Animated Logical Graphs

[Jon Awbrey]

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

[Jon Awbrey]

Cf: Animated Logical Graphs • 46
http://inquiryintoinquiry.com/2020/11/15/animated-logical-graphs-46/

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/

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

[Jon Awbrey]

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

[Jon Awbrey]

Cf: Animated Logical Graphs • 48
http://inquiryintoinquiry.com/2020/11/30/animated-logical-graphs-48/

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/

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

[Jon Awbrey]

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

[Jon Awbrey]

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/

Re: Animated Logical Graphs
30. https://inquiryintoinquiry.com/2019/08/25/animated-logical-graphs-30/

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

[Jon Awbrey]

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

[Jon Awbrey]

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]

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.

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

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 )

• 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

[Jon Awbrey]

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

[Jon Awbrey]

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

All,

I updated my last Survey page on Animated Logical Graphs
and added links to the series of posts on CSP, GSB, & Me.

https://inquiryintoinquiry.com/2017/07/19/charles-sanders-peirce-george-spencer-brown-and-me/
https://inquiryintoinquiry.com/2017/07/20/charles-sanders-peirce-george-spencer-brown-and-me-1/
https://inquiryintoinquiry.com/2017/07/21/charles-sanders-peirce-george-spencer-brown-and-me-2/
https://inquiryintoinquiry.com/2017/07/31/charles-sanders-peirce-george-spencer-brown-and-me-3/
•••
https://inquiryintoinquiry.com/2017/08/25/charles-sanders-peirce-george-spencer-brown-and-me-10/
https://inquiryintoinquiry.com/2021/01/18/charles-sanders-peirce-george-spencer-brown-and-me-11/
https://inquiryintoinquiry.com/2021/01/25/charles-sanders-peirce-george-spencer-brown-and-me-12/
https://inquiryintoinquiry.com/2021/01/26/charles-sanders-peirce-george-spencer-brown-and-me-13/

Regards,

Jon

[Jon Awbrey]

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]

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]

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

[Jon Awbrey]

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]

Cf: Animated Logical Graphs • 58
http://inquiryintoinquiry.com/2021/02/11/animated-logical-graphs-58/

Re: Laws of Form
https://groups.io/g/lawsofform/topic/animated_logical_graphs/79952098

:: 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]

Cf: Animated Logical Graphs • 59
http://inquiryintoinquiry.com/2021/02/21/animated-logical-graphs-59/

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

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.

Cf: Animated Logical Graphs • 30
https://inquiryintoinquiry.com/2019/08/25/animated-logical-graphs-30/

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

[Jon Awbrey]

Cf: Animated Logical Graphs • 60
http://inquiryintoinquiry.com/2021/02/21/animated-logical-graphs-60/

Re: Laws of Form
https://groups.io/g/lawsofform/topic/animated_logical_graphs/79952098
::: Lyle Anderson

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]

Cf: Animated Logical Graphs • 61
http://inquiryintoinquiry.com/2021/02/26/animated-logical-graphs-61/

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/2021/02/21/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

[Jon Awbrey]

Cf: Animated Logical Graphs • 62
http://inquiryintoinquiry.com/2021/02/28/animated-logical-graphs-62/

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/2021/02/21/animated-logical-graphs-59/
https://inquiryintoinquiry.com/2021/02/21/animated-logical-graphs-60/

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

[Jon Awbrey]

Cf: Animated Logical Graphs • 63
http://inquiryintoinquiry.com/2021/03/02/animated-logical-graphs-63/

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/2021/02/21/animated-logical-graphs-59/
https://inquiryintoinquiry.com/2021/02/21/animated-logical-graphs-60/
https://inquiryintoinquiry.com/2021/02/26/animated-logical-graphs-61/

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.

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

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

[Jon Awbrey]

Cf: Animated Logical Graphs • 64
https://inquiryintoinquiry.com/2021/03/04/animated-logical-graphs-64/

Re: Laws of Form
https://groups.io/g/lawsofform/topic/animated_logical_graphs/79952098

::: 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

[Jon Awbrey]

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

Re: Laws of Form
https://groups.io/g/lawsofform/topic/animated_logical_graphs/79952098

::: 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]

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

[Jon Awbrey]

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]

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

[Jon Awbrey]

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]

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

[Jon Awbrey]

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)”.

