Animated Logical Graphs

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Jon Awbrey

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Nov 13, 2020, 9:54:45 AM11/13/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 45
http://inquiryintoinquiry.com/2020/11/12/animated-logical-graphs-45/

Re: Richard J. Lipton ( https://rjlipton.wordpress.com/about-me/ )
::: The Art Of Math ( https://rjlipton.wordpress.com/2020/11/12/the-art-of-math/ )

There's a nice interplay between geometric and logical dualities
in C.S. Peirce's graphical systems of logic, rooted in his discovery
of the “amphecks” NAND and NNOR and flowering in his logical graphs
for propositional and predicate calculus. Peirce's logical graphs bear
the dual interpretations he dubbed “entitative” and “existential” graphs.

Here's a Table of Boolean Functions on Two Variables, using an
extension of Peirce's graphs from trees to cacti, illustrating
the duality so far as it affects propositional calculus.

Boolean Functions on Two Variables
https://inquiryintoinquiry.files.wordpress.com/2020/11/boolean-functions-on-two-variables.png

Resources
=========

• Logic Syllabus ( https://oeis.org/wiki/Logic_Syllabus )

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

• Cactus Language ( https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview )

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

• Minimal Negation Operators ( https://oeis.org/wiki/Minimal_negation_operator )

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

• Survey of Animated Logical Graphs
( https://inquiryintoinquiry.com/2019/05/22/survey-of-animated-logical-graphs-2/ )

• Propositional Equation Reasoning Systems
( https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems )

• Applications • Constraint Satisfaction Problems
https://www.academia.edu/4727842/Applications_of_a_Propositional_Calculator_Constraint_Satisfaction_Problems

Regards,

Jon

inquiry into inquiry: https://inquiryintoinquiry.com/
academia: https://independent.academia.edu/JonAwbrey
oeiswiki: https://www.oeis.org/wiki/User:Jon_Awbrey
facebook page: https://www.facebook.com/JonnyCache
Boolean Functions on Two Variables.png

Jon Awbrey

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Nov 16, 2020, 11:30:45 AM11/16/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 46
http://inquiryintoinquiry.com/2020/11/15/animated-logical-graphs-46/
Re: Animated Logical Graphs
30: https://inquiryintoinquiry.com/2019/08/25/animated-logical-graphs-30/
45: https://inquiryintoinquiry.com/2020/11/12/animated-logical-graphs-45/

Dear All,

Another way of looking at Peirce duality is given by the following Table,
which shows how logical graphs denote boolean functions under entitative
and existential interpretations.

Column 1 shows the logical graphs for the sixteen boolean functions on two variables.
Column 2 shows the boolean functions denoted under the entitative interpretation and
Column 3 shows the boolean functions denoted under the existential interpretation.

Table. Logical Graphs : Entitative and Existential Interpretations
https://inquiryintoinquiry.files.wordpress.com/2020/11/logical-graphs-e280a2-entitative-and-existential-interpretations.png

Regards,

Jon
Logical Graphs • Entitative and Existential Interpretations.png

Jon Awbrey

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Nov 27, 2020, 2:00:25 PM11/27/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 47
http://inquiryintoinquiry.com/2020/11/27/animated-logical-graphs-47/

Re: Richard J. Lipton ( https://rjlipton.wordpress.com/about-me/ )
::: The Art Of Math ( https://rjlipton.wordpress.com/2020/11/12/the-art-of-math/ )
Re: Animated Logical Graphs
30. https://inquiryintoinquiry.com/2019/08/25/animated-logical-graphs-30/
45. https://inquiryintoinquiry.com/2020/11/12/animated-logical-graphs-45/
46. https://inquiryintoinquiry.com/2020/11/15/animated-logical-graphs-46/

A logical concept represented by a boolean variable has its “extension”,
the cases it covers in a designated universe of discourse, and its
“comprehension” (or “intension”), the properties it implies in a
designated hierarchy of predicates. The formulas and graphs tabulated
in previous posts are well-adapted to articulating the syntactic and
intensional aspects of propositional logic. But their very tailoring
to those tasks tends to slight the extensional and therefore empirical
applications of logic. Venn diagrams, despite their unwieldiness as the
number of logical dimensions increases, are indispensable in providing
the visual intuition with a solid grounding in the extensions of logical
concepts. All that makes it worthwhile to reset our table of boolean
functions on two variables to include the corresponding venn diagrams.

Table. Venn Diagrams and Logical Graphs on Two Variables
https://inquiryintoinquiry.files.wordpress.com/2020/11/venn-diagrams-and-logical-graphs-on-two-variables.png

Regards,

Jon
Venn Diagrams and Logical Graphs on Two Variables.png

Jon Awbrey

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Nov 30, 2020, 2:40:17 PM11/30/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 48
http://inquiryintoinquiry.com/2020/11/30/animated-logical-graphs-48/
47. https://inquiryintoinquiry.com/2020/11/27/animated-logical-graphs-47/

A more graphic picture of Peirce duality is given by the next Table,
which shows how logical graphs map to venn diagrams under entitative
and existential interpretations.

Column 1 shows the logical graphs for the sixteen boolean functions on two variables.
Column 2 shows the venn diagrams associated with the entitative interpretation and
Column 3 shows the venn diagrams associated with the existential interpretation.

Table. Logical Graphs • Entitative and Existential Venn Diagrams
https://inquiryintoinquiry.files.wordpress.com/2020/11/logical-graphs-e280a2-entitative-and-existential-venn-diagrams.png

Regards,

Jon
Logical Graphs • Entitative and Existential Venn Diagrams.png

Jon Awbrey

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Dec 3, 2020, 2:56:26 PM12/3/20
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Cf: Animated Logical Graphs • 49
http://inquiryintoinquiry.com/2020/12/03/animated-logical-graphs-49/

Re: Richard J. Lipton • The Art Of Math
::: https://rjlipton.wordpress.com/2020/11/12/the-art-of-math/

Dualities are symmetries of order two and symmetries bear on
complexity by reducing its measure in proportion to their order.
The inverse relationship between symmetry and all those dissymmetries
from dispersion and diversity to entropy and uncertainty is governed
in cybernetics by the Law of Requisite Variety, the medium of which
exchanges C.S. Peirce invested in his formula:

• Information = Comprehension × Extension
https://oeis.org/wiki/Information_%3D_Comprehension_%C3%97_Extension

The duality between entitative and existential interpretations of
logical graphs is one example of a mathematical symmetry but it's not
unusual to find symmetries within symmetries and it's always rewarding
to find them where they exist. To that end let's take up our Table of
Venn Diagrams and Logical Graphs on Two Variables and sort the rows to
bring together diagrams and graphs having similar shapes. What defines
their similarity is the action of a mathematical group whose operations
transform the elements of each class among one another but intermingle
no dissimilar elements. In the jargon of transformation groups these
classes are called “orbits”. We find the sixteen rows partition into
seven orbits, as shown below.

