Differential Logic, Dynamic Systems, Tangent Functors

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Jon Awbrey

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Sep 5, 2018, 3:00:23 PM9/5/18
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People interested in category theory as applied to systems
may wish to check out the following article, reporting work
I carried out while engaged in a systems engineering program
at Oakland University.

The problem addressed is a longstanding one, that of building bridges
to negotiate the gap between qualitative and quantitative descriptions
of complex phenomena, like those we meet in analyzing and engineering
systems, especially intelligent systems endowed with a capacity for
processing information and acquiring knowledge of objective reality.

One of the ways this problem arises has to do with describing change
in logical, qualitative, or symbolic terms, long before we grasp the
reality beneath the appearances firmly enough to cast it in measured,
quantitative, real number form.

Development on the quantitative shore got no further than a Sisyphean
beachhead until the discovery/invention of differential calculus by
Leibniz and Newton, after which things advanced by leaps and bounds.

And there's our clue what we need to do on the qualitative shore, namely,
to discover/invent the missing logical analogue of differential calculus.

With that pre-ramble ...

• Differential Logic and Dynamic Systems 2.0
http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0

“This article develops a differential extension of propositional calculus
and applies it to a context of problems arising in dynamic systems. The
work pursued here is coordinated with a parallel application that focuses
on neural network systems, but the dependencies are arranged to make the
present article the main and the more self-contained work, to serve as
a conceptual frame and a technical background for the network project.”

Regards,

Jon

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joseph simpson

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Sep 5, 2018, 5:01:04 PM9/5/18
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Jon:

Interesting approach.

I have scanned this work before (no detailed study by me).

My initial impression is that the discrete logic presented here is based on the classical "intersection rule combination" approach to binary logic.  

William Combs created a "union rule combination" approach for binary logic which he also integrated into fuzzy logic.

There seems to be many strong similarities between these two bodies of work.

Please see:


Take care, be good to yourself and have fun,

Joe

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--
Joe Simpson

“Reasonable people adapt themselves to the world. 

Unreasonable people attempt to adapt the world to themselves. 

All progress, therefore, depends on unreasonable people.”

George Bernard Shaw

Jon Awbrey

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Dec 12, 2018, 11:11:53 AM12/12/18
to Cybernetic Communications, Ontolog Forum, SysSciWG, Structural Modeling
All & Sundry —

Various discussions in various places have brought back to mind this thread
from early this fall, prompting me to make a try at continuing it. Here's
a series of blog posts where I kept track of a few points along the way:

• Differential Logic, Dynamic Systems, Tangent Functors • 1
https://inquiryintoinquiry.com/2018/09/05/differential-logic-dynamic-systems-tangent-functors-%e2%80%a2-1/

• Differential Logic, Dynamic Systems, Tangent Functors • Discussion 1
https://inquiryintoinquiry.com/2018/10/01/differential-logic-dynamic-systems-tangent-functors-%e2%80%a2-discussion-1/

• Differential Logic, Dynamic Systems, Tangent Functors • Discussion 2
https://inquiryintoinquiry.com/2018/10/05/differential-logic-dynamic-systems-tangent-functors-%e2%80%a2-discussion-2/

• Differential Logic, Dynamic Systems, Tangent Functors • Discussion 3
https://inquiryintoinquiry.com/2018/12/12/differential-logic-dynamic-systems-tangent-functors-%e2%80%a2-discussion-3/

> Another thing to keep in mind here is the difference between General Systems Theory, following on Bertalanffy et al.,
> and what is known as Dynamical Systems Theory (DST) or Mathematical Systems Theory (MST). GST spends a lot of time
> studying part-whole hierarchies while DST/MST deals with the state space of a system and the possible trajectories of
> the system through it.
>
> Category theory is especially useful in the latter application, abstracting or generalizing as it does the concepts
> of mathematical objects, functions, and transformations.
>
> For my part I have come to take the DST/MST approach as more fundamental since it starts with fewer assumptions about
> the anatomy or architecture of the as-yet hypothetical agent, making it one of the first and continuing tasks of the
> agent to discover its own boundaries, potentials, and structures.

