For any past event E, there is a date E(D) at which it occurred
> (or an interval of time over which it occurred).
That should be D(E).
Neil Nelson
I wrote:
>>(It could easily be that many or not a majority of mathematicians
>>would hold that Church's Thesis identifies a boundary between,
>>say, practical or computable math and what many would hold as more
>>theoretical than computable and I would disagree, but for the
>>current discussion we are only interested in computable. In, say,
>>physics we need to be able to compute our prediction values at
>>some point.)
I see that I have made a typo in the above and 'many or not a majority
of mathematicians' should be 'many and very likely a majority' without
the 'not' so it would read as
>>(It could easily be that many and very likely a majority of mathematicians
>>would hold that Church's Thesis identifies a boundary between,
>>say, practical or computable math and what many would hold as more
>>theoretical than computable and I would disagree, but for the
>>current discussion we are only interested in computable. In, say,
>>physics we need to be able to compute our prediction values at
>>some point.)
I look forward to any intelligent, studied argument of substance
about the subject you may care to make. If you need some assistance
as to where and how an opposing argument might be made, I could give
you a hand up.
Neil Nelson
Pat
If they are not concerned with reality, that's bad news. They ought to be.
So should you.
> if there is any real application for the logic or mathematics they are
> thinking about. And further along this line, if you say to a logician
> or mathematician, "What does this have to do with reality?" they are
> likely to think that you have wandered into the wrong department with
> no concept of what mathematics or logic is about. And just so that we
If you are not concerned with reality, Neil, I think you have wondered into
a wrong department and cannot find your way out.
> do not get confused with the Platonist reality common in Mathematics,
> the reality you seem to be referring to is a physical or empiricist
> reality.
>
> Now that we are not in Kansas (reality) anymore, and if you can begin
> to understand where you are, the realm of mathematics and logic, then
> you may also begin to understand why mathematics in particular and
> logic in part identifies certain absolutes that cannot be avoided no
> matter what language, model, or method you happen to choose or create.
Your theocentric logic and mathematics confirm my worst fears. Any
lumberjack in Kansas would be able to better defend his ground. Your
high-faluting absolutes are human constructions, too.
>
> You say mathematics is on par with physics or social sciences and such
> but mathematics is not on par with anything else. Ultimately there is
> only mathematics as a language that we can not avoid using when
> communicating whatever it may be: physics, social sciences, or anything
> that we can possibly communicate and particularly to some degree of
> precision. Language is not an artifact in your meaning in that we
> need to first have language in order speak about artifacts. You may
> think that there artifacts even though you do not speak of them, but
> language is a rather fundamental and unique artifact, if we care to
> call it that, quite separate from the remainder.
Again, your theocentric view of language and mathematics makes you sound
like a religious freak rather than a sound logician. I can communicate
perfectly well without using any mathematics. You can convert my
communicative mode into mathematics if you like, but that is a
transformation of it, a model so to speak.
> You can choose not to use language but unless those around you feel
> there is a good reason why you are not using language you may end up
> in an asylum. But usually we use language as a very expected behavior
> of our species and once you commit to using language, every sentence,
> symbol, word, formula or whatever you can express in language is part
> of or contained within the domain of what languages are and can do.
With this I agree with you. Especially the point where you say that one may
end up in an asylum.
> Arithmetic in particular is a kind of universal language for precise
> expression. It is precise because it is made of functions over a
> discrete domain. It is universal because of Church's Thesis that
> observes that every symbolic language of precision has been shown
> to be able to be translated to all or some part of arithmetic. From
> what anyone has been able to tell for the last 65+ years, there is
> only one most expressive language for precision in the sense that
> whatever language you choose you could just as well use arithmetic
> or any other language that may be translated to arithmetic (or
> arithmetic to these other languages).
This is your theocentric view of arithmetic (and I am not confusing Church
with the insitution of the same name). I think your defence is misdirected.
I can translate social science to artihmetic any time. With few of my own
primitives and assumptions I can create an internally consistent system of
logic which you cannot translate into arithmetic, because thinking in
arithmetics you would not be able to understand it.
>
> It is these two points of the necessary use of language and one of
> precision, and the universal nature of arithmetic that you are not
> appreciating. This is not the corn fields of Kansas, this is all
> languages anywhere, anytime, about anything anyone may possibly
> think of. It is perhaps the most universal and fundamental aspect of
> mathematics and hence of humans as the species that uses language.
> (It could easily be that many or not a majority of mathematicians
> would hold that Church's Thesis identifies a boundary between,
> say, practical or computable math and what many would hold as more
> theoretical than computable and I would disagree, but for the
> current discussion we are only interested in computable. In, say,
> physics we need to be able to compute our prediction values at
> some point.)
Hmmmmmm.....
> Now before you can speak about reality with precision you need a
> precise language. That is, we are not yet dealing with reality, we
> are figuring out what it means to have a precise language so that
> if we want to speak about anything else, and one of those things may
> be reality, we can do it with some precision.
Again I agree with you. But you fail to see my point. Read it again and
compare your notes.
>
> Now we would seem to agree that some language of precision is
> required for these various areas of reality you want to speak about.
> What likely needs to be done next is that you understand what
> Church's Thesis is, and that should be done from the mathematical
> perspective and not from a philosophical perspective. That is,
> before worrying about speaking about reality we need to agree on
> how to be speaking about anything at all. And then you might
> realize how Church's Thesis fundamentally affects how we can speak
> about things whatever those things may be. And we are not yet in
> Kansas.
