Survey of Animated Logical Graphs

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Jon Awbrey

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May 23, 2019, 10:54:13 AM5/23/19
to Ontolog Forum, Structural Modeling, SysSciWG, Cybernetic Communications, Laws Of Form Group
Survey of Animated Logical Graphs
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

inquiry into inquiry: https://inquiryintoinquiry.com/
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Jon Awbrey

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May 30, 2019, 9:16:16 AM5/30/19
to Cybernetic Communications, Laws Of Form Group
Re: Survey of Animated Logical Graphs : 2
At: https://inquiryintoinquiry.com/2019/05/22/survey-of-animated-logical-graphs-%e2%80%a2-2/

Dear CybCom, LoF Groups,

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.

See: 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 will be
working on converting the figures and tables and trying
to make the presentation more understandable.

Jon Awbrey

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Jul 1, 2019, 1:20:15 PM7/1/19
to Cybernetic Communications, Laws Of Form Group, SysSciWG, Structural Modeling, Ontolog Forum
Cf: Animated Logical Graphs : 15
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
> 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.
>
> See: 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 will be
> working on converting the figures and tables and trying
> to make the presentation more understandable.
>
> Regards,
>
> Jon

--
Box (A(A)).jpg
Box (A(A)) Series.jpg
Box (A(A))= .jpg

Jon Awbrey

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Jul 8, 2019, 5:18:35 PM7/8/19
to Ontolog Forum, Structural Modeling, SysSciWG, Cybernetic Communications, Laws Of Form Group
| I'm still waiting to hear from the T-bird Hive Mind why
| 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
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:

: 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:

: not (A o1 B) = (not A) o2 (not B)

We already know two solutions to this "operator equation", namely,
(o1, o2) = (and, or) and (o1, o2) = (or, and). 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

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Jul 9, 2019, 9:10:57 AM7/9/19
to Cybernetic Communications, Laws Of Form Group
Dear CybCom and LoF groups,

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:
Animated Logical Graphs
(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

Jon Awbrey

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Jul 10, 2019, 10:10:09 AM7/10/19
to Cybernetic Communications, Laws Of Form Group
Cf: Animated Logical Graphs : 17
At: https://inquiryintoinquiry.com/2019/07/09/animated-logical-graphs-%e2%80%a2-17/

Dear CybCom and LoF groups,

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
Box (A).jpg
Box (),(()).jpg

Jon Awbrey

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Jul 11, 2019, 7:52:18 AM7/11/19
to Cybernetic Communications, Laws Of Form Group
Cf: Animated Logical Graphs : 18
At: https://inquiryintoinquiry.com/2019/07/10/animated-logical-graphs-%e2%80%a2-18/

Dear CybCom and LoF groups,

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 "( )".
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)"?

Clearly, a variation between the absence and the presence of
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?
Box (A)={(),(())}.jpg
Box A que={A,(A)}.jpg

Jon Awbrey

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Jul 11, 2019, 10:48:54 AM7/11/19
to Ontolog Forum, Structural Modeling, SysSciWG, Cybernetic Communications, Laws Of Form Group
We have encountered the question of how to extend our
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
Box Q que P={Q,(Q)}.jpg

Jon Awbrey

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Jul 11, 2019, 5:30:09 PM7/11/19
to Ontolog Forum, Structural Modeling, SysSciWG, Cybernetic Communications, Laws Of Form Group
Cf: Animated Logical Graphs
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
Box Q qua P.jpg

Jon Awbrey

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Jul 12, 2019, 2:00:30 PM7/12/19
to Ontolog Forum, Structural Modeling, SysSciWG, Cybernetic Communications, Laws Of Form Group
Cf: Animated Logical Graphs : 21
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
Box A quo B.jpg
Table (A,B) Space Cross.png

Jon Awbrey

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Jul 22, 2019, 8:12:16 PM7/22/19
to Ontolog Forum, Structural Modeling, SysSciWG, Cybernetic Communications, Laws Of Form Group
Cf: Animated Logical Graphs : 22
At: https://inquiryintoinquiry.com/2019/07/22/animated-logical-graphs-%e2%80%a2-22/

The step of controlled reflection we just took can be iterated as far
as we wish to take it, as suggested by the following series of forms:

Figure 11. Reflective Series (a) to (a, b, c, d)
https://inquiryintoinquiry.files.wordpress.com/2019/07/reflective-series-a-to-abcd.jpg

Written inline, we have the series "(a)", "(a, b)", "(a, b, c)", "(a, b, c, d)",
and so on, whose general form is "(x_1, x_2, ..., x_k)". With this move we have
passed beyond the graph-theoretical form of rooted trees to what graph theorists
know as "rooted cacti".

I will discuss this "cactus language" and its logical interpretations next.
Reflective Series (A) to (A,B,C,D).jpg

Jon Awbrey

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Jul 24, 2019, 2:09:01 PM7/24/19
to Ontolog Forum, Structural Modeling, SysSciWG, Cybernetic Communications, Laws Of Form Group
Cf: Animated Logical Graphs : 23
At: https://inquiryintoinquiry.com/2019/07/23/animated-logical-graphs-%e2%80%a2-23/

The following Table will suffice to show how the "streamer-cross" forms
C.S. Peirce used in his essay on "Qualitative Logic" and Spencer Brown
used in his "Laws of Form", as they get extended by successive steps
of controlled reflection, translate into syntactic strings and
rooted cactus graphs:

Table. Syntactic Correspondences
https://inquiryintoinquiry.com/syntactic-correspondences/
Syntactic Correspondences.jpg

Jon Awbrey

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Jul 26, 2019, 11:45:56 AM7/26/19
to Ontolog Forum, Structural Modeling, SysSciWG, Cybernetic Communications, Laws Of Form Group
Cf: Animated Logical Graphs : 24
At: https://inquiryintoinquiry.com/2019/07/26/animated-logical-graphs-%e2%80%a2-24/

Re: Joseph Simpson
At: https://groups.google.com/d/msg/ontolog-forum/wF03K5KG1vQ/ZRlQdavJBgAJ

Boolean functions f : B^k -> B and different ways of contemplating their
complexity are definitely the right ballpark, or at least the right planet,
for field-testing logical graphs.