Table. Venn Diagrams and Logical Graphs on Two Variables • Orbit Order
https://inquiryintoinquiry.files.wordpress.com/2020/12/venn-diagrams-and-logical-graphs-on-two-variables-e280a2-orbit-order.png
Venn Diagrams and Logical Graphs on Two Variables • Orbit Order.png

Jon Awbrey

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Dec 5, 2020, 4:56:36 PM12/5/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 50
http://inquiryintoinquiry.com/2020/12/05/animated-logical-graphs-50/

Re: Richard J. Lipton • The Art Of Math
::: https://rjlipton.wordpress.com/2020/11/12/the-art-of-math/

48. https://inquiryintoinquiry.com/2020/11/30/animated-logical-graphs-48/
49. https://inquiryintoinquiry.com/2020/12/03/animated-logical-graphs-49/

In our last of six ways of looking at the Peirce duality between
entitative and existential interpretations, here is the previous
Table of Logical Graphs and Venn Diagrams sorted in Orbit Order.

Table. Logical Graphs • Entitative and Existential Venn Diagrams • Orbit Order
https://inquiryintoinquiry.files.wordpress.com/2020/12/logical-graphs-e280a2-entitative-and-existential-venn-diagrams-e280a2-orbit-order.png
)

Regards,

Jon
Logical Graphs • Entitative and Existential Venn Diagrams • Orbit Order.png

Jon Awbrey

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Dec 20, 2020, 4:24:22 PM12/20/20
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Cf: Animated Logical Graphs • 51
http://inquiryintoinquiry.com/2020/12/20/animated-logical-graphs-51/

Fig. 1. Animated Proof of Peirce's Law
https://inquiryintoinquiry.files.wordpress.com/2012/01/peirces-law-2-0-animation.gif

Synchronicity being what it is, a long-running discussion
on the Peirce List just gave me a handy bridge to a topic
I've been meaning to take up in several other connections.
So I'm adding my comment to this series, along with links
to additional resources.

Re: Peirce List
https://list.iupui.edu/sympa/arc/peirce-l/2020-11/thrd1.html#00022
https://list.iupui.edu/sympa/arc/peirce-l/2020-11/thrd2.html#00037
https://list.iupui.edu/sympa/arc/peirce-l/2020-12/thrd1.html#00004
https://list.iupui.edu/sympa/arc/peirce-l/2020-12/thrd3.html#00063

Pursuing the discussion of many things:
of laws — and graphs — and reasoning —
of contradictions — and abducations —
and why the third is given not —
and whether figs have wings —

It might not be non sequitur to remember that place in Peirceland
where we walk the line between classical and intuitionistic logic,
namely, the boundary marked by the principle we have come to call
Peirce's Law.

Here's links to bits of fol-de-rule, with graphs and everything —

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

Resources
=========

• Logic Syllabus ( https://oeis.org/wiki/Logic_Syllabus )
• Ampheck ( https://oeis.org/wiki/Ampheck )

• Logical Graphs ( https://oeis.org/wiki/Logical_Graphs )
• One ( https://inquiryintoinquiry.com/2008/07/29/logical-graphs-1/ )
• Two ( https://inquiryintoinquiry.com/2008/09/19/logical-graphs-2/ )

• Propositions As Types Analogy
https://oeis.org/wiki/Propositions_As_Types_Analogy

• Survey of Animated Logical Graphs
https://inquiryintoinquiry.com/2020/08/23/survey-of-animated-logical-graphs-3/

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

• Survey of Abduction, Deduction, Induction, Analogy, Inquiry
https://inquiryintoinquiry.com/2020/12/16/survey-of-abduction-deduction-induction-analogy-inquiry-2/

Regards,

Jon
Peirce's Law 2.0 Animation.gif

Jon Awbrey

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Dec 22, 2020, 3:04:14 PM12/22/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 52
http://inquiryintoinquiry.com/2020/12/21/animated-logical-graphs-52/

Re: Richard J. Lipton ( https://rjlipton.wordpress.com/about-me/ )
::: The Future Of Mathematics?
https://rjlipton.wordpress.com/2020/12/10/the-future-of-mathematics/
::: Is The End Near?
https://rjlipton.wordpress.com/2020/12/12/is-the-end-near/

Re: Peirce List
https://list.iupui.edu/sympa/arc/peirce-l/2020-12/thrd3.html#00063
::: Jon Alan Schmidt
https://list.iupui.edu/sympa/arc/peirce-l/2020-12/msg00065.html

Peirce’s explorations in logic and the theory of signs opened several
directions of generalization from logics of complete information (LOCI)
to theories of partial information (TOPI). Naturally we hope these
avenues of approach will eventually converge on a unified base camp
from which greater heights of understanding may be reached, but that
is still a work in progress, at least for me.

Any passage from logic as a critical, formal, or normative theory
of controlled semiotic conduct to the descriptive study of signs
“in the wild” involves relaxing logical norms to statistical norms.

One of the headings under which Peirce expands the scope of logic to
something more general — whether keeping or losing the name of “logic”
is a secondary consideration — is found in his study of Generality and
Vagueness as affecting signs not fully primed for logical use. There's
a bit about that at the following places.

• C.S. Peirce • Collected Papers (CP 5.448)
https://oeis.org/wiki/User:Jon_Awbrey/EXCERPTS#Excerpt_6._Peirce_.28CP_5.448.29

• FOM List • C.S. Peirce on “General” and “Vague”
https://cs.nyu.edu/pipermail/fom/2009-March/thread.html#13437
1. https://cs.nyu.edu/pipermail/fom/2009-March/013437.html
2. https://cs.nyu.edu/pipermail/fom/2009-March/013446.html
3. https://cs.nyu.edu/pipermail/fom/2009-March/013448.html

As you can see, in this direction of generalization Peirce considers relaxing
both the principle of contradiction and the principle of excluded middle.