>
Sorry about my terse notes. I think too much verbosity will bring up too
many issues to be dealt with simultaneously. I think the basic difference
between your view and mine is that I refuse to isolate logic and mathematics
in general from other human concepts and artifacts, whereas you look at them
from the viewpoint of some kind of universal background flow for the rest of
the phenomena we experience. You bring up Church's Thesis - I would call it
a Church's opinion: it has no validity as it is unprovable. Church's Thesis
seem to fundamentally affect the way you speak, but it doesn't affect the
way I speak. I guess I will soon have to send you some examples of Kansas
logic.
Neil Nelson
> Sorry about my terse notes. I think too much verbosity will bring up too
> many issues to be dealt with simultaneously. I think the basic difference
> between your view and mine is that I refuse to isolate logic and mathematics
> in general from other human concepts and artifacts, whereas you look at them
> from the viewpoint of some kind of universal background flow for the rest of
> the phenomena we experience. You bring up Church's Thesis - I would call it
> a Church's opinion: it has no validity as it is unprovable. Church's Thesis
> seem to fundamentally affect the way you speak, but it doesn't affect the
> way I speak. I guess I will soon have to send you some examples of Kansas
> logic.
I do not think I am missing your point. Church's Thesis is not provable
but there are no serious counter-examples to it and if you would study
the subject you may understand why there will not likely be any counter-
examples. And perhaps I should explain that now.
Look at Revesz _Introduction to Formal Languages_, p. 53, which is
chapter 5, titled, "Unrestricted Phrase Structure Languages". It shows
the generalized Kuroda form in Theorem 5.1. An unrestricted phrase
structure language or type 0 grammar is what we would also call a
Turing complete language. The theorem is:
Theorem 5.1 for every type 0 grammar there is an equivalent grammar
G' in which every rule has any of the following forms:
And then the theorem lists 7 transformation rules.
Now there are many kinds of languages for precision or languages
taking one discrete state to another. And in symbolic languages
or in a physics experiment we have functions, though in common
language not commonly functions of the required precision, taking
some set of expressions that are discrete values, as in the case of
physics, to some other set of expressions of discrete values. That
is whatever computation method or physical experiment we wish to
have, we will have a mapping of one string of characters to another,
or we can make it that way by numbering our states, and of course
numbering the same states the same, and using the numbers as the
required strings. And to be clear on a physics experiment, you
need to record your starting and ending values. And then in order
to predict, an essential requirement of physics, you need a computable
function taking those representations in a language to another set
of representations.
Now we have the conditions with which we can apply the above
Theorem 5.1 and we are done.
Now that we are getting close to Kansas, at some point we need to
agree that we either need to use a language of precision or not
in speaking about reality. That is, do you intend to be speaking
about reality as opposed to not speaking, and do you intend to be
speaking with some precision or definiteness, or are we to be
talking about reality in vague generalities where we might agree,
disagree whenever, and be for all purposes indistinguishable from
being confused, perhaps this is where the inconsistencies come to
play.
Assuming that we intend to be speaking about reality with some
precision, and we then take the physics example, we can quickly
see that physics under the requirement of communicating physics
within a precise language must conform to Church's Thesis in
the condition where we are speaking about physics. That is,
though reality may in fact be more expressive than a Turing-
complete language, our necessity to speak about it confines it
to a Turing-complete language. Sure, we can speak in vague
generalities and note how we can get into inconsistencies by
doing so, but if you want to have a useful physics whose primary
purpose is for physical prediction, taking an initial physical
state to another, and inconsistencies and vague generalities
are not going to assist that, then you will be restricted to
some Turing-complete language.
Reality, if you intend to be talking about it, is then what you
say it is. That is, if I say that reality is such-and-such then
I am saying what reality is. And since you are saying, using a
language, you cannot avoid the restrictions that using a language
impose.
Now that we are talking about Kansas there is a bit more we can
put on the nature of reality with regard to Church's Thesis.
Neil Nelson
Chris Menzel
<ull...@math.okstate.edu> said:
> >> The truth that seems to me is a logical truth would be something of
> >> the form "assuming [insert large number of axioms and definitions]
> >> then arithmetic is inconsistent" - I assumed that's what you really
> >> meant, and it seems to me that that statement _is_ a logical truth.
> >
> >Well, sure, *that* statement is. But I wouldn't identify that with the
> >incompleteness.
>
> Note that "inconsistent" was a typo for "incomplete".
Yes, of course. I was talking about the intended conditional.
Giuseppe
[...]
>Logic, like physics or arithmetics is an artifact, a construct if you like,
>or a model. As an artifact, I put logic at par with any other explanation of
>reality, not above mathematics or physics or social sciences to that matter.
>It can be improved to reflect the reality, the state of things, better. But
>it will remain a model. It will never be the real thing. Yes, we are forced
>to tolerate deficiencies which may seem small because that's how far our
>knowledge reaches at any one moment. That's the limit we have reached with
>our current state of knowledge. That's where we come across many
>inconcistencies, and that's where we need to improve our model be it logic,
>nanotechnology or animal husbandry. I get the feeling that many people think
>that logic is somehow above everything else, not a human artifact but a
>God-given revelation. The Arabic world gave us the words for algebra,
>alchemy, alcohol etc., all very useful artefacts, especially the last one.
>There is another thing today's science should adapt from today's Arabs,
>especially those calling themselves logicians: When you create an artifact,
>leave some room for imperfection, because only GOD himself is perfect. God,
>I think, could just as well refer to reality we try to model. (Previously I
>thought I must be an atheist or an agnostic, but now I notice I have this
>religious streak which pops up in very unexpected quarters).
