Re: Semantic vs logical truth
On 03 Dec 2012, at 21:55, meekerdb wrote:
On 12/3/2012 10:03 AM, Bruno Marchal wrote:
On 03 Dec 2012, at 00:04, meekerdb wrote:
On 12/2/2012 7:27 AM, Bruno Marchal wrote:
The 1p truth of the machine is not coded in the machine. Some
actual machines knows already that, and can justified that If
there are machine (and from outside we can know this to correct)
then the 1p-truth is not codable. The 1p truth are more related
to the relation between belief and reality (not necessarily
physical reality, except for observation and sensation).
Even the simple, and apparently formal Bp & p is NOT codable.
Most truth about machine, including some that they can know, are
not codable.
Many things true about us is not codable either.
Let me see if I understand that. I think you are saying that p,
i.e. that "p" describes a fact about the world, a meta-level above
the coding of a machine.
No, p is for some statement at the base level, like 1+1 = 2.
Yes, I understand that. I didn't express myself clearly. p is a 0-
level statement. "That p" (i.e. that p is true, that "p"
describes a fact) is a 1-level statement.
p is "that p". When the machine asserts 1+1=2, she meant that it is
the case that "1+1=2", independently of the truth or falsity of the
assertion. But we mimit ourself to correct machine, so in this case
"p" and "p is true" are equivalent, and so we can model True("p"),
which is not expressible in the language of the machine, by
"p" (asserted by the machine).
That the Mars Rover believes it is south of it's landing point is
implicit in its state and might be inferred from its behavior, but
there is no part of the state corresponding to "I *believe* I am
south of my landing point."
Then Mars Rover is not Löbian. But I am not even sure that Mars
Rover is Turing universal, or that it exploits its Turing
universality.
Well not the current Mars Rover, but a Mars Rover could be, it's
just a matter of program. So the Rover could not only encode p,
also encode that it believed p.
OK.
But PA and ZF can represent "I believe". So we can study the logic
of a new 'knowledge" operator defined (at the meta level, for each
arithmetical proposition) by Bp & p. For example if p is "1+1=2",
it is
Believe"("1+1=2") & 1+1 = 2.
I don't understand the significance of the unpaired quote marks?
read:
Believe("1+1=2") & 1+1 = 2
(Sorry).
We cannot define such operator in arithmetic. We would need
something like Believe"("1+1=2") & True("1+1 = 2"), but True, in
general cannot defined in arithmetic. Yet, we can metadefine it and
study its logic, which obeys a soprt of temporal intuionistic logic
(interpreting the S4Grz logic obtained).
One could include such second-level states (which one might want
to communicate to Pasadena) but then that state would be just
another first-level state. Right?
Not sure I see what you mean. The meta, available by the machine is
in the "I believe". It is the 3-I. The presentation of myself to
myself. The 1-I will be the non definable operator above. We
connect the believer to the truth. It is easy to do for the sound
correct machine.
What I mean is that if you programmed the Rover to be Lobian and it
not only thought p but also though Bp, both of those would be just
be similar physical states within its computer memory - their
hierarchical relation would just be that encoded in the Lobian
program.
?
"s(0) + s(0) = s(s(0))" is far more shorter than the coding of
"Beweisbar ("s(0) + s(0) = s(s(0))")"
So a physical error in the computer could change Bp to ~Bp, yet this
would have no effect on the performance of the Rover except in
reporting what it believed.
To survive, such self-referential falsity will not help.
To do "philosophy of mind", such false reports can be disastrous.
But for the correct machine, by definition we don't have that problem.
Bp & p is strictly equivalent with Bp. Now the correct machine can
never prove ("for all p") such an equivalence. In fact by Löb theorem,
if ever the machine proves Bp -> p, she will always been able to prove
p.
G* proves ((Bp & p) <-> Bp)
G does not prove that. Indeed the machine can't prove Bf -> f.