Jon Awbrey

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Jan 22, 2021, 1:15:42 PMJan 22
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 53
http://inquiryintoinquiry.com/2021/01/22/animated-logical-graphs-53/

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

All,

A good deal of the work I've been doing on the C.S. Peirce/Spencer Brown
approach to “the mathematical hypostases laying the grounds for logic”
currently flies under the banner of Animated Logical Graphs. There's
a Survey of related resources I update from time to time at the
following location.
( https://oeis.org/wiki/Propositions_As_Types_Analogy )

• Propositional Equation Reasoning Systems
( https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems )

• Survey of Abduction, Deduction, Induction, Analogy, Inquiry
( https://inquiryintoinquiry.com/2020/12/16/survey-of-abduction-deduction-induction-analogy-inquiry-2/ )

Regards,

Jon
Praeclarum Theorema 2.0 Animation.gif

Jon Awbrey

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Jan 27, 2021, 4:30:25 PMJan 27
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 54
http://inquiryintoinquiry.com/2021/01/27/animated-logical-graphs-54/

Re: Peter Cameron
https://cameroncounts.wordpress.com/about/
::: Doing Research
https://cameroncounts.wordpress.com/2009/11/11/doing-research/

Re: Gil Kalai
https://gilkalai.wordpress.com/about/
::: Chomskian Linguistics
https://gilkalai.wordpress.com/2009/09/29/chomskian-linguistics/

Speaking of dreams, the night before last I had a dream where
I was listening to a lecturer and something he said made me think
of a logical formula having the form “if if if a, b, c, d”, which
I visualized as the Peircean logical graph shown below.

If If If
https://inquiryintoinquiry.files.wordpress.com/2021/01/if-if-if.jpg

I knew I had seen something the day before prompting that fragment
and a search through my browser history turned up Gil Kalai's post
on Chomskian Linguistics where I'd read the phrase “anti anti anti
missile missile missile missile”.

Regards,

Jon
If If If.jpg

Jon Awbrey

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Jan 29, 2021, 5:15:39 PMJan 29
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Jon Awbrey

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Jan 30, 2021, 12:45:40 PMJan 30
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 55
http://inquiryintoinquiry.com/2021/01/30/animated-logical-graphs-55/

Re: Laws of Form
https://groups.io/g/lawsofform/topic/sociological_reading_of_lof/79753680
::: William Bricken ( https://groups.io/g/lawsofform/message/19 )

<QUOTE WB:>

Kauffman’s 2001 piece on Peirce (title is “The Mathematics of Charles Sanders Peirce
( http://homepages.math.uic.edu/~kauffman/Form.html )”) is IMO fundamental to this
discussion.

Here’s a brief excerpt from a piece I did in 2005:
“Boundary Logic and Alpha Existential Graphs (AEG)”

4.3 LoF and Alpha Graphs Compared

AEG applies the diagrammatic structure of enclosure specifically to logic.
The representations of LoF and AEG are isomorphic, while the systems of
transformation rules are remarkably close to being the same. ...

</QUOTE>

Dear William,

Many thanks for your excerpt. It highlights many of the most critical points
in comparing the systems of Peirce and Spencer Brown so I'll take up the topic
of duality first. Over the years I've always found that to be one of the stickier
wickets in the whole field. I'll discuss it in my Animated Logical Graphs series
as that's where I've recently redoubled my efforts to explain the issue and why
it's important.

Regards,

Jon

Jon Awbrey

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Feb 6, 2021, 11:54:36 AMFeb 6
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 56
http://inquiryintoinquiry.com/2021/02/06/animated-logical-graphs-56/

Re: Re: Animated Logical Graphs • 55
https://inquiryintoinquiry.com/2021/01/30/animated-logical-graphs-55/

Re: Laws of Form
https://groups.io/g/lawsofform/topic/animated_logical_graphs/79952098
::: William Bricken
https://groups.io/g/lawsofform/message/78

<QUOTE WB:>

Weird how we’ve been doing this for so many years! I look forward to what you have to say. Dunno if you’ve seen this,
may be of interest.

We built some stuff similar to logic graphs, we called distinction networks (d-nets), in the deep past. Here’s some
implementation details (1995) for asynchronous d-net computation. Ran it first on an Intel Hypercube with 16 nodes
(ugh, course-grain parallelism — a technical abstract (1987) at “The Losp Parallel Deduction Engine” (PDF) (
http://wbricken.com/pdfs/01bm/05arch/01dnets/03para-eng-ijcai.pdf ) ) and eventually migrated to a distributed network
architecture in which each node was an independent operating system, more for the convenience of doing VR than for the
elegance of fine-grain logic parallelism.

Distinction Networks
====================

Abstract. Intelligent systems can be modeled by organizationally closed networks of interacting agents. An interesting
step in the evolution from agents to systems of agents is to approach logic itself as a system of autonomous elementary
processes called distinctions. Distinction networks are directed acyclic graphs in which links represent logical
implication and nodes are autonomous agents which act in response to changes in their local environment of connectivity.
Asynchronous communication of local decisions produces global computational results without global coordination.
Biological/environmental programming uses environmental semantics, spatial syntax, and boundary transformation to
produce strongly parallel logical deduction.

Reference
=========

• Bricken, W. (July 1995), “Distinction Networks” (PDF)
( http://wbricken.com/pdfs/01bm/05arch/01dnets/04distinction-networks.pdf ) .

</QUOTE>

Dear William,

Thanks for the readings. Maybe I've just got McCulloch on the brain right now
but the things I'm reading in several groups lately keep flashing me back to
themes from his work. What you wrote on distinction networks took me back
to the beginnings of my interest in AI, especially as approached from
logical directions. There's a couple of posts on my blog where I made
an effort to point up what I regard as critical issues. I'll reshare
those next and see if I can throw more light on what's at stake.

Regards,

Jon

Jon Awbrey

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Feb 6, 2021, 4:00:23 PMFeb 6
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 30
https://inquiryintoinquiry.com/2019/08/25/animated-logical-graphs-30/

All,

This upgrades an earlier post where I began to focus more
even-handedly on the dual interpretations of Peirce's and
Spencer Brown's graphical calculi for propositional logic.

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.

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 (see also attached image)
https://inquiryintoinquiry.files.wordpress.com/2020/11/boolean-functions-on-two-variables.png

Regards,

Jon
Boolean Functions on Two Variables.png

Jon Awbrey

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Feb 11, 2021, 10:48:32 AMFeb 11
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Cf: Animated Logical Graphs • 57
http://inquiryintoinquiry.com/2021/02/11/animated-logical-graphs-57/

| All other sciences without exception depend upon
| the principles of mathematics; and mathematics
| borrows nothing from them but hints.
|
| C.S. Peirce • “Logic of Number”

| A principal intention of this essay is to separate
| what are known as algebras of logic from the subject
| of logic, and to re-align them with mathematics.
|
| G. Spencer Brown • Laws of Form

The duality between entitative and existential interpretations
of logical graphs tells us something important about the relation
between logic and mathematics. It tells us that the mathematical
forms giving structure to reasoning are deeper and more abstract
at once than their logical interpretations.

