Interpretive Duality in Logical Graphs • 1
• https://inquiryintoinquiry.com/2024/04/22/interpretive-duality-in-logical-g…
All,
The duality between Entitative and Existential interpretations
of logical graphs is a good example of a mathematical symmetry,
in this case a symmetry of order two. Symmetries of this and
higher orders give us conceptual handles on excess complexity
in the manifold of sensuous impressions, making it well worth
the effort to seek them out and grasp them where we find them.
Both Peirce and Spencer Brown understood the significance of
the mathematical unity underlying the dual interpretation of
logical graphs. Peirce began with the Entitative option and
later switched to the Existential choice while Spencer Brown
exercised the Entitative option in his “Laws of Form”.
In that vein, here's a Rosetta Stone to give us a grounding in
the relationship between boolean functions and our two readings
of logical graphs.
Boolean Functions on Two Variables
• https://inquiryintoinquiry.com/wp-content/uploads/2020/11/boolean-functions…
Regards,
Jon
cc: https://www.academia.edu/community/5REb1n
Operator Variables in Logical Graphs • 1
• https://inquiryintoinquiry.com/2024/04/06/operator-variables-in-logical-gra…
All,
In lieu of a field study requirement for my bachelor's degree I spent
two years in various state and university libraries reading everything
I could find by and about Peirce, poring most memorably through reels
of microfilmed Peirce manuscripts Michigan State had at the time, all
in trying to track down some hint of a clue to a puzzling passage in
Peirce's “Simplest Mathematics”, most acutely coming to a head with
that bizarre line of type at CP 4.306, which the editors of Peirce's
“Collected Papers”, no doubt compromised by the typographer's reluctance
to cut new symbols, transmogrified into a script more cryptic than even
the manuscript's original hieroglyphic.
I found one key to the mystery in Peirce's use of “operator variables”,
which he and his students Christine Ladd–Franklin and O.H. Mitchell
explored in depth. I will shortly discuss that theme as it affects
logical graphs but it may be useful to give a shorter and sweeter
explanation of how the basic idea typically arises in common
logical practice.
Consider De Morgan's rules:
• ¬(A ∧ B) = ¬A ∨ ¬B
• ¬(A ∨ B) = ¬A ∧ ¬B
The common form exhibited by the two rules could be captured in a single
formula by taking “o₁” and “o₂” as variable names ranging over a family
of logical operators, then asking what substitutions for o₁ and o₂ would
satisfy the following equation.
• ¬(A o₁ B) = ¬A o₂ ¬B
We already know two solutions to this “operator equation”, namely,
(o₁, o₂) = (∧, ∨) and (o₁, o₂) = (∨, ∧). Wouldn't it be just
like Peirce to ask if there are others?
Having broached the subject of “logical operator variables”,
I will leave it for now in the same way Peirce himself did:
❝I shall not further enlarge upon this matter at this point,
although the conception mentioned opens a wide field; because
it cannot be set in its proper light without overstepping the
limits of dichotomic mathematics.❞ (Peirce, CP 4.306).
Further exploration of operator variables and operator invariants
treads on grounds traditionally known as second intentional logic
and “opens a wide field”, as Peirce says. For now, however, I will
tend to that corner of the field where our garden variety logical
graphs grow, observing the ways in which operative variations and
operative themes naturally develop on those grounds.
Regards,
Jon
cc: https://www.academia.edu/community/Lxn1Ww
cc: https://mathstodon.xyz/@Inquiry/112225263055943815
Dima,
Yes, they were in the same field as George Miller (psychology). But they also hung out with enough neuroscientists that some of the blood and guts rubbed off on them. Right now, the major research on the topic depends on neuroscience.
That is one among many reasons why I prefer to use the term 'Cognitive Science'. The subject is so complex that collaboration among the different fields is essential.
John
----------------------------------------
From: "Dima, Alden A. (Fed)' via ontolog-forum" <ontolog-forum(a)googlegroups.com>
Hi John,
A certain large language model tells me that Alan Baddeley and Graham Hitch were psychologists and not neuroscientists.