Bruno
http://iridia.ulb.ac.be/~marchal/
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To post to this group, send email to everything-list@googlegroups.com.
To unsubscribe from this group, send email to
everything-list+unsubscr...@googlegroups.com.
For more options, visit this group at
http://groups.google.com/group/everything-list?hl=en.
Re: Semantic vs logical truth
On 12/3/2012 10:03 AM, Bruno Marchal wrote:
On 03 Dec 2012, at 00:04, meekerdb wrote:
On 12/2/2012 7:27 AM, Bruno Marchal wrote:
The 1p truth of the machine is not coded in the machine. Some actual machines knows
already that, and can justified that If there are machine (and from outside we can
know this to correct) then the 1p-truth is not codable. The 1p truth are more related
to the relation between belief and reality (not necessarily physical reality, except
for observation and sensation).
Even the simple, and apparently formal Bp & p is NOT codable.
Most truth about machine, including some that they can know, are not codable.
Many things true about us is not codable either.
Let me see if I understand that. I think you are saying that p, i.e. that "p"
describes a fact about the world, a meta-level above the coding of a machine.
No, p is for some statement at the base level, like 1+1 = 2.
Yes, I understand that. I didn't express myself clearly. p is a 0-level statement.
"That p" (i.e. that p is true, that "p" describes a fact) is a 1-level statement.
That the Mars Rover believes it is south of it's landing point is implicit in its state
and might be inferred from its behavior, but there is no part of the state
corresponding to "I *believe* I am south of my landing point."
Then Mars Rover is not Löbian. But I am not even sure that Mars Rover is Turing
universal, or that it exploits its Turing universality.
Well not the current Mars Rover, but a Mars Rover could be, it's just a matter of
program. So the Rover could not only encode p, also encode that it believed p.
But PA and ZF can represent "I believe". So we can study the logic of a new 'knowledge"
operator defined (at the meta level, for each arithmetical proposition) by Bp & p. For
example if p is "1+1=2", it is
Believe"("1+1=2") & 1+1 = 2.
I don't understand the significance of the unpaired quote marks?
We cannot define such operator in arithmetic. We would need something
like Believe"("1+1=2") & True("1+1 = 2"), but True, in general cannot defined in
arithmetic. Yet, we can metadefine it and study its logic, which obeys a soprt of
temporal intuionistic logic (interpreting the S4Grz logic obtained).
One could include such second-level states (which one might want to communicate to
Pasadena) but then that state would be just another first-level state. Right?
Not sure I see what you mean. The meta, available by the machine is in the "I believe".
It is the 3-I. The presentation of myself to myself. The 1-I will be the non definable
operator above. We connect the believer to the truth. It is easy to do for the sound
correct machine.
What I mean is that if you programmed the Rover to be Lobian and it not only thought p but
also though Bp, both of those would be just be similar physical states within its computer
memory - their hierarchical relation would just be that encoded in the Lobian program. So
a physical error in the computer could change Bp to ~Bp, yet this would have no effect on
the performance of the Rover except in reporting what it believed.
Brent
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To post to this group, send email to everything-list@googlegroups.com.
To unsubscribe from this group, send email to
everything-list+unsubscr...@googlegroups.com.
For more options, visit this group at
http://groups.google.com/group/everything-list?hl=en.
Re: Semantic vs logical truth
On 12/3/2012 4:42 AM, Roger Clough wrote:
Hi meekerdb
1p is not a fiction. Your 1p is what is reading this page.
Where did I refer to 1p? Where did I call anything a 'fiction' Are you replying to the
voices in your head? Don't put words in my mouth.