Regards,

Jon

Lenard Troncale

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Dec 12, 2018, 1:04:14 PM12/12/18
to structura...@googlegroups.com, Cybernetic Communications, Ontolog Forum, SysSciWG
Jon,

I appreciate your characterization of GST but think that it is very much outdated or limited to just your experience. For example, the SPT (Systems Processes Theory) candidate GST has “hierarchy” considerations as just ONE of its 110 (or a more conservative list of 80) isomorphies not to even mention its many Linkage Propositions defining influences among the isomorphies. Each of those isomorphies has many dozens of books in its key references list, and literally million or more research articles on their development and elucidation. So just stating that GST is limited to hierarchy is simply not true. Although I do appreciate your focus on one approach and lineage as being a bit more static and the other more dynamic if your purview is limited to just the one isomorphy.

Systems Engineer Duane Hybertson and I were going to produce a paper roughly entitled “Model of Models” at one time that focused on the individual models of individual Isomorphies being literally a library of potential piecemeal models that could be assembled for any model of any complex system. But we diverged so much in our outline that we never produced it.

Otherwise, I would like to say that I have appreciated your many and detailed responses to the Simpson(s) projects and to the Category Theory debates with Ken Lloyd. Your comments have been detailed and informative to the limits of my ability to understand them and your participation superior. Just please expand your view of GST by seeing these websites (still under construction).
albertgwilson.com   (just the systems pull down menu)
as the years go by they will be more heavily populated with articles and exposition on these matters.

Dr. Len Troncale
Professor Emeritus and Past Chair
Dept. of Biological Sciences,
Founding Director Emeritus, Inst. for Advanced Systems Studies,
Currently Lecturer, Master in Systems Engineering
California State Polytechnic University
3801 W. Temple Ave.
Pomona, California 91768
University Office: (94-290) 909-869-2045
Claremont Office: 232 Harrison, Suite B: 909-541-5095


On Dec 12, 2018, at 8:11 AM, Jon Awbrey <jaw...@att.net> wrote:

All & Sundry —

Various discussions in various places have brought back to mind this thread
from early this fall, prompting me to make a try at continuing it.  Here's
a series of blog posts where I kept track of a few points along the way:

• Differential Logic, Dynamic Systems, Tangent Functors • 1


• Differential Logic, Dynamic Systems, Tangent Functors • Discussion 1


• Differential Logic, Dynamic Systems, Tangent Functors • Discussion 2


• Differential Logic, Dynamic Systems, Tangent Functors • Discussion 3

“This article develops a differential extension of propositional calculus and applies it to a context of problems
arising in dynamic systems.  The work pursued here is coordinated with a parallel application that focuses on neural
network systems, but the dependencies are arranged to make the present article the main and the more self-contained
work, to serve as a conceptual frame and a technical background for the network project.”
Regards,
Jon

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joseph simpson

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Dec 12, 2018, 4:46:42 PM12/12/18
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Jon:

Very interesting work, perspective and presentation.

It appears that the Theme One computer program is designed to support this type of differential logic.

We have settled back into our home and have just a few minor outstanding issues to address.

Now I will have more time to consider how differential logic compares to fuzzy logic and "The Combs Method For Rapid Inference."  See:


In any case, I hope to make more progress in this area.

Take care, be good to yourself and have fun,

Joe

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Joe Simpson

“Reasonable people adapt themselves to the world. 

Unreasonable people attempt to adapt the world to themselves. 

All progress, therefore, depends on unreasonable people.”

George Bernard Shaw

Jon Awbrey

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Dec 17, 2018, 9:40:33 AM12/17/18
to SysSciWG, Lenard Troncale, Structural Modeling, Cybernetic Communications, Ontolog Forum
Dear Len,

Thanks for the links. I have signed on to the LifeWork blog
at http://lentroncale.com and will explore it as time goes by.