[Mauro Bertani]

Dear Jon,

Maybe (a) = 0 = false.

 a = n = true?

Regards

Mauro

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[Mauro Bertani]

Dear Jon,

Maybe in logic win the singularity, in arithmetic the compound

Regards

Màuro

[Mauro Bertani]

Dear Jon,

Or maybe the compound in logic is similar at the law of time

Regards

Mauro

[Jon Awbrey]

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]

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

[Mauro Bertani]

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

--
All contributions to this forum are covered by an open-source license.
For information about the wiki, the license, and how to subscribe or
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[Jon Awbrey]

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

[Mauro Bertani]

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]

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]

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

[Jon Awbrey]

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

[Mauro Bertani]

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]

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

[Jon Awbrey]

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)

[Jon Awbrey]

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

[Jon Awbrey]

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

[Jon Awbrey]

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

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

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

[Jon Awbrey]

Cf: Animated Logical Graphs • 24
https://inquiryintoinquiry.com/2019/07/26/animated-logical-graphs-24/

Re: Ontolog Forum • Joseph Simpson
https://groups.google.com/g/ontolog-forum/c/wF03K5KG1vQ/m/ZRlQdavJBgAJ

Dear Joe,

Boolean functions f : Bⁿ → 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 happened to run across an enlightening article about it just
yesterday and I did a while ago begin an exploration into what
appears to be a related question, Péter 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/2014/09/07/frankl-my-dear-1/ •••
https://inquiryintoinquiry.com/2014/11/24/frankl-my-dear-12/

R.J. Lipton and K.W. Regan • Discrepancy Games and Sensitivity
https://rjlipton.wpcomstaging.com/2019/07/25/discrepancy-games-and-sensitivity/

Regards,

Jon

[Jon Awbrey]

Cf: Animated Logical Graphs • 25
https://inquiryintoinquiry.com/2019/07/26/animated-logical-graphs-25/

Re: Animated Logical Graphs • 23
https://inquiryintoinquiry.com/2019/07/23/animated-logical-graphs-23/

All,

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.files.wordpress.com/2021/04/formal-operation-table-a-b-c-e280a2-variant-1.png

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.files.wordpress.com/2021/04/formal-operation-table-a-b-c-e280a2-variant-2.png

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

[Jon Awbrey]

Cf: Animated Logical Graphs • 26
https://inquiryintoinquiry.com/2019/07/28/animated-logical-graphs-26/

All,

This post and the next wrap up the Themes and Variations
section of my speculation on Futures Of Logical Graphs.

( https://oeis.org/wiki/Futures_Of_Logical_Graphs#Themes_and_variations )

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

Regards,

Jon

[Jon Awbrey]

Cf: Animated Logical Graphs • 27
https://inquiryintoinquiry.com/2019/07/31/animated-logical-graphs-27/

Re: Animated Logical Graphs • 26
https://inquiryintoinquiry.com/2019/07/28/animated-logical-graphs-26/

All,

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

Regards,

Jon

[Jon Awbrey]

Cf: Animated Logical Graphs • 28
https://inquiryintoinquiry.com/2019/08/03/animated-logical-graphs-28/

Re: R.J. Lipton and K.W. Regan • Discrepancy Games and Sensitivity
https://rjlipton.wpcomstaging.com/2019/07/25/discrepancy-games-and-sensitivity/

Re: Ontolog Forum • Joseph Simpson
https://groups.google.com/g/ontolog-forum/c/wF03K5KG1vQ/m/ZRlQdavJBgAJ

Re: Animated Logical Graphs • 24
https://inquiryintoinquiry.com/2019/07/26/animated-logical-graphs-24/

All,

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.

