Changes

==Note 23==

==Note 23==

I want to continue developing the basic tools of differential logic, which arose from exploring the connections between dynamics and logic, 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 see these systems as agents or cultures, individuals or species, organisms or organizations.

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 there are and those 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 see these systems as agents or cultures,

individuals or species, organisms or organizations.

A cybernetic system has goals and actions for reaching them.

A cybernetic system has goals and actions for reaching them. It has a state space <math>X,\!</math> giving us all of the states that the system can be in, plus it has a goal space <math>G \subseteq X,</math> 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 <math>\mathcal{T}</math> of all possible actions, each of which is a transformation of the form <math>T : X \to X,</math> 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 <math>X,\!</math> 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.

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

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&nbsp;23.

remind us of these fundamental features of the cybernetic system even

as things get far more complex, is afforded by Figure 23.

o---------------------------------------------------------------------o

o---------------------------------------------------------------------o

|                                                                    |

|                                                                    |

Figure 23.  Elements of a Cybernetic System

Figure 23.  Elements of a Cybernetic System

</pre>

</pre>

==Note 24==

==Note 24==