I wasn't trying to define a whole field but merely describe my experience in forums
like these, where it took me a while to realize that when I use the word “system”
a great many people are not thinking what I'm thinking when I use it. The first
thing in my mind is almost always a state space X and the possible trajectories
of a representative point through it. But a lot of people will be thinking of
a “system”, like the word says, as a collection of parts “standing together”.
Naturally I'd like to reach the point of discussing such things, it's just
that it takes me a while, and considerable analysis of X, to get there.

It goes without saying this has to do with the boundaries of my own experience
and the emphases of my teachers and other influencers in systems, the early ones
taking their ground in Ashby, Wiener, and the MIT school, the later ones stressing
optimal control and learning organizations, but mostly it has to do with my current
objectives and the species of intelligent systems, Inquiry Driven Systems, I want to
understand and help to build.

Regards,

Jon

--
On 12/12/2018 1:04 PM, Lenard Troncale wrote:
> Jon,
>
> I appreciate your characterization of GST but think that it is very much outdated or limited to just your experience.
> For example, the SPT (Systems Processes Theory) candidate GST has “hierarchy” considerations as just ONE of its 110
> (or a more conservative list of 80) isomorphies not to even mention its many Linkage Propositions defining influences
> among the isomorphies. Each of those isomorphies has many dozens of books in its key references list, and literally
> million or more research articles on their development and elucidation. So just stating that GST is limited to
> hierarchy is simply not true. Although I do appreciate your focus on one approach and lineage as being a bit more
> static and the other more dynamic if your purview is limited to just the one isomorphy.
>
> Systems Engineer Duane Hybertson and I were going to produce a paper roughly entitled “Model of Models” at one time
> that focused on the individual models of individual Isomorphies being literally a library of potential piecemeal
> models that could be assembled for any model of any complex system. But we diverged so much in our outline that we
> never produced it.
>
> Otherwise, I would like to say that I have appreciated your many and detailed responses to the Simpson(s) projects
> and to the Category Theory debates with Ken Lloyd. Your comments have been detailed and informative to the limits of
> my ability to understand them and your participation superior. Just please expand your view of GST by seeing these
> websites (still under construction). lentroncale.com<http://lentroncale.com>
> systemsprocessestheory.com<http://systemsprocessestheory.com> ISSP.org<http://ISSP.org>
> albertgwilson.com<http://albertgwilson.com> (just the systems pull down menu) is-ge.com<http://is-ge.com> as the
> years go by they will be more heavily populated with articles and exposition on these matters.
>
> Dr. Len Troncale Professor Emeritus and Past Chair Dept. of Biological Sciences, Founding Director Emeritus, Inst.
> for Advanced Systems Studies, Currently Lecturer, Master in Systems Engineering California State Polytechnic
> University 3801 W. Temple Ave. Pomona, California 91768 University Office: (94-290) 909-869-2045 Claremont Office:
> 232 Harrison, Suite B: 909-541-5095 Cell: 909-374-6115 Email: lrtro...@cpp.edu<mailto:lrtro...@cpp.edu>
>

Jon Awbrey

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Dec 17, 2018, 2:48:15 PM12/17/18
to SysSciWG, joseph simpson, Structural Modeling, Cybernetic Communications, Ontolog Forum
Dear Joe, All —

Yes, all these strands are strongly entangled.

I had already spent a full decade wrestling with the works of
Charles Sanders Peirce and George Spencer Brown before my need
to figure out what they were talking about in the way of logical
graphs and math in general drove me to the extremes of enrolling
in a graduate math program. I found much diversion and measures
of enlightenment there but soon encountered questions I just had
to know the answers to. One of my office mates suggested I devote
some effort to developing a theorem prover to assist with the task.
Theorem provers in those days were so primitive everyone we knew was
hacking out their own, so that is what I set to. Naturally I turned
to my previous tillage of logical graphs as a seed bed for my system.

Need to beak here ... to be continued ...

Lenard Troncale

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Dec 17, 2018, 4:04:49 PM12/17/18
to Jon Awbrey, SysSciWG, Structural Modeling, Cybernetic Communications, Ontolog Forum
Jon,

I have been following, with some difficulty due to my limitations, your conversations with the Simpsons et. al. and their excellent series of Saturday discussions. Really good work with rare continuity. Thank you for signing on to the Troncale Lifework. You might also be interested in the Wilson Lifework website. albertgwilson.com
because it covers many of the same things. We now have seven interlocking websites.

