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Directory:Jon Awbrey/Papers/Differential Logic : Introduction (view source)
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, 12:57, 23 June 2009→Note 23: → →Applications
{{DISPLAYTITLE:Differential Logic : Introduction}}
{{DISPLAYTITLE:Differential Logic : Introduction}}
<div class="nonumtoc">__TOC__</div>
'''Differential logic''' is the component of logic whose object is the description of variation — for example, the aspects of change, difference, distribution, and diversity — in [[universes of discourse]] that are subject to logical description. In formal logic, differential logic treats the principles that govern the use of a ''differential logical calculus'', that is, a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.
'''Differential logic''' is the component of logic whose object is the description of variation — for example, the aspects of change, difference, distribution, and diversity — in [[universes of discourse]] that are subject to logical description. In formal logic, differential logic treats the principles that govern the use of a ''differential logical calculus'', that is, a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.
The differential field <math>\operatorname{D}(pq)</math> specifies the changes that need to be made from each point of <math>X\!</math> in order to feel a change in the felt value of the field <math>pq.\!</math>
The differential field <math>\operatorname{D}(pq)</math> specifies the changes that need to be made from each point of <math>X\!</math> in order to feel a change in the felt value of the field <math>pq.\!</math>
==Note 23==
==Note 24==
Now that we've introduced the field picture as an aid to thinking about propositions and their analytic series, a very pleasing way of picturing the relationships among a proposition <math>f : X \to \mathbb{B},</math> its enlargement or shift map <math>\operatorname{E}f : \operatorname{E}X \to \mathbb{B},</math> and its difference map <math>\operatorname{D}f : \operatorname{E}X \to \mathbb{B}</math> can now be drawn.
To illustrate this possibility, let's return to the differential analysis of the conjunctive proposition <math>f(p, q) = pq,\!</math> giving the development a slightly different twist at the appropriate point.
Figure 24-1 shows the proposition <math>pq\!</math> once again, which we now view as a scalar field — analogous to a ''potential hill'' in physics, but in logic tantamount to a ''potential plateau'' — where the shaded region indicates an elevation of 1 and the unshaded region indicates an elevation of 0.
{| align="center" cellpadding="10" style="text-align:center"
| [[Image:Field Picture PQ Conjunction.jpg|500px]]
|-
| <math>\text{Figure 24-1. Proposition}~ pq : X \to \mathbb{B}</math>
|}
Given a proposition <math>f : X \to \mathbb{B},</math> the ''tacit extension'' of <math>f\!</math> to <math>\operatorname{E}X</math> is denoted <math>\varepsilon f : \operatorname{E}X \to \mathbb{B}</math> and defined by the equation <math>\varepsilon f = f,</math> so it's really just the same proposition residing in a bigger universe. Tacit extensions formalize the intuitive idea that a function on a particular set of variables can be extended to a function on a superset of those variables in such a way that the new function obeys the same constraints on the old variables, with a "don't care" condition on the new variables.
Figure 24-2 shows the tacit extension of the scalar field <math>pq : X \to \mathbb{B}</math> to the differential field <math>\varepsilon (pq) : \operatorname{E}X \to \mathbb{B}.</math>
{| align="center" cellpadding="10" style="text-align:center; width:90%"
{| align="center" cellpadding="10" style="text-align:center"
| [[Image:Field Picture PQ Tacit Extension Conjunction.jpg|500px]]
|-