Brent
[Roger Clough], [rclo...@verizon.net] <mailto:rclo...@verizon.net]>
12/3/2012
"Forever is a long time, especially near the end." -Woody Allen
- Receiving the following content -
*From:* meekerdb <mailto:meeke...@verizon.net>
*Receiver:* everything-list <mailto:everything-list@googlegroups.com>
*Time:* 2012-12-02, 18:04:38
*Subject:* Re: Semantic vs logical truth
On 12/2/2012 7:27 AM, Bruno Marchal wrote:
The 1p truth of the machine is not coded in the machine. Some actual
machines knows
already that, and can justified that If there are machine (and from outside
we can
know this to correct) then the 1p-truth is not codable. The 1p truth are
more
related to the relation between belief and reality (not necessarily physical
reality, except for observation and sensation).
Even the simple, and apparently formal Bp & p is NOT codable.
Most truth about machine, including some that they can know, are not
codable.
Many things true about us is not codable either.
Let me see if I understand that. I think you are saying that p, i.e. that
"p"
describes a fact about the world, a meta-level above the coding of a
machine. That
the Mars Rover believes it is south of it's landing point is implicit in
its state
and might be inferred from its behavior, but there is no part of the state
corresponding to "I *believe* I am south of my landing point." One could
include
such second-level states (which one might want to communicate to Pasadena)
but then
that state would be just another first-level state. Right?
Brent
No virus found in this message.
Checked by AVG - www.avg.com <http://www.avg.com>
Version: 2012.0.2221 / Virus Database: 2634/5433 - Release Date: 12/02/12
--
You received this message because you are subscribed to the Google Groups "Everything
List" group.
To post to this group, send email to everything-list@googlegroups.com.
To unsubscribe from this group, send email to
everything-list+unsubscr...@googlegroups.com.
For more options, visit this group at
http://groups.google.com/group/everything-list?hl=en.
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To post to this group, send email to everything-list@googlegroups.com.
To unsubscribe from this group, send email to
everything-list+unsubscr...@googlegroups.com.
For more options, visit this group at
http://groups.google.com/group/everything-list?hl=en.
Re: Semantic vs logical truth
On 03 Dec 2012, at 00:04, meekerdb wrote:
On 12/2/2012 7:27 AM, Bruno Marchal wrote:
The 1p truth of the machine is not coded in the machine. Some
actual machines knows already that, and can justified that If there
are machine (and from outside we can know this to correct) then the
1p-truth is not codable. The 1p truth are more related to the
relation between belief and reality (not necessarily physical
reality, except for observation and sensation).
Even the simple, and apparently formal Bp & p is NOT codable.
Most truth about machine, including some that they can know, are
not codable.
Many things true about us is not codable either.
Let me see if I understand that. I think you are saying that p,
i.e. that "p" describes a fact about the world, a meta-level above
the coding of a machine.
No, p is for some statement at the base level, like 1+1 = 2.
That the Mars Rover believes it is south of it's landing point is
implicit in its state and might be inferred from its behavior, but
there is no part of the state corresponding to "I *believe* I am
south of my landing point."
Then Mars Rover is not Löbian. But I am not even sure that Mars Rover
is Turing universal, or that it exploits its Turing universality.
But PA and ZF can represent "I believe". So we can study the logic of
a new 'knowledge" operator defined (at the meta level, for each
arithmetical proposition) by Bp & p. For example if p is "1+1=2", it is
Believe"("1+1=2") & 1+1 = 2.
We cannot define such operator in arithmetic. We would need something
like Believe"("1+1=2") & True("1+1 = 2"), but True, in general cannot
defined in arithmetic. Yet, we can metadefine it and study its logic,
which obeys a soprt of temporal intuionistic logic (interpreting the
S4Grz logic obtained).
One could include such second-level states (which one might want to
communicate to Pasadena) but then that state would be just another
first-level state. Right?
Not sure I see what you mean. The meta, available by the machine is in
the "I believe". It is the 3-I. The presentation of myself to myself.
The 1-I will be the non definable operator above. We connect the
believer to the truth. It is easy to do for the sound correct machine.
Bruno
Brent
--
You received this message because you are subscribed to the Google
Groups "Everything List" group.