I don't know much about the Boolean Sensitivity Conjecture but I did run across
an enlightening article about it just yesterday and I did once begin an exploration
of what appears to be a related question, Peter Frankl's "Union-Closed Sets Conjecture".
See the resource pages linked below.

At any rate, now that we've entered the ballpark, or standard orbit, of boolean functions,
I can skip a bit of dancing around and jump to the next blog post I have on deck.

Resources
=========

* Frankl Conjecture
( https://inquiryintoinquiry.com/category/frankl-conjecture/ )

* R.J. Lipton and K.W. Regan : Discrepancy Games and Sensitivity
( https://rjlipton.wordpress.com/2019/07/25/discrepancy-games-and-sensitivity/ )

Jon Awbrey

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Jul 27, 2019, 10:20:32 AM7/27/19
to Ontolog Forum, Structural Modeling, SysSciWG, Cybernetic Communications, Laws Of Form Group
Cf: Animated Logical Graphs : 25
At: https://inquiryintoinquiry.com/2019/07/26/animated-logical-graphs-%e2%80%a2-25/

Let's examine the formal operation table for the third in our series
of reflective forms to see if we can elicit the general pattern:

Formal Operation Table (a, b, c) Variant 1
https://inquiryintoinquiry.com/formal-operation-table-a-b-c-variant-1/

Or, thinking in terms of the corresponding cactus graphs,
writing "o" for a blank node and "|" for a terminal edge,
we get the following Table:

Formal Operation Table (a, b, c) Variant 2
https://inquiryintoinquiry.com/formal-operation-table-a-b-c-variant-2/

Evidently, the rule is that "(a, b, c)" denotes the value denoted by "o"
if and only if exactly one of the variables a, b, c has the value denoted
by "|", otherwise "(a, b, c)" denotes the value denoted by "|". Examining
the whole series of reflective forms shows this is the general rule.

* In the Entitative Interpretation (En), where o = false and | = true,
"(x_1, ..., x_k)" translates as "not just one of the x_j is true".

* In the Existential Interpretation (Ex), where o = true and | = false,
"(x_1, ..., x_k)" translates as "just one of the x_j is not true".

Regards,

Jon
Formal Operation Table (a, b, c) Variant 1.png
Formal Operation Table (a, b, c) Variant 2.png

Jon Awbrey

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Jul 29, 2019, 3:24:18 PM7/29/19
to Ontolog Forum, Structural Modeling, SysSciWG, Cybernetic Communications, Laws Of Form Group
Cf: Animated Logical Graphs : 26
At: https://inquiryintoinquiry.com/2019/07/28/animated-logical-graphs-%e2%80%a2-26/

This post and the next wrap up the Themes and Variations section
of my speculation on Futures Of Logical Graphs. I made an effort
to "show my work", reviewing the steps I took to arrive at the
present perspective on logical graphs, whistling past the least
productive of the blind alleys, cul-de-sacs, detours, and forking
paths I explored along the way. It can be useful to tell the story
that way, partly because others may find things I missed down those
roads, but it does call for a recap of the main ideas I would like
readers to take away.

Partly through my reflections on Peirce's use of operator variables I was led
to what I called the "reflective extension of logical graphs", or what I now
call the "cactus language", after its principal graph-theoretic data structure.
This graphical formal language arises from generalizing the negation operator "( )"
in a particular direction, treating "( )" as the "controlled", "moderated", or
"reflective" negation operator of order 1, and adding another operator for each
integer parameter greater than 1. This family of operators is symbolized by
bracketed argument lists of the forms "( )", "( , )", "( , , )", and so on,
where the number of places is the order of the reflective negation operator
in question.

Two rules suffice for evaluating cactus graphs:

* The rule for evaluating a k-node operator,
corresponding to an expression of the form
"x_1 x_2 ... x_{k-1} x_k", is as follows:

Figure 12. Node Evaluation Rule
https://inquiryintoinquiry.files.wordpress.com/2019/07/box-xj-node-evaluation-rule.jpg

* The rule for evaluating a k-lobe operator,
corresponding to an expression of the form
"(x_1, x_2, ..., x_{k-1}, x_k)", is as follows:

Figure 13. Lobe Evaluation Rule
https://inquiryintoinquiry.files.wordpress.com/2019/07/box-xj-lobe-evaluation-rule.jpg

References
==========

Futures Of Logical Graphs
https://oeis.org/wiki/Futures_Of_Logical_Graphs

Themes and Variations
https://oeis.org/wiki/Futures_Of_Logical_Graphs#Themes_and_variations

Regards,

Jon

Box Xj Node Evaluation Rule.jpg
Box Xj Lobe Evaluation Rule.jpg

Jon Awbrey

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Aug 1, 2019, 10:48:17 AM8/1/19
to Ontolog Forum, Structural Modeling, SysSciWG, Cybernetic Communications, Laws Of Form Group
Cf: Animated Logical Graphs : 27
At: https://inquiryintoinquiry.com/2019/07/31/animated-logical-graphs-%e2%80%a2-27/

The rules given in the previous post for evaluating cactus graphs were
given in purely formal terms, that is, by referring to the mathematical
forms of cacti without mentioning their potential for logical meaning.
As it turns out, two ways of mapping cactus graphs to logical meanings
are commonly found in practice. These two mappings of mathematical
structure to logical meaning are formally dual to each other and known
as the Entitative and Existential interpretations respectively. The
following Table compares the entitative and existential interpretations
of the primary cactus structures, from which the rest of their semantics
can be derived.