I'm not a good person but I have to admit that
thoughts in math (or in real logic) are too complex and perfects,
they are not humans work, they could come only from reality (but
someone had to make them)
So we have: God --> reality --> math and real logic.
Pat
> Pat wrote:
>
>
> > Sorry about my terse notes. I think too much verbosity will bring up too
> > many issues to be dealt with simultaneously. I think the basic
difference
> > between your view and mine is that I refuse to isolate logic and
mathematics
> > in general from other human concepts and artifacts, whereas you look at
them
> > from the viewpoint of some kind of universal background flow for the
rest of
> > the phenomena we experience. You bring up Church's Thesis - I would call
it
> > a Church's opinion: it has no validity as it is unprovable. Church's
Thesis
> > seem to fundamentally affect the way you speak, but it doesn't affect
the
> > way I speak. I guess I will soon have to send you some examples of
Kansas
> > logic.
>
>
> I do not think I am missing your point. Church's Thesis is not provable
> but there are no serious counter-examples to it and if you would study
> the subject you may understand why there will not likely be any counter-
> examples. And perhaps I should explain that now.
This means that it is a hypothesis among other hypotheses (a construct). It
is not hard to find some serious counter-examples to Church's Thesis. You
must be aware for example of Bringsjord's Narrational Case Against Church's
Thesis (http://www.rpi.edu/~brings/SELPAP/CT/ct/ct.html). There are others,
but perhaps not in the web.
> Look at Revesz _Introduction to Formal Languages_, p. 53, which is
> chapter 5, titled, "Unrestricted Phrase Structure Languages". It shows
> the generalized Kuroda form in Theorem 5.1. An unrestricted phrase
> structure language or type 0 grammar is what we would also call a
> Turing complete language. The theorem is:
>
> Theorem 5.1 for every type 0 grammar there is an equivalent grammar
> G' in which every rule has any of the following forms:
I don't have that book at hand so it must wait until I get hold of it. I
basically view Phrase structure grammars as grammars constrained internally
not by transformations but by their own structure. Converting from one
grammar to another is a mapping exercise (functions, transformations) and
adds little to the basic structure. That basic structure is an artifact.
You are getting close here. You need a language, which is an imperfect
model, to talk about anything. But, as you seem to admit now, you cannot
talk about reality without introducing inconcistencies.
> Assuming that we intend to be speaking about reality with some
> precision, and we then take the physics example, we can quickly
> see that physics under the requirement of communicating physics
> within a precise language must conform to Church's Thesis in
> the condition where we are speaking about physics. That is,
> though reality may in fact be more expressive than a Turing-
> complete language, our necessity to speak about it confines it
> to a Turing-complete language. Sure, we can speak in vague
> generalities and note how we can get into inconsistencies by
> doing so, but if you want to have a useful physics whose primary
> purpose is for physical prediction, taking an initial physical
> state to another, and inconsistencies and vague generalities
> are not going to assist that, then you will be restricted to
> some Turing-complete language.
I think you are rephrasing the point I made. (Except the Church bit, that
is) Sure we need some Turing-complete languages but it does not do away with
the point I made.
> Reality, if you intend to be talking about it, is then what you
> say it is. That is, if I say that reality is such-and-such then
> I am saying what reality is. And since you are saying, using a
> language, you cannot avoid the restrictions that using a language
> impose.
Are you saying that the language you use constraints reality? A rather bold
statement.
>
Patrick O'Hooligan The WiseGUY
Pat
"Giuseppe" <d...@rt.yy> wrote in message news:3d8df173...@news.tin.it...
Some quasi-religious people argue that Science (Maths and Logic) originated
from the occasion when Satan told Eve to eat from the tree of Knowledge.
This would give us two formulas:
God --> Reality and Kansas Logic
Satan ---> Maths and Logic (That's the side Neil is representing)
I leave it to Neil to find a Turing engine transformation between these two
formulae.
Neil Nelson
> This means that it is a hypothesis among other hypotheses (a construct). It
> is not hard to find some serious counter-examples to Church's Thesis. You
> must be aware for example of Bringsjord's Narrational Case Against Church's
> Thesis (http://www.rpi.edu/~brings/SELPAP/CT/ct/ct.html). There are others,
> but perhaps not in the web.
An hypothesis but one that is well accepted by mathematicians. The
reasons are that it was early on realized that many of the formal
languages being used could be translated to each other, that no
formal language has been shown to be more expressive than one of
these Turing complete languages and then we have arguments like
the one in Revesz that suggest that there will not be any more
expressive languages.
As for Bringsjord's argument which appears to be primarily on page
http://www.rpi.edu/~brings/SELPAP/CT/ct/node4.html, I do not know
what to say. An obviously intelligent person is putting forth an
argument that essentially depends on the notion of 'interesting'.
It is obvious to me that what is interesting to one person may not
be interesting to another and vice versa and so we are apparently
being given an argument with a primary premise of which we do not
have much of a clue. And of the several arguments I have seen
attempting to avoid Church's Thesis, this lack if definiteness or
precision in the premises seems to be a common theme. And though
I do not want to dwell on this more than necessary, but commonly
when giving an argument a person attempts to define their premises
to a sufficiency for the argument at hand.