To post to this group, send email to everything-list@googlegroups.com.
To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com
.
For more options, visit this group at http://groups.google.com/group/everything-list?hl=en
.
http://iridia.ulb.ac.be/~marchal/
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To post to this group, send email to everything-list@googlegroups.com.
To unsubscribe from this group, send email to
everything-list+unsubscr...@googlegroups.com.
For more options, visit this group at
http://groups.google.com/group/everything-list?hl=en.
Re: Re: Semantic vs logical truth
Hi meekerdb 1p is not a fiction. Your 1p is what is reading this page. [Roger Clough], [rclo...@verizon.net] 12/3/2012 "Forever is a long time, especially near the end." -Woody Allen - Receiving the following content - From: meekerdb Receiver: everything-list Time: 2012-12-02, 18:04:38 Subject: Re: Semantic vs logical truth On 12/2/2012 7:27 AM, Bruno Marchal wrote: The 1p truth of the machine is not coded in the machine. Some actual machines knows already that, and can justified that If there are machine (and from outside we can know this to correct) then the 1p-truth is not codable. The 1p truth are more related to the relation between belief and reality (not necessarily physical reality, except for observation and sensation). Even the simple, and apparently formal Bp & p is NOT codable. Most truth about machine, including some that they can know, are not codable. Many things true about us is not codable either. Let me see if I understand that. I think you are saying that p, i.e. that "p" describes a fact about the world, a meta-level above the coding of a machine. That the Mars Rover believes it is south of it's landing point is implicit in its state and might be inferred from its behavior, but there is no part of the state corresponding to "I *believe* I am south of my landing point." One could include such second-level states (which one might want to communicate to Pasadena) but then that state would be just another first-level state. Right? Brent -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Re: Semantic vs logical truth
Hi Richard Ruquist You still think everything's objective. But nobody know how to code 1p because it's subjective. 3p is objective, 1p is subjective. [Roger Clough], [rclo...@verizon.net] 12/3/2012 "Forever is a long time, especially near the end." -Woody Allen - Receiving the following content - From: Richard Ruquist Receiver: everything-list Time: 2012-12-02, 08:16:03 Subject: Re: Semantic vs logical truth Roger, Computers will do 1p truth when their results become emergent in which case they will be doing the coding as well so to speak. Richard On Sun, Dec 2, 2012 at 7:11 AM, Roger Clough wrote: > Hi Bruno Marchal > > Semantic truth I think is 1p (personal, private) truth, > which mnakes it contingent, while logical truth is necessary > as well as public or 3p truth. I think > comnputers have problems with 1p truth because > for one thing the coding is done by someone outside. > > > [Roger Clough], [rclo...@verizon.net] > 12/2/2012 > "Forever is a long time, especially near the end." -Woody Allen > > > - Receiving the following content - > From: Bruno Marchal > Receiver: everything-list > Time: 2012-12-02, 04:07:39 > Subject: Re: Numbers in the Platonic Realm > > > On 30 Nov 2012, at 21:28, meekerdb wrote: > > On 11/30/2012 10:02 AM, Roger Clough wrote: > > And a transcendent truth could be arithmetic truth or > the truth of necessary logic. > > > True in logic and formal mathematics is just marker "T" that is preserved by > the rules of inference. > > > This makes no sense. You confuse the propositional constant T, with the > semantical notion of truth. The first is expressible/definable formally > (indeed by T, or by "0 = 0" in arithmetic), the second is not (Tarski > theorem). When we say that truth is preserved by the rules of inference, we > are concerned with the second notion. > > > > In applications it is interpreted as if it were the correspondence meaning > of 'true'. > > > Like in arithmetic. Truth of "ExP(x)" means that it exists a n such that > P(n), at the "metalevel", which is the bare level in logic (that explains > many confusion). > > > > > But like all applications of mathematics, it may be only approximate. > > > Yes, but for arithmetic it is pretty clear, as we share our intuition on the > so-called standard finite numbers. > > Bruno > > > http://iridia.ulb.ac.be/~marchal/ > > > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To post to this group, send email to everything-list@googlegroups.com. > To unsubscribe from this group, send email to > everything-list+unsubscr...@googlegroups.com. > For more options, visit this group at > http://groups.google.com/group/everything-list?hl=en. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Semantic vs logical truth