<QUOTE JA:>
Just by way of a general observation, concepts like discrepancy,
influence, sensitivity, and the like 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.
</QUOTE>

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 Survey of related resources on the
following page.

• Survey of Differential Logic
https://inquiryintoinquiry.com/2020/02/08/survey-of-differential-logic-2/

Regards,

Jon

[Jon Awbrey]

Cf: Animated Logical Graphs • 29
https://inquiryintoinquiry.com/2019/08/11/animated-logical-graphs-29/

Re: Animated Logical Graphs • 21
https://inquiryintoinquiry.com/2019/07/12/animated-logical-graphs-21/

Re: Ontolog Forum • Joseph Simpson

https://groups.google.com/g/ontolog-forum/c/wF03K5KG1vQ/m/ssNdCXssEgAJ

<QUOTE JS:>

I tend to view equivalence and distinction as relationships
as opposed to operations. I do not know if this makes any
significant difference in this context.

</QUOTE>

Dear Joe,

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 k be a positive integer.
Then B^k is the set of k-tuples of elements of B.

• A “k-variable boolean function” f is a mapping from B^k to B.
A function of that type is typically notated as f : B^k → B.

• A “k-place boolean relation” L is a subset of B^k.
A relation of that type is notated as L ⊆ 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” (or “characteristic function”) f_L : B^k → B
defined by the rule that f_L (x) = 1 if x is in L and f_L (x) = 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).

Resources
=========

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

• Relation Theory
https://oeis.org/wiki/Relation_theory

• Boolean Function
https://oeis.org/wiki/Boolean_function

• Boolean-Valued Function
https://oeis.org/wiki/Boolean-valued_function

Regards,

Jon

[Jon Awbrey]

Cf: Animated Logical Graphs • 71
http://inquiryintoinquiry.com/2021/04/18/animated-logical-graphs-71/

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/2021/03/02/animated-logical-graphs-63/
https://inquiryintoinquiry.com/2021/03/09/animated-logical-graphs-66/
https://inquiryintoinquiry.com/2021/03/30/animated-logical-graphs-69/
https://inquiryintoinquiry.com/2021/04/07/animated-logical-graphs-70/

All,

Our investigation has brought us to the point of seeing both
a transformation group and a triadic sign relation in the duality
between entitative and existential interpretations of logical graphs.

Given the level of the above abstractions it helps to anchor them in
concrete structural experience. In that spirit we’ve been pursuing
the case of a group action T : X → X and a sign relation L ⊆ O×X×X
where O is the set of boolean functions on two variables and X is
a set of logical graphs denoting those functions. We drew up a
Table combining the aspects of both structures and sorted it
according to the orbits T induces on X and consequently on O.

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

Resources
=========

[Jon Awbrey]

Cf: Animated Logical Graphs • 72
http://inquiryintoinquiry.com/2021/04/22/animated-logical-graphs-72/

Re: Animated Logical Graphs
https://inquiryintoinquiry.com/2021/02/28/animated-logical-graphs-62/
https://inquiryintoinquiry.com/2021/03/02/animated-logical-graphs-63/
•••
https://inquiryintoinquiry.com/2021/04/07/animated-logical-graphs-70/
https://inquiryintoinquiry.com/2021/04/18/animated-logical-graphs-71/

Turning again to our Table of Orbits let's see what we can learn
about the structure of the sign relational system in view.

We saw in Episode 62 that the transformation group T = {1, t} partitions
the set X of 16 logical graphs and also the set O of 16 boolean functions
into 10 orbits: 4 orbits of size 1 each and 6 orbits of size 2 each.

Points in singleton orbits are called “fixed points” of the
transformation group X → X since they are left unchanged, or
changed into themselves, by all group actions. Viewed in the
frame of the sign relation L ⊆ O×X×X , where the transformations
in T are literally “translations” in the linguistic sense, the
fixed points are the signs in X which Existential Interpreters
and Entitative Interpreters all use alike to denote the same
objects in O.