On page
http://www.rpi.edu/~brings/SELPAP/CT/ct/node8.html
he seems to want to avoid my objection, but saying that because
you can choose some not well defined predicate that it has
bivalent properties and hence may be applied with logic has
little to do with the point. The point is that if you want to
show a counterexample to Church's Thesis you need a language
of precision and not of imprecision. Saying that you can add
a necessarily vague premise and have a conclusion beyond Church's
Thesis is a somewhat obvious but useless result.
> I don't have that book at hand so it must wait until I get hold of it. I
> basically view Phrase structure grammars as grammars constrained internally
> not by transformations but by their own structure. Converting from one
> grammar to another is a mapping exercise (functions, transformations) and
> adds little to the basic structure. That basic structure is an artifact.
I am not seeing much of an objection here. Your use of the word
'artifact' might be something, but I guess we could call anything
that we would call a 'thing' an 'artifact' and hence little is
implied by that word.
> You are getting close here. You need a language, which is an imperfect
> model, to talk about anything. But, as you seem to admit now, you cannot
> talk about reality without introducing inconcistencies.
I have nowhere said that we need to introduce inconsistences at
any point. I have been characterizing the opposing position as
as often associated with inconsistencies, a condition which often
attends vague premises as in the case of 'interesting' above.
> I think you are rephrasing the point I made. (Except the Church bit, that
> is) Sure we need some Turing-complete languages but it does not do away with
> the point I made.
The point apparently being that whatever language is selected, it will
be inaccurate or we might say inconsistent in the area of inaccuracy
with repect to the reality it is modelling. And that is how I would
usefully interpret what you are saying.
But what I am saying is that however inaccurate the model may be
assuming that the inaccuracies may be attributed to the requirement
for a computable language, there are no other known or expected
alternatives. And then I think we need to analyse this area if
inaccuracy a bit for what possible kinds there may be. That is,
at the moment I am not saying that such models are necessarily
inaccurate beyond the usual assumptions that models of reality are
assumed to be approximations for reasons having little to do with
the language. We have common assumptions that would include the
language but we also have assumptions that do not include the
language and I am looking at the non-language portion before we
get to the other. That is, until we can sort these two portions
out we cannot assert that one or the other or both is/are the
cause.
>>Reality, if you intend to be talking about it, is then what you
>>say it is. That is, if I say that reality is such-and-such then
>>I am saying what reality is. And since you are saying, using a
>>language, you cannot avoid the restrictions that using a language
>>impose.
>>
>
> Are you saying that the language you use constraints reality? A rather bold
> statement.
My point, and I think it is about time you understood it, is that
if you want to speak about reality you will need to use a language.
There is no great difficulty in that point.
Do you intend to be speaking about reality or not?
If you intend to be speaking about reality, which would seem rather
hard to avoid, then the implication that others who speak about
reality have essential deficiences would be applied to you if you
wish to be speaking about reality. You do not gain anything by
making that point.
Neil Nelson
David C. Ullrich
wrote:
>On Sun, 22 Sep 2002 11:43:08 GMT, David C. Ullrich
Yes of course back at you - didn't mean to be suggesting that
if you knew what I meant you would have said something different,
just correcting the typo "for the record" - someone else thought
I meant "inconsistent".
David C. Ullrich
David C. Ullrich
wrote:
>On Sun, 22 Sep 2002 11:43:08 GMT, David C. Ullrich
Yes of course back at you - didn't mean to be suggesting that
Pat
That's correct. I have never denied it: what applies to others applies to
me too. You have given me a lot of food for thought. I think I will need the
next week to digest it properly and to check up Church, Revesz and
Bringsjord . I made up some spurious arguments to be sure, but also one
basic point. I think I need to stop at this point for the week.
With the words of a good friend of mine (Mr. Yudkowsky)
see the original at: http://www.aleph.se/Trans/Cultural/Fun/zombie.html
No-one screams in horror when I stagger through a room.
They think I'm drunk or stupid, not an instrument of doom.
I drooled blood on one of them (Neil Nelson), and all he did was grin.
I tried to bite his head off, but my teeth would not sink in.
Patrick O'Hooligan Grave Consultant for the Grim Reaper
George
> > David C. Ullrich wrote:
> >
> >
> > > The truth that seems to me is a logical truth would be something of
> > > the form "assuming [insert large number of axioms and definitions]
> > > then arithmetic is inconsistent" - I assumed that's what you really
> > > meant, and it seems to me that that statement _is_ a logical truth.
> >
> >
> > being inconsistent and logical truth. I am not aware of any argument
> > for the inconsistency of arithmetic and can rather think of a good
> > argument for the consistency of any Turing complete language of which
> > Arithmetic sometimes is held to be, which is that such a language is
> > comprised only of functions. And I think there are related arguments
> > by Goedel and Shoenfield. And so it must have been a typo I have
> > overreacted to.
> >
> > Perhaps what is meant instead of logical truth is that the statement
> > would have logical form and then if it were held or asserted would
> > be held or asserted true. That is, a person if they held that
> > argument would not then say that they did not hold a truth without
> > logic (though at the moment it would be illogical to do so).
> >
> > Neil Nelson
>
> The 3 'R's: aRithemtic, Rhetoric and Religion are all consistent in their
> own world. They start becoming inconsistent when you look at them from
> outside. Consistency is a viewpoint delimiting the universe around it.
> Marvin Minsky used Gödel's incompleteness theorem to point out that logic
> itself is inconsistent, as you must be well aware. Take the following
simple
> construct:
>
> A-->B-->C (Read it: If A then B and if B then C). Transitivity in logic
> dictates then that: If A then C.