A formal duality points to a more encompassing unity, founding
a calculus of forms whose expressions can be read in alternate
ways by switching the meanings assigned to a pair of primitive
terms. Spencer Brown’s mathematical approach to Laws of Form
and the whole of Peirce’s work on the mathematics of logic
shows both thinkers were deeply aware of this principle.

Peirce explored a variety of dualities in logic which he treated on
analogy with the dualities in projective geometry. This gave rise to
formal systems where the initial constants, and thus their geometric and
graph-theoretic representations, had no uniquely fixed meanings but could be
given dual interpretations in logic.

It was in this context that Peirce’s systems of logical graphs developed,
issuing in dual interpretations of the same formal axioms which Peirce
referred to as “entitative graphs” and “existential graphs”, respectively.
He developed only the existential interpretation to any great extent, since
the extension from propositional to relational calculus appeared more natural
in that case, but whether there is any logical or mathematical reason for the
symmetry to break at that point is a good question for further research.

Resources
=========

• Duality Indicating Unity
https://inquiryintoinquiry.com/2013/01/31/duality-indicating-unity-1/

• C.S. Peirce • Logic of Number
https://inquiryintoinquiry.com/2012/09/01/c-s-peirce-logic-of-number-ms-229/

• C.S. Peirce • Syllabus • Selection 1
https://inquiryintoinquiry.com/2014/08/24/c-s-peirce-syllabus-selection-1/

References
==========

• Peirce, C.S., [Logic of Number — Le Fevre] (MS 229),
in Carolyn Eisele (ed., 1976), The New Elements of
Mathematics by Charles S. Peirce, vol. 2, 592–595.

• Spencer Brown, G. (1969), Laws of Form,
George Allen and Unwin, London, UK.

Regards,

Jon

Jon Awbrey

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Feb 11, 2021, 1:33:04 PMFeb 11
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:: Lyle Anderson
https://groups.io/g/lawsofform/message/109

Re: Brading, K., Castellani, E. and Teh, N, (2017),
“Symmetry and Symmetry Breaking”, The Stanford Encyclopedia
of Philosophy (Winter 2017), Edward N. Zalta (ed.). Online
https://plato.stanford.edu/archives/win2017/entries/symmetry-breaking/

Dear Lyle,

Thanks for the link to the article on symmetry and its breaking. I did once
take a Master's in Mathematics, specializing in combinatorics, graph theory,
and group theory. As far as the applications to logical graphs and the
calculus of indications goes, it will take careful attention to the details
of the relationship between the two interpretations recognized by Peirce and
Spencer Brown.

Both Peirce and Spencer Brown recognized the relevant duality, if they differed
in what they found most convenient to use in their development and exposition,
and most of us will emphasize one interpretation or the other as a matter of
taste or facility in a chosen application, so it requires a bit of effort to
keep the underlying unity in focus. I recently made another try at taking
a more balanced view, drawing up a series of tables in parallel columns the
way one commonly does with dual theorems in projective geometry, so I will
shortly share more of that work.

Regards,

Jon

Jon Awbrey

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Feb 21, 2021, 9:45:17 AMFeb 21
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Re: Richard J. Lipton
https://rjlipton.wordpress.com/about-me/
::: The Art Of Math
https://inquiryintoinquiry.com/2021/02/11/animated-logical-graphs-57/
https://inquiryintoinquiry.com/2021/02/11/animated-logical-graphs-58/

All,

Returning to the theme of duality and more general group-theoretic
symmetries in logical graphs, here's an improved version of the
introduction I gave two years ago.
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.

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. Boolean Functions on Two Variables (see also attached)
https://inquiryintoinquiry.files.wordpress.com/2020/11/boolean-functions-on-two-variables.png

Resources
=========

• Logic Syllabus
( https://oeis.org/wiki/Logic_Syllabus )

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

• Duality Indicating Unity
( https://inquiryintoinquiry.com/2013/01/31/duality-indicating-unity-1/ )

Regards,

Jon
Boolean Functions on Two Variables.png

Jon Awbrey

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Feb 21, 2021, 7:30:22 PMFeb 21
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Cf: Animated Logical Graphs • 60
http://inquiryintoinquiry.com/2021/02/21/animated-logical-graphs-60/
https://groups.io/g/lawsofform/message/139

<QUOTE LA:>

Definition 1. A group (G, ∗) is a set G together
with a binary operation ∗ : G × G → G satisfying
the following three conditions.

1. Associativity. For any x, y, z ∈ G,
we have (x ∗ y) ∗ z = x ∗ (y ∗ z).

2. Identity. There is an identity element e ∈ G
such that ∀ g ∈ G, we have e ∗ g = g ∗ e = g.

3. Inverses. Each element has an inverse, that is,
for each g ∈ G, there is some h ∈ G such that
g ∗ h = h ∗ g = e.

</QUOTE>

Dear Lyle,

Thanks for supplying that definition of a mathematical group.
It will afford us a wealth of useful concepts and notations as we
proceed. As you know, the above three axioms define what is properly
called an “abstract group”. Over the course of group theory’s history
this definition was gradually abstracted from the more concrete examples
of permutation groups and transformation groups initially arising in the
theory of equations and their solvability.

As it happens, the application of group theory I’ll be developing
over the next several posts will be using the more concrete type
of structure, where a transformation group G is said to “act on”
a set X by permuting its elements among themselves. In the work
we do here, each group G we contemplate will be acting on a set X
which may be taken as either one of two things, either a canonical
set of expressions in a formal language or the mathematical objects
denoted by those expressions.

What you say about deriving arithmetic, algebra, group theory,
and all the rest from the calculus of indications may well be
true, but it remains to be shown if so, and that’s aways down
the road from here.

Regards,

Jon

Jon Awbrey

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Feb 26, 2021, 2:48:13 PMFeb 26
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Cf: Animated Logical Graphs • 61
http://inquiryintoinquiry.com/2021/02/26/animated-logical-graphs-61/

Re: Richard J. Lipton • The Art Of Math
https://inquiryintoinquiry.com/2021/02/21/animated-logical-graphs-59/
https://inquiryintoinquiry.com/2021/02/21/animated-logical-graphs-60/

All,

Anything called a “duality” is naturally associated with
a transformation group of order 2, say a group G acting on
a set X. Transformation groupies generally refer to X as
a set of “points” even when the elements have additional
structure of their own, as they often do. A group of order
two has the form G = {1, t}, where 1 is the identity element
and the remaining element t satisfies the equation t² = 1,
being on that account self-inverse.

A first look at the dual interpretation of logical graphs from
a group-theoretic point of view is provided by the Table below.