Table 1. Peirce Duality as Sign Relation • Orbit Order

[Jon Awbrey]

Cf: Animated Logical Graphs • 73
https://inquiryintoinquiry.com/2021/04/24/animated-logical-graphs-73/

Re: Animated Logical Graphs
https://inquiryintoinquiry.com/2021/04/22/animated-logical-graphs-72/

All,

Last time we took up the four singleton orbits in the action
of T on X and saw each consists of a single logical graph which
T fixes, preserves, or transforms into itself. On that account
these 4 logical graphs are said to be “self-dual” or “T-invariant”.

In general terms, it is useful to think of the entitative and
existential interpretations as two formal languages which happen
to use the same set of signs, each in its own way, to denote the
same set of formal objects. Then T defines the translation between
languages and the self-dual logical graphs are the points where the
languages coincide, where the same signs denote the same objects in
both. Such constellations of “fixed stars” are indispensable to
navigation between languages, as every argot-naut discovers in time.

Returning to the case at hand, where T acts on a selection of 16
logical graphs for the 16 boolean functions on two variables, the
following Table shows the values of the denoted boolean function
f : B × B → B for each of the self-dual logical graphs.

Table 1. Self-Dual Logical Graphs
https://inquiryintoinquiry.files.wordpress.com/2021/04/self-dual-logical-graphs.png

The functions indexed here as f₁₂ and f₁₀ are known as
the “coordinate projections” (x, y) ↦ x and (x, y) ↦ y
on the first and second coordinates, respectively, and
the functions indexed as f₃ and f₅ are the negations
(x, y) ↦ ¬x and (x, y) ↦ ¬y of those projections,
respectively.

Resources
=========

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

Truth Tables
https://oeis.org/wiki/Truth_table

Zeroth Order Logic
https://oeis.org/wiki/Zeroth_order_logic

[Jon Awbrey]

Cf: Animated Logical Graphs • 74
http://inquiryintoinquiry.com/2021/04/30/animated-logical-graphs-74/

Re: Animated Logical Graphs
https://inquiryintoinquiry.com/2021/04/07/animated-logical-graphs-70/
https://inquiryintoinquiry.com/2021/04/18/animated-logical-graphs-71/
https://inquiryintoinquiry.com/2021/04/22/animated-logical-graphs-72/
https://inquiryintoinquiry.com/2021/04/24/animated-logical-graphs-73/

All,

After the four orbits of self-dual logical graphs we come to six orbits
of dual pairs. In no particular order of importance, we may start by
considering the following two.

• The logical graphs for the “constant functions” f₁₅ and f₀
are dual to each other.

• The logical graphs for the “ampheck functions” f₇ and f₁
are dual to each other.

The values of the constant and ampheck functions for each (x, y)
in B × B and the text expressions of their logical graphs are
given in the following Table.

Table 1. Constants and Amphecks
https://inquiryintoinquiry.files.wordpress.com/2021/04/constants-and-amphecks.png

Resources
=========

• Logic Syllabus ( https://oeis.org/wiki/Logic_Syllabus )
• Amphecks ( https://oeis.org/wiki/Ampheck )
• Truth Tables ( https://oeis.org/wiki/Truth_table )
• Zeroth Order Logic ( https://oeis.org/wiki/Zeroth_order_logic )

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

Regards,

Jon

[Jon Awbrey]

Cf: Survey of Animated Logical Graphs • 4
http://inquiryintoinquiry.com/2021/05/01/survey-of-animated-logical-graphs-4/

All,

I've updated my Survey of resources related to Logical Graphs.
Please see the blog post linked above for the complete list —
I'll give just the headings and a selection of links below.