On 12/2/2012 7:27 AM, Bruno Marchal wrote: The 1p truth of the machine is not coded in the machine. Some actual machines knows already that, and can justified that If there are machine (and from outside we can know this to correct) then the 1p-truth is not codable. The 1p truth are more related to the relation between belief and reality (not necessarily physical reality, except for observation and sensation). Even the simple, and apparently formal Bp & p is NOT codable. Most truth about machine, including some that they can know, are not codable. Many things true about us is not codable either. Let me see if I understand that. I think you are saying that p, i.e. that "p" describes a fact about the world, a meta-level above the coding of a machine. That the Mars Rover believes it is south of it's landing point is implicit in its state and might be inferred from its behavior, but there is no part of the state corresponding to "I *believe* I am south of my landing point." One could include such second-level states (which one might want to communicate to Pasadena) but then that state would be just another first-level state. Right? Brent -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Semantic vs logical truth
On 02 Dec 2012, at 13:11, Roger Clough wrote: Hi Bruno Marchal Semantic truth I think is 1p (personal, private) truth, Truth is always semantical, and it can be 3p (it usually is). Subjective truth, like with feeling, sensations, ... is 1p, but it is only a personal relative view on something bigger. You might be conflating God and the inner God, somehow. Semantic is a sub-branch of mathematical logic, usually known as model theory. Note that logician use "model" in the sense of the painters. the model is the reality, not the theory used to describe the reality. Logicians studied explicitly the relation between theories and their models. The 1p comes when *some* representable beliefs are true. A bit like when you plunge a map in the territory described by the map: there will be a fixed point which is the same in the territory and in reality (the famous: "you are here" point). which mnakes it contingent, while logical truth is necessary Even logical truth can be said contingent on the logic chosen. There are many logics. Now for a platonist, classical logic is single out, but not as the most true one, but as the one which is the simpler to make sense on ... the other logics. as well as public or 3p truth. I think comnputers have problems with 1p truth because for one thing the coding is done by someone outside. The 1p truth of the machine is not coded in the machine. Some actual machines knows already that, and can justified that If there are machine (and from outside we can know this to correct) then the 1p- truth is not codable. The 1p truth are more related to the relation between belief and reality (not necessarily physical reality, except for observation and sensation). Even the simple, and apparently formal Bp & p is NOT codable. Most truth about machine, including some that they can know, are not codable. Many things true about us is not codable either. Bruno [Roger Clough], [rclo...@verizon.net] 12/2/2012 "Forever is a long time, especially near the end." -Woody Allen - Receiving the following content - From: Bruno Marchal Receiver: everything-list Time: 2012-12-02, 04:07:39 Subject: Re: Numbers in the Platonic Realm On 30 Nov 2012, at 21:28, meekerdb wrote: On 11/30/2012 10:02 AM, Roger Clough wrote: And a transcendent truth could be arithmetic truth or the truth of necessary logic. True in logic and formal mathematics is just marker "T" that is preserved by the rules of inference. This makes no sense. You confuse the propositional constant T, with the semantical notion of truth. The first is expressible/definable formally (indeed by T, or by "0 = 0" in arithmetic), the second is not (Tarski theorem). When we say that truth is preserved by the rules of inference, we are concerned with the second notion. In applications it is interpreted as if it were the correspondence meaning of 'true'. Like in arithmetic. Truth of "ExP(x)" means that it exists a n such that P(n), at the "metalevel", which is the bare level in logic (that explains many confusion). But like all applications of mathematics, it may be only approximate. Yes, but for arithmetic it is pretty clear, as we share our intuition on the so-called standard finite numbers. Bruno http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com . For more options, visit this group at http://groups.google.com/group/everything-list?hl=en . http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Semantic vs logical truth
