Survey of Animated Logical Graphs
Jon Awbrey
https://inquiryintoinquiry.com/2019/05/22/survey-of-animated-logical-graphs-%e2%80%a2-2/
The blog post linked above updates my Survey of Resources having to do with
Animated Logical Graphs. There you will find links to basic expositions and
extended discussions of the graphs themselves, deriving from the Alpha Graphs
C.S. Peirce used for propositional logic, more recently revived and augmented
by G. Spencer Brown in his Laws of Form. What I added was the extension from
tree-like forms to what graph theorists know as cacti, and thereby hangs many
a tale yet to be told. I hope to add more proof animations as time goes on.
Regards,
Jon
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joseph simpson
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“Reasonable people adapt themselves to the world.
Unreasonable people attempt to adapt the world to themselves.
All progress, therefore, depends on unreasonable people.”
- George Bernard Shaw
- Git Hub link:
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Jon Awbrey
At: https://inquiryintoinquiry.com/2019/05/22/survey-of-animated-logical-graphs-%e2%80%a2-2/
Joe, All ???
One of the things I added to the Survey this time around was an
earlier piece of work titled ???Futures Of Logical Graphs??? (FOLG),
which takes up a number of difficult issues in more detail than
I've found the ability or audacity to do since. In particular,
it gives an indication of the steps I took from trees to cacti
in the graph-theoretic representation of logical propositions
and boolean functions, along with the factors that forced me
to make that transition.
??? https://oeis.org/wiki/Futures_Of_Logical_Graphs
A lot of the text goes back to the dusty Ascii days of the old discussion lists
where I last shared it, so I'll be working on converting the figures and tables
and trying to make the presentation more understandable.
Regards,
Jon
joseph simpson
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Jon Awbrey
At: https://inquiryintoinquiry.com/2019/06/30/animated-logical-graphs-%e2%80%a2-15/
In George Spencer Brown's "Laws of Form" the relation between the primary arithmetic
and the primary algebra is founded on the idea that a variable name appearing as an
operand in an algebraic expression indicates the contemplated absence or presence of
any expression in the arithmetic, with the understanding that each appearance of the
same variable name indicates the same state of contemplation with respect to the same
expression of the arithmetic.
For example, consider the following expression:
Figure 1. Cactus Graph (a(a))
https://inquiryintoinquiry.files.wordpress.com/2019/06/box-aa.jpg
We may regard this algebraic expression as a general expression
for an infinite set of arithmetic expressions, starting like so:
Figure 2. Cactus Graph Series (a(a))
https://inquiryintoinquiry.files.wordpress.com/2019/06/box-aa-series.jpg
Now consider what this says about the following algebraic law:
Figure 3. Cactus Graph Equation (a(a)) = <blank>
https://inquiryintoinquiry.files.wordpress.com/2019/06/box-aa-.jpg
It permits us to understand the algebraic law as saying, in effect, that every one
of the arithmetic expressions of the contemplated pattern evaluates to the very same
canonical expression as the upshot of that evaluation. This is, as far as I know,
just about as close as we can come to a conceptually and ontologically minimal way
of understanding the relation between an algebra and its corresponding arithmetic.
To be continued ...
Regards,
Jon
On 5/30/2019 9:16 AM, Jon Awbrey wrote:
> Re: Survey of Animated Logical Graphs : 2
> At: https://inquiryintoinquiry.com/2019/05/22/survey-of-animated-logical-graphs-%e2%80%a2-2/
> One of the things I added to the Survey this time around was an
> in the graph-theoretic representation of logical propositions
> and boolean functions, along with the factors that forced me
> to make that transition.
>
> A lot of the text goes back to the dusty Ascii days of the
> to make the presentation more understandable.
>
> Regards,
>
> Jon
Ravi Sharma
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Ravi
(Dr. Ravi Sharma)
Jon Awbrey
At: https://inquiryintoinquiry.com/2019/07/07/animated-logical-graphs-%e2%80%a2-16/
In lieu of a field study requirement for my bachelor's degree I spent
a couple years in a host of state and university libraries reading
everything I could find by and about Peirce, poring most memorably
through the 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 the ''Collected Papers'', no doubt compromised by the
typographer's resistance to cutting 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 this 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.
