Cf: Differential Propositional Calculus • Overview
At: http://inquiryintoinquiry.com/2020/02/16/differential-propositional-calculus-%e2%80%a2-overview/

|| The most fundamental concept in cybernetics is that of "difference",
|| either that two things are recognisably different or that one thing
|| has changed with time.
||
|| W. Ross Ashby : An Introduction to Cybernetics
|| ( http://pespmc1.vub.ac.be/books/IntroCyb.pdf )

Here's the outline of I sketch I wrote on differential propositional calculi,
which extend propositional calculi by adding terms for describing aspects of
change and difference, for example, processes taking place in a universe of
discourse or transformations mapping a source universe to a target universe.
I wrote this as an intuitive introduction to differential logic, which is
my best effort so far at dealing with ancient and persistent problems of
dealing with diversity and mutability in logical terms. I'll be looking
at ways to improve this draft as I serialize it to my blog.

Part 1
======
( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_1 )

Casual Introduction
===================
( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_1#Casual_Introduction )

Cactus Calculus
===============
( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_1#Cactus_Calculus )

Part 2
======
( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2 )

Formal_Development
==================
( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2#Formal_Development )

Elementary Notions
==================
( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2#Elementary_Notions )

Special Classes of Propositions
===============================
( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2#Special_Classes_of_Propositions )

Differential Extensions
=======================
( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2#Differential_Extensions )

Appendices
==========
( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Appendices )

References
==========
( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_References )

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,
While I will look at this in more detail later, as I read the introduction, I was immediately reminded of my studied in Statistical Thermodynamics. 
https://en.wikipedia.org/wiki/Partition_function_%28statistical_mechanics%29

73,
Lyle
 

Jon,
Another connection/application of differential propositional calculus is to random walks and partial differential equations. A propagating difference is a signal.  The CotU started with three books: text, numbers, and communications.

"Under various conditions we obtain partial differential equations of various types, and where appropriate complete the initial value problems by specifying initial or boundary conditions. These equations all result from taking limits of difference equations that serve as models for the discrete random walk problems studied. Following the approach of S. Goldstein we also study partial difference equations for correlated random walks, leading to variants of the telegraph equation, an equation that governs the propagation of signals on telegraph lines. Finally in the self assembling of particles, we apply the random walk concept to model, simulate, and characterize cluster growth and form."

https://ui.adsabs.harvard.edu/abs/2010PhDT........16A/abstract

Cf: Differential Propositional Calculus • Discussion 3
https://inquiryintoinquiry.com/2021/03/15/differential-propositional-calculus-discussion-3/

| That mathematics, in common with other art forms, can lead us
| beyond ordinary existence, and can show us something of the
| structure in which all creation hangs together, is no new idea.
| But mathematical texts generally begin the story somewhere in the
| middle, leaving the reader to pick up the thread as best he can.
| Here the story is traced from the beginning.
|
| G. Spencer Brown • Laws of Form

Re: Laws of Form
https://groups.io/g/lawsofform/topic/differential_propositional/81172986
::: Lyle Anderson
(1) https://groups.io/g/lawsofform/message/162
(2) https://groups.io/g/lawsofform/message/163

Dear Lyle,

Charles S. Peirce, with his x-ray vision, revealed for the first time
in graphic detail the mathematical forms structuring our logical organon.
Spencer Brown broadened that perspective in two directions, tracing more
clearly than Peirce’s bare foreshadowings the infrastructure of primary
arithmetic and hypothesizing the existence of imaginary logical values
in a larger algebraic superstructure.

Spencer Brown explored the algebraic extension of the boolean domain B
to a superset equipped with logical imaginaries, operating on analogy
with the algebraic extension of the real line R to the complex plane C.
Seeing as how complex variables are frequently used to model time domains
in physics and engineering, that will continue to be a likely and natural
direction of exploration.

My own work, however, led me in a different direction.
There are many different ways of fruitfully extending
a given domain. Aside from the above class of algebraic
extensions there is a class of differential extensions and,
when that proverbial road diverged, I took the differential one.

