Bruno
I am trying to visualize Lob formula as a block diagram to be
implemented either in neural net, as computer program or as a digital
cicuit. Digital circuits have the advantage of being very simple
(binary) so let's try to express Lob's formula as a truth table that
could be implemented with NAND gates.
Let's write Lob's formula as B2(B1p -> p) -> B1p
where B1, B2, and p are binary variables.
Note that B1 applies to p and B2 applies to the implication (B1p ->
p). (Should I have done this differently?)
Let
~ = NOT
+ = OR
. = AND
We can convert the implication B1p -> p to ~(B1.p) + p
The Boolean equivalent to Lob is
~B2(~(B1p)+ p) + B1p
The truth table is
B2 B1 p B1p ~B1p+p ~B2(~(B1p)+ p)) ~B2(~(B1p)+ p) +
B1p
0 0 0 0 1 1
1
0 0 1 0 1 1
1
0 1 0 0 1 1
1
0 1 1 1 1 1
1
1 0 0 0 1 0
0
1 0 1 0 1 0
0
1 1 0 0 1 0
0
1 1 1 1 1 0
1
I am now confused. The fifth column ~B1p+p surprisingly is all 1's. The
last column ~B2(~(B1p)+ p) + B1p which is Lob's statement and which I
expected to be all 1's is not. I have rechecked this table and I don't
see anything wrong. Is there something wrong?
It may be that Boolean algebra is not adequate to express Lob. The
question is how can Lob's formula be expressed simply by a digital
circuit a block diagram or a neural net?
George
Bruno Marchal wrote:
A long time
ago (1987), a french logician (a student at that time),
Philippe Balbiani, who did attend a talk I made
on the logic of self-reference (G) in Toulouse send me a letter
where he proposes informally to interpret the Lobian formula
(that is B(Bp->p)->Bp) as a form of closure for the french
self-persuading strategy known as "la méthode Coué" (la methode
Coue)).
I must confess I was not really convinced. I thought this would be
somehow to beautiful to be true. My mind will slightly evolve on
that
question when I will understand, in part through Smullyan's FU
(Forever Undecided) that the Lob formula does indeed capture,
at least formally, a form of self-fulfilling nature of machine's
belief.
The Lob formula does indeed say that if a machine believes Bp->p
for
some proposition p, then the machine will believe p.
This is very astonishing, and still quite mysterious to me. My
thesis
has never been based directly on Lob formula, except that through
Solovay's theorem Lob formula formalize the entire discourse of
the self-referentially correct machine.
Then recently, when I was just explaining the Lob formula
in my Amsterdam paper, John Mikes send me the message below
which shows experimental evidence on the working of the placebo
effect (quite similar to the methode Coue). I have download many
papers on the placebo and eventually conclude that Lob formula
could indeed provide a formal explanation of the working of
that placebo phenomenon.
This makes reality still more "psychological" like if the
universe(s)
was the product of a form of wishful thinking! It also
vindicates
in a deeper way the similarity between the Grand-Mother
psychology and the Lobian machine psychology. Thanks to John.
With the Knight Knaves Island Lob's theorem is not difficult
to explain and we can go back to that (but apparently some KK
posts are missing in the archive, and I don't know how to proceed,
and I will think the how and why for awhile).
A lot of physicians say the placebo effect is *subversive* with
respect to traditional science. What is clear is that it forces
even the therapist to address (at least) the mind body relation,
and this in some novel way (with respect to Aristotle).
Bruno
John Mikes wrote:
Bruno, your topic, maybe
interesting novelty (I doubt). IMO the brain can encode data in el-chem
perception, no indication so far how the qualia-gap is transcended into
thought context. Not even in picture/music/taste apperceptions. The
neuronal brain is a TOOL and the ongoing reductionist research stops at
phenomenology of "the tool does it so the tool does it all".
(Philosophy of "kill the messenger").
I hold the complexity to which "human" belongs unseparable in
its functions unless one is a faithful dualist with a soul. Even then:
does the 'soul' think?
John
----- Original Message -----
From: Robert Karl
Stonjek
To: A Group
MindBrain
Sent: Monday, August 02, 2004 5:50 PM
Subject: [Mind and Brain] Article: New Views On Mind-Body
Connection
New Views On Mind-Body
Connection
Studies into placebo effect and empathy
suggest how
the brain encodes subjective experience |
By Eugene
Russo
Courtesy of Fabrizio Benedetti
![bkg_spacer.gif]()
![access.jpg]() ![bkg_spacer.gif]()
![captionarrow.gif]() UNPRECEDENTED
ACCESS: During a deep brain stimulation clinical trial,
researchers
detected elements of the placebo effect. The pre-placebo neuron was
recorded from the left subthalamic nucleus as a control. The
post-placebo
neuron was recorded from the right subthalamic nucleus. Other neurons
demonstrated a similiar decrease in activity.
![200px_dash.gif]()
![bkg_spacer.gif]()
Revealing the complexities of the pain experience may offer a window
into
the mind-body interaction. Several recent studies into the placebo
effect, human empathy, and their apparent interconnectedness are
providing insight into the human subjective experience.
Such investigations, says Jon-Kar Zubieta, associate professor in
psychiatry and radiology at the University of Michigan, help scientists
understand the intersection of physical and emotional states. "The
placebo effect gets at the core of how individuals react and modulate
environmental events, whether positive or negative in nature," he
says. If harnessed, the regulatory mechanisms involved could point to
better treatments for pain, depression, and stress.
In earlier work, University of Turin physiology professor Fabrizio
Benedetti showed that administering an opioid-blocking drug could
reverse
the psychological placebo effect.1 "People started
believing there was something real there," says Columbia University
assistant professor Tor Wager, lead author of a recent placebo effect
study on functional magnetic resonance imaging (fMRI) .
Wager's group took a different tack, uncovering regions of the brain
that
showed decreased activity during the placebo effect.2 In one
trial, they told subjects that they were administering a powerful
analgesic cream. In another, the subjects received the same cream but
were told it has no effect. When subjects were experiencing the placebo
effect, a subset of known pain-sensitive brain regions showed a signal
reduction of 20% to 25%
In a subsequent study, Benedetti's group observed patterns of neuronal
firing, not visible via neuroimaging, that corresponded with Wager's
findings.3 His group performed single-neuron recording in
patients with Parkinson disease who had been administered a sham
treatment.
Read the rest at The Scientist
http://www.the-scientist.com/yr2004/aug/research2_040802.html
Posted by
Robert Karl Stonjek
http://iridia.ulb.ac.be/~marchal/
|