Table. Logical Interpretations of Cactus Structures
https://inquiryintoinquiry.files.wordpress.com/2019/07/logical-interpretations-of-cactus-structures-en-ex.jpg

This concludes the "New Cacti for Old Trees" episode
of my prospectus on Futures Of Logical Graphs. I've
got to take care of some procrastinations about home
and garden and then try to catch up with the comments
and questions that have accumulated in the meantime.

Regards,

Jon
Logical Interpretations of Cactus Structures En Ex.jpg

Jon Awbrey

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Aug 3, 2019, 11:15:42 AM8/3/19
to Ontolog Forum, Structural Modeling, SysSciWG, Cybernetic Communications, Laws Of Form Group
Cf: Animated Logical Graphs : 28
At: https://inquiryintoinquiry.com/2019/08/03/animated-logical-graphs-%e2%80%a2-28/

I will have to focus on other business for a couple of weeks ?
so just by way of reminding myself what we were talking about
at this juncture where logical graphs and differential logic
intersect, here's my comment on R.J. Lipton and K.W. Regan's
blog post about Discrepancy Games and Sensitivity.

At: https://rjlipton.wordpress.com/2019/07/25/discrepancy-games-and-sensitivity/

<QUOTE>
Just by way of a general observation, concepts like discrepancy,
influence, sensitivity, etc. are differential in character, so
I tend to think the proper grounds for approaching them more
systematically will come from developing the logical analogue
of differential geometry.

I took a few steps in this direction some years ago in connection
with an effort to understand a certain class of intelligent systems
as dynamical systems. There's a motley assortment of links here:

* Survey of Differential Logic
https://inquiryintoinquiry.com/2015/05/11/survey-of-differential-logic-%E2%80%A2-1/
</QUOTE>

Resources
=========

* Logical Graphs
https://oeis.org/wiki/Logical_Graphs

* Differential Logic
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Introduction

Regards,

Jon

Yeu Wen

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Aug 6, 2019, 4:34:04 PM8/6/19
to cyb...@googlegroups.com
Hi all,

I wonder if anyone here has found the model useful and if so, where was it applied and what were the learnings that you could share with us?

Thank you so much for your kind help and guidance.

Yeu Wen

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Jason the Goodman

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Aug 6, 2019, 4:54:11 PM8/6/19
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Hi Yue Wen,
I suggest you using "4U Test": Is it Useful? Does it Work? Is it Used?
Many academics fall into traps of "misplaced concerns"... we need to guard ourselves not fall victim of that. However, identifying the potential usefulness of any given theory is itself a big challenge...
---------------------------------
Jason Jixuan Hu, Ph.D.
Independent Research Scholar & Discussant, Club of REMY
---------------------------------------------------


Krippendorff, Klaus

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Aug 6, 2019, 5:49:32 PM8/6/19
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To put the two conceptualizers on one page is a sign of ignorance. 

Korzybski was committed to a representational theory of language which presumes objective knowledge of the world of references as distinct from signs. He had s dualist conception of the world, now largely dismissed. 

Bateson pointed out the evolutionary nature of Ashby’s cybernetics: One can never know whether one’s conceptions of the world are correct. One can only experience their failures when enacting them. 

Klaus

Sent from my iPhone

Jason the Goodman

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Aug 6, 2019, 6:11:43 PM8/6/19
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Dear Klaus, could you share a source URL of that Bateson point you just quoted? Many thanks - Jason

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Krippendorff, Klaus

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Aug 6, 2019, 10:50:31 PM8/6/19
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Hi Jason,
One source is Bateson’s “Cybernetic explanation” pages 405-515 in his Steps to an ecology of mind. You would gain further insight in some of Ross Ashby’s writing. 
I recently contributed a paper for Tomas Fischer’s design cybernetics book, titled “the cybernetics of design and the design of cybernetics.”

Klaus

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Loet Leydesdorff

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Aug 7, 2019, 1:21:53 AM8/7/19
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Thanks, Klaus. I'll read. 

Best,
Loet


Loet Leydesdorff

Professor emeritus, University of Amsterdam
Amsterdam School of Communication Research (ASCoR)

lo...@leydesdorff.net ; http://www.leydesdorff.net/
Associate Faculty, SPRU, University of Sussex;

Guest Professor Zhejiang Univ., Hangzhou; Visiting Professor, ISTIC, Beijing;

Visiting Fellow, Birkbeck, University of London;

http://scholar.google.com/citations?user=ych9gNYAAAAJ&hl=en

Jason the Goodman

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Aug 7, 2019, 2:10:01 PM8/7/19
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Thank you Klaus! Is your paper available online? If yes please share the link. Best regards - Jason

ben sweeting

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Aug 7, 2019, 6:53:18 PM8/7/19
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Available as an e-book:
Fischer, T., & Herr, C. M. (Eds.). (2019). Design Cybernetics: Navigating the new. Cham: Springer.

