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 <mailto:stonjek.domain.name.hidden>
>> To: A Group MindBrain <mailto:MindBrain.domain.name.hidden>
>> 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
>> <mailto:erusso.domain.name.hidden>
>>
>>
>>
>>
>> Courtesy of Fabrizio Benedetti
>>
>>
>> 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.
>>
>>
>>
>> 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/ <http://iridia.ulb.ac.be/%7Emarchal/>
>
Received on Sat Aug 28 2004 - 01:47:32 PDT