Table. Peirce Duality as Group Symmetry
https://inquiryintoinquiry.files.wordpress.com/2021/02/peirce-duality-as-group-symmetry.png

The sixteen boolean functions f : B × B → B on two variables
are listed in Column 1.

Column 2 lists the elements of the set X, specifically,
the sixteen logical graphs γ giving canonical expression
to the boolean functions in Column 1.

Column 2 shows the graphs in existential order but the
order is arbitrary since only the transformations of
the set X into itself are material in this setting.

Column 3 shows the result 1γ of the group element 1 acting
on each graph γ in X, which is of course the same graph γ
back again.

Column 4 shows the result tγ of the group element t acting
on each graph γ in X, which is the entitative graph dual to
the existential graph in Column 2.

The last Row of the Table displays a statistic of considerable
interest to transformation group theorists. It is the total
incidence of fixed points, in other words, the number of points
in X left invariant or unchanged by the various group actions.
I’ll explain the significance of the fixed point parameter
next time.

Regards,

Jon
Peirce Duality as Group Symmetry.png

Jon Awbrey

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Feb 28, 2021, 12:00:24 PMFeb 28
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Cf: Animated Logical Graphs • 62
http://inquiryintoinquiry.com/2021/02/28/animated-logical-graphs-62/
https://inquiryintoinquiry.com/2021/02/26/animated-logical-graphs-61/

All,

Another way of looking at the dual interpretation of logical graphs
from a group-theoretic point of view is provided by the following Table.
In this arrangement we have sorted the rows of the previous Table to
bring together similar graphs γ belonging to the set X, the similarity
being determined by the action of the group G = {1, t}. Transformation
group theorists refer to the corresponding similarity classes as “orbits”
of the group action under consideration. The orbits are defined by the
group acting “transitively” on them, meaning elements of the same orbit
can always be transformed into one another by some group operation while
elements of different orbits cannot.

Table. Peirce Duality as Group Symmetry • Orbit Order
https://inquiryintoinquiry.files.wordpress.com/2021/02/peirce-duality-as-group-symmetry-e280a2-orbit-order.png

Scanning the Table we observe the 16 points of X fall into 10 orbits
total, divided into 4 orbits of 1 point each and 6 orbits of 2 points
each. The points in singleton orbits are called “fixed points” of the
transformation group since they are not moved, or mapped into themselves,
by all group actions. The bottom row of the Table tabulates the total
number of fixed points for the group operations 1 and t respectively.
The group identity 1 always fixes all points, so its total is 16.
The group action t fixes only the four points in singleton orbits,
giving a total of 4.

I leave it as an exercise for the reader to investigate the relationship
between the group order |G| = 2, the number of orbits 10, and the total
number of fixed points 16 + 4 = 20.

Regards,

Jon
Peirce Duality as Group Symmetry • Orbit Order.png

Jon Awbrey

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Mar 2, 2021, 1:36:25 PMMar 2
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Cf: Animated Logical Graphs • 63
http://inquiryintoinquiry.com/2021/03/02/animated-logical-graphs-63/
https://inquiryintoinquiry.com/2021/02/28/animated-logical-graphs-62/

All,

We’ve been using the duality between entitative and existential
interpretations of logical graphs to get a handle on the mathematical
forms pervading logical laws. A few posts ago we took up the tools
of groups and symmetries and transformations to study the duality
and we looked to the space of 2-variable boolean functions as a
basic training grounds. On those grounds the translation between
interpretations presents as a group G of order two acting on
a set X of sixteen logical graphs denoting boolean functions.

Last time we arrived at a Table showing how the group G partitions
the set X into ten orbits of logical graphs. Here again is that Table.
I invited the reader to investigate the relationship between the
group order |G| = 2, the number of orbits 10, and the total number
of fixed points 16 + 4 = 20. In the present case the product of the
group order (2) and the number of orbits (10) is equal to the sum of
the fixed points (20) — Is that just a fluke? If not, why so? And
does it reflect a general rule?

We can make a beginning toward answering those questions by inspecting
the “incidence relation” of fixed points and orbits in the Table above.
Each singleton orbit accumulates two hits, one from the group identity
and one from the other group operation. But each doubleton orbit also
accumulates two hits, since the group identity fixes both of its two
points. Thus all the orbits are double-counted by counting the
incidence of fixed points and orbits. In sum, dividing the total
number of fixed points by the order of the group brings us back
to the exact number of orbits.

Regards,

Jon
Peirce Duality as Group Symmetry • Orbit Order.png

Jon Awbrey

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Mar 4, 2021, 9:42:29 AMMar 4
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Cf: Animated Logical Graphs • 64
https://inquiryintoinquiry.com/2021/03/04/animated-logical-graphs-64/
::: John M.
https://groups.io/g/lawsofform/message/152

Dear John,

It occurred to me a picture might save a few thousand words.
A good place to start is the following Table from an earlier
post on my blog.

The smart way to deal with parens + character strings in computing
is to parse them into graph-theoretic data structures and then work
on those instead of the strings themselves. Usually one gets some
sort of tree structures for the “parse graphs”. In my work on logical
graphs I eventually came to use the more general species of structure
graph theorists call “cactus graphs” or “cacti”.

Referring to the Table —

Table. Logical Graphs • Entitative and Existential Venn Diagrams
https://inquiryintoinquiry.files.wordpress.com/2020/11/logical-graphs-e280a2-entitative-and-existential-venn-diagrams.png

• Column 1 shows the logical graphs I use for the sixteen boolean
functions on two variables, with the string forms underneath.
The “cactus string” obtained by “traversing” the cactus graph
uses parens + commas + characters in forms like “(x, y)” and
“((x, y))”.

• Column 2 shows the venn diagram associated with the “entitative
interpretation” of the graph in Column 1. This is the interpretation
C.S. Peirce used in his earlier work on “enitative graphs” and the one
Spencer Brown used in his Laws of Form.

• Column 3 shows the venn diagram associated with the “existential
interpretation” of the graph in Column 1. This is the interpretation
C.S. Peirce used in his later work on “existential graphs”.

