Differential Logic and Dynamic Systems

[Jon Awbrey]

Cf: Differential Logic and Dynamic Systems : Overview
At: https://inquiryintoinquiry.com/2019/09/10/differential-logic-and-dynamic-systems-%e2%80%a2-overview/

In modeling intelligent systems, whether we are trying to understand
a natural system or engineer an artificial system, there has long been
a tension or trade-off between dynamic paradigms and symbolic paradigms.
Dynamic models take their cue from physics, using quantitative measures
and differential equations to model the evolution of a system's state
through time. Symbolic models use logical methods to describe systems
and their agents in qualitative terms, deriving logical consequences of
a system's description or an agent's state of information. Logic-based
systems have tended to be static in character, largely because we have
lacked a proper logical analogue of differential calculus. The line of
work laid out in this series of posts is intended to address that lack.

Overview : https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview
Part 1 : https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1
Part 2 : https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2
Part 3 : https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_3
Part 4 : https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_4
Appendices : https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Appendices
References : https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_References
Doc History : https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Document_History

Note: This is work I carried out under the auspices of a Systems Engineering program
at Oakland University. I will spend the next few weeks revising the text, redrawing
the graphics, and reformatting the LaTeX, as the ravages of time and the innovations
of MediaWiki developers oblivious to the civilities of upward compatibility have not
been kind. Meanwhile, I'll also experiment with serializing the content to my blog.

Regards,

Jon

inquiry into inquiry: https://inquiryintoinquiry.com/
academia: https://independent.academia.edu/JonAwbrey
oeiswiki: https://www.oeis.org/wiki/User:Jon_Awbrey
isw: http://intersci.ss.uci.edu/wiki/index.php/JLA
facebook page: https://www.facebook.com/JonnyCache

[Jon Awbrey]

Cf: Differential Logic and Dynamic Systems • Discussion 1
http://inquiryintoinquiry.com/2021/05/18/differential-logic-and-dynamic-systems-discussion-1/

❝It is understandable that an engineer should be completely
absorbed in his speciality, instead of pouring himself out
into the freedom and vastness of the world of thought, even
though his machines are being sent off to the ends of the
earth; for he no more needs to be capable of applying to
his own personal soul what is daring and new in the soul of
his subject than a machine is in fact capable of applying
to itself the differential calculus on which it is based.
The same thing cannot, however, be said about mathematics;
for here we have the new method of thought, pure intellect,
the very well-spring of the times, the fons et origo of an
unfathomable transformation.❞

❧ Robert Musil • The Man Without Qualities
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_3#Transformations_of_Discourse

Re: Michael Harris • Does Mathematics “Progress”?
https://mathematicswithoutapologies.wordpress.com/2021/05/05/does-mathematics-progress/

All,

Just off-hand, by way of getting grounded and oriented,
there are a few questions I'd have to ask first.

• Is mathematics a science, or a form of scientific inquiry?

• If so, what is the place of mathematics within the sciences?

• Does science progress?

•• (I mean progress in the sense of progress toward a goal,
not necessarily progressive jazz or progressions in music
generally.)

• If so, what is the goal of science?

• If science has a goal, does mathematics serve it?

• If so, how?

By “science” or “scientific inquiry” I meant to focus on species
of goal-directed activity with the specific goal of “knowledge”
and to ask whether mathematical arts and crafts and rites fall
within that ballpark. We come back to the antic Socratic question
of whether we put another quarter in the machine for the sake of
an external gain or simply to continue the play for its own sake.

<QUOTE MH:>

In this post I was less concerned with philosophy than with philology —
which, by the way, is another example of a term that was once seen as
hopelessly antiquated and musty but that has been revived recently.

But when you introduce “knowledge” you have to grapple with “truth” and
“objective reality”. Philosophy has been so unsuccessful at pinning those
down that some would prefer to give up on them altogether. We can choose
to call what mathematics generates “knowledge” but that just leads to the
(philological) question of what this has to do with what other practices
call “knowledge”.

</QUOTE>

Not that I don't love wisdom and words, however often their stars may cross,
but I invoked Musil rather for the way he syzygied the way of the engineer,
the way of the mathematician, and the recursive point where their ways diverge.
It's in this frame I think of the word “entelechy”, which I got from readings
in Aristotle, Goethe, and Peirce and promptly gave a personal gloss as “end
in itself”, partly on the influence of Conway's game theory.