I greatly admire your statement below that shows great sensitivity to the parameters of your personal neural nets (brain evolution). Very few people are so aware of themselves. I, myself, am severely limited by almost no learning in math over all my years. That limits me greatly. But it also colors my ability to regard systems as you say. So I also have MY version of "systemness.” Mostly from the viewpoint of particular systems as understood by scientific experiments. That is considerably different from math versions, but not necessarily to be discounted. Comparing across many case studies is one way to find regularities of pattern before they are expressed in formalisms, but not the only way. Presumably the patterns observed will eventually be formalized.

Len


On Dec 17, 2018, at 6:40 AM, Jon Awbrey <jaw...@att.net> wrote:

Dear Len,

Thanks for the links.  I have signed on to the LifeWork blog


I wasn't trying to define a whole field but merely describe my experience in forums
like these, where it took me a while to realize that when I use the word “system”
a great many people are not thinking what I'm thinking when I use it.  The first
thing in my mind is almost always a state space X and the possible trajectories
of a representative point through it.  But a lot of people will be thinking of
a “system”, like the word says, as a collection of parts “standing together”.
Naturally I'd like to reach the point of discussing such things, it's just
that it takes me a while, and considerable analysis of X, to get there.

It goes without saying this has to do with the boundaries of my own experience
and the emphases of my teachers and other influencers in systems, the early ones
taking their ground in Ashby, Wiener, and the MIT school, the later ones stressing
optimal control and learning organizations, but mostly it has to do with my current
objectives and the species of intelligent systems, Inquiry Driven Systems, I want to
understand and help to build.

Regards,

Jon

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On 12/12/2018 1:04 PM, Lenard Troncale wrote:
Jon,
I appreciate your characterization of GST but think that it is very much outdated or limited to just your experience.
For example, the SPT (Systems Processes Theory) candidate GST has “hierarchy” considerations as just ONE of its 110
(or a more conservative list of 80) isomorphies not to even mention its many Linkage Propositions defining influences
among the isomorphies. Each of those isomorphies has many dozens of books in its key references list, and literally
million or more research articles on their development and elucidation. So just stating that GST is limited to
hierarchy is simply not true. Although I do appreciate your focus on one approach and lineage as being a bit more
static and the other more dynamic if your purview is limited to just the one isomorphy.
Systems Engineer Duane Hybertson and I were going to produce a paper roughly entitled “Model of Models” at one time
that focused on the individual models of individual Isomorphies being literally a library of potential piecemeal
models that could be assembled for any model of any complex system. But we diverged so much in our outline that we
never produced it.
Otherwise, I would like to say that I have appreciated your many and detailed responses to the Simpson(s) projects
and to the Category Theory debates with Ken Lloyd. Your comments have been detailed and informative to the limits of
my ability to understand them and your participation superior. Just please expand your view of GST by seeing these
websites (still under construction). lentroncale.com<https://na01.safelinks.protection.outlook.com/?url=http%3A%2F%2Flentroncale.com&amp;data=02%7C01%7Clrtroncale%40cpp.edu%7Ca708118ddc734157ccb408d6642d9945%7C164ba61e39ec4f5d89ffaa1f00a521b4%7C0%7C0%7C636806544338545047&amp;sdata=dd5jnxGbc4QRqk8zyxcMq%2Be%2FnjtJfKHPCvHlnRfoW9c%3D&amp;reserved=0> systemsprocessestheory.com<https://na01.safelinks.protection.outlook.com/?url=http%3A%2F%2Fsystemsprocessestheory.com&amp;data=02%7C01%7Clrtroncale%40cpp.edu%7Ca708118ddc734157ccb408d6642d9945%7C164ba61e39ec4f5d89ffaa1f00a521b4%7C0%7C0%7C636806544338555052&amp;sdata=cnTJNz2WUJasApiVInR9Ox3kEIgGHcPjoeQNcAgrL%2Fg%3D&amp;reserved=0> ISSP.org<https://na01.safelinks.protection.outlook.com/?url=http%3A%2F%2FISSP.org&amp;data=02%7C01%7Clrtroncale%40cpp.edu%7Ca708118ddc734157ccb408d6642d9945%7C164ba61e39ec4f5d89ffaa1f00a521b4%7C0%7C0%7C636806544338555052&amp;sdata=f8WTXALBC9nkd73ujVEFuAn7tTbljz8BflNgfo8r0UM%3D&amp;reserved=0> albertgwilson.com<https://na01.safelinks.protection.outlook.com/?url=http%3A%2F%2Falbertgwilson.com&amp;data=02%7C01%7Clrtroncale%40cpp.edu%7Ca708118ddc734157ccb408d6642d9945%7C164ba61e39ec4f5d89ffaa1f00a521b4%7C0%7C0%7C636806544338555052&amp;sdata=PSgGd6vuKO5MTQH%2FT1tMpV0jDOlNCKB68%2BudBM%2BueSw%3D&amp;reserved=0>   (just the systems pull down menu) is-ge.com<https://na01.safelinks.protection.outlook.com/?url=http%3A%2F%2Fis-ge.com&amp;data=02%7C01%7Clrtroncale%40cpp.edu%7Ca708118ddc734157ccb408d6642d9945%7C164ba61e39ec4f5d89ffaa1f00a521b4%7C0%7C0%7C636806544338555052&amp;sdata=rUQSXGJ5VnFqROJdlumpmITdB4tsjpBB%2BN14dHUMyY8%3D&amp;reserved=0> as the