Take a gander at all that and I’ll discuss more tomorrow …

Regards,

Jon
Logical Graphs • Entitative and Existential Venn Diagrams.png

Jon Awbrey

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Mar 4, 2021, 6:00:18 PMMar 4
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Cf: Animated Logical Graphs • 65
http://inquiryintoinquiry.com/2021/03/04/animated-logical-graphs-65/

| Thus, what looks to us like a sphere of scientific knowledge
| more accurately should be represented as the inside of a highly
| irregular and spiky object, like a pincushion or porcupine, with
| very sharp extensions in certain directions, and virtually no
| knowledge in immediately adjacent areas. If our intellectual
| gaze could shift slightly, it would alter each quill's direction,
| and suddenly our entire reality would change.
|
| Herbert J. Bernstein • “Idols of Modern Science”
| https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#The_Cactus_Patch
::: Lyle Anderson https://groups.io/g/lawsofform/message/155

Re: Richard J. Lipton • The Art Of Math
https://rjlipton.wordpress.com/2020/11/12/the-art-of-math/

Dear Lyle,

Thanks for the link to the Wikipedia article on Cactus Graphs
( https://en.wikipedia.org/wiki/Cactus_graph ), which I found
surprisingly good for that venue. I was pleased to see it
mentioned the role my own first teacher in graph theory,
Frank Harary ( https://en.wikipedia.org/wiki/Frank_Harary ),
played in the history of cactus graphs. Frank co-authored
Graphical Enumeration and many papers with Ed Palmer, my
second teacher in graph theory and later my advisor in
grad school.

Synchronicity being what it is, one the jobs I worked on between
my undergrad decade and my first crack at grad school was scanning
and measuring particle interactions on bubble-chamber filmstrips in
a high-energy physics lab, so I got a gadshillion gammas burned in
my brain from that time.

Regards,

Jon

Resources
=========

• Logic Syllabus ( https://oeis.org/wiki/Logic_Syllabus )

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

Jon Awbrey

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Mar 9, 2021, 3:40:18 PMMar 9
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Cf: Animated Logical Graphs • 66
http://inquiryintoinquiry.com/2021/03/09/animated-logical-graphs-66/

Re: Richard J. Lipton • The Art Of Math
https://rjlipton.wordpress.com/2020/11/12/the-art-of-math/
Re: Animated Logical Graphs
https://inquiryintoinquiry.com/2021/02/26/animated-logical-graphs-61/
https://inquiryintoinquiry.com/2021/02/28/animated-logical-graphs-62/
https://inquiryintoinquiry.com/2021/03/02/animated-logical-graphs-63/

All,

Once we bring the dual interpretations of logical graphs to the
same Table and relate their parleys to the same objects, it is
clear we are dealing with a triadic sign relation of the sort
taken up in C.S. Peirce’s “semiotics” or theory of signs.

A “sign relation” L ⊆ O × S × I, as a set L embedded in a
cartesian product O × S × I, tells how the “signs” in S
and the “interpretant signs” in I correlate with the
“objects” or objective situations in O.

There are many ways of using sign relations to model
various types of sign-theoretic situations and processes.
The following cases are often seen.

• Some sign relations model co‑referring signs or transitions
between signs within a single language or symbol system.
In that event L ⊆ O × S × I has S = I.

• Other sign relations model translations between different
languages or different interpretations of the same language,
in other words, different ways of referring the same set of
signs to a shared object domain.

The next Table extracts the sign relation L ⊆ O × S × I involved
in switching between existential and entitative interpretations
of logical graphs.

Table. Peirce Duality as Sign Relation (also attached)
https://inquiryintoinquiry.files.wordpress.com/2021/03/peirce-duality-as-sign-relation.png

• Column 1 shows the object domain O as the
set of 16 boolean functions on 2 variables.

• Column 2 shows the sign domain S as a representative set
of logical graphs denoting the objects in O according to
the existential interpretation.

• Column 3 shows the interpretant domain I as the same set
of logical graphs denoting the objects in O according to
the entitative interpretation.

Resources
=========

C.S. Peirce • Logic and Signs
https://inquiryintoinquiry.com/2012/06/01/c-s-peirce-on-the-definition-of-logic/

C.S. Peirce • Logic as Semiotic
https://inquiryintoinquiry.com/2012/06/04/c-s-peirce-logic-as-semiotic/

Sign Relation
https://oeis.org/wiki/Sign_relation

Triadic Relation
https://oeis.org/wiki/Triadic_relation

Regards,

Jon

Peirce Duality as Sign Relation.png

Jon Awbrey

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Mar 21, 2021, 12:00:32 PMMar 21
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Cf: Animated Logical Graphs • 14
https://inquiryintoinquiry.com/2019/05/28/animated-logical-graphs-14/

All,

One of the things I recently added to the Survey of Animated Graphs
( https://inquiryintoinquiry.com/2020/08/23/survey-of-animated-logical-graphs-3/ )
is an earlier piece of work titled “Futures Of Logical Graphs” (FOLG)
( https://oeis.org/wiki/Futures_Of_Logical_Graphs ) which takes up a
number of difficult issues in more detail than I’ve found the ability or
audacity to do since. Among other things, 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 forces driving
that transition.

Regards,

Jon

Jon Awbrey

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Mar 24, 2021, 7:40:14 AMMar 24
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Cf: Animated Logical Graphs • 15
https://inquiryintoinquiry.com/2019/06/30/animated-logical-graphs-15/

All,

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

Jon Awbrey

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Mar 24, 2021, 1:00:12 PMMar 24
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Cf: Animated Logical Graphs • 16
https://inquiryintoinquiry.com/2019/07/07/animated-logical-graphs-16/

All,

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, 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:

<QUOTE CSP:>

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.

Regards,

Jon

Jon Awbrey

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Mar 25, 2021, 10:00:39 AMMar 25
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Animated Logical Graphs • 17
https://inquiryintoinquiry.com/2019/07/09/animated-logical-graphs-17/

All,

To get a clearer view of the relation between primary arithmetic
and primary algebra consider the following extremely simple
algebraic expression:

Figure 1. 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 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 formal value, and of which values
we know but two. Thus, the given algebraic expression varies
between these two choices:

Figure 2. 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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Mar 25, 2021, 3:28:32 PMMar 25
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Cf: Animated Logical Graphs • 17
https://inquiryintoinquiry.com/2019/07/09/animated-logical-graphs-17/

All,

Note. There were a few clumsy and potentially misleading
constructions in my last email. I've fixed them in the
blog post linked above and in the transcript below.



To get a clearer view of the relation between primary arithmetic
and primary algebra consider the following extremely simple
algebraic expression.

Figure 1. 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.

In effect, 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 formal value, and of which values we know but two. Putting it all
together, the algebraic expression “(a)” varies between the following
two choices.

