Logical Graphs

[Jon Awbrey]

Cf: Logical Graphs • Discussion 1
https://inquiryintoinquiry.com/2021/07/28/logical-graphs-discussion-1/

Re: Laws of Form
https://groups.io/g/lawsofform/topic/software_for_lof/84501600
::: John Mingers ( https://groups.io/g/lawsofform/message/559

<QUOTE JM:>
I find it very frustrating not to be able to draw crosses and
expressions within emails or Word documents. Does anyone know
of any software or apps that can do this? If not, with so many
computer scientists on this group, could someone produce something?
</QUOTE>

Dear John, All …

People with backgrounds in computing, combinatorics, or graph theory
would immediately recognize Spencer Brown's expressions are isomorphic
to what graph theorists know and love as “trees”, more specifically
“rooted trees”, with a particular manner of attaching letters to the
nodes to be described later. In those fields there's a standard way
of mapping trees to strings of parentheses and letters. This operation
is called “traversing the tree” when one passes from trees to strings
and the reverse operation is called “parsing” when one passes from
strings to “parse trees”.

The transformation of Spencer Brown's simple closed figures in the plane,
or his formal expressions of “crosses”, into rooted trees, together with
the further transformation of those two forms to “pointer data structures”
in computer memory is discussed in the following post on my blog.

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

There's a more formal presentation of logical graphs, working from
the axioms or “initials” I borrowed with modifications from Peirce
and Spencer Brown, in the following blog post.

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

Those two pieces are combined and extended in the following article.

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

The program I developed all through the 80s using those data structures
in its logic module is documented so far as I've managed at this point
on the following page.

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

Regards,

Jon

[Jon Awbrey]

Cf: Logical Graphs • Discussion 2
https://inquiryintoinquiry.com/2021/07/29/logical-graphs-discussion-2/

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/245453493

<QUOTE CN:>
Recently a few of us have been using the “cartesian bicategories of
relations” of Carboni and Walters, in particular their string diagrams,
as syntax for relations. The string diagrams in question are more or
less a directed version of Peirce's lines of identity. They're usually
described in terms of commutative special frobenius algebras. I suspect
the reason we keep finding commutative special frobenius algebras is that
they support lines of identity in this way.
</QUOTE>

Dear Chad, Henry, …

Chaos rules my niche of the world right now so I'll just
break a bit of the ice by sharing the following links to
my ongoing study of Peirce's 1870 Logic Of Relatives.

• Peirce's 1870 LOR • Overview
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Overview
• Part 1
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1
• Part 2
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2
• Part 3
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_3
• References
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_References

See especially the following paragraph.

* Peirce • CP 3.93
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_3#Selection_B

To my way of thinking the above paragraph is one of the most radical
passages in the history of logic, relativizing traditional assumptions
of an absolute distinction between generals (universals) and individuals.
Among other things, it pulls the rug out from under any standing for
nominalism as opposed to realism about universals.

Regards,

Jon

[Jon Awbrey]

Cf: Logical Graphs • Discussion 3
https://inquiryintoinquiry.com/2021/07/31/logical-graphs-discussion-3/

Re: Peirce List
https://list.iupui.edu/sympa/arc/peirce-l/2021-07/thrd4.html#00111
::: John Sowa
https://list.iupui.edu/sympa/arc/peirce-l/2021-07/msg00111.html
•••
::: Robert Marty
https://list.iupui.edu/sympa/arc/peirce-l/2021-07/msg00144.html

Dear John, Robert, Edwina,

This discussion reminds me of the time I spent the big bucks buying a copy of
Stjernfelt's Diagrammatology ( https://www.springer.com/gp/book/9781402056512 )
which ran to over 500 pages with many sections in very small print and had just
over 50 diagrams in the whole book.

So I think the real “versus” being dealt with here is not so much the
difference between “thinking in diagrams” and “thinking in words” as
the difference between “thinking in words about thinking in diagrams”
and “thinking in words about thinking in words”.

Those of us, the very few, who have actually been working on “moving pictures”
from the very get-go, have learned to see things somewhat differently.

Peirce Syllabus • Metaphysics, Normative Science, Phenomenology, Mathematics
http://inquiryintoinquiry.files.wordpress.com/2014/08/peirce-syllabus.jpg

“Normative science rests largely on phenomenology and on mathematics;
metaphysics on phenomenology and on normative science.”

❧ Charles Sanders Peirce • Collected Papers, CP 1.186 (1903)
Syllabus • Classification of Sciences (CP 1.180–202, G-1903-2b)
http://web.archive.org/web/20111105121054/http://www.princeton.edu/~batke/peirce/cl_o_sci_03.htm

Regardez,

Jon

[Jon Awbrey]

Cf: Logical Graphs • Discussion 4
https://inquiryintoinquiry.com/2021/08/01/logical-graphs-discussion-4/

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/245454945

<QUOTE HS:>
Evan Patterson's “Knowledge Representation in Bicategories of Relations”
( https://arxiv.org/abs/1706.00526 ) is also drawn up in terms of string
diagrams, as a way of explaining the W3C RDF and OWL standards. So it
looks like we have a nice route from Peirce to RDF via string diagrams.
Or the other way around: whichever route one prefers to travel.
</QUOTE>

Dear Henry,

I opened a topic on Zulip | Relation Theory
https://categorytheory.zulipchat.com/#narrow/stream/233104-theory.3A-logic/topic/relation.20theory
to discuss the logic of relative terms and the mathematics of relations
as they develop from Peirce’s first breakthroughs (1865–1870). As I keep
telling people, there are radical innovations in this work, probing deeper
strata of logic and mathematics than ever mined before and thus undermining
the fundamental nominalism of First Order Logic as we know it.

Regards.

Jon