Think of De Morgan???s rules:
: ??(A ??? B) = ??A ??? ??B
: ??(A ??? B) = ??A ??? ??B
We could capture the common form of these two rules in a single formula
by taking "o1" and "o2" as variable names ranging over a set of logical
operators, and then by 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:
<QUOTE>
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. (Collected Papers, CP 4.306).
</QUOTE>
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 operative variations and operative
themes naturally develop on those grounds.
Jon Awbrey
| everything got so iffy with unicodes a couple months ago,
| and what to do about it. In the meantime here's plaintext
| version of that last post. As always, there's a better
| formatted version at the blog post linked below.
Cf: Animated Logical Graphs : 16
In lieu of a field study requirement for my bachelor's degree I spent
a couple years in a host of state and university libraries reading
everything I could find by and about Peirce, poring most memorably
through the 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 the ''Collected Papers'', no doubt compromised by the
typographer's resistance to cutting new symbols, transmogrified into
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 this 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.
: not (A and B) = (not A) or (not B)
: not (A or B) = (not A) and (not B)
We could capture the common form of these two rules in a single formula
by taking "o1" and "o2" as variable names ranging over a set of logical
operators, and then by asking what substitutions for o1 and o2 would
satisfy the following equation:
We already know two solutions to this "operator equation", namely,
Having broached the subject of logical operator variables,
I will leave it for now in the same way Peirce himself did:
<QUOTE>
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. (Collected Papers, CP 4.306).
</QUOTE>
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 operative variations and operative
themes naturally develop on those grounds.
joseph simpson
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Jon Awbrey
This material is coming from the section of my "Futures Of Logical Graphs" (FOLG)
titled "Themes and Variations" where I explain how I came down from logical trees
and learned to love logical cacti (ouch) --
https://oeis.org/wiki/Futures_Of_Logical_Graphs#Themes_and_variations
I spent the last month upgrading the ancient ascii graphics to jpegs and the text
will hopefully get less rambling and clearer as I serialize it to my inquiry blog.
Previous Installments:
Survey of Animated Logical Graphs
https://inquiryintoinquiry.com/2019/05/22/survey-of-animated-logical-graphs-%e2%80%a2-2/
(13) https://inquiryintoinquiry.com/2019/05/24/animated-logical-graphs-%e2%80%a2-13/
(14) https://inquiryintoinquiry.com/2019/05/28/animated-logical-graphs-%e2%80%a2-14/
(15) https://inquiryintoinquiry.com/2019/06/30/animated-logical-graphs-%e2%80%a2-15/
(16) https://inquiryintoinquiry.com/2019/07/07/animated-logical-graphs-%e2%80%a2-16/
Regards,
Jon
joseph simpson
Jon Awbrey
At: https://inquiryintoinquiry.com/2019/07/09/animated-logical-graphs-%e2%80%a2-17/
To get a clearer view of the relation between
primary arithmetic and primary algebra consider
the following extremely simple algebraic expression:
Figure 4. Cactus Graph (a)
https://inquiryintoinquiry.files.wordpress.com/2019/07/box-a.jpg
In this expression the variable name "a" appears as an "operand name".
In functional terms, "a" is called an "argument name", but it's best
to avoid the potentially confusing connotations of the word "argument"
here, since it also refers in logical discussions to a more or less
specific pattern of reasoning.
As we've discussed, the algebraic variable name indicates the
contemplated absence or presence of any arithmetic expression
taking its place in the surrounding template, which expression
is proxied well enough by its logical value, and of which values
we know but two. Thus, the given algebraic expression varies
between these two choices:
Figure 5. Cactus Graph Set (),(())
https://inquiryintoinquiry.files.wordpress.com/2019/07/box-.jpg
The above selection of arithmetic expressions is what it means
to contemplate the absence or presence of the operand "a" in
the algebraic expression "(a)". But what would it mean to
contemplate the absence or presence of the operator "( )"
in the algebraic expression "(a)"?
That is the question I'll take up next.
Regards,
Jon
Jon Awbrey
At: https://inquiryintoinquiry.com/2019/07/10/animated-logical-graphs-%e2%80%a2-18/
We had been contemplating the penultimately simple
algebraic expression "(a)" as a name for a set of
arithmetic expressions, namely, (a) = { () , (()) },
taking the equality sign in the appropriate sense.