Figure 2. 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 arithmetic constant “( )”
in the place of the operand “a” in the algebraic expression “(a)”.
Box (A).jpg
Box (),(()).jpg

Mauro Bertani

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Mar 25, 2021, 3:39:07 PMMar 25
to ontolo...@googlegroups.com
Dear Jon,
Maybe (a) = 0 = false.
 a = n = true?
Regards
Mauro

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Mauro Bertani

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Mar 25, 2021, 3:45:45 PMMar 25
to ontolo...@googlegroups.com
Dear Jon,
Maybe in logic win the singularity, in arithmetic the compound
Regards
Màuro

Mauro Bertani

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Mar 25, 2021, 3:56:26 PMMar 25
to ontolo...@googlegroups.com
Dear Jon,
Or maybe the compound in logic is similar at the law of time
Regards
Mauro

Jon Awbrey

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Mar 25, 2021, 4:02:56 PMMar 25
to Mauro Bertani, Ontolog Forum
Dear Mauro,

Sorry, I'm not sure what you're saying. Here I'm using
a parenthesized rendition of the forms Spencer Brown and
Peirce used in their logical graphs. In contexts where
I have better formatting I use a different typeface for
the logical parentheses to avoid confusing them with the
ordinary sort.

For example, in the so-called existential interpretation
Peirce eventually settled on for propositional calculus:

" " is "true".

"( )" is "false".

"a b" is "a and b"

"(a)" is "not a"

"((a)(b)) is "a or b"

"(a(b))" is "a implies b"

etc.

See the following article:

https://inquiryintoinquiry.com/2008/09/19/logical-graphs-2/

Regards,

Jon

Jon Awbrey

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Mar 26, 2021, 1:51:02 PMMar 26
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 18
https://inquiryintoinquiry.com/2019/07/10/animated-logical-graphs-18/

Re: Animated Logical Graphs • 17
https://inquiryintoinquiry.com/2019/07/09/animated-logical-graphs-17/

All,

Last time we contemplated 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 1. 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 arithmetic constant “( )”
in the place 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)”?

Evidently, a variation between the absence and the presence of the
operator “( )” in the algebraic expression “(a)” refers to a variation
between the algebraic expression “a” and the algebraic expression “(a)”,
somewhat as pictured below.

Figure 2. 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?

Regards,

Jon
Box (A)={(),(())}.jpg
Box A que={A,(A)}.jpg

Mauro Bertani

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Mar 26, 2021, 4:16:52 PMMar 26
to ontolo...@googlegroups.com
Dear Jon,
The duality is to write logic with and and not or with or and not. This is what are you saying¿
Regards
Mauro

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Jon Awbrey

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Mar 27, 2021, 10:42:35 AMMar 27
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 19
https://inquiryintoinquiry.com/2019/07/11/animated-logical-graphs-19/

All,

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 1. Cactus Graph (q)_p = {q,(q)}
https://inquiryintoinquiry.files.wordpress.com/2019/07/box-q-que-pqq.jpg

It was obvious from the outset 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

Mauro Bertani

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Mar 27, 2021, 12:04:05 PMMar 27
to ontolo...@googlegroups.com, Cybernetic Communications, Laws of Form, Peirce List, Structural Modeling, SysSciWG
Dear Jon, 
I read something similar in https://g.co/kgs/beC6Jq. But here it is only sketchy. The variabile is the result of an evaluation of an expression. It 's not so obviuos. The base of the logic
Regards
Mauro

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Jon Awbrey

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Mar 27, 2021, 4:16:13 PMMar 27
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 67
http://inquiryintoinquiry.com/2021/03/27/animated-logical-graphs-67/

Re: Differential Propositional Calculus • Discussion 4
https://inquiryintoinquiry.com/2021/03/20/differential-propositional-calculus-discussion-4/

Re: Laws of Form • Lyle Anderson
https://groups.io/g/lawsofform/message/198

Re: Peirce List • Mauro Bertani
https://list.iupui.edu/sympa/arc/peirce-l/2021-03/msg00134.html

Dear Lyle,

Yes, the ability to work with functions as “first class citizens”,
as we used to say, is one of the things making lambda calculus at
the theoretical level and Lisp at the practical level so nice.
All of which takes us straight into Curry-Howard-ville ...

Dear Mauro,

That is the right ball park, functional calculi and all that.
I haven't been taking time out to mention all the players apart
from Peirce and Spencer Brown — Boole, Frege, Schönfinkel, Curry,
Howard, and others — because I'm still in the middle of tackling
Helmut Raulien's question about the link between cactus graphs
and differential logic. At any rate I'll be focused on that
for a while longer.

Regards,

Jon

Jon Awbrey

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Mar 28, 2021, 10:40:45 AMMar 28
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Animated Logical Graphs • 20
https://inquiryintoinquiry.com/2019/07/11/animated-logical-graphs-20/

All,

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 1. Transitional Form (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 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 this point on.

Regards,

Jon
Box Q qua P.jpg

Jon Awbrey

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Mar 28, 2021, 4:18:50 PMMar 28
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 21
https://inquiryintoinquiry.com/2019/07/12/animated-logical-graphs-21/

All,

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 1. Control Form (a)_b
https://inquiryintoinquiry.files.wordpress.com/2019/07/box-a-quo-b.jpg

What happened here is this. Our contemplation of a constant operator
as being potentially variable gave rise to the contemplation of a newly
introduced but otherwise quite ordinary operand variable, albeit in a
newly-fashioned 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.

We may regard this development as marking a form of “controlled reflection”,
or a form of “reflective control”. From here on out we'll use the inline
syntax “(a, b)” for the corresponding operation on two variables, whose
operation table is given below.

Table 1. Operation Table for (a, b)
https://inquiryintoinquiry.files.wordpress.com/2019/07/table-ab-space-cross.png

• The Entitative Interpretation (En),
where Space = False and Cross = True,
calls this operation “equivalence”.

• The Existential Interpretation (Ex),
where Space = True and Cross = False,
calls this operation “distinction”.

Regards,

Jon
Box A quo B.jpg
Table (A,B) Space Cross.png

Mauro Bertani

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Mar 29, 2021, 1:10:24 PM (13 days ago) Mar 29
to ontolo...@googlegroups.com, Cybernetic Communications, Laws of Form, Peirce List, Structural Modeling, SysSciWG
Dear Jon,
By those I have understood this is a logic of differences. If a and b are the same the evaluation (a,b) is true but if they are different it is false ( in En and the opposite in Ex). My answer is: a and b are the same concept ( evaluation of the same equation p) in different time ( for example p at t0 and t1) or different concept ( evaluation of different equation p1 and p2). And if they are different concept what does it mean the relation ( a,b) ?
Thanks in advance
Regards
Mauro


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Jon Awbrey

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Mar 29, 2021, 6:40:44 PM (13 days ago) Mar 29
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 68
http://inquiryintoinquiry.com/2021/03/29/animated-logical-graphs-68/

Re: Ontolog Forum
https://groups.google.com/g/ontolog-forum/c/ru_NmgntJq0
::: Mauro Bertani
https://groups.google.com/g/ontolog-forum/c/ru_NmgntJq0/m/RIwyFDXoBgAJ

Dear Mauro,

Let's take a another look at the Table we reached at the end of Episode 21.
( https://inquiryintoinquiry.com/2019/07/12/animated-logical-graphs-21/ ) .