And that's where I remember all those two-bit pieces I gave up to pinball
machines in the early 70s and the critical point in my own trajectory when
I realized I would never beat those machines, not that way, not ever, and
I turned to the more collaboratory ends of teaching machines how to learn
and reason.

♩ On A Related Note ♪ The Music Of The Primes ♫
( https://oeis.org/wiki/Riffs_and_Rotes )

Now there's a progression of progressions I could enjoy, musically speaking,
ad infinitum, and yet this pilgrim would consider it progress, mathematically
speaking, if he could understand why the sequentiae should be sequenced as
they are. Would that understanding add to my enjoyment? On jugera …

Resources
=========

• Differential Logic and Dynamic Systems • Overview
https://inquiryintoinquiry.com/2019/09/10/differential-logic-and-dynamic-systems-overview/

• Survey of Differential Logic
https://inquiryintoinquiry.com/2021/05/15/survey-of-differential-logic-3/

Regards,

Jon

[Jon Awbrey]

Cf: Differential Logic and Dynamic Systems • Overview
https://inquiryintoinquiry.com/2019/09/10/differential-logic-and-dynamic-systems-overview/

All,

Here's an update of the anchor post on this subject.

In modeling intelligent systems, whether we are trying to understand
a natural system or engineer an artificial system, there has long been
a tension or trade-off between dynamic paradigms and symbolic paradigms.
Dynamic models take their cue from physics, using quantitative measures
and differential equations to model the evolution of a system's state
through time. Symbolic models use logical methods to describe systems
and their agents in qualitative terms, deriving logical consequences of
a system's description or an agent's state of information. Logic-based
systems have tended to be static in character, largely because we have

lacked a proper logical analogue of differential calculus. The work

laid out in this series of posts is intended to address that lack.

Overview : https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview
Part 1 : https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1
Part 2 : https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2
Part 3 : https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_3
Part 4 : https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_4

Part 5 : https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_5

Appendices : https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Appendices
References : https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%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 Awbrey]

Cf: Differential Logic and Dynamic Systems • Discussion 2
http://inquiryintoinquiry.com/2021/05/20/differential-logic-and-dynamic-systems-discussion-2/

Re: Michael Harris • Does Mathematics “Progress”? • Comment
https://mathematicswithoutapologies.wordpress.com/2021/05/05/does-mathematics-progress/#comment-21380

All,

In several places I can't find right now I described formalization as an arrow. A related idea occurs in a paper by
Susan Awbrey and myself where we discussed “a dimension of increasing formalization in our mental models of the world”
as an obstacle to integrating knowledge across various styles of inquiry. An excerpt follows.

<QUOTE>

Conceptual Barriers to Creating Integrative Universities
========================================================
https://www.academia.edu/1266492/Conceptual_Barriers_to_Creating_Integrative_Universities

The Trivializing of Integration
===============================

From reviewing its philosophical sources, we can see that the trivialization of integration hypothesis presents
barriers to creating an integrated learning environment. Below we focus on three closely interrelated problematics and
the bearing that the triviality of integration hypothesis has on them.

Problematic 1 is the tension that arises along a dimension of increasing formalization in our mental models of the
world, between what we may call the ‘informal context’ of real-world practice and the ‘formal context’ of specialized study.

Problematic 2 is the difficulty in communication that is created by differing mental models of the world, in other
words, by the tendency among groups of specialists to form internally coherent but externally disparate systems of
mental images.

Problematic 3 is a special type of communication difficulty that commonly arises between the ‘Two Cultures’ of the
scientific and the humanistic disciplines. A significant part of the problem derives from the differential emphasis
that each group places on its use of symbolic and conceptual systems, limiting itself to either the denotative or the
connotative planes of variation, but seldom integrating the two.

</QUOTE>

References
==========

• Awbrey, S.M., and Awbrey, J.L. (May 2001), “Conceptual Barriers to Creating Integrative Universities”, Organization :
The Interdisciplinary Journal of Organization, Theory, and Society 8(2), Sage Publications, London, UK, 269–284.
Abstract ( https://org.sagepub.com/cgi/content/abstract/8/2/269 ) . Online (
https://www.academia.edu/1266492/Conceptual_Barriers_to_Creating_Integrative_Universities ) .