Figure 6. Cactus Graph Equation (a) = {(),(())}
https://inquiryintoinquiry.files.wordpress.com/2019/07/box-a-1.jpg
Then we asked the corresponding question about the operator "( )".
to contemplate the absence or presence of the operand "a" in
the algebraic expression "(a)". But what would it mean to
contemplate the absence or presence of the operator "( )"
in the algebraic expression "(a)"?
the operator "( )" in the algebraic expression "(a)" refers to
a variation between the algebraic expressions "a" and "(a)",
respectively, somewhat as pictured below:
Figure 7. Cactus Graph Equation ?a? = {a,(a)}
https://inquiryintoinquiry.files.wordpress.com/2019/07/box-a-queaa.jpg
But how shall we signify such variations in a coherent calculus?
(end of season cliff-hanger ...)
Regards,
Jon
Jon Awbrey
formal calculus to take account of operator variables.
In the days when I scribbled these things on the backs of computer punchcards,
the first thing I tried was drawing big loopy script characters, placing some
inside the loops of others. Lower case alphas, betas, gammas, deltas, and
so on worked best. Graphics like these conveyed the idea that a character-
shaped boundary drawn around another space can be viewed as absent or present
depending on whether the formal value of the character is unmarked or marked.
The same idea can be conveyed by attaching characters directly to the edges
of graphs.
Here is how we might suggest an algebraic expression of the form "(q)"
where the absence or presence of the operator "( )" depends on the value
of the algebraic expression "p", the operator "( )" being absent whenever
p is unmarked and present whenever p is marked.
Figure 8. Cactus Graph (q)_p = {q,(q)}
https://inquiryintoinquiry.files.wordpress.com/2019/07/box-q-que-pqq.jpg
It was obvious to me from the outset that this sort of tactic would need
a lot of work to become a usable calculus, especially when it came time
to feed those punchcards back into the computer.
Regards,
Jon
Jon Awbrey
19: https://inquiryintoinquiry.com/2019/07/11/animated-logical-graphs-%e2%80%a2-19/
20: https://inquiryintoinquiry.com/2019/07/11/animated-logical-graphs-%e2%80%a2-20/
Another tactic I tried by way of porting operator variables into logical graphs and
laws of form was to hollow out a leg of Spencer-Brown's crosses, gnomons, markers,
whatever you wish to call them, as shown below:
Figure 9. Transitional (q)_p = {q,(q)}
https://inquiryintoinquiry.files.wordpress.com/2019/07/box-q-qua-p.jpg
The initial idea I had in mind was the same as before, that the operator
over q would be counted as absent when p evaluates to a space and present
when p evaluates to a cross.
However, much in the same way that operators with a shade of negativity
tend to be more generative than the purely positive brand, it turned out
more useful to reverse this initial polarity of operation, letting the
operator over q be counted as absent when p evaluates to a cross and
present when p evaluates to a space.
So that is the convention I'll adopt from here on.
Regards,
Jon
joseph simpson
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Jon Awbrey
At: https://inquiryintoinquiry.com/2019/07/12/animated-logical-graphs-%e2%80%a2-21/
A funny thing just happened. Let's see if we can tell where.
We started with the algebraic expression "(a)", in which the
operand "a" suggests the contemplated absence or presence of
any arithmetic expression or its value, then we contemplated
the absence or presence of the operator "( )" in "(a)" to be
indicated by a cross or a space, respectively, for the value
of a newly introduced variable, "b", placed in a new slot of
a newly extended operator form, as suggested by this picture:
Figure 10. Control Form (a)_b
https://inquiryintoinquiry.files.wordpress.com/2019/07/box-a-quo-b.jpg
What happened here is this. Our contemplation of an operator variable
just as quickly transformed into the contemplation of a newly introduced
but otherwise quite ordinary operand variable, fitting into a new form of
formula. In its interpretation for logic the newly formed operation may be
viewed as an extension of ordinary negation, one in which the negation of the
first variable is "controlled" by the value of the second variable. Thus, we
may regard this development as marking a form of "controlled reflection", or a
form of "reflective control". From this point on we will use the inline syntax
"(a , b)" for the associated operation on two variables, whose operation table
is given below:
Operation Table for (a , b)
https://inquiryintoinquiry.files.wordpress.com/2019/07/table-ab-space-cross.png
: The Entitative Interpretation (En), for which Space = False and Cross = True,
calls this operation "equivalence".