Alden
Doug,
The central executive controls all the processes that are controllable by the human ego. But the term 'executive' should be considered the equivalent of what the chief executive officer (CEO) of a business does in managing a corporation. There are intermediaries at various points.
Baddeley & Hitch wrote their initial article in 1974. They wrote that in response to George Miller's "Magic Number 7, plus or minus 2." They realized that there was much more to short-term memory than just words and phonemes. They called Miller's storage "the phonological loop" and they added a visuo-spatial scratchpad for short-term imagery and feelings. And they continued to revise and extend their hypotheses for another 20 or 30 years. Other neuroscientists, who are specialists in different aspects, have been working on related issues.
The idea is an important one that the Generative AI gang has not yet latched onto. But some AI people are starting to take notice, and I believe that they are on the right track. In summary, there is more to come. See the references I cited, and do whatever googling and searching you like.
John
----------------------------------------
From: "doug foxvog" <doug(a)foxvog.org>
John,
Baddeley & Hitch's "central executive" (CE) is described as an attentional
controlling system. I have just briefly glanced at it, but it seems that
the point is coordinating and accessing memory through an episodic buffer,
phonological loop, and visio-spatial "sketchpad". The hypothesized CE
deals with information, language, memory, imagery, & spatial awareness.
That covers a lot, and i assume it would also cover conscious actions and
processes.
But i don't see it covering neurohormone production or things like
heartrate. Lower level processes like basal signaling between neurons
would have no need of a central executive, as they are just basal
processes.
It's the word "all" in "all processes" that indicates to me that the claim
is excessive.
FWIW, i note that sharks also have brains -- as do "higher" orders of
invertebrates.
-- doug f
> On Wed, April 10, 2024 18:38, John F Sowa wrote:
> Doug,
>
> The central executive was proposed by the neuroscientists Baddeley &
> Hitch, not by AI researchers. There is nothing "machine-like" in the
> idea, by itself. Without something like it, there is no way to explain
> how a huge tangle of neurons could act together and coordinate their
> efforts to support a common effort.
>
> It reminds me of a neighboring town (to my residence in Croton on Hudson,
> NY), which was doing some major developments without hiring a general
> contractor. They thought that their local town employees could schedule
> all the processes. It turned out to be a total disaster. All the
> subcontractors did their tasks in a random order, each one interfering
> with some of the others, and causing a major mess. There were lawsuits
> back and forth, and the town management was found guilty and had losses
> that were many times greater than the cost of hiring a general contractor.
>
> It is certainly true that there is a huge amount of computation going on
> in the brain that is below conscious awareness. Most of that is done by
> the cerebellum (little brain), which is physically much smaller than the
> cerebral cortex. But it contains over four times the number of neurons.
> In effect, the cerebellum behaves like a GPU (Graphics Processing Unit)
> which is a superfast, highly specialized processor for all the perception
> and action that takes place without conscious awareness.
>
> For example, when you're walking down the street talking on your cell
> phone, the cerebellum is monitoring your vision, muscles, and strides --
> until you step off the curb and get run over by a bus. That's why you need
> a central controller to monitor and coordinate all the processes.
>
> Sharks and dolphins are about the same size and they eat the same kind of
> prey. Sharks have a huge cerebellum and a small lump for a cerebellum.
> Dolphins have a huge cerebral cortex and a huge cerebellum. They are as
> agile as sharks, but they can plan, communicate, and coordinate their
> activities. When the food is plentiful, they can both eat their fill.
> But when it's scarce, the dolphins are much more successful.
>
> Please look at the citations in my previous note and the attached
> Section7.pdf. The cycle of abduction, induction, testing, and induction
> depends on a central executive that is responsible for planning,
> coordinating, and integrating those steps of conscious feeling, thinking,
> reasoning, and acting. With a central executive, an AI system would be
> more intelligent. But much, much more R & D would be required before
> anything could be called "Artificial General Intelligence" (AGI). That's
> why I have very little faith in anything called AGI.
