[Inquiry] Re: Dynamics And Logic

[Jon Awbrey]

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DAL.  Note 23

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Continuing a thread that was initially spun off at these two sites:

DAL. http://stderr.org/pipermail/inquiry/2004-May/thread.html#1400
DAL. http://forum.wolframscience.com/showthread.php?threadid=420

I want to continue developing the basic tools of differential logic,
which arose out of many years of thinking about the connections
between dynamics and logic -- those that there are and those
that there ought to be -- but I also wanted to give some
hint of the applications that have motivated this work
all along.  One of these applications is to cybernetic
systems, whether we view them as agents or cultures,
individuals or species, organisms or organizations.

A cybernetic system has goals and actions for reaching them.
It has a state space X, giving us all of the states that the
system can be in, plus it has a goal space G c X, the set of
states that the system "likes" to be in, in other words, the
distinguished subset of possible states where the system is
regarded as living, surviving, or thriving, depending on the
type of goal that one has in mind for the system in question.
As for actions, there is to begin with the full set !T! of all
possible actions, each of which is a transformation of the form
T : X -> X, but a given cybernetic system will most likely have
but a subset of these actions available to it at any given time.
And even if we begin by thinking of actions in very general and
very global terms, as arbitrarily complex transformations acting
on the whole state space X, we quickly find a need to analyze and
approximate them in terms of simple transformations acting locally.
The preferred measure of "simplicity" will of course vary from one
paradigm of research to another.

A generic enough picture at this stage of the game, and one that will
remind us of these fundamental features of the cybernetic system even
as things get far more complex, is afforded by Figure 23.

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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` X ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `o-------------------o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` /`````````````````````\ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `/```````````````````````\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` /`````````````````````````\ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `/```````````````````````````\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` /`````````````````````````````\ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `/```````````````````````````````\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` /`````````````````````````````````\ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `o```````````````````````````````````o` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `|```````````````````````````````````|` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `|```````````````````````````````````|` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `|```````````````````````````````````|` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `|`````````````````G`````````````````|` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `|```````````````````````````````````|` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `|```````````````````````````````````|` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `|```````````````````````````````````|` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `o```````````````````````````````````o` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` \`````````````````````````````````/ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `\```````````````````````````````/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` \```````````````````````````T`/ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\`````````````o<------------/-------------o` ` ` ` ` ` ` ` ` |
| ` ` ` ` \`````````````````````````/ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\```````````````````````/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` \`````````````````````/ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `o-------------------o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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Figure 23.  Elements of a Cybernetic System

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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