Who knows? maybe some where in that undergrowth the roads converge again …

Resources
=========

Differential Logic • Introduction
https://inquiryintoinquiry.com/2008/07/29/differential-logic/

Differential Propositional Calculus
https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_1
https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2

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

Re: Differential Propositional Calculus
https://inquiryintoinquiry.com/2020/02/16/differential-propositional-calculus-overview/
::: Discussion 3
https://inquiryintoinquiry.com/2021/03/15/differential-propositional-calculus-discussion-3/

Re: Peirce List
https://list.iupui.edu/sympa/arc/peirce-l/2021-03/thrd1.html#00020
::: Helmut Raulien
https://list.iupui.edu/sympa/arc/peirce-l/2021-03/msg00095.html

<QUOTE HR:>

1. I think I like very much your Cactus Graphs. Meaning that I
am in the process of understanding them, and finding it much
better not to have to draw circles, but lines.

2. Less easy for me is the differential calculus.
Where is the consistency between (x,y) and (x, y, z)?
(x, y) means that x and y are not equal and (x, y, z)
means that one of them is false.
Unequality and truth/falsity for me are two concepts
so different I cannot think them together or see
a consistency between them.

3. What about (w, x, y, z)?

4. Can you give a grammar, like, what does a comma mean,
what do brackets mean, what does writing letters following
each other with an empty space but no comma mean, and so on?

5. Same with Cactus Graphs, though I think, they might be
self-explaining for me — everything is self-explaining,
depending on intellectual capacity, but mine is limited.

</QUOTE>

Dear Helmut,

Many thanks for your detailed comments and questions. They help me see
the places where more detailed explanations are needed. I added numbers
to your points above for ease of reference and possible future reference
in case I can't get to them all in one pass.

Cactus Graphs
=============

I'm glad you found the cactus graphs to your liking.
It was a critical transition for me when I passed from
trees to cacti in my graphing and programming and it
came about by recursively applying a trick of thought
I learned from Peirce himself. These days I call it a
“Meta-Peircean Move” to apply one of Peirce's heuristics
of choice or standard operating procedures to the state
resulting from previous applications. All that makes for
a longer story I made a start at telling in the following
series of posts.

Animated Logical Graphs
https://inquiryintoinquiry.com/2019/05/28/animated-logical-graphs-14/

[Links omitted for this email. See the blog post linked at the top for the list.]

https://inquiryintoinquiry.com/2019/07/31/animated-logical-graphs-27/

Well, the clock in the hall struck time for lunch some time ago,
so I think I'll answer its call, break here, and continue later …

Regards,

Jon

Jon,

You mean, I think, “equality/unequality and truth/falsity.” Yes?

In the Notes to Ch.2, GSB writes:

When we attempt to realize a piece of music composed by another person, we do so by *illustrating*, to ourselves, with a musical instrument of some kind, the composer's commands. Similarly, if we are to realize a piece of mathematics, we find a way of illustrating, to ourselves, the commands of the mathematician.
and

It is not necessary for the reader to confine his illustrations to the commands in the text. He may wander at will, inventing his own illustrations, either consistent or inconsistent with the textual commands.

Along these lines, any set of illustrations of the injunctions in LOF, if those illustrations are consistent with the injunctions, will necessarily correspond in their form to the form of distinction, the laws of which are illustrated in LOF as words and figures, as discourse for the most part.

Every interpretation of LOF that maintains consistency with the injunctions indicated by the text can be seen to be ruled by the laws of form.
(Interpretation = illustration)

Every interpretation, imaginable or unimaginable, that exhibits the form of distinction conforms to the laws of form.

If equality/inequality and truth/falsity are each exhaustive distinctions of some space, they each conform to the laws of form. That is the consistency between them.

Pretty much everything that is possible for us to think is based on distinction. Everything we can think of follows the laws of form.
(At least, everything we can talk about, everything we can think of discursively.)

All interpretations of the calculus of indications are formally equivalent to each other.
(Duh!)

Jerry

Cf: Differential Propositional Calculus • Discussion 5
http://inquiryintoinquiry.com/2021/03/21/differential-propositional-calculus-discussion-5/

Re: Peirce List
https://list.iupui.edu/sympa/arc/peirce-l/2021-03/thrd1.html#00020
::: Helmut Raulien
https://list.iupui.edu/sympa/arc/peirce-l/2021-03/msg00095.html

Re: Ontolog Forum
https://groups.google.com/g/ontolog-forum/c/T8DAe6shKFw
::: Mauro Bertani
https://groups.google.com/g/ontolog-forum/c/T8DAe6shKFw/m/zKS50rJAAwAJ

<QUOTE HR:>

1. I think I like very much your Cactus Graphs. Meaning that I
am in the process of understanding them, and finding it much
better not to have to draw circles, but lines.