Yeu Wen

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Aug 8, 2019, 6:18:02 AM8/8/19
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Thank you all for your guidance!

I will take a while to go through all the wonderful references.

Please bear with me.  I am sure I am going to come back with more questions to seek more help.

Thank you all once again!

Yeu Wen

Jon Awbrey

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Aug 13, 2019, 5:15:36 PM8/13/19
to Cybernetic Communications, Laws Of Form Group
Cf: Animated Logical Graphs : 29
At: https://inquiryintoinquiry.com/2019/08/11/animated-logical-graphs-%e2%80%a2-29/

Re: Animated Logical Graphs : 21
At: https://inquiryintoinquiry.com/2019/07/12/animated-logical-graphs-%e2%80%a2-21/

I invoked the general concepts of equivalence and distinction at this point
in order to keep the wider backdrop of ideas in mind but since we've been
focusing on boolean functions to coordinate the semantics of propositional
calculi we can get a sense of the links between operations and relations
by looking at their relationship in a boolean frame of reference.

Let B = {0, 1} and let k be a positive integer.
Then B^k is the set of k-tuples of elements of B.

A "k-variable boolean function" is a mapping B^k -> B.

A "k-place boolean relation" is a subset of B^k.

The correspondence between boolean functions
and boolean relations may be articulated as follows:

Any k-place relation L, as a subset of B^k, has a corresponding
"indicator function" (also called a "characteristic function")
f_L : B^k -> B defined such that f_L (x) = 1 if x is in L and
f_L = 0 if x is not in L.

Any k-variable function f : B^k -> B is the indicator function of
a k-place relation L_f consisting of all the x in B^k where f(x) = 1.
The set L_f is called the "fiber of 1" or the "pre-image of 1" in B^k
and is commonly notated as [f^(-1)](1).

Regards,

Jon

Jon Awbrey

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Aug 25, 2019, 3:54:32 PM8/25/19
to Ontolog Forum, Structural Modeling, SysSciWG, Cybernetic Communications, Laws Of Form Group
Cf: Animated Logical Graphs : 30
At: https://inquiryintoinquiry.com/2019/08/25/animated-logical-graphs-%e2%80%a2-30/

The duality between Entitative and Existential interpretations
of logical graphs is a basic example of a mathematical symmetry,
in this case a symmetry of order 2. Symmetries of this and
higher orders give us conceptual handles on excess complexities
in the manifold of sensuous impressions, making it well worth
our trouble to seek them out and grasp them where we find them.

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:

| Table too big to get a good screen shot.
| Please see the blog post linked above.

Regards,

Jon

Jon Awbrey

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Sep 2, 2019, 8:56:23 AM9/2/19
to syss...@googlegroups.com, Aleksandar Malečić, Ontolog Forum, Structural Modeling, Cybernetic Communications, Laws Of Form Group
Cf: Animated Logical Graphs : 31
At: https://inquiryintoinquiry.com/2019/09/01/animated-logical-graphs-%e2%80%a2-31/

On 7/13/2019 5:33 AM, Aleksandar Male??i?? wrote:
> 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, All,

The larger questions asked above -- interdisciplinary inquiry, the
interest in integration, the synthesis of ideas across isolated silos
of specialization, and what it might mean for the future -- are issues
Susan Awbrey and I addressed from a pragmatic semiotic perspective:

* Awbrey, S.M., and Awbrey, J.L. (2001),
"Conceptual Barriers to Creating Integrative Universities", Organization :
The Interdisciplinary Journal of Organization, Theory, and Society 8(2),
Sage Publications, London, UK, 269-284.
Abstract ( https://journals.sagepub.com/doi/abs/10.1177/1350508401082013 )
Online ( https://www.academia.edu/1266492/Conceptual_Barriers_to_Creating_Integrative_Universities )

* Awbrey, S.M., and Awbrey, J.L. (1999),
"Organizations of Learning or Learning Organizations : The Challenge
of Creating Integrative Universities for the Next Century", Second
International Conference of the Journal 'Organization', Re-Organizing
Knowledge, Trans-Forming Institutions : Knowing, Knowledge, and the
University in the 21st Century, University of Massachusetts, Amherst, MA.
Online ( http://www.iupui.edu/~arisbe/menu/library/aboutcsp/awbrey/integrat.htm )

From that vantage point, what I'm about here is just a subgoal of a subgoal,
panning what bits of elemental substrate can be found ever nearer that elusive
"rock bottom reality".

Regards,

Jon

Jon Awbrey

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Jun 14, 2020, 10:25:12 AM6/14/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 32
At: http://inquiryintoinquiry.com/2020/06/14/animated-logical-graphs-%e2%80%a2-32/

Re: R.J. Lipton and K.W. Regan
https://rjlipton.wordpress.com/about-me/
::: Proof Checking
https://rjlipton.wordpress.com/2020/06/13/proof-checking-not-line-by-line/

Here's a place where I explore several different shapes of proofs within propositional calculus deriving from the
graphical systems of Charles S. Peirce and G. Spencer Brown.

• Propositional Equation Reasoning Systems • Analysis of Contingent Propositions
https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems#Analysis_of_contingent_propositions

I don't know whether that helps any with P ≟ NP but it does supply a lot of nice pictures to contemplate.