> Step outside the world of logic and to consistent world of language and
> interpret the sign "-->" as meaning "next to" and reinterpret the
statement
> in question. A implies B which implies C. A implies that B is next to it
and
> B implies that C is next to it. Does A then imply that C is next to it?
> Certainly not. I know there is modal logic or something else to that
effect
> to deal with cases like this and that Implication in formal logic is also
a
> two-barreled gun. My point is: whenever the subject area becomes
> inconsistent you need to go outside of it and redefine. As long as you
stick
> within, Bob's your uncle. Stray outside, and all the hell gets loose.
>
> Patrick O'Hooligan, the WiseGuy
I know what your point is PsychoPat boy and I know you are
setting me up. You are talking about data organization and
what it means to disturb a system. How is it most efficient to
distribute new information without constructing need for deep
level hierarchical recursively. The best is a pyramid free
approach and everything should be inconsistent, smooth
and normal. Build pyramidal foundations around an ideology
and one gets into a rock hard religion. So I guess this
defines that John Lennon was right and that it is better
to stay away from fundamental central models as they can
cause a heavy block with needs for recursive "insanity"
if by any chance daring to try to introduce new ideas.
This of course defines all political parties and central
governments wrong. Well, capitalism is about building
empirical superpowers. So of course it is dumb. Just like
a mob. Huge! Ouch. Headshake. Burp.
Hoo Hoo
Pat
"George" <george...@hotmail.com> wrote in message
news:luKj9.41$ew4....@news.uswest.net...
the Universe and the recipe for it. I know we both agree that people are
living within a bloody huge computer and that the program has run amok and
there is no-one in the engine room who could do anything about it because
the bleeding code wasn't documented properly.
According to the Bible, God created the universe in approximately seven days
without any time spent on requirements gathering or specification not to
speak of documentation. Design is emphasised throughout the Genesis, but the
estimation of results is generally of the form: "...and God saw that it was
good." It is no surprise that some serious bugs were found later on. An
error committed by Eve was caught, terminating the Paradise loop. Real bugs
were found later on, and the Great Flood was implemented to restart the
whole program. But the program seems to have more holes than Windows, so I
guess it was abandonded and let to run its course.
If you wish, I can post the details of the program later on and the
algorithm that should do the job properly.
John
Unfortunately, the verdict that set theory does not belong to logic
has been handed down by the very crew who stand most to benefit
by it: those who ground their notion of *set* in the
iterative hierarchy. Not uncoincidentally, for this crew
(whose students include Mike Oliver), no axioms--including
mine--are worthy of note that trade on a logical conception of set.
Perhaps your unconcern for the fact that principles of pairing,
union and power set are *theorems* in my system--they are not
theorems in standard systems--reflects such a bias.
--John
Chris Menzel
> cme...@philebus.tamu.edu (Chris Menzel) wrote in message news:<slrnaomv47....@philebus.tamu.edu>...
> > relation is a part of pure logic at all. Rather, it's part of the
> > subject matter of a more specialized abstract discipline that studies a
> > certain species of mathematical object, and hence, in my view, is not
> > sufficiently general to count as part of pure logic. This is, of
> > course, not a judgment of value, but simply a (far from universal) view
> > about what counts as pure logic and what doesn't.
>
> Unfortunately, the verdict that set theory does not belong to logic
> has been handed down by the very crew who stand most to benefit
> by it: those who ground their notion of *set* in the
> iterative hierarchy. Not uncoincidentally, for this crew
> (whose students include Mike Oliver), no axioms--including
> mine--are worthy of note that trade on a logical conception of set.
But surely that's OK, if the structure the "crew" in question is
interested in *is* the iterative hierarchy. You're just interested in
something *else*.
> Perhaps your unconcern for the fact that principles of pairing,
> union and power set are *theorems* in my system--they are not
> theorems in standard systems--reflects such a bias.
I don't think so. I would suggest that the intuitions behind your
logical conception of set are actually intuitions about *properties*
and, more generally, relations, which I would agree are plausibly
thought of as part of the primitive ontology of logic. But classes
won't fit the bill (so sez I, anyway), as they are extensional and
properties and relations ain't.
Chris Menzel
John
> On 26 Sep 2002 14:02:25 -0700, John <john_...@excite.ca> said:
> > cme...@philebus.tamu.edu (Chris Menzel) wrote in message news:<slrnaomv47....@philebus.tamu.edu>...
> > > Perhaps more relevantly, though, I do not think that the set membership
> > > relation is a part of pure logic at all. Rather, it's part of the
> > > subject matter of a more specialized abstract discipline that studies a
> > > certain species of mathematical object, and hence, in my view, is not
> > > sufficiently general to count as part of pure logic. This is, of
> > > course, not a judgment of value, but simply a (far from universal) view
> > > about what counts as pure logic and what doesn't.
> >
> > Unfortunately, the verdict that set theory does not belong to logic
> > has been handed down by the very crew who stand most to benefit
> > by it: those who ground their notion of *set* in the
> > iterative hierarchy. Not uncoincidentally, for this crew
> > (whose students include Mike Oliver), no axioms--including
> > mine--are worthy of note that trade on a logical conception of set.
>
> But surely that's OK, if the structure the "crew" in question is
> interested in *is* the iterative hierarchy. You're just interested in
> something *else*.
Nevertheless, every model of N is a model of C (and conversely).