>
> John
>
> ----------------------------------------
> From: "doug foxvog" <doug(a)foxvog.org>
> Subject: Re: [ontolog-forum] The central executive
>
> On Wed, April 10, 2024 14:07, John F Sowa wrote:
>> In today's ZOOM meeting, I objected to the term 'neuro-symbolic hybrid'
>> of
>> artificial neural networks (ANNs) with symbols. Hybrids simply relate
>> two
>> (sometimes more) distinctly different things. But all the processes in
>> the mind and brain are integrated, and they all operate continuously in
>> different parts of the brain, which are all monitored and controlled by
>> a
>> central executive. ...
>
> This seems to me to be modeling the body as a machine and not an accurate
> description.
>
> There are a wide variety of processes in the mind and brain -- many
> processes in the brain occur independently without being integrated either
> with each other or with the mind. I am excluding standard cellular level
> processes that go on in every cell and the processes of the circulatory
> system in the brain. Every neuron regularly chemically interacts with
> adjacent neurons & passes electrical signals along its surface.
>
> As far as i understand, much that goes on in the brain we are unaware of,
> neurohormone production, for example. Sensory input processing does not
> seem to be integrated with a number of other processes. I have seen no
> evidence of a central executive in the brain that monitors and controls
> all the other processes. I'm not sure how such a central executive could
> have evolved.
>
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Doug,
The central executive was proposed by the neuroscientists Baddeley & Hitch, not by AI researchers. There is nothing "machine-like" in the idea, by itself. Without something like it, there is no way to explain how a huge tangle of neurons could act together and coordinate their efforts to support a common effort.
It reminds me of a neighboring town (to my residence in Croton on Hudson, NY), which was doing some major developments without hiring a general contractor. They thought that their local town employees could schedule all the processes. It turned out to be a total disaster. All the subcontractors did their tasks in a random order, each one interfering with some of the others, and causing a major mess. There were lawsuits back and forth, and the town management was found guilty and had losses that were many times greater than the cost of hiring a general contractor.
It is certainly true that there is a huge amount of computation going on in the brain that is below conscious awareness. Most of that is done by the cerebellum (little brain), which is physically much smaller than the cerebral cortex. But it contains over four times the number of neurons. In effect, the cerebellum behaves like a GPU (Graphics Processing Unit) which is a superfast, highly specialized processor for all the perception and action that takes place without conscious awareness.
For example, when you're walking down the street talking on your cell phone, the cerebellum is monitoring your vision, muscles, and strides -- until you step off the curb and get run over by a bus. That's why you need a central controller to monitor and coordinate all the processes.
Sharks and dolphins are about the same size and they eat the same kind of prey. Sharks have a huge cerebellum and a small lump for a cerebellum. Dolphins have a huge cerebral cortex and a huge cerebellum. They are as agile as sharks, but they can plan, communicate, and coordinate their activities. When the food is plentiful, they can both eat their fill. But when it's scarce, the dolphins are much more successful.
Please look at the citations in my previous note and the attached Section7.pdf. The cycle of abduction, induction, testing, and induction depends on a central executive that is responsible for planning, coordinating, and integrating those steps of conscious feeling, thinking, reasoning, and acting. With a central executive, an AI system would be more intelligent. But much, much more R & D would be required before anything could be called "Artificial General Intelligence" (AGI). That's why I have very little faith in anything called AGI.
John
----------------------------------------
From: "doug foxvog" <doug(a)foxvog.org>
Subject: Re: [ontolog-forum] The central executive
On Wed, April 10, 2024 14:07, John F Sowa wrote:
> In today's ZOOM meeting, I objected to the term 'neuro-symbolic hybrid' of
> artificial neural networks (ANNs) with symbols. Hybrids simply relate two
> (sometimes more) distinctly different things. But all the processes in
> the mind and brain are integrated, and they all operate continuously in
> different parts of the brain, which are all monitored and controlled by a
> central executive. ...
This seems to me to be modeling the body as a machine and not an accurate
description.