2. Less easy for me is the differential calculus.
Where is the consistency between (x,y) and (x, y, z)?
(x, y) means that x and y are not equal and (x, y, z)
means that one of them is false.
Unequality and truth/falsity for me are two concepts
so different I cannot think them together or see
a consistency between them.

3. What about (w, x, y, z)?

</QUOTE>

<QUOTE MB:>

So, if I want to transform a circle into a line
I have to use a function f : Bⁿ → B ? This is the
base of temporal logic? I’m using f : Nⁿ → N.

</QUOTE>

Dear Mauro,

Here I think Helmut is describing the transition from forms
of enclosure on a plane sheet of paper, such as those used
by Peirce and Spencer Brown, to their topological duals in
the form of rooted trees. There is more detail about this
transformation at the following sites.

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

Logical Graphs • Duality : Logical and Topological
https://oeis.org/wiki/Logical_Graphs#Duality:_logical_and_topological

This is the first step in the process of converting planar maps
to graph-theoretic data structures. Further transformations take
us from trees to the more general class of cactus graphs, which
implement a highly efficient family of logical primitives called
“minimal negation operators”. These are described in the following
article.

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

Regards,

Jon

Jerry,
Thanks for reminding me of this GSB note!  I have added it to the Preface of The Unified Theory of Everything.

Understanding a concept or communicating it to someone else cannot really be done by simply describing the finished concept or thing, but by communicating a series of rules, steps on how the object of the concept is made. All of mathematics is communicated through a series of rules. I can still hear my 10th Grade Geometry teacher encouraging us to make the rules of Euclidean Geometry our own by saying, “You must know your rules!” George Spencer-Brown made the point so completely in the Notes to Chapter 2 of Laws, that I am just going to put it here. It may be helpful at this stage to realize that the primary form of mathematical communication is not description, but injunction. In this respect it is comparable with practical art forms like cookery, in which the taste of a cake, although literally indescribable, can be conveyed to a reader in the form of a set of injunctions called a recipe. Music is a similar art form, the composer does not even attempt to describe the set of sounds he has in mind, much less the set of feelings occasioned through them, but writes down a set of commands which, if they are obeyed by the reader, can result in a reproduction, to the reader, of the composer's original experience. The CotU has written down in the Holy Scriptures, a series of commands which, if they are obeyed by an individual person, can result in that person experiencing the fullness of the part that person was intended to play in the story of the Universe. https://www.scribd.com/document/470732777/Unified-Theory-of-Everything

Cf: Differential Propositional Calculus • Discussion 6
http://inquiryintoinquiry.com/2021/03/22/differential-propositional-calculus-discussion-6/

Re: Differential Propositional Calculus • Discussions
https://inquiryintoinquiry.com/2021/03/15/differential-propositional-calculus-discussion-3/
https://inquiryintoinquiry.com/2021/03/20/differential-propositional-calculus-discussion-4/
https://inquiryintoinquiry.com/2021/03/21/differential-propositional-calculus-discussion-5/

<QUOTE HR:>

1. I think I like very much your Cactus Graphs. Meaning that I
am in the process of understanding them, and finding it much
better not to have to draw circles, but lines.

2. Less easy for me is the differential calculus.
Where is the consistency between (x,y) and (x, y, z)?
(x, y) means that x and y are not equal and (x, y, z)
means that one of them is false.
Unequality and truth/falsity for me are two concepts
so different I cannot think them together or see
a consistency between them.

3. What about (w, x, y, z)?

</QUOTE>

Dear Helmut,

Table 1 shows the cactus graphs, the corresponding cactus expressions in
“traversal string” or plain text form, their logical meanings under the
“existential interpretation”, and their translations into conventional
notations for a number of common propositional forms. I’ll change
variables to {x, a, b, c} instead of {w, x, y, z} at this point
simply because I already had a Table like that on hand.

As far as the consistency between (a, b) and (a, b, c) goes,
that’s easy enough to see — if exactly one of two boolean
variables is false then the two must have different values.

Out of time for today, so I’ll get to the rest of your questions next time.

Table 1. Syntax and Semantics of a Calculus for Propositional Logic (also attached)
https://inquiryintoinquiry.files.wordpress.com/2021/03/syntax-and-semantics-of-a-calculus-for-propositional-logic-3.0.png

Regards,

Jon