Accordingly, my attention to the axioms of "baby set theory"
(Herbert Enderton's term) has yielded a system which is equivalent
in power to NBG minus Choice, infinity, and Replacement. (Concerning
how powerful a system must be to encode most of twentieth century
mathematics, see John Sowa's letter to Pat Hayes--follows my sig.)
Moreover, C3 and C4 guarantee pair classes, unions and power
classes. So C3 and C4 capture what it is about classes qua
classes that grounds these principles, which ZFC/NBG
Separation/Classification and Extensionality do not.
> > Perhaps your unconcern for the fact that principles of pairing,
> > union and power set are *theorems* in my system--they are not
> > theorems in standard systems--reflects such a bias.
>
> I don't think so. I would suggest that the intuitions behind your
> logical conception of set are actually intuitions about *properties*
> and, more generally, relations, which I would agree are plausibly
> thought of as part of the primitive ontology of logic. But classes
> won't fit the bill (so sez I, anyway), as they are extensional and
> properties and relations ain't.
Don't C1,C2,C4 make C extensional?
C1 AxAy[xIy -> 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
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}]
Extensionality
>
> Chris Menzel
--John
John Sowa's letter to Pat Hayes re Twentieth Century Mathematics
http://www-ksl.stanford.edu/email-archives/interlingua.messages/308.html
I just want to emphasize how totally disjoint Cantor's diagonal is from
anything in mathematics that has any useful applicability to anything.
NOTHING, absolutely NOTHING, in analysis depends in any way on Cantor's
diagonal proof. All the work on epsilon's and delta's, not to mention
infinitesimals, was completed long before by Cauchy, Riemann, Weirstrass,
etc. They never made any assumptions about orders of inifinity, and in
fact, they never even talked about infinity as a completed whole. All
of modern analysis follows directly in their footsteps and does not in
any way depend on anything that Cantor did or said about uncountable sets.
Some comments on your comments:
> ... You referred rather
> casually to 'Cantor's paradise', which is usually taken to refer to
> modern mathematics....
That was Hilbert's phrase, and he used it only to refer to Cantor's
work on transfinite numbers (cardinals and ordinals). Very little
of "modern mathematics" deals with uncountable sets.
> You cite the diagonalisation argument as though it were a mere
> bywater, but its consequences permeate contemporary mathematics.
> The definition of the real line depends on it: all of analysis
> would have to be reworked if we reject this. The ideas of limits
> of infinite series, used fundamentally throughout physics and
> chaos theory and topology would need to be revised. It's very
> basic.
No. None of those subfields of mathematics depend in any way on
Cantor's diagonal "proof". As I pointed out above, all of analysis
was very thoroughly established long before Cantor's work. Limits,
infinite series, etc., never refer to notions of uncountability in
any way. The term "real line", by the way, is an interesting case
in point. A number of mathematicians, including Kurt Goedel, by
the way, have observed some dubious properties about that identification
of the real numbers with points on a line. Suppose, for example, that
you take a line segment of length 1 and break it in half. Intuitively,
you might imagine that you would get two identical segments of length
0.5 -- at least that is what everyone from Aristotle up to the late
nineteenth century would have assumed. But if you "identify" that
line segment with the reals from 0.0 to 1.0, you have to ask what
happens to the point at 0.5 -- which "half" does it belong to?
If it stays with only one side, then you don't have two identical
line segments. Instead, you have one segment that is closed at
both ends and another segment that is open at one end. This is a
very unpleasant consequence that Goedel was not at all pleased about.
My hero, C. S. Peirce, had a solution: he did not "identify" the
real numbers with points on a line. In fact, he did not believe that
a line could be identified with a collection of points; instead, he
believed that points were determined by intersecting lines -- they
were not the constituents that made up the material of which the line
was constructed. Anyway, that is getting off on another tangent,
but it does illustrate the point that there are a lot of serious
questions that remain unanswered, and even someone like Goedel who
worked long and hard in Cantor's paradise was still bothered by them.
> Your citing Whitehead as having worries about the construction
> is revealing. Indeed, when these results were new they were
> controversial, and many older philosophers and mathematicians
> had doubts and reservations. But no convincing arguments were ever
> produced to suggest that the argument is faulty: and after
> nearly a century of critical testing and analysis, 'Cantor's
> paradise' (Hilbert's phrase) seems to be pretty secure. Certainly
> no coherent alternative has ever been constructed (except the
> intuitionist's smaller place.) So again, your rejection of this
> established and well-understood piece of basic mathmatics,
> citing the opinions of somone considered even in his day a
> bit out in left field is, well, idiosyncratic.
No. Whitehead didn't start discussing his doubts until after he
finished his work on the Principia with Russell -- over 30 years
after Cantor's work. And I don't want to go into another litany
of "Big Names" or more "amateur scholarly" citations, but this whole
area has never been one that is anywhere near as solid as the rest
of mathematics. Frank Ramsey, for example, whom Russell considered
one of the most brilliant of his students, made a number of contributions
to the theory of large cardinals. But after thinking about the whole
subject in greater depth, he began to have doubts about it (he was a
friend of Wittgenstein's and I don't know who influenced whom on this
point). If Ramsey hadn't met his tragic death at age 28, he would
probably have debunked the whole theory long ago, and we wouldn't have
to be arguing about it now.
> But you don't give any argument against it, and you ignore all the
> thinking that has been done about it.
On the contrary, there hasn't been much thinking about it at all.