joseph simpson

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Dec 17, 2018, 8:37:11 PM12/17/18
to Jon Awbrey, SysSciWG, Structural Modeling, Cybernetic Communications, Ontolog Forum
Jon:

Very interesting.. I look forward to the continuing story...

A common theme that runs through "expert systems", large group discussions, and system design discussions is the idea of "cognitive complexity" overload.

William Combs addressed the rule explosion associated with binary expert systems using a different type of logical rule base (union rule combination.)

Lotfi Zadeh addressed a similar issue associated with binary logic using fuzzy logic (continuous logical value set.)

Warfield addressed a similar issue in large scale system component composition using the concept of controlling dimensions (describe a space - enumerate the objects in the space.)

Your work with differential logic appears to blend discrete values and continuous values  depending on the context of interest.

I would like to see if I can apply your deferential logic to the insurance problem outlined by Combs.  I am looking for a simple, concrete example that I can use to explore your concepts.

One area of interest is the idea of definition.  Warfield outlines four types of definitions:
  1: Definition by naming (weakest type of definition.)
  2: Definition by extension
  3: Definition by intension
  4: Definition by relationship (strongest form of definition.)

Definition by relationship creates natural clusters and conceptual dimensions.

When we engage and unknown area and start to define this new unknown area (or system), we are engaged in the activity of definition.

As pointed out by Kevin Dye, last Saturday, current structural modeling techniques have a natural complexity barrier.  The structural modeling process, for large scale activities, needs to incorporate the concepts associated with dimensions, quads and tapestries (all concepts developed by Warfield.). 

Groups of objects are encapsulated in a quad (identifiable objects.)

Quads may be "woven" into tapestries that depict and define the area of interest.

I am working on a new set of code for the January 5th structural modeling session.

Some of these more advanced techniques will hopefully make it into one of the future versions of the open source structural modeling software.

Lenard Troncale

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Dec 18, 2018, 1:59:18 PM12/18/18
to syss...@googlegroups.com, Jon Awbrey, Structural Modeling, Cybernetic Communications, Ontolog Forum
Ken,

Please define the acronym NAFKI for those less experienced.
Thanks. Len

On Dec 18, 2018, at 8:50 AM, kall...@gmail.com wrote:

This report may be of interest:
COLLABORATIONS
of
CONSEQUENCE
NAKFI’S 15 YEARS
IGNITING INNOVATION AT THE
INTERSECTIONS OF DISCIPLINES
 
 
Notice the empty quadrant …
 
<image002.png>
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Jon Awbrey

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Dec 19, 2018, 11:11:05 AM12/19/18
to SysSciWG, Structural Modeling, Cybernetic Communications, Ontolog Forum
Dear Joe,

You are racing way ahead of me, and maybe down a couple different roads,
but I'll be making a dogged effort to stick with my math-bio-graphical
narrative this time around, and try to tell how I came to climb down
from logical trees and learned to love logical cacti.