• Awbrey, S.M., and Awbrey, J.L. (September 1999), “Organizations of Learning or Learning Organizations : The Challenge
of Creating Integrative Universities for the Next Century”, Second International Conference of the Journal
‘Organization’, Re‑Organizing Knowledge, Trans‑Forming Institutions : Knowing, Knowledge, and the University in the 21st
Century, University of Massachusetts, Amherst, MA. Online (
https://arisbe.sitehost.iu.edu//menu/library/aboutcsp/awbrey/integrat.htm ) .

Resources
=========

• Differential Logic and Dynamic Systems • Overview
https://inquiryintoinquiry.com/2019/09/10/differential-logic-and-dynamic-systems-overview/

[Jon Awbrey]

Cf: Differential Logic and Dynamic Systems • Discussion 3
https://inquiryintoinquiry.com/2021/05/21/differential-logic-and-dynamic-systems-discussion-3/

Re: FB | Differential Logic
https://www.facebook.com/DifferentialLogic/
::: Rajib Hossain Pavel
https://www.facebook.com/JonnyCache/posts/10223983051737669?comment_id=10223983444667492

<QUOTE RHP:>

How can Differential Logic find Optimal Condition in a Game setting for an
Individual Player (Individual Choice) and Overall System (Social Welfare)?

</QUOTE>

Dear Rajib,

Differential logic is a general framework for analyzing aspects of change
and difference in systems amenable to qualitative description, for example,
processes taking place in a universe of discourse or transformations mapping
a source universe to a target universe. To apply its concepts and methods to
a concrete case one has to define the state space and the objective function
one desires to optimize. This may indeed be the hardest part of the problem.
It helps to break ground if one can think up a simple example from the class
of systems one has in mind.

Resources
=========

• Survey of Differential Logic
https://inquiryintoinquiry.com/2021/05/15/survey-of-differential-logic-3/

• Differential Logic and Dynamic Systems • Overview
https://inquiryintoinquiry.com/2019/09/10/differential-logic-and-dynamic-systems-overview/

Regards,

Jon

[Jon Awbrey]

Cf: Differential Logic and Dynamic Systems • Discussion 4
https://inquiryintoinquiry.com/2021/05/26/differential-logic-and-dynamic-systems-discussion-4/

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

Dear Lyle,

I've been meaning to get back to your comments linked above — the connections you observed to finite difference
calculus, ordinary and partial differential equations, and differential geometry are very apt — but preparing the ground
for a smooth transition to differential logic takes time, plus I needed to deal with the details of the En ↔ Ex logical
graph duality I've been wanting to give their due for decades.

The flat out fastest key to the highway of differential logic is still the “Casual Introduction” I wrote for
“Differential Propositional Calculus”. It affords direct access to the basic intuitions and motivations of the
subject but stops short of the syntactic mechanics needed to really take off.

The jump from this point to the more aggressive approaches of “Differential Logic” and “Differential Logic and Dynamic
Systems” has long been a challenge. I went looking for materials to bridge the gap and was pleased to find a few old
writings I had almost forgotten but wrote when I myself was passing through a similar transition. Perhaps one of these
will help the intrepid reader hit the ground running in this field.

I'll take up one of those pieces next time.

To be continued …

Resources
=========

Differential Propositional Calculus • 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

Differential Logic
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Overview

Differential Logic and Dynamic Systems

https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview

Regards,

Jon

[Jon Awbrey]

Cf: Differential Logic and Dynamic Systems • Discussion 5
https://inquiryintoinquiry.com/2021/06/13/differential-logic-and-dynamic-systems-discussion-5/

Re: Differential Logic and Dynamic Systems • Discussion 4

https://inquiryintoinquiry.com/2021/05/26/differential-logic-and-dynamic-systems-discussion-4/
Re: Laws of Form
https://groups.io/g/lawsofform/topic/differential_propositional/81172986

::: Lyle Anderson ( https://groups.io/g/lawsofform/message/163 )

All,

Last time I mentioned a few old writings I thought might serve to smooth the
transition between “Differential Propositional Calculus” [1] and the steeper
climbs of “Differential Logic” [2] & “Differential Logic and Dynamic Systems” [3].