Resources
=========

• Logical Graphs
https://oeis.org/wiki/Logical_Graphs

• Propositional Equation Reasoning Systems
https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems
facebook page: https://www.facebook.com/JonnyCache

Jon Awbrey

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Jun 18, 2020, 3:12:06 PM6/18/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 33
At: http://inquiryintoinquiry.com/2020/06/18/animated-logical-graphs-%e2%80%a2-33/

A reader's request for more examples of animated logical graphs
prompted me to look again at the User Guide for my Theme One Program,
whose exposition develops a series of logical graphs increasing in
complexity from extremely simple to more complex and interesting
than any I've posted online so far.

• Theme One Program • User Guide
https://www.academia.edu/5211369/Theme_One_Program_User_Guide

I'm thinking now it may be worthwhile to look at those examples again
and see if they're suitable for recycling as a series of blog posts.

Resources
=========

• Logical Graphs
https://oeis.org/wiki/Logical_Graphs

• Proof Animations
https://oeis.org/wiki/User:Jon_Awbrey/ANIMATION#Proof_Animations

• Survey of Theme One Program
https://inquiryintoinquiry.com/2018/02/25/survey-of-theme-one-program-%e2%80%a2-2/

• Survey of Animated Logical Graphs
https://inquiryintoinquiry.com/2019/05/22/survey-of-animated-logical-graphs-%e2%80%a2-2/

Jon Awbrey

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Aug 11, 2020, 11:20:21 AM8/11/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 34
http://inquiryintoinquiry.com/2020/08/11/animated-logical-graphs-34/

Re: Ontolog Forum ( https://groups.google.com/d/topic/ontolog-forum/7iwyulzDpFA/overview )
::: John Sowa ( https://groups.google.com/d/msg/ontolog-forum/7iwyulzDpFA/dwlvIiMaBwAJ )
Re: Peirce List ( https://list.iupui.edu/sympa/arc/peirce-l/2020-08/thrd2.html#00051 )
::: John Sowa ( https://list.iupui.edu/sympa/arc/peirce-l/2020-08/msg00051.html )

All,

This adds a few resource links to an earlier reply on
the Ontolog Forum and the Peirce List. I added it to
my blog series on Animated Logical Graphs mostly just
by way of reminding myself to get back to that.

Dear John,

I can't imagine why anyone would bother with Peirce's logic
if it's just Frege and Russell in a different syntax, which
has been the opinion I usually get from FOL fans. But the fact
is Peirce's 1870 "Logic of Relatives" is already far in advance
of anything we'd see again for a century, in principle in most
places, in practice in many others, chock full of revolutionary
ideas, not all of which he developed fully in subsequent work.
Although I studied the 1870 Logic from early on I did not realize
how far ahead of its time it was until I began reading approaches
to logic from category-theoretic and computation-theoretic angles
in the 1970s and 1980s. An indication of Peirce's innovations can
be found in the series of selections and commentary I started on
the 1870 Logic of Relatives.

Here's the work in progress so far on the OEIS Wiki.

* Peirce's 1870 Logic Of Relatives
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Overview
* Part 1
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1
* Part 2
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2
* Part 3
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_3

Here's the overview for a parallel series of blog posts.

* Peirce's 1870 Logic Of Relatives • Overview
https://inquiryintoinquiry.com/2019/09/24/peirces-1870-logic-of-relatives-%e2%80%a2-overview/

Resources
=========

* Logical Graphs
https://oeis.org/wiki/Logical_Graphs

* Proof Animations
https://oeis.org/wiki/User:Jon_Awbrey/ANIMATION#Proof_Animations

* Survey of Theme One Program
https://inquiryintoinquiry.com/2018/02/25/survey-of-theme-one-program-%e2%80%a2-2/

* Survey of Animated Logical Graphs
https://inquiryintoinquiry.com/2019/05/22/survey-of-animated-logical-graphs-%e2%80%a2-2/

* Propositional Equation Reasoning Systems
https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems

Regards,

Jon

Jon Awbrey

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Aug 19, 2020, 10:40:09 AM8/19/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs ??? 35
http://inquiryintoinquiry.com/2020/08/19/animated-logical-graphs-35/

Re: Richard J. Lipton
https://rjlipton.wordpress.com/about-me/
::: Logical Complexity Of Proofs
https://rjlipton.wordpress.com/2020/08/19/logical-complexity-of-proofs/

The smoothest way I know to do propositional calculus
is by using minimal negation operators as primitives,
parsing propositional formulas into (painted and rooted)
cactus graphs, and using the appropriate extension of
the axiom set from Charles S. Peirce's logical graphs
and G. Spencer Brown's laws of form.

There's a quick link here:

* Cactus Language for Propositional Logic
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Cactus_Language_for_Propositional_Logic

Resources
=========

* Logic Syllabus
https://oeis.org/wiki/Logic_Syllabus
* Futures Of Logical Graphs
https://oeis.org/wiki/Futures_Of_Logical_Graphs

* Minimal Negation Operators
https://oeis.org/wiki/Minimal_negation_operator

Jon Awbrey

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Aug 21, 2020, 11:00:24 AM8/21/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs ??? 36
http://inquiryintoinquiry.com/2020/08/21/animated-logical-graphs-36/
Dear Dick,

You asked, "Is this measure, the logical flow of a proof, of any interest?"

I wasn't quite clear how you define the measure of flow in a proof --
it seemed to have something to do with the number of implication arrows
in the argument structure?

But this does bring up interesting issues of "proof style" ...