There are a wide variety of processes in the mind and brain -- many
processes in the brain occur independently without being integrated either
with each other or with the mind. I am excluding standard cellular level
processes that go on in every cell and the processes of the circulatory
system in the brain. Every neuron regularly chemically interacts with
adjacent neurons & passes electrical signals along its surface.
As far as i understand, much that goes on in the brain we are unaware of,
neurohormone production, for example. Sensory input processing does not
seem to be integrated with a number of other processes. I have seen no
evidence of a central executive in the brain that monitors and controls
all the other processes. I'm not sure how such a central executive could
have evolved.
In today's ZOOM meeting, I objected to the term 'neuro-symbolic hybrid' of artificial neural networks (ANNs) with symbols. Hybrid's imply two (sometimes more) distinctly different things. But all the processes in the mind and brain are integrated, and they all operate continuously in different parts of the brain, which are all monitored and controlled by a central executive. For AI, integration is the goal, and a hybrid stage is something that needs to be replaced with a tighter integration. I believe that the final document should emphasize the dangers that Gary Marcus and I discussed in March.
And for that matter, artificial neural networks are not new. William James suggested the telephone network as a model, and more detailed mathematical models were developed in the 1940s. In fact, Marvin Minsky, one of the founders of AI, wrote his PhD thesis at Princeton on a mathematical model of neural networks in the early 1950s.
Research in the cognitive sciences involves a collaboration of all the sciences that study any and every aspect of cognition: philosophy, psychology, logic, artificial intelligence, neuroscience, and anthropology. As an overview of the methods of integration, I attach a copy (Section 7) of an article that is in press: Phaneroscopy: The Science of Diagrams.
In that section, I show how the theories of C. S. Peirce and recent developments in the cognitive sciences support, illustrate, and explain the issues. In particular, they go far beyond just a hybrid of two approaches. They treat the brain and the mind it supports as an integrated system. The key to integration is a central executive, located in the frontal lobes that relates and controls every component in the cerebral cortex, the cerebellum, and the brain stem.
Note the loop with a photo of Peirce standing in the center. It shows how the four steps of abduction, deduction, observation, and induction work together. Every iteration -- from milliseconds to hours to days -- involves guessing, reasoning, observing, and learning. These processes are not separated. They operate continuously.
For details, see Section 7, which also contains several references to articles and slides with more detail. I also recommend The Central Executive Network (CEN): https://www.o8t.com/blog/central-executive-network#:~:text=Since%20its%20in…
John
Peirce's 1885 “Algebra of Logic” • Selection 1.1
• https://inquiryintoinquiry.com/2024/03/24/peirces-1885-algebra-of-logic-sel…
All,
I'm laying down a few source materials
in preparation for a later discussion.
Selection from C.S. Peirce, “On the Algebra of Logic :
A Contribution to the Philosophy of Notation” (1885)
❝§1. Three Kinds Of Signs❞
❝Any character or proposition either concerns one subject,
two subjects, or a plurality of subjects. For example, one
particle has mass, two particles attract one another, a particle
revolves about the line joining two others. A fact concerning
two subjects is a dual character or relation; but a relation
which is a mere combination of two independent facts concerning
the two subjects may be called “degenerate”, just as two lines
are called a degenerate conic. In like manner a plural character
or conjoint relation is to be called degenerate if it is a mere
compound of dual characters.
❝A sign is in a conjoint relation to the thing denoted and to the mind.
If this triple relation is not of a degenerate species, the sign is
related to its object only in consequence of a mental association,
and depends upon a habit. Such signs are always abstract and general,
because habits are general rules to which the organism has become
subjected. They are, for the most part, conventional or arbitrary.
They include all general words, the main body of speech, and any
mode of conveying a judgment. For the sake of brevity I will call
them “tokens”. [Note. Peirce more frequently calls these “symbols”.]
Regards,
Jon
cc: https://www.academia.edu/community/LpeZP7
cc: https://mathstodon.xyz/@Inquiry/112156450035935700