There are three kinds of mathematicians: people who work in fields
like analysis, algebra, chaos theory, etc., who don't depend in any
way on the existence of uncountable sets; students whose professors
gave them a classical theorem in mathematics and told them to generalize
it to a higher order of infinity for their PhD theses; and really
profound thinkers like Goedel, Wittgenstein, Ramsey, etc., who have
worried about it. Of the third group, Wittgenstein represents the
skeptic, Goedel represents the true believer who wrestled with doubts
but overcame them, and Ramsey represents an ardent disciple who later
lost his faith. I consider myself an agnostic who admires Goedel,
but sympathizes with Wittgenstein and Ramsey.
> That is the conclusion of a very pointed and convincing argument, not
> just a vague feeling that things aren't right (which is all that you
> offer here).(I share your gut feeling, and would be interested in finding
> out how you propose to get past Cantor's argument.)
Your reference is vague. Whose argument did you consider "very
pointed and convincing"? Mine or Cantor's? I am not claiming to
be a better mathematician than Cantor, but I am claiming to have the
benefit of the work by Turing and others on noncomputability. If
Turing's work had preceded Cantor's, I doubt that anyone would have
considered it to be anything more than another variation of a
noncomputability theorem.
In any case, I have been cleaning out my old office at IBM, and I
unearthed a collection of volumes of the Journal of Symbolic Logic,
which I used to subscribe to until I got fed up with some large
cardinal number of dissertations on large cardinals. If you know
of anyone who is fond of such things, I'll send them to anyone who
is willing to pay the freight.
John
Chris Menzel
> cme...@philebus.tamu.edu (Chris Menzel) wrote in message
> news:<slrnap8qu8....@philebus.tamu.edu>...
> > On 26 Sep 2002 14:02:25 -0700, John <john_...@excite.ca> said:
> > > cme...@philebus.tamu.edu (Chris Menzel) wrote in message
> > > news:<slrnaomv47....@philebus.tamu.edu>...
> > > > Perhaps more relevantly, though, I do not think that the set
> > > > membership relation is a part of pure logic at all. Rather,
> > > > it's part of the subject matter of a more specialized abstract
> > > > discipline that studies a certain species of mathematical
> > > > object, and hence, in my view, is not sufficiently general to
> > > > count as part of pure logic. This is, of course, not a judgment
> > > > of value, but simply a (far from universal) view about what
> > > > counts as pure logic and what doesn't.
> > >
> > > Unfortunately, the verdict that set theory does not belong to logic
> > > has been handed down by the very crew who stand most to benefit
> > > by it: those who ground their notion of *set* in the
> > > iterative hierarchy. Not uncoincidentally, for this crew
> > > (whose students include Mike Oliver), no axioms--including
> > > mine--are worthy of note that trade on a logical conception of set.
> >
> > But surely that's OK, if the structure the "crew" in question is
> > interested in *is* the iterative hierarchy. You're just interested in
> > something *else*.
>
> Nevertheless, every model of N is a model of C (and conversely).
The point here is lost on me, I'm afraid.
> > > Perhaps your unconcern for the fact that principles of pairing,
> > > union and power set are *theorems* in my system--they are not
> > > theorems in standard systems--reflects such a bias.
> >
> > I don't think so. I would suggest that the intuitions behind your
> > logical conception of set are actually intuitions about *properties*
> > and, more generally, relations, which I would agree are plausibly
> > thought of as part of the primitive ontology of logic. But classes
> > won't fit the bill (so sez I, anyway), as they are extensional and
> > properties and relations ain't.
>
> Don't C1,C2,C4 make C extensional?
>
> C1 AxAy[xIy -> 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
> C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}]
> Extensionality
Actually, I was wrong before; your classes aren't (provably)
extensional, as you can't say anything about their identity at all.
Perhaps this gives you a foothold for the claim that they can plausibly
be taken to be properties (should you want to).
Chris Menzel
John
Modulo D1 and D2, every theorem of N is a theorem of C, and conversely.
D1 "x=y" means "Az(z in x <-> z in y)"
D2 "xIy" means "x=y & set x & set y"
So C guarantees all and only the sets/proper classes that N does--
including the class of sets, the class of all non-self-membered
sets, and the class of self-identicals. True, in C the class of
sets and the class of self-identicals are equi-membered but distinct,
as would be the corresponding properties. However, in C there
is only one empty class--and this is a set. Therefore, in C there
is no way to model distinct empty properties--such as the property of being
Vulcan and the property of being the largest prime.
It remains to establish that FOL+ and FOL= are formally equivalent
(modulo appropriate definitions). Once I've done this, I hope you'll
have a look at the system which extends FOL+, which is a 'tweaked'
version of C.
--John
John
> john_...@excite.ca (John) wrote in message news:<70f94e16.02090...@posting.google.com>...
In the thread "Proofs" I established the formal equivalence--modulo
D1, D2 & D3--of C1-C9 and N1-N8 (these follow my sig).
I'll now show the same relation holds (modulo appropriate definitions)
between FOL= and FOL+. The FOL= I have in mind--call it Correy FOL=--is
obtained from FOL by adding axioms (1,=2,=3):
1. AxAy[Az(x R z <-> y R z) -> Az(z R x <-> z R y)]
=2. AxAy[Az(x R z <-> y R z) -> x = y]
=3. AxAyAz[x = y -> (x R z <-> y R z)]
(1,=2,=3) establish = as symmetric, transitive and reflexive and
identicals as indiscernible. Moreover, axiom (=2) also lays down
a sufficient condition for the identity of x and y: x and y are
identical if x bears R to z whenever y does (and conversely), for
all z.