| <math>\text{Figure 24-2. Tacit Extension}~ \varepsilon (pq) : \operatorname{E}X \to \mathbb{B}</math>
|-
|
|
<pre>
<math>\begin{array}{rcccccc}
\varepsilon (pq)
& = &
p & \cdot & q & \cdot &
\texttt{(} \operatorname{d}p \texttt{)}
\texttt{(} \operatorname{d}q \texttt{)}
\\[4pt]
& + &
p & \cdot & q & \cdot &
\texttt{(} \operatorname{d}p \texttt{)}
\texttt{~} \operatorname{d}q \texttt{~}
\\[4pt]
& + &
p & \cdot & q & \cdot &
\texttt{~} \operatorname{d}p \texttt{~}
\texttt{(} \operatorname{d}q \texttt{)}
\\[4pt]
& + &
p & \cdot & q & \cdot &
\texttt{~} \operatorname{d}p \texttt{~}
\texttt{~} \operatorname{d}q \texttt{~}
\end{array}</math>
</pre>
|}
|}
==Note 24==
==Note 25==
Continuing with the example <math>pq : X \to \mathbb{B},</math> Figure 25-1 shows the enlargement or shift map <math>\operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}</math> in the same style of differential field picture that we drew for the tacit extension <math>\varepsilon (pq) : \operatorname{E}X \to \mathbb{B}.</math>
{| align="center" cellpadding="10" style="text-align:center"
{| align="center" cellpadding="10" style="text-align:center"
| [[Image:Field Picture PQ Conjunction.jpg|500px]]
| [[Image:Field Picture PQ Enlargement Conjunction.jpg|500px]]
|-
|-
| <math>\text{Figure 24-1. Proposition}~ pq : X \to \mathbb{B}</math>
| <math>\text{Figure 25-1. Enlargement Map}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}</math>
|-
|-
|
|
<math>\begin{array}{rcccccc}
<math>\begin{array}{rcccccc}
\varepsilon (pq)
\operatorname{E}(pq)
& = &
& = &
p & \cdot & q & \cdot &
p
& \cdot &
q
& \cdot &
\texttt{(} \operatorname{d}p \texttt{)}
\texttt{(} \operatorname{d}p \texttt{)}
\texttt{(} \operatorname{d}q \texttt{)}
\texttt{(} \operatorname{d}q \texttt{)}
\\[4pt]
\\[4pt]
& + &
& + &
p & \cdot & q & \cdot &
p
& \cdot &
\texttt{(} q \texttt{)}
& \cdot &
\texttt{(} \operatorname{d}p \texttt{)}
\texttt{(} \operatorname{d}p \texttt{)}
\texttt{~} \operatorname{d}q \texttt{~}
\texttt{~} \operatorname{d}q \texttt{~}
\\[4pt]
\\[4pt]
& + &
& + &
p & \cdot & q & \cdot &
\texttt{(} p \texttt{)}
& \cdot &
q
& \cdot &
\texttt{~} \operatorname{d}p \texttt{~}
\texttt{~} \operatorname{d}p \texttt{~}
\texttt{(} \operatorname{d}q \texttt{)}
\texttt{(} \operatorname{d}q \texttt{)}
\\[4pt]
\\[4pt]
& + &
& + &
p & \cdot & q & \cdot &
\texttt{(} p \texttt{)}
& \cdot &
\texttt{(} q \texttt{)}
& \cdot &
\texttt{~} \operatorname{d}p \texttt{~}
\texttt{~} \operatorname{d}p \texttt{~}
\texttt{~} \operatorname{d}q \texttt{~}
\texttt{~} \operatorname{d}q \texttt{~}
|}
|}
A very important conceptual transition has just occurred here, almost tacitly, as it were. Generally speaking, having a set of mathematical objects of compatible types, in this case the two differential fields <math>\varepsilon f</math> and <math>\operatorname{E}f,</math> both of the type <math>\operatorname{E}X \to \mathbb{B},</math> is very useful, because it allows us to consider these fields as integral mathematical objects that can be operated on and combined in the ways that we usually associate with algebras.
In this case one notices that the tacit extension <math>\varepsilon f</math> and the enlargement <math>\operatorname{E}f</math> are in a certain sense dual to each other. The tacit extension <math>\varepsilon f</math> indicates all the arrows out of the region where <math>f\!</math> is true and the enlargement <math>\operatorname{E}f</math> indicates all the arrows into the region where <math>f\!</math> is true. The only arc they have in common is the no-change loop <math>\texttt{(} \operatorname{d}p \texttt{)(} \operatorname{d}q \texttt{)}</math> at <math>pq.\!</math> If we add the two sets of arcs in mod 2 fashion then the loop of multiplicity 2 zeroes out, leaving the 6 arrows of <math>\operatorname{D}(pq) = \varepsilon(pq) + \operatorname{E}(pq)</math> that are illustrated in Figure 25-2.