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
• Logical Graphs
• Minimal Negation Operators
• Propositional Equation Reasoning Systems

Examples
=========

• Peirce’s Law
https://inquiryintoinquiry.com/2008/10/06/peirces-law/

• Praeclarum Theorema
https://inquiryintoinquiry.com/2008/10/05/praeclarum-theorema/

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

Excursions
==========

• Cactus Language
• Futures Of Logical Graphs

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
https://oeis.org/wiki/User:Jon_Awbrey/Exploratory_Qualitative_Analysis_of_Sequential_Observation_Data

• Differential Analytic Turing Automata
https://oeis.org/wiki/Differential_Analytic_Turing_Automata_%E2%80%A2_Overview

• Survey of Theme One Program
https://inquiryintoinquiry.com/2020/08/28/survey-of-theme-one-program-3/

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

• Animated Logical Graphs (1) ••• (74)

Anamnesis
=========

• CSP, GSB, & Me (1) ••• (15)

Regards,

Jon

[Jon Awbrey]

Cf: Animated Logical Graphs • 75
https://inquiryintoinquiry.com/2021/05/10/animated-logical-graphs-75/

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/2021/04/22/animated-logical-graphs-72/
https://inquiryintoinquiry.com/2021/04/24/animated-logical-graphs-73/
https://inquiryintoinquiry.com/2021/04/30/animated-logical-graphs-74/

All,

Continuing down the rows of the Table from Episode 72, the next
two orbits contain the logical graphs for the boolean functions
f₂, f₁₁, f₄, f₁₃, in that order. A first glance shows these two
orbits have surprisingly intricate structures and relationships
to each other — let’s isolate this section for a closer look.

Table 1. Peirce Duality • Subtractions and Implications
https://inquiryintoinquiry.files.wordpress.com/2021/05/peirce-duality-e280a2-subtractions-and-implications.png

• The boolean functions f₂ and f₄ are called “subtraction functions”.
• The boolean functions f₁₁ and f₁₃ are called “implication functions”.

• The logical graphs for f₂ and f₁₁ are dual to each other.
• The logical graphs for f₄ and f₁₃ are dual to each other.

The values of the subtraction and implication functions for each
(x, y) in B × B and the text expressions for their logical graphs

are given in the following Table.

Table 2. Truth Tables • Subtractions and Implications
https://inquiryintoinquiry.files.wordpress.com/2021/05/subtractions-and-implications.png

Resources
=========

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

• Logical Implication
( https://oeis.org/wiki/Logical_implication )

• Truth Tables
( https://oeis.org/wiki/Truth_table )

• Zeroth Order Logic
( https://oeis.org/wiki/Zeroth_order_logic )

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

Regards,

Jon

[Jon Awbrey]

Cf: Animated Logical Graphs • 76
https://inquiryintoinquiry.com/2021/06/06/animated-logical-graphs-76/

Taking from our wallets an old schedule of orbits from Episode 72
( https://inquiryintoinquiry.com/2021/04/22/animated-logical-graphs-72/ ),
let's review the classes of logical graphs we've covered so far.

Self-Dual Logical Graphs
========================

Four orbits of “self-dual logical graphs”, x, y, (x), (y), were discussed in Episode 73.
( https://inquiryintoinquiry.com/2021/04/24/animated-logical-graphs-73/ )

Table. Self-Dual Logical Graphs
https://inquiryintoinquiry.files.wordpress.com/2021/04/self-dual-logical-graphs.png

The logical graphs x, y, (x), (y) denote the boolean functions f₁₂, f₁₀, f₃, f₅,
in that order, and the value of each function f at each point (x, y) in B × B
is shown in the Table above.

Constants and Amphecks
======================

Two orbits of logical graphs called “constants” and “amphecks” were discussed in Episode 74.
( https://inquiryintoinquiry.com/2021/04/30/animated-logical-graphs-74/ )

Table. Constants and Amphecks
https://inquiryintoinquiry.files.wordpress.com/2021/04/constants-and-amphecks.png

The constant logical graphs denote the constant functions

• f₀ : B × B → 0,

• f₁₅ : B × B → 1.

• Under Ex the logical graph whose text form is “ ” denotes the function f₁₅
and the logical graph whose text form is “( )” denotes the function f₀.

• Under En the logical graph whose text form is “ ” denotes the function f₀
and the logical graph whose text form is “( )” denotes the function f₁₅.

The ampheck logical graphs denote the ampheck functions

• f₁(x, y) = NNOR(x, y),

• f₇(x, y) = NAND(x, y).

