Logical Graphs • Discussion

[Jon Awbrey]

Logical Graphs • Discussion 5
• https://inquiryintoinquiry.com/2023/08/28/logical-graphs-discussion-5/

Re: Logical Graphs • First Impressions
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/
Re: Facebook • Daniel Everett
•
https://www.facebook.com/permalink.php?story_fbid=pfbid026jD3t75k69Wbs9q3qaCAvTA6zb1GXCqwu4ZWfssxgGGd1er7Wwz8PyygiQUmF6t3l&id=100093271525294

DE: Nice discussion. Development of icon-based reasoning.

My Comment —

As it happens, even though Peirce's systems of logical graphs
do have iconic features, their real power over other sorts of
logical diagrams (like venn diagrams) is due to their deeper
symbolic character. Thereby will hang many tales to come …

Regards,

Jon

cc: https://www.academia.edu/community/VjWX8L
cc: https://mathstodon.xyz/@Inquiry/110967659414575933

[Jon Awbrey]

Logical Graphs • Discussion 6
• https://inquiryintoinquiry.com/2023/08/29/logical-graphs-discussion-6/

Re: Logical Graphs • First Impressions
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Logical Graphs • Figures 1 and 2
• https://inquiryintoinquiry.files.wordpress.com/2023/08/logical-graph-figures-1-2-framed.png

Re: Academia.edu • Robert Appleton
• https://www.academia.edu/community/lavbw5?c=Q4jlVy

RA:
❝As a professional graphic designer and non-mathematician reading your
two diagrams, I need to ask for a simpler statement of their purpose.
What do Fig 1 and Fig 2 represent to you? And what insight do they
provide us?❞

My Comment —

Figures 1 and 2 are really just a couple of “in medias res” pump‑primers
or ice‑breakers. This will all be explained in the above linked blog post,
where I'm revising the text and upgrading the graphics of some work I first
blogged in 2008 based on work I did even further back. I'll be taking a fresh
look at that as I serialize it here.

Those two Figures come from George Spencer Brown's 1969 book Laws of Form,
where he called them the Law of Calling and the Law of Crossing. GSB revived
and clarified central aspects of Peirce's systems of logical graphs and I find
it helpful to integrate his work into my exposition of Peirce. For now you can
think of those as exemplifying two core formal principles which go to the root
of the mathematical forms underlying logical reasoning.

Regards,

Jon

cc: https://www.academia.edu/community/L2OGMl
cc: https://mathstodon.xyz/@Inquiry/110967659414575933

[Jon Awbrey]

Logical Graphs • Discussion 7
• https://inquiryintoinquiry.com/2023/10/01/logical-graphs-discussion-7/

Re: Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-2/
Re: Laws of Form • Alex Shkotin
• https://groups.io/g/lawsofform/message/2461

<QUOTE AS:>
When we look at undirected graph it is usual, before
describing a rules of graph transformation, to describe
exactly what kind of graphs we are working with ...
</QUOTE>

Hi Alex,

I am traveling this week, with limited internet.
There's a quickie glossary under the heading
“Painted And Rooted Cacti” on the following
blog page.

Theme One Program • Exposition 2
• https://inquiryintoinquiry.com/2022/06/16/theme-one-program-exposition-2-2/

Regards,

Jon

P.S. Back home now ... with access to books ...
will attempt to fill in some of the blanks
in last week's sketchy vacation messages.
—JA

[Jon Awbrey]

Logical Graphs • Discussion 8
• https://inquiryintoinquiry.com/2023/10/02/logical-graphs-discussion-8/

Re: Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-2/
Re: Laws of Form • Alex Shkotin

• https://groups.io/g/lawsofform/message/2468

Hi Alex,

I got my first brush with graph theory in a course on the Foundations of Mathematics Frank Harary taught at the
University of Michigan in 1970. Frank was the don, founder, mover, and shaker of what we affectionately called the
“MiGhTy” school of graph theory, spawned at U of M, Michigan State, Illinois, Indiana, and eventually spreading to other
hotbeds of research in the Midwest and beyond. Later I took my first graduate course in graph theory from Ed Palmer at
Michigan State, using Harary's “Graph Theory” as the text of choice.

Definitions of graphs vary in style and substance according to the level of abstraction befitting a particular approach
or application. The following is a classic formulation, one which covers the essential ideas in a very short space, and
one whose elegance and power I've come to appreciate more and more as time goes by.

Harary (1969):
❝A “graph” G consists of a finite nonempty set V = V(G) of p “points” together with a prescribed set X of q unordered
pairs of distinct points of V. Each pair x = {u, v} of points in X is a “line” of G, and x is said to “join” u and v.
We write x = uv and say that u and v are “adjacent points” (sometimes denoted u adj v); point u and line x are
“incident” with each other, as are v and x. If two distinct lines x and y are incident with a common point, then they
are “adjacent lines”. A graph with p points and q lines is called a (p, q) graph. The (1, 0) graph is “trivial”.❞
(Harary, Graph Theory, p. 9).

I'll be hewing fairly close to that definition and terminology, though most graph theorists are used to the more common
variations, like “nodes” instead of “points” and “edges” instead of “lines” — all but for the notion of “painted graphs”
where I had to invent a new term on account of the fact that “labels” and “colors” were already taken for other uses.

References —

• Harary, F. (1969), Graph Theory, Addison-Wesley, Reading, MA.

• Harary, F., and Palmer, E.M. (1973), Graphical Enumeration, Academic Press, New York, NY.

• Palmer, E.M. (1985), Graphical Evolution : An Introduction to the Theory of Random Graphs, John Wiley and Sons, New
York, NY.

Regards,

Jon

[Jon Awbrey]

Logical Graphs • Discussion 9
• http://inquiryintoinquiry.com/2023/10/12/logical-graphs-discussion-9/

Re: Logical Graphs • Formal Development

• https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-a/
Re: Laws of Form • Lyle Anderson
• https://groups.io/g/lawsofform/message/2511

LA:
❝The Gestalt Switch from parenthesis to graphs is stimulating.
There are probably things in Laws of Form that we didn't see
because we were blinded by the crosses.❞

Hi Lyle,

That has been my experience. Viewing a space of mathematical objects
from a new angle and changing the basis of representation can bring out
new and surprising aspects of their form and even expand the field of view
to novel directions of generalization.

One of the first things I learned in the early years of computing with
logical graphs is how essential it is to “slip the surly bonds” of the
planar embedding and work with free trees in a space of their own.

Regards,

Jon