As far as the logical ballpark goes, this is all just classical propositional
logic, what my old circle used to call “zeroth order logic”, alluding to its
basemental status for every storey built on it. (But I have since found that
others use that term for other things, so usage varies as it usually does.)

When it comes to semantics, the class of formal or mathematical objects residing
among the referents of our propositional signs, I'm content for most purposes to
say they're all the same, namely, Boolean functions of abstract type f : B^k → B,
where B = {0, 1} and k is a non-negative integer. Although we're likely to have
other sorts of meanings in mind, this class of models suffices for a ready check
on logical consistency and serves us well, especially in practical applications.

The upshot is — I'm aiming for innovation solely in the syntactic sphere,
the end being only to discover/invent a better syntax for the same realm
of logical objects.

To be continued ...
On 12/17/2018 8:36 PM, joseph simpson wrote:
> Jon:
>

joseph simpson

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Dec 19, 2018, 2:09:18 PM12/19/18
to Ontolog Forum @ GG, SysSciWG, Structural Modeling, Cybernetic Communications
Jon:

Thanks for the thoughtful, detailed reply.

Your careful, measured approach is the best.

A number of candidate relationships, based on my past experience, were presented in an effort to populate the broader conceptual space.

Your statement:

"I'm aiming for innovation solely in the syntactic sphere,
the end being only to discover/invent a better syntax for the same realm
of logical objects."

Helps me better understand your objective.

Syntax is associated with a specific language type.

"Syntax: the arrangement of words and phrases to create well-formed sentences in a language."

Any given item of interest can be described in at least three kinds of languages;
  -- Natural language (informal )
  -- Structured Graphics (could be informal or formal)
  -- Mathematics (formal)

One goal of the structural modeling project is the establishment of the capability to express the same system (item of interest) isomorphically in all three languages.

Each language has its advantages and disadvantages, a combination of languages appears to have more expressive power that a single language.

Exploration of formal, mathematical syntax is an excellent goal, do not let me distract you.

You are correct in stating that I am combining language types. 

In the case of fuzzy logic, natural language and mathematics are combined.

In the case of the Union Rule Configuration, only mathematics is considered.  However, the application to discrete and fuzzy logic extends the exploration to natural language.

In the case of "Warfield dimensional space," mathematics and natural language are both addressed, but at a somewhat limited level and scale.

Racing ahead -- guilty as charged.

I will try to stay a little closer to the proposed path.
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Jon Awbrey

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Dec 20, 2018, 1:54:32 PM12/20/18
to SysSciWG, Ontolog Forum, Structural Modeling, Cybernetic Communications
Joe, All —

Let me stand back from the picture and say how the dimensions of
syntax, semantics, and pragmatics look from a pragmatic semiotic
or triadic sign relational perspective.

O is an object domain, a set of elements under view in a given discussion.
Depending on the application we might be calling it a universe of discourse,
a population, a sample space, a state space, or any number of other things.

S and I are sets of signs that are related to O by means of a triadic relation,
L ⊆ O × S × I. If the triadic relation L satisfies a set of further conditions
that we call a definition of a sign relation then we say L is a sign relation.

Peirce's best definitions of a sign relation are pretty minimal in what they
demand and cover a wide range of cases from barely formed to highly structured.

Let's skip to the more structured types of sign relations that form our usual interest.

In a typical case like that, S is a formal language defined by a formal grammar.

Generally speaking, we might think of I as being more loosely defined in its own right,
but when it comes down to the brass syn-tacks of formal investigations, the so-called
interpretant sign domain I will also be a formal language. Here the cases divide into
two broad sorts.