• Under Ex the logical graph (xy) denotes the function f₇(x, y) = NAND(x, y)
and the logical graph (x)(y) denotes the function f₁(x, y) = NNOR(x, y).

• Under En the logical graph (xy) denotes the function f₁(x, y) = NNOR(x, y)
and the logical graph (x)(y) denotes the function f₇(x, y) = NAND(x, y).

Subtractions and Implications
=============================

The logical graphs called “subtractions” and “implications” were discussed in Episode 75.
( https://inquiryintoinquiry.com/2021/05/10/animated-logical-graphs-75/ )

Table. Subtractions and Implications
https://inquiryintoinquiry.files.wordpress.com/2021/05/subtractions-and-implications.png

The subtraction logical graphs denote the subtraction functions

• f₂(x, y) = y ¬ x,

• f₄(x, y) = x ¬ y.

The implication logical graphs denote the implication functions

• f₁₁(x, y) = x ⇒ y

• f₁₃(x, y) = y ⇒ x.

Under the action of the En ↔ Ex duality the
logical graphs for the subtraction f₂ and the
implication f₁₁ fall into one orbit while the
logical graphs for the subtraction f₄ and the
implication f₁₃ fall into another orbit, making
these 2 partitions of the 4 functions “orthogonal”
or “transversal” to each other.

Regards,

Jon

[Jon Awbrey]

Cf: Animated Logical Graphs • 77
https://inquiryintoinquiry.com/2021/07/03/animated-logical-graphs-77/

Cf: Category Theory
https://categorytheory.zulipchat.com/#narrow/stream/233104-theory.3A-logic/topic/logical.20graphs
::: Jon Awbrey
https://categorytheory.zulipchat.com/#narrow/stream/233104-theory.3A-logic/topic/logical.20graphs /near/244807090 )

All,

| I opened a topic in the “logic” stream of “category theory.zulipchat”
| to discuss logical graphs in a category theoretic environment and
| began by linking a few basic resources. Here goes ...

A place for exploring animated forms of visual inference
inspired by the work of C.S. Peirce and Spencer Brown.

Resources
=========

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

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

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

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

• Zeroth Order Logic ( https://oeis.org/wiki/Zeroth_order_logic )

• Survey of Animated Logical Graphs

( https://inquiryintoinquiry.com/2021/05/01/survey-of-animated-logical-graphs-4/ )

Regards,
Jon

[Jon Awbrey]

Cf: Animated Logical Graphs • 78
http://inquiryintoinquiry.com/2021/07/04/animated-logical-graphs-78/

Cf: Category Theory
https://categorytheory.zulipchat.com/#narrow/stream/233104-theory.3A-logic/topic/logical.20graphs
::: Jon Awbrey

https://categorytheory.zulipchat.com/#narrow/stream/233104-theory.3A-logic/topic/logical.20graphs/near/244812352

All,

As far as the “animated” part goes, I lost my klutz-friendly animation app in my last platform change and then got
occupied with other things, so it may be a while before I get back to that, but two examples of animated proofs in a
CSP∫GSB-style propositional calculus should give a hint of how things might develop.

Peirce's Law
https://inquiryintoinquiry.com/2008/10/06/peirces-law/

Figure 1. Peirce's Law • Proof Animation
https://inquiryintoinquiry.files.wordpress.com/2012/01/peirces-law-2-0-animation.gif

Praeclarum Theorema
https://inquiryintoinquiry.com/2008/10/05/praeclarum-theorema/

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

Resources
=========

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

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

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

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

Zeroth Order Logic
https://oeis.org/wiki/Zeroth_order_logic

Survey of Animated Logical Graphs

https://inquiryintoinquiry.com/2021/05/01/survey-of-animated-logical-graphs-4/

Regards,

Jon

[Jon Awbrey]

Cf: Animated Logical Graphs • 79
https://inquiryintoinquiry.com/2021/07/05/animated-logical-graphs-79/