[1] https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Overview
[2] https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Overview
[3] https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview

I was pleased to discover LibreOffice made nice PDFs of my ancient Word Docs,
so I can make a first installment on my promissory note by linking to a couple
of those documents below.

❧ Theme One • A Program Of Inquiry
https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Overview

❧ Introduction • Program Overview and User Guide (pdf)
https://inquiryintoinquiry.files.wordpress.com/2021/05/theme-one-guide.pdf

❧ Linear Topics • Differential Theory of Qualitative Equations (pdf)
https://inquiryintoinquiry.files.wordpress.com/2021/05/theme-one-guide-e280a2-linear-topics.pdf

Those are samples of the documentation I wrote for the Theme One Program
I developed all through the 1980s and eventually submitted in fulfillment
of the thesis requirement for my Master’s in Psychology at Michigan State (1989).
In time I’ll upgrade the formatting and serialize the relevant parts to my blog.

Resources
=========

• Survey of Differential Logic
https://inquiryintoinquiry.com/2021/05/15/survey-of-differential-logic-3/

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

Regards,

Jon

[Jon Awbrey]

Cf: Differential Logic and Dynamic Systems • Discussion 6
http://inquiryintoinquiry.com/2021/06/14/differential-logic-and-dynamic-systems-discussion-6/

Re: Differential Logic and Dynamic Systems • Discussions
https://inquiryintoinquiry.com/2021/05/26/differential-logic-and-dynamic-systems-discussion-4/
https://inquiryintoinquiry.com/2021/06/13/differential-logic-and-dynamic-systems-discussion-5/
Re: Laws of Form
https://groups.io/g/lawsofform/topic/differential_logic_and/82914915
::: Lyle Anderson ( https://groups.io/g/lawsofform/message/316 )

<QUOTE LA:>
Have you ever done any statistical analysis of data, or any of the statistical sciences such as The theory of Error,
Experimental Psychology, Thermodynamics, Quantum Theory? These handle what you are calling differential logic using
probability distribution functions of various types over various knowledge domains. If you are heading toward logic
based on vague or imprecise statements, then there is the well established science and application of Fuzzy Logic.
</QUOTE>

Dear Lyle,

Yes, been there done that. On top of the usual theory courses
in Probability and Statistics, Measure Theory, etc. contributing
to a master's in math (1980) and the canonical year of “Advanced
Psychometrics”, really a quaint legacy term for what had long since
morphed into statistical methods for experimental design and analysis,
sine qua non for a master's in psych (1989), there was a raft of short
courses on statistical computing packages which always seemed to get me a
lot more bucks for the head-banging than anything else as the years wore on.

Regards,

Jon

[Jon Awbrey]

Cf: Differential Logic and Dynamic Systems • Discussion 7
http://inquiryintoinquiry.com/2022/02/09/differential-logic-and-dynamic-systems-discussion-7/

Re: Differential Logic and Dynamic Systems

https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1
::: Intentional Propositions
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1#Intentional_Propositions

Re: FB | Differential Logic
https://www.facebook.com/DifferentialLogic/posts/2868375713219794
::: Marius V. Constantin
https://www.facebook.com/JonnyCache/posts/10225410016490896?comment_id=10225410104973108

All,

Marius Constantin asks about the logical value of intentions which are not carried out.

<QUOTE MVC:>
I have in my intention to give like, but I didn't.
What is the value (logic) of this proposition?
</QUOTE>

Dear Marius,

A difference between an expected state and an observed state
is called a Surprise. A surprise calls for an explanation.

A difference between an intended state and an observed state
is called a Problem. A problem calls for a plan of action.

There's more discussion in the following essay and section.

Prospects for Inquiry Driven Systems
https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems

Topos : Rudiments and Immediate Resources
https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems#Topos_:_Rudiments_and_Immediate_Resources

Resources
=========

Survey of Differential Logic
https://inquiryintoinquiry.com/2021/05/15/survey-of-differential-logic-3/

Survey of Inquiry Driven Systems
https://inquiryintoinquiry.com/2020/12/27/survey-of-inquiry-driven-systems-3/

Regards,

Jon