(S = I). We use this case to discuss transitions in time from one sign to the next.
(S ≠ I). We use this case to discuss translations from one language to another.

Have to break here ...

Jon

On 12/19/2018 2:09 PM, joseph simpson wrote:
> Jon:
>

joseph simpson

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Dec 20, 2018, 8:07:41 PM12/20/18
to structura...@googlegroups.com, SysSciWG, Ontolog Forum, Cybernetic Communications
Jon:

Thanks for the additional information and detail.

The detail creates an expanded view of the discussion context.

Any further comments will be reserved for a while to allow the narrative an unencumbered space to expand and grow. 

Take care, be good to yourself and have fun,

Joe


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Jon Awbrey

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Dec 24, 2018, 12:05:03 PM12/24/18
to SysSciWG, Structural Modeling, Cybernetic Communications, Ontolog Forum
Len, All —

Let me go back to the top of this thread and expand on the motivations
I expressed there, with particular reference to the points you raised.

Differential Logic, Dynamic Systems, Tangent Functors • 1
https://inquiryintoinquiry.com/2018/09/05/differential-logic-dynamic-systems-tangent-functors-%e2%80%a2-1/

> People interested in category theory as applied to systems
> may wish to check out the following article, reporting work
> I carried out while engaged in a systems engineering program
> at Oakland University.

Category theory, as working mathematicians understand it, is one of the chief
conceptual frameworks for the development of mathematics today, the other being
set theory. Every graduate math course I ever took began with a two- or three-week
review of set theory and category theory before launching into the main subject matter.
From a logical point of view, however, category theory has a history stretching back to
Aristotle.

I once started writing a sketch on the “Precursors of Category Theory”, collecting
a sample of historical landmarks along the way, from Aristotle up through its modern
mathematical avatars. Here's a link to a survey page on my blog:

Survey of Precursors Of Category Theory • 1
https://inquiryintoinquiry.com/2015/05/15/survey-of-precursors-of-category-theory-%e2%80%a2-1/

I'll continue from here next time ...

Jon

--
> The problem addressed is a longstanding one, that of building bridges
> to negotiate the gap between qualitative and quantitative descriptions
> of complex phenomena, like those we meet in analyzing and engineering
> systems, especially intelligent systems endowed with a capacity for
> processing information and acquiring knowledge of objective reality.
>
> One of the ways this problem arises has to do with describing change
> in logical, qualitative, or symbolic terms, long before we grasp the
> reality beneath the appearances firmly enough to cast it in measured,
> quantitative, real number form.
>
> Development on the quantitative shore got no further than a Sisyphean
> beachhead until the discovery/invention of differential calculus by
> Leibniz and Newton, after which things advanced by leaps and bounds.
>
> And there’s our clue what we need to do on the qualitative shore, namely,
> to discover/invent the missing logical analogue of differential calculus.
>
> With that pre-ramble ...
>
> Differential Logic and Dynamic Systems
> intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0
>
> This article develops a differential extension of propositional calculus
> and applies it to a context of problems arising in dynamic systems. The
> work pursued here is coordinated with a parallel application that focuses
> on neural network systems, but the dependencies are arranged to make the
> present article the main and the more self-contained work, to serve as a
> conceptual frame and a technical background for the network project.

On 12/17/2018 4:04 PM, Lenard Troncale wrote:> Jon,
>
> I have been following, with some difficulty due to my limitations,
> your conversations with the Simpsons et. al. and their excellent
> series of Saturday discussions. Really good work with rare continuity.
> Thank you for signing on to the Troncale Lifework. You might also be
> interested in the Wilson Lifework website. < http://albertgwilson.com >
> because it covers many of the same things. We now have seven interlocking
> websites.
>
> I greatly admire your statement below that shows great sensitivity
> to the parameters of your personal neural nets (brain evolution).
> Very few people are so aware of themselves. I, myself, am severely
> limited by almost no learning in math over all my years. That limits
> me greatly. But it also colors my ability to regard systems as you say.
> So I also have MY version of "systemness”. Mostly from the viewpoint
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