--John
C1 AxAyAz[xIy -> (z in x <-> z in y)] LL1
C2 AxAyAz[Az(z in x <-> z in y) <-> (Az(x in z <-> y in z))] LL2
C3 EyAx[x in y <-> Et(x in t) & P(x)] (with y not free in P(x))
Classification
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}] Weak
Extensionality
C5 AaAbAy(Ax[x in y <-> (xIa v xIb)] -> yIy) Axiom of Pairs
C6 AaAy(Ax[x in y -> x in a] -> (aIa -> yIy)) Axiom of Subsets
C7 AaAy[Ax(x in y <-> Ew(w in a & x in w)) -> (aIa -> yIy)] Axiom of Unions
C8 Aa(Ay[Ax(x in y <-> Aw(w in x -> w in a))) -> (aIa -> yIy]) Axiom of Power
Sets
N1 AxAyAz[x=y -> (x in z <-> y in z)] LL 1
N2 AxAyAz[x=y -> (z in x <-> z in y)] LL 2
N3 EyAx[x in y <-> (Et(x in t) & P(x))] (with y not free in (x))
Classification
N4 AxAy[Az(z in x <-> z in y) -> x = y] Extensionality
N5 AaAb[set a & set b -> Ey(set y & Ax(x in y <-> x=a v x=b))] Axiom of Pairs
N6 Aa[set a -> Ay(Ax(x in y -> x in a) -> set y)] Axiom of Subsets
N7 Aa[set a -> Ey(set y & Ax(x in y <-> Ew(w in a & x in w)))] Axiom of
Unions
N8 Aa[set a -> Ey(set y & Ax(x in y <-> Aw(w in x -> w in a)))] Axiom of
Power Sets
N9 Ex(set x & Ay~(y in x)) Empty Set Axiom
D1: "x=y" means "Az(z in x <-> z in y)"
D2 "xIy" means "x=y & set x & set y"
D3 "set x" means "Et(x in t)"
John
> john_...@excite.ca (John) wrote in message news:<70f94e16.02091...@posting.google.com>...
> > john_...@excite.ca (John) wrote in message news:<70f94e16.02090...@posting.google.com>...
>
> In the thread "Proofs" I established the formal equivalence--modulo
> D1, D2 & D3--of C1-C9 and N1-N8 (these follow my sig).
>
> I'll now show the same relation holds (modulo appropriate definitions)
> between FOL= and FOL+. The FOL= I have in mind--call it Correy FOL=--is
> obtained from FOL by adding axioms (1,=2,=3):
>
> 1. AxAy[Az(x R z <-> y R z) -> Az(z R x <-> z R y)]
> =2. AxAy[Az(x R z <-> y R z) -> x = y]
> =3. AxAyAz[x = y -> (x R z <-> y R z)]
>
> (1,=2,=3) establish = as symmetric, transitive and reflexive and
> identicals as indiscernible. Moreover, axiom (=2) also lays down
> a sufficient condition for the identity of x and y: x and y are
> identical if x bears R to z whenever y does (and conversely), for
> all z.
>
> --John
There was a typo in the axioms I gave for C. These should have been:
C1 AxAyAz[xIy -> (z in x <-> z in y)] LL1
C2 AxAyAz[Az(z in x <-> z in y) -> (Az(x in z <-> y in z))] LL2
C3 EyAx[x in y <-> Et(x in t) & P(x)] (with y not free in P(x))
Classification
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}] Weak
Extensionality
C5 AaAbAy(Ax[x in y <-> (xIa v xIb)] -> yIy) Axiom of Pairs
C6 AaAy(Ax[x in y -> x in a] -> (aIa -> yIy)) Axiom of Subsets
C7 AaAy[Ax(x in y <-> Ew(w in a & x in w)) -> (aIa -> yIy)] Axiom of Unions
C8 Aa(Ay[Ax(x in y <-> Aw(w in x -> w in a))) -> (aIa -> yIy]) Axiom of Power
Sets
The axioms for N--and definitions--are:
John
Concerning the natural numbers--and (to a lesser extent) for sets,
Aatu Koskensilta writes:
> the original intention of
> the axiomatisation was to characterise precisely (up to isomorphism)
> what the structure we study is and what properties it haves by
> virtue of being just that structure. Thus the structure comes first,
> and its axiomatic description comes later.
news:<aa503d8.02092...@posting.google.com>
In a similar vein, concerning the inconsistent (1,2,3)
>> 1) AxAy[Az(z in x <-> z in y) -> x = y]
> >> 2) EyAx(x in y <-> Et(x in t) & Px) (whatever the P)
> >> 3) AtAwEyAx[x in y <-> (x = t v x = w)]
a correspondent of mine wrote:
> However what You write suggests that you have in mind ONE model of
> classes and want to know which of 1), 2) or 3) are true in that model.
> I too have in mind ONE such model and in my model 1) and 3)
> and the negation of 2) for Px <=> (x = x), are all true (They constitute a
> part of the description of that model). This model of mine (The axioms of
> ZFC constitute a fuller description of that model) is accepted today by
> all working mathematicians (with the exception of some odd characters,
> e.g. a group called the intuitionists, the latter are particularly
> unclear, and intuitionism is quite a misnomer).
If I have understood him correctly, once again the point is: the
model/structure comes first, the axioms come later.
In view of the foregoing, let me try to make my point again. Granted
that every model of C is a model of N (and conversely), if the sole
criterion for choosing between sets of axioms is the models that
these axioms have, what basis is there for preferring N to C, granted
that these have the same models?
--John