Propositional calculus as a formal language and boolean functions
as an object domain form an instructive microcosm for many issues
of logic writ large. The relation between proof theory and model
theory is one of those issues, despite, or maybe in virtue of,
propositional logic's status as a special case.

Folks who pursue the CSP-GSB line of development
in graphical syntax for propositional calculus are
especially likely to notice the following dimensions
of proof style.

Formal Duality
==============
This goes back to Peirce's discovery of the "amphecks"
( https://oeis.org/wiki/Ampheck ) and the duality between
Not Both (nand ( https://oeis.org/wiki/Logical_NAND ) ) and
Both Not (nnor ( https://oeis.org/wiki/Logical_NNOR ) ).
The same duality is present in Peirce's graphical systems
for propositional calculus. It is analogous to the duality
in projective geometry and it means we are always proving
two theorems for the price of one. That's a reduction in
complexity -- it raises the question of how many such
group-theoretic reductions we can find.

To be continued ...

Resources
=========

* Cactus Language
https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview

Applications
============

* Applications of a Propositional Calculator ??? Constraint Satisfaction Problems
https://www.academia.edu/4727842/Applications_of_a_Propositional_Calculator_Constraint_Satisfaction_Problems

* Exploratory Qualitative Analysis of Sequential Observation Data
http://web.archive.org/web/20180828161616/http://intersci.ss.uci.edu/wiki/index.php/Exploratory_Qualitative_Analysis_of_Sequential_Observation_Data


Regards,

Jon

Jon Awbrey

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Aug 22, 2020, 2:20:21 PM8/22/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 37
http://inquiryintoinquiry.com/2020/08/22/animated-logical-graphs-37/
Another dimension of proof style has to do with how much information
is kept or lost as the argument develops. For the moment let's focus
on classical deductive reasoning at the propositional level. Then we
can distinguish between "equational inferences", which keep all the
information represented by the input propositions, and "implicational
inferences", which permit information to be lost as the proof proceeds.

Information-Preserving vs. Information-Reducing Inferences
==========================================================

Implicit in Peirce's systems of logical graphs is the ability to use
equational inferences. Spencer Brown drew this out and turned it to
great advantage in his revival of Peirce's graphical forms. As it
affects "logical flow" this allows for bi-directional or reversible
flows, you might even say a "logical equilibrium" between two states
of information.

It is probably obvious when we stop to think about it, but
seldom remarked, that all the more familiar inference rules,
like modus ponens and resolution or transitivity, entail in
general a loss of information as we traverse their arrows or
turnstiles.

For example, the usual form of modus ponens takes us from knowing
p and p => q to knowing q but in fact we know more, we actually know
p and q. With that in mind we can formulate two variants of modus ponens,
one reducing and one preserving the actual state of information, as shown
in the following figure.

Modus Ponens Variants
https://inquiryintoinquiry.files.wordpress.com/2020/08/modus-ponens-variants.png

There's more discussion of this topic at the following location.

* Propositional Equation Reasoning Systems : Computation and Inference as Semiosis
https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems#Computation_and_inference_as_semiosis

To be continued ...

Regards,

Jon
Modus Ponens Variants.png

Jon Awbrey

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Aug 24, 2020, 10:04:08 AM8/24/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Survey of Animated Logical Graphs • 3
http://inquiryintoinquiry.com/2020/08/23/survey-of-animated-logical-graphs-3/

All,

I just updated my Survey of blog and wiki posts relating to
Animated Logical Graphs. A great many links went missing
when my old worksite, the InterSciWiki, went offline so
I've been repairing those as I run across them.

Beginnings
==========

Logical Graphs : Introduction
https://inquiryintoinquiry.com/2008/07/29/logical-graphs-1/

Logical Graphs : Formal Development
https://inquiryintoinquiry.com/2008/09/19/logical-graphs-2/

Elements
========

Logic Syllabus
https://oeis.org/wiki/Logic_Syllabus

Logical Graphs
https://oeis.org/wiki/Logical_Graphs
Propositional Equation Reasoning Systems
https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems

Examples
========

Peirce's Law
* This Blog
https://inquiryintoinquiry.com/2008/10/06/peirce-s-law/
* OEIS Wiki
https://oeis.org/wiki/Peirce%27s_law

Praeclarum Theorema
* This Blog
https://inquiryintoinquiry.com/2008/10/05/praeclarum-theorema/
* OEIS Wiki
https://oeis.org/wiki/Logical_Graphs#Praeclarum_theorema
Excursions
==========

Cactus Language
https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview
Applications
============

Applications of a Propositional Calculator :
Differential Analytic Turing Automata
https://oeis.org/wiki/Differential_Analytic_Turing_Automata

Survey of Theme One Program
https://inquiryintoinquiry.com/2018/02/25/survey-of-theme-one-program-2/

Blog Dialogs
============

Animated Logical Graphs
https://inquiryintoinquiry.com/2015/01/08/animated-logical-graphs-1/
https://inquiryintoinquiry.com/2015/01/14/animated-logical-graphs-2/
https://inquiryintoinquiry.com/2015/01/26/animated-logical-graphs-3/
...
https://inquiryintoinquiry.com/2020/08/11/animated-logical-graphs-34/
https://inquiryintoinquiry.com/2020/08/19/animated-logical-graphs-35/
https://inquiryintoinquiry.com/2020/08/21/animated-logical-graphs-36/
https://inquiryintoinquiry.com/2020/08/22/animated-logical-graphs-37/

Regards,

Jon

Jon Awbrey

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Aug 25, 2020, 4:01:07 PM8/25/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 38
http://inquiryintoinquiry.com/2020/08/25/animated-logical-graphs-38/
Three examples of propositional proofs in logical graphs using
equational inference rules can be found at the following location.