Re: Category Theory
https://categorytheory.zulipchat.com/#narrow/stream/233104-theory.3A-logic/topic/logical.20graphs
::: Henry Story
https://categorytheory.zulipchat.com/#narrow/stream/233104-theory.3A-logic/topic/logical.20graphs/near/244813007

<QUOTE HS:>
I think in this 2020 Applied Category Theory talk by Rocco Gangle, A Generic Figures Reconstruction of Peirce's
Existential Graphs (Alpha) ( https://www.youtube.com/watch?v=j7Bp6_uiFaQ ), he is looking at showing how Peirce's work
can be expressed in terms of Category Theory.
</QUOTE>

I looked at that once, I think, seem to recall he is still using the planar maps which I consider the mark of a novice,
but I will give it another look.

Okay, I see he introduces forests about half-way through, that's a good thing, but he's not up to cacti yet, which is
something I found necessary early on for the sake of both conceptual and computational efficiency. So there's a few
things I will need to explain …

I started working on logical graphs early in my undergrad years, after my encounter with Peirce's Collected Papers,
quickly followed by my study of Spencer Brown's Laws of Form, from the outset trying what I could hack by way of syntax
handlers and theorem provers in every mix of languages and machines I got my hands on. That combination of forces and
media summed to form my current direction.

Peirce broke ground and laid the groundwork, Spencer Brown shored up the infrastructure of primary arithmetic and
leveled the proving grounds to facilitate equational inference, and a host of computers supplied the real-world
recalcitrance of matter, the resistance to facile simplicity, and the rebuke of all too facile reductionism.

Resources
=========

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

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

Regards,

Jon

[Jon Awbrey]

Cf: Animated Logical Graphs • 80
http://inquiryintoinquiry.com/2021/07/09/animated-logical-graphs-80/

Re: Category Theory
https://categorytheory.zulipchat.com/#narrow/stream/233104-theory.3A-logic/topic/logical.20graphs

::: Chad Nester
https://categorytheory.zulipchat.com/#narrow/stream/233104-theory.3A-logic/topic/logical.20graphs/near/245408483

<QUOTE CN:>
Re: Categorical Treatments of Existential Graphs
Cf: N. Haydon and P. Sobociński • “Compositional Diagrammatic First-Order Logic”
( https://www.ioc.ee/~pawel/papers/peirce.pdf )

Thanks, Chad, for that extremely nice treatment of Peirce's existential graphs at the
β level, tantamount to predicate calculus or first order logic as we know it today.

The logic of relatives and the mathematics of relations appear in a different light
from the perspective of Peirce's own standpoint on logic, evolving as it does out of
distinctive pragmatic and semiotic insights. The reflections of Spencer Brown afford
a few angles Peirce anticipated but in a glass, darkly. And my own time tumbling
recalcitrant calculi toward more ready tools for inquiry may add a few more wrinkles,
with luck to more than my own brow. All that will develop as we go.

Regards,

Jon

[Jon Awbrey]

Cf: Animated Logical Graphs • 81
https://inquiryintoinquiry.com/2021/08/26/animated-logical-graphs-81/

Re: R.J. Lipton and K.W. Regan • A Negative Comment On Negations
https://rjlipton.wpcomstaging.com/2021/08/24/a-negative-comment-on-negations/

Minsky and Papert's “Perceptrons” was the work that nudged me over the
limen from gestalt psychology, psychophysics, relational biology, etc.
and made me believe AI could fly. I later found out a lot of people
thought it had thrown cold water on the subject but that was not
my sense of it.

The real reason Rosenblatt's perceptrons short-shrift XOR and EQ among the
sixteen boolean functions on two variables is the adoption of a particular
role for neurons in the activity of the brain and a particular model of how
neurons serve computation, namely, as threshold activation devices. It is
as if we tried to do mathematics using only the inequality “ ≤ ” instead of
using equations. Sure, you can circumlocute it, but why? Of course, x ≤ y
for boolean variables x, y is equivalent to x ⇒ y so this fits right in with
the weakness of implicational inference compared to equational inference rules.

But there are other models for the role neurons play in the
activity of the brain and the work they do in computation.

Regards,

Jon