Roger, Computers will do 1p truth when their results become emergent in which case they will be doing the coding as well so to speak. Richard On Sun, Dec 2, 2012 at 7:11 AM, Roger Clough wrote: > Hi Bruno Marchal > > Semantic truth I think is 1p (personal, private) truth, > which mnakes it contingent, while logical truth is necessary > as well as public or 3p truth. I think > comnputers have problems with 1p truth because > for one thing the coding is done by someone outside. > > > [Roger Clough], [rclo...@verizon.net] > 12/2/2012 > "Forever is a long time, especially near the end." -Woody Allen > > > - Receiving the following content - > From: Bruno Marchal > Receiver: everything-list > Time: 2012-12-02, 04:07:39 > Subject: Re: Numbers in the Platonic Realm > > > On 30 Nov 2012, at 21:28, meekerdb wrote: > > On 11/30/2012 10:02 AM, Roger Clough wrote: > > And a transcendent truth could be arithmetic truth or > the truth of necessary logic. > > > True in logic and formal mathematics is just marker "T" that is preserved by > the rules of inference. > > > This makes no sense. You confuse the propositional constant T, with the > semantical notion of truth. The first is expressible/definable formally > (indeed by T, or by "0 = 0" in arithmetic), the second is not (Tarski > theorem). When we say that truth is preserved by the rules of inference, we > are concerned with the second notion. > > > > In applications it is interpreted as if it were the correspondence meaning > of 'true'. > > > Like in arithmetic. Truth of "ExP(x)" means that it exists a n such that > P(n), at the "metalevel", which is the bare level in logic (that explains > many confusion). > > > > > But like all applications of mathematics, it may be only approximate. > > > Yes, but for arithmetic it is pretty clear, as we share our intuition on the > so-called standard finite numbers. > > Bruno > > > http://iridia.ulb.ac.be/~marchal/ > > > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To post to this group, send email to everything-list@googlegroups.com. > To unsubscribe from this group, send email to > everything-list+unsubscr...@googlegroups.com. > For more options, visit this group at > http://groups.google.com/group/everything-list?hl=en. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Semantic vs logical truth
Hi Bruno Marchal Semantic truth I think is 1p (personal, private) truth, which mnakes it contingent, while logical truth is necessary as well as public or 3p truth. I think comnputers have problems with 1p truth because for one thing the coding is done by someone outside. [Roger Clough], [rclo...@verizon.net] 12/2/2012 "Forever is a long time, especially near the end." -Woody Allen - Receiving the following content - From: Bruno Marchal Receiver: everything-list Time: 2012-12-02, 04:07:39 Subject: Re: Numbers in the Platonic Realm On 30 Nov 2012, at 21:28, meekerdb wrote: On 11/30/2012 10:02 AM, Roger Clough wrote: And a transcendent truth could be arithmetic truth or the truth of necessary logic. True in logic and formal mathematics is just marker "T" that is preserved by the rules of inference. This makes no sense. You confuse the propositional constant T, with the semantical notion of truth. The first is expressible/definable formally (indeed by T, or by "0 = 0" in arithmetic), the second is not (Tarski theorem). When we say that truth is preserved by the rules of inference, we are concerned with the second notion. In applications it is interpreted as if it were the correspondence meaning of 'true'. Like in arithmetic. Truth of "ExP(x)" means that it exists a n such that P(n), at the "metalevel", which is the bare level in logic (that explains many confusion). But like all applications of mathematics, it may be only approximate. Yes, but for arithmetic it is pretty clear, as we share our intuition on the so-called standard finite numbers. Bruno http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