: The Existential Interpretation (Ex), for which Space = True and Cross = False,
calls this operation "distinction".
Regards,
Jon
joseph simpson
Each step on its own, as far as I can follow them, makes sense. You are, if I understand it correctly, trying to figure out something fundamental, the rock bottom reality. When can we expect that results of such a research to become "applicable to more than one of the traditional departments of knowledge" (http://isss.org/world/about-the-isss)? What kinds of tragedy, disaster, misunderstanding, mismanagement, or failure would/will be preventable by your approach?Aleksandar
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bruces...@cox.net
Thanks for these comments, Joe – thanks to Jon for exploring the connection with Laws of Form. Thanks for linking to a few other conversations.
I just want to mention that G. Spencer Brown and LOF had a significant influence on me as well. This book was popular in the late 1960’s or early 1970’s, and it was very up my line. I was exploring topological approaches to logic, and Brown’s book was very striking and influential.
I did not really understand his graphic symbolism – the notion of “crossing”, etc. – and I think a lot of people were mystified by it, but yet intrigued. It just so happened that in 1973, I met a couple of times with the famous anthropologist/epistemologist Gregory Bateson (author of the very influential Steps to an Ecology of Mind, which introduced the theme of “The Pattern That Connects” into the new-thought conversation). He was in residence at UC Santa Cruz at the time, and one of the things we talked about was Laws of Form. Like me, Bateson was intrigued – but asked me if I thought the book was a trick, a mystifying joke being played on a gullible public. He found some citation in the index that he thought was a clue. https://link.springer.com/chapter/10.1007/978-1-4020-6706-8_14
But for me, this question was just a distraction. The idea that stuck with me – and shaped everything I’ve been doing since, including a lot of messages posted to Ontolog – has to do with the concept of “distinction”, which LOF first put in my head.
“The first command – draw a distinction”
That’s something I’ve been feeling for a long time.
In what I am doing right now, this idea is at the core –
“Draw a distinction in a distinction”
There’s a lot we can say about this.
In Dr. Susan Carey’s “The Origin of Concepts”, she talks about “Quinian Bootstrapping” (W.V.O. Quine) – which I think relates to this kind of mysterious coalescence of concept and form from a mysterious figure/ground tension. She is talking about concept formation in children. Something organic drives an emerging fuzzy notion that becomes explicitly codified.
What I want to suggest is that there is a organic drive dynamic that pushes conceptual form out of the continuum (Tao, real number line, unit interval, etc.) , under the force of some kind of “local” motivation. A distinction gets drawn in a distinction. Maybe this relates to Helen Keller’s powerful experience with the concept/word W-A-T-E-R as drawn on her hand….
In any case, thank you.
It feels to me like there is a powerful transcendental theorem in the air right now. We are figuring this out. Something wants to explode through the keyhole.
For those who want to decode this mystery, there’s a lot to consider right here in this excerpt from Wikipedia.
https://en.wikipedia.org/wiki/Laws_of_Form
Bruce Schuman
Santa Barbara CA USA
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bruces...@cox.net
Thanks Joe.
I am continuing to bump along on my project for “integral ontology”, and I now have a growing list of sources that includes LOF – along with a bunch of other things, including Conceptual Structures by JS.
Another thing I have done is develop a way to build a list or system of “principles” – guiding design ideas, etc. – which I start off with a series of quotes from JS, that were (and are) very influential on me. I have a way to comment on each quote or principle individually – and something like this could be opened to group dialogue. What are we trying to do, what are the strongest insights for helping us do it?
On LOF – this is some deep stuff, a little bit hard to figure, but I can feel my way into it in ways that seem significant.
There are a few areas or “themes” that seem relevant:
- Where do concepts come from (how do they emerge from nowhere?)
- How do abstract notions become codified into symbolic structures (symbolic structures that then become defined as “states in a medium”, the content of database cells, etc.)
- How does this subject relate to the continuum and “the foundations of mathematics”?