Table 1. Formal Operation Table (a, b)
https://inquiryintoinquiry.files.wordpress.com/2021/03/formal-operation-table-a-b.png

I call it a Formal Operation Table — rather than, say, a Truth Table —
because it describes the operation of mathematical forms preceding the
stage of logical interpretation. I know the word “formal” tends to get
overworked past the point of semantic fatigue but I can still hope to
revive it a little. We'll use other labels for Table entries at other
times but I tried this time to mitigate interpretive bias by choosing
a mix of senses from both Peirce and Spencer Brown.

Entering the stage of logical interpretation,
we arrive at the following two options.

The entitative interpretation of (a, b)
produces the truth table for “logical equality”.

Table 2. En (a, b)
https://inquiryintoinquiry.files.wordpress.com/2021/03/truth-table-en-a-b.png

The existential interpretation of (a, b)
produces the truth table for “logical inequality”,
also known as “exclusive disjunction”.

Table 3. Ex (a, b)
https://inquiryintoinquiry.files.wordpress.com/2021/03/truth-table-ex-a-b.png

Resources
=========

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
Formal Operation Table (a, b).png
Truth Table En (a, b).png
Truth Table Ex (a, b).png

Jon Awbrey

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Mar 31, 2021, 12:36:19 PM (11 days ago) Mar 31
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 69
https://inquiryintoinquiry.com/2021/03/30/animated-logical-graphs-69/

Re: Richard J. Lipton • The Art Of Math
https://rjlipton.wpcomstaging.com/2020/11/12/the-art-of-math/
Re: Animated Logical Graphs
https://inquiryintoinquiry.com/2021/02/11/animated-logical-graphs-57/
• • •
https://inquiryintoinquiry.com/2021/03/09/animated-logical-graphs-66/

All,

It's been a while, so a bit of review and refocus may be in order ...

“I know what you mean but I say it another way” — it’s a thing
I find myself saying often enough, if only under my breath, to
rate an acronym for it ☞ IKWYMBISIAW ☜ and not too coincidentally
it’s a rubric of relevance to many situations in semiotics where
sundry manners of speaking and thinking converge, more or less,
on the same patch of pragmata.

We encountered just such a situation in our exploration of the duality
between entitative and existential interpretations of logical graphs.
The two interpretations afford distinct but equally adequate ways of
reasoning about a shared objective domain. To cut our teeth on a
simple but substantial example of an object domain, we picked the
space of boolean functions or propositional forms on two variables.
This brought us to the following Table, highlighting the sign relation
L ⊆ O × S × I involved in switching between existential and entitative
interpretations of logical graphs.

Table 1. Peirce Duality as Sign Relation (also attached)
Peirce Duality as Sign Relation.png

Jon Awbrey

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Mar 31, 2021, 4:08:12 PM (11 days ago) Mar 31
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 22
https://inquiryintoinquiry.com/2019/07/22/animated-logical-graphs-22/

The step of controlled reflection we took in the previous post
( https://inquiryintoinquiry.com/2019/07/12/animated-logical-graphs-21/ )
can be repeated at will, as suggested by the following series of forms.

Figure 1. 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 “(x₁, x₂, …, xₖ)”. 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.

Regards,

Jon
Reflective Series (A) to (A,B,C,D).jpg

Jon Awbrey

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Apr 2, 2021, 9:16:41 AM (9 days ago) Apr 2
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 23
https://inquiryintoinquiry.com/2019/07/23/animated-logical-graphs-23/

All,

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 are extended through successive steps
of controlled reflection, translate into syntactic strings and rooted
cactus graphs.

Table 1. Syntactic Correspondences
https://inquiryintoinquiry.files.wordpress.com/2019/07/syntactic-correspondences.png

Regards,

Jon

Syntactic Correspondences.png

Jon Awbrey

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Apr 7, 2021, 10:00:17 AM (4 days ago) Apr 7
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Animated Logical Graphs • 70
http://inquiryintoinquiry.com/2021/04/07/animated-logical-graphs-70/

Re: Richard J. Lipton • The Art Of Math
https://rjlipton.wpcomstaging.com/2020/11/12/the-art-of-math/
Re: Animated Logical Graphs
https://inquiryintoinquiry.com/2021/03/09/animated-logical-graphs-66/
https://inquiryintoinquiry.com/2021/03/30/animated-logical-graphs-69/

All,

Our study of the duality between entitative and existential
interpretations of logical graphs has brought to light its fully
sign-relational character. We can see this in the sign relation
linking an object domain with two sign domains, whose signs denote
the objects in two distinct ways. We illustrated the general principle
using an object domain consisting of the sixteen boolean functions on
two variables and a pair of sign domains consisting of representative
logical graphs for those functions, as shown in the following Table.

Table 1. Peirce Duality as Sign Relation
https://inquiryintoinquiry.files.wordpress.com/2021/03/peirce-duality-as-sign-relation.png

• Column 1 shows the object domain O as the
set of 16 boolean functions on 2 variables.

• Column 2 shows the sign domain S as a representative set
of logical graphs denoting the objects in O according to
the existential interpretation.

• Column 3 shows the interpretant domain I as the same set
of logical graphs denoting the objects in O according to
the entitative interpretation.

Additional aspects of the sign relation's structure
can be brought out by sorting the Table in accord with
the orbits induced on the object domain by the action of
the transformation group inherent in the dual interpretations.
Performing that sort produces the following Table.

Table 2. Peirce Duality as Sign Relation • Orbit Order
https://inquiryintoinquiry.files.wordpress.com/2021/03/peirce-duality-as-sign-relation-e280a2-orbit-order.png

That's enough bytes to chew on for one post — we'll
extract more information from these Tables next time.

Resources
=========

Logic Syllabus
https://oeis.org/wiki/Logic_Syllabus
Survey of Animated Logical Graphs
https://inquiryintoinquiry.com/2020/08/23/survey-of-animated-logical-graphs-3/

Survey of Semiotics, Semiosis, Sign Relations
https://inquiryintoinquiry.com/2019/10/29/survey-of-semiotics-semiosis-sign-relations-1/

Regards,

Jon
Peirce Duality as Sign Relation.png
Peirce Duality as Sign Relation • Orbit Order.png
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