* Propositional Equation Reasoning Systems • Exemplary Proofs
https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems#Exemplary_proofs

By way of a quick overview, here are the animated proofs of
these examples, along with links to their detailed descriptions.

Peirce's Law
============
https://inquiryintoinquiry.com/2008/10/06/peirces-law/
https://oeis.org/wiki/Peirce%27s_law

https://inquiryintoinquiry.files.wordpress.com/2012/01/peirces-law-2-0-animation.gif

Praeclarum Theorema
===================
https://inquiryintoinquiry.com/2008/10/05/praeclarum-theorema/
https://oeis.org/wiki/Logical_Graphs#Praeclarum_theorema

https://inquiryintoinquiry.files.wordpress.com/2012/01/praeclarum-theorema-2-0-animation.gif

Two-Thirds Majority Function
============================
https://oeis.org/wiki/Logical_Graphs#Two-thirds_majority_function
https://oeis.org/wiki/Futures_Of_Logical_Graphs#Two-thirds_majority_function
https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems#Two-thirds_majority_function

https://inquiryintoinquiry.files.wordpress.com/2020/08/two-thirds-majority-function-500-x-250-animation.gif

Regards,

Jon
Peirce's Law 2.0 Animation.gif
Praeclarum Theorema 2.0 Animation.gif
Two-Thirds Majority Function 500 x 250 Animation.gif

Jon Awbrey

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Sep 10, 2020, 3:12:44 PM9/10/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 39
http://inquiryintoinquiry.com/2020/09/10/animated-logical-graphs-39/

Happy Peirce's Birthday, Everyone !!!
We've been discussing aspects of proof style arising in connection with
the complexity of proofs. In previous posts we took up (1) the aspect of
formal duality, reflecting in passing on the prospect of higher symmetries,
and (2) the spectrum ranging from information-reducing to information-preserving
inference rules. Here's a quick recap --

* Animated Logical Graphs
=========================
https://inquiryintoinquiry.com/2020/08/25/animated-logical-graphs-38/

A third aspect of proof style arising in this connection is the degree
of insight demanded and demonstrated in the performance of a proof.
Generally speaking, the same endpoint can be reached in many different
ways from given starting points, by paths ranging from those exhibiting
appreciable insight to those exercising little more than persistence in
sticking to a set routine.

A modicum of insight suffices to suggest the quality of "insight" resists
pinning down in a succinct definition but we do tend to recognize it when
we see it, so let me inch forward by highlighting its salient features in
a graded series of examples.

Jon Awbrey

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Sep 26, 2020, 7:48:16 PM9/26/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 40
http://inquiryintoinquiry.com/2020/09/26/animated-logical-graphs-40/

| Note: as always, follow the above link for better formatting,
| especially if the unicode math symbols fail to get through.

One way to see the difference between insight proofs and routine proofs
is to pick a single example of a theorem in propositional calculus and
prove it two ways, one more insightful and one more routine.

The praeclarum theorema, or splendid theorem, is a theorem
of propositional calculus noted and named by G.W. Leibniz,
who stated and proved it in the following manner.

<QUOTE>

If a is b and d is c, then ad will be bc.
This is a fine theorem, which is proved in this way:
a is b, therefore ad is bd (by what precedes),
d is c, therefore bd is bc (again by what precedes),
ad is bd, and bd is bc, therefore ad is bc. Q.E.D.

— Leibniz • Logical Papers, p. 41.

</QUOTE>

Expressed in contemporary logical notation,
the theorem may be written as follows.

Expressed in contemporary logical notation, the theorem may be written as follows.

((a ⇒ b) ∧ (d ⇒ c)) ⇒ ((a ∧ d) ⇒ (b ∧ c))

Using teletype parentheses ( ... ) for the logical negation (p)
of a proposition p and simple concatenation pq for the logical
conjunction of propositions p, q permits the theorem to be
written in the following in-line and lispish ways.

Inline Syntax
=============

( (a (b)) (d (c)) ( (ad (bc)) ))

Lispish ("pretty-printed")
==========================

( (a (b)) (d (c))
( (ad (bc))
))

Jon Awbrey

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Sep 29, 2020, 11:40:13 AM9/29/20
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Cf: Animated Logical Graphs • 41
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All,

Last time we looked at a formula of propositional logic Leibniz
called a Praeclarum Theorema (PT). We don't concur it's a theorem,
of course, until there's a proof it's identically true and Leibniz
gave an argument to demonstrate that. Written out in one of our
more current formalisms, PT takes the following form.

((a ⇒ b) ∧ (d ⇒ c)) ⇒ ((a ∧ d) ⇒ (b ∧ c))

Somewhat in the spirit of Reduced Instruction Set Computing,
we reformulated PT in a propositional calculus using just
two primitive operations, writing the logical negation of
a proposition p as (p) and the logical conjunction of two
propositions p, q as pq. That gave us a text string in
teletype parentheses and proposition letters, formatted
two ways below.

Figure 1. Praeclarum Theorema Text Strings
https://inquiryintoinquiry.files.wordpress.com/2020/09/praeclarum-theorema-text-strings.png

Our next transformation of the theorem’s expression exploits a
standard correspondence in combinatorics and computer science
between parenthesized symbol strings and trees with symbols
attached to the nodes.