- What if anything does this have to do with the “figure/ground” relationship?
So – you write:
Joe S
> "We now see that the first distinction, the mark, and the observer are not only interchangeable, but, in the form, identical."
This is trippy, maybe confusing, maybe confused – but I would say it’s a sincere hard push towards something deep. Like maybe think about a “holon” – a “two-sided” (“Janus-faced”) concept, that appears as both a part or a whole, depending on how it is viewed – i.e., whether Janus is “looking UP the hierarchy” of part/whole relationships, so that part appears as a whole – or DOWN the hierarchy from the point of view of the whole, so “the same thing” now appears as “a part”. Think of an automobile. Is a carburetor a part or a whole? It’s both.
So I think this goes to the mystery of concept formation – which I would say GSB is try to demystify.
Joe
> The statement above, to me, indicates that the formal language representations associated with the "Laws Of Form," are highly restricted.
I am not sure exactly what you are saying, Joe –“highly restricted” – how? They are restricted to a logic defined only within LOF?
But I do absolutely agree that the wording is confusing. “The mark and the observer of the mark are 1) interchangeable, and 2) identical.”
Huh?
But lets give the guy some space. “The observer is one with what he (she) observes” That makes sense – with a charitable reading.
> No one would say that a mark on a sheet of paper is an observer, in real life.
Yes. So that is confusing – maybe (?) just badly conceptualized. But I sense that there is a mysterious “emergent” kind of process, where we are kind of feeling our way into something – where we are “kind of” making a distinction – but what IS that, what is that distinction made IN – and HOW is it made? BY what? For what reason? How do we codify it, or remember it?
I just purchased the book The Origin of Concepts, by Susan Carey, a child psychology professor at Harvard, where she talks about “bootstrapping” at the foundation of concept formation. I think we are seeing a kind of bootstrapping, and LOF might (?) be a very early precursor to this idea.
Something is making a distinction in something
All very blurry and very primal.
> The challenge is to find a suitable natural language that properly applies the Laws Of Form to a range of real situations.
Or maybe a “suitable interpretation in natural language” that can model/describe what LOF is talking about…..
I’d say that is the route I am taking.
> The space, state or contents associated with the distinction is a key consideration in the proper application of the Laws Of Form.
And what I probably want to do – is to define the “dimensionality” of all these intersecting elements. “The distinction intersects something and makes a distinction in it”.
In broader terms, talking taxonomy – we might say that “something makes a distinction in a genus, and that distinction forms a species”.
> The relationships associated with the real space, state or contents are not as restricted as the relationships associated with the Laws Of Form.
“the real space, state or content” -- i.e. the reality in which we are making a distinction
If it is a “state” – I would way we are making a distinction in an abstraction – since “state” is an abstract concept, maybe a variable or the values of a variable
“Content” might be bounded in some way – like “the content of a matrix cell” – but with no internal differentiation. It’s just a unit, a whole.
But if we seem to detect some variation in units that are supposed to be “identical” – we might start analyzing that difference, and trying to “draw a distinction”
How does that happen? Did we really see a “difference” between these two “identical units”? How can we conceptualize that.
Get out the surgical knife, make a cut, draw a distinction…. And then give it a name and call it something – or if you cut one thing into two things, name them both, and figure out why they are different and how
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On related subject –I really want to ground all this stuff in measurement.
I think these themes are very related. Are “distinctions orthogonal to the thing they are making a distinction in?”
(“are species orthogonal to their genus”?)
I like this carpenter square. It looks a lot like GSB’s primary mark. It’s a lot of distinctions in one place.
Something like this, I think – is how we bring this stuff down to earth. How we make it real. How we make it matter.
Thanks.
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joseph simpson
Joe S
> "We now see that the first distinction, the mark, and the observer are not only interchangeable, but, in the form, identical."
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What I wrote was a quote from Laws of Form.
The "form" is an abstract representation of a specific distinction made by the "scribe" (the agent that made the mark.)
The scribe could be the observer or the observer could be a different agent.
The utility of the Laws of Form, in my opinion, is directly related to the value generated by encoding a distinction into a collection of marks.
What are the specific benefits of encoding distinctions using marks and the Laws of Form?
The answer to this question is what guides my interest in this topic.