{| align="center" cellpadding="10" style="text-align:center"
{| align="center" cellpadding="10" style="text-align:center"
| [[Image:Field Picture PQ Enlargement Conjunction.jpg|500px]]
| [[Image:Field Picture PQ Difference Conjunction.jpg|500px]]
|-
|-
| <math>\text{Figure 25-1. Enlargement Map}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}</math>
| <math>\text{Figure 25-2. Difference Map}~ \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}</math>
|-
|-
|
|
<math>\begin{array}{rcccccc}
<math>\begin{array}{rcccccc}
\operatorname{E}(pq)
\operatorname{D}(pq)
& = &
& = &
p
p
q
q
& \cdot &
& \cdot &
\texttt{(}
\texttt{(} \operatorname{d}p \texttt{)}
\texttt{(} \operatorname{d}p \texttt{)}
\texttt{(} \operatorname{d}q \texttt{)}
\texttt{(} \operatorname{d}q \texttt{)}
\texttt{)}
\\[4pt]
\\[4pt]
& + &
& + &
\texttt{(} q \texttt{)}
\texttt{(} q \texttt{)}
& \cdot &
& \cdot &
\texttt{~}
\texttt{(} \operatorname{d}p \texttt{)}
\texttt{(} \operatorname{d}p \texttt{)}
\texttt{~} \operatorname{d}q \texttt{~}
\texttt{~} \operatorname{d}q \texttt{~}
\texttt{~}
\\[4pt]
\\[4pt]
& + &
& + &
q
q
& \cdot &
& \cdot &
\texttt{~}
\texttt{~} \operatorname{d}p \texttt{~}
\texttt{~} \operatorname{d}p \texttt{~}
\texttt{(} \operatorname{d}q \texttt{)}
\texttt{(} \operatorname{d}q \texttt{)}
\texttt{~}
\\[4pt]
\\[4pt]
& + &
& + &
\texttt{(} p \texttt{)}
\texttt{(} p \texttt{)}
& \cdot &
& \cdot &
\texttt{(} q \texttt{)}
\texttt{(}q \texttt{)}
& \cdot &
& \cdot &
\texttt{~}
\texttt{~} \operatorname{d}p \texttt{~}
\texttt{~} \operatorname{d}p \texttt{~}
\texttt{~} \operatorname{d}q \texttt{~}
\texttt{~} \operatorname{d}q \texttt{~}
\texttt{~}
\end{array}</math>
\end{array}</math>
|}
|}
==Tangent and Remainder Maps==
If we follow the classical line that singles out linear functions as ideals of simplicity, then we may complete the analytic series of the proposition <math>f = pq : X \to \mathbb{B}</math> in the following way.
Figure 26-1 shows the differential proposition <math>\operatorname{d}f = \operatorname{d}(pq) : \operatorname{E}X \to \mathbb{B}</math> that we get by extracting the cell-wise linear approximation to the difference map <math>\operatorname{D}f = \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}.</math> This is the logical analogue of what would ordinarily be called ''the'' differential of <math>pq,\!</math> but since I've been attaching the adjective ''differential'' to just about everything in sight, the distinction tends to be lost. For the time being, I'll resort to using the alternative name ''tangent map'' for <math>\operatorname{d}f.\!</math>
{| align="center" cellpadding="10" style="text-align:center"
{| align="center" cellpadding="10" style="text-align:center"
| [[Image:Field Picture PQ Difference Conjunction.jpg|500px]]
| [[Image:Field Picture PQ Differential Conjunction.jpg|500px]]
|-
|-
| <math>\text{Figure 26-1. Tangent Map}~ \operatorname{d}(pq) : \operatorname{E}X \to \mathbb{B}</math>
|}
Just to be clear about what's being indicated here, it's a visual way of summarizing the following data:→
{| align="center" cellpadding="10" style="text-align:center"
|
|
<math>\begin{array}{rcccccc}
<math>\begin{array}{rcccccc}
\operatorname{D}(pq)
\operatorname{d}(pq)
& = &
& = &
p
p & \cdot & q & \cdot &
& \cdot &
\texttt{(} \operatorname{d}p \texttt{,} \operatorname{d}q \texttt{)}
q
& \cdot &
\texttt{(} \operatorname{d}p \texttt{)}
\texttt{)}
\\[4pt]
\\[4pt]
& + &
& + &
p
p & \cdot & \texttt{(} q \texttt{)} & \cdot &
& \cdot &
\operatorname{d}q
\texttt{(} q \texttt{)}
& \cdot &
\\[4pt]
\\[4pt]
& + &
& + &
\texttt{(} p \texttt{)}
\texttt{(} p \texttt{)} & \cdot & q & \cdot &
& \cdot &
\operatorname{d}p
q
& \cdot &
\\[4pt]
\\[4pt]
& + &
& + &
\texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & 0
\texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & 0
\end{array}</math>
\end{array}</math>
|}
|}
In short, <math>\operatorname{r}(pq)</math> is a constant field, having the value <math>\operatorname{d}p~\operatorname{d}q</math> at each cell.
In short, <math>\operatorname{r}(pq)</math> is a constant field, having the value <math>\operatorname{d}p~\operatorname{d}q</math> at each cell.
==Applications==
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.
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.
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.
{| align="center" cellpadding="10" style="text-align:center; width:90%"
|
<pre>
o---------------------------------------------------------------------o
| |
| X |
| o-------------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| / \ |
| / \ |
| / \ |
| o G o |
| | | |
| | | |
| | | |
| | o<---------T---------o |
| | | |
| | | |
| | | |
| o o |
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| o-------------------o |
| |
| |
o---------------------------------------------------------------------o
Figure 23. Elements of a Cybernetic System
</pre>
|}
==Further Reading==
==Further Reading==