Figure 2. Praeclarum Theorema Parse Graph
https://inquiryintoinquiry.files.wordpress.com/2020/09/praeclarum-theorema-parse-graph-2.0.png

We can see the correspondence between text and tree in the case of PT
by starting at the root of the tree and reading off the characters of
the text string as we traverse the edges and nodes of the tree in the
following manner. The initial "(" tells us to ascend the first edge,
the next "(" tells us to ascend the next edge on the left, where we
find the letter "a" from the string checks with the letter "a" attached
to the node of the tree where we are. Another "(" takes us up another
edge, where we find the letter "b" from the string checks with the
letter "b" on the current tree node. Reading the first ")" on the
string entitles us to descend an edge and reading another ")" gives us
licence to descend another. The way of things is most likely clear by
this point — at any rate, I leave the exercise to the reader.

On the scene of the general correspondence between formulas and graphs
the action may be summed up as follows. The tree, called a "parse tree"
or "parse graph", is constructed in the process of checking whether the
text string is syntactically well-formed, in other words, whether it
satisfies the prescriptions of the associated formal grammar and is
therefore a member in good standing of the prescribed formal language.
If the text string checks out, grammatically speaking, we call it a
"traversal string" of the corresponding parse graph, because it can be
reconstructed from the graph by a process like that illustrated above
called "traversing" the graph.

To be continued …

Regards,

Jon

Praeclarum Theorema Text Strings.png
Praeclarum Theorema Parse Graph 2.0.png

Jon Awbrey

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Oct 3, 2020, 4:16:12 PM10/3/20
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Cf: Animated Logical Graphs • 42
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Figure 1. Praeclarum Theorema Parse Graph
https://inquiryintoinquiry.files.wordpress.com/2020/09/praeclarum-theorema-parse-graph-2.0.png

Now that our propositional formula is cast in the form of a graph
its evaluation proceeds as a sequence of graphical transformations
where each graph in turn belongs to the same formal equivalence class
as its predecessor and thus of the first. The sequence terminates in
a canonical graph making it manifest whether the initial formula is
identically true by virtue of its form or not.

To be continued ...

Reference
=========

* Leibniz, Gottfried W. (1679–1686?),
“Addenda to the Specimen of the Universal Calculus”,
pp. 40–46 in G.H.R. Parkinson (ed., trans., 1966),
Leibniz : Logical Papers, Oxford University Press,
London, UK.

Praeclarum Theorema
===================

* Inquiry Blog
https://inquiryintoinquiry.com/2008/10/05/praeclarum-theorema/

* Metamath Proof Explorer
http://us.metamath.org/mpegif/mmset.html
• Praeclarum Theorema
http://us.metamath.org/mpegif/prth.html

* Frithjof Dau ( http://dr-dau.net/ )
• Computer Animated Proof of Leibniz's Praeclarum Theorema
http://dr-dau.net/pc.shtml

Regards,

Jon
Praeclarum Theorema Parse Graph 2.0.png

Jon Awbrey

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Oct 5, 2020, 2:36:44 PM10/5/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 43
http://inquiryintoinquiry.com/2020/10/05/animated-logical-graphs-43/

Re: FB | Ecology Of Systems Thinking
https://www.facebook.com/groups/ecologyofsystemsthinking/permalink/3377906762288444/
::: Richard Saunders
https://www.facebook.com/groups/ecologyofsystemsthinking/permalink/3377906762288444/?comment_id=3379444682134652 )
RS: “I wonder if we might find such graphs in the physical microstructures of brains, cells, proteins, etc.”

Dear Richard,

You are reading my mind. See the following message I posted
back in the day on the old Standard Upper Ontology List.

* SUO: Re: Propositional Equation Reasoning Systems (PERS)
http://web.archive.org/web/20060717224355/http://suo.ieee.org/email/msg04525.html )

I took up a simple example of a contingent proposition and proceeded in the
medium of logical graphs to prove its equivalence to a simpler proposition.
Reflecting on the form of the proof gone before, I concluded with the
following remark.

JA: “For some reason I always think of that as the way that our DNA would prove it.”

There's further discussion of that example at the following location.

* Propositional Equation Reasoning Systems • Analysis of Contingent Propositions
https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems#Analysis_of_contingent_propositions

Regards,

Jon
Praeclarum Theorema Parse Graph 2.0.png

Jon Awbrey

unread,
Oct 6, 2020, 10:16:00 AM10/6/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 44
http://inquiryintoinquiry.com/2020/10/06/animated-logical-graphs-44/

Re: FB | Ecology Of Systems Thinking
https://www.facebook.com/groups/ecologyofsystemsthinking/permalink/3377906762288444/
::: Richard Saunders
https://www.facebook.com/groups/ecologyofsystemsthinking/permalink/3377906762288444/?comment_id=3380578865354567

RS: DNA and proteins might be good places to look for logical graphs in nature since our tech for mapping those
structures has become fairly proficient lately. Do you think we could train some kind of neural net to find the
patterns? Might that lead to a real breakthrough in computational microbiology?

Dear Richard,

Models of neural nets are extremely various. I don't especially cotton to the ones based on threshold computation, as I
think they're bound to remain rather dumb. I view all those blinking neurons as something like a night view of the
earth's cities from space. What we see is only a measure of the raw power consumption occurring in the cities,
buildings, and homes, not anything like the actual processes going on inside those sites.

Regards,

Jon
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