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{{DISPLAYTITLE:Dynamics And Logic}}
{{DISPLAYTITLE:Dynamics And Logic}}
'''Note.''' Many problems with the sucky MathJax on this page. The parser apparently reads 4 tildes inside math brackets the way it would in the external wiki environment, in other words, as signature tags. [[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 18:00, 5 December 2014 (UTC)
==Note 1==
==Note 1==
:* DATA. http://forum.wolframscience.com/showthread.php?threadid=228
:* DATA. http://forum.wolframscience.com/showthread.php?threadid=228
One of the first things that you can do, once you have a moderately functional calculus for boolean functions or propositional logic, whatever you choose to call it, is to start thinking about, and even start computing, the differentials of these functions or propositions.
One of the first things that you can do, once you have a moderately efficient calculus for boolean functions or propositional logic, whatever you choose to call it, is to start thinking about, and even start computing, the differentials of these functions or propositions.
Let us start with a proposition of the form <math>p ~\operatorname{and}~ q</math> that is graphed as two labels attached to a root node:
Let us start with a proposition of the form <math>p ~\operatorname{and}~ q</math> that is graphed as two labels attached to a root node:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10"
| [[Image:Cactus Graph Existential P And Q.jpg|500px]]
| p and q |
|}
|}
Written as a string, this is just the concatenation "<math>p~q</math>".
Written as a string, this is just the concatenation <math>p~q</math>.
The proposition <math>pq\!</math> may be taken as a boolean function <math>f(p, q)\!</math> having the abstract type <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B},</math> where <math>\mathbb{B} = \{ 0, 1 \}</math> is read in such a way that <math>0\!</math> means <math>\operatorname{false}</math> and <math>1\!</math> means <math>\operatorname{true}.</math>
The proposition <math>pq\!</math> may be taken as a boolean function <math>f(p, q)\!</math> having the abstract type <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B},</math> where <math>\mathbb{B} = \{ 0, 1 \}</math> is read in such a way that <math>0\!</math> means <math>\operatorname{false}</math> and <math>1\!</math> means <math>\operatorname{true}.</math>
In this style of graphical representation, the value <math>\operatorname{true}</math> looks like a blank label and the value <math>\operatorname{false}</math> looks like an edge.
In this style of graphical representation, the value <math>\operatorname{true}</math> looks like a blank label and the value <math>\operatorname{false}</math> looks like an edge.
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10"
| [[Image:Cactus Graph Existential True.jpg|500px]]
| true |
|}
|}
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10"
| [[Image:Cactus Graph Existential False.jpg|500px]]
| |
| false |
|}
|}
Back to the proposition <math>pq.\!</math> Imagine yourself standing in a fixed cell of the corresponding venn diagram, say, the cell where the proposition <math>pq\!</math> is true, as shown here:
Back to the proposition <math>pq.\!</math> Imagine yourself standing in a fixed cell of the corresponding venn diagram, say, the cell where the proposition <math>pq\!</math> is true, as shown in the following Figure:
{| align="center" cellpadding="10"
{| align="center" cellpadding="10"
Don't think about it — just compute:
Don't think about it — just compute:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10"
| [[Image:Cactus Graph (P,dP)(Q,dQ).jpg|500px]]
| |
|}
|}
The cactus formula <math>\texttt{(p, dp)(q, dq)}</math> and its corresponding graph arise by substituting <math>p + \operatorname{d}p</math> for <math>p\!</math> and <math>q + \operatorname{d}q</math> for <math>q\!</math> in the boolean product or logical conjunction <math>pq\!</math> and writing the result in the two dialects of cactus syntax. This follows from the fact that the boolean sum <math>p + \operatorname{d}p</math> is equivalent to the logical operation of exclusive disjunction, which parses to a cactus graph of the following form:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10"
| [[Image:Cactus Graph (P,dP).jpg|500px]]
|}
|}
|}
Next question: What is the difference between the value of the
Next question: What is the difference between the value of the
proposition <math>pq\!</math> "over there" and the value of the proposition <math>pq\!</math> where you are, all expressed in the form of
proposition <math>pq\!</math> over there, at a distance of <math>\operatorname{d}p</math> and <math>\operatorname{d}q,</math> and the value of the proposition <math>pq\!</math> where you are standing, all expressed in the form of a general formula, of course? Here is the appropriate formulation:
a general formula, of course? Here is the appropriate formulation:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10"
| [[Image:Cactus Graph ((P,dP)(Q,dQ),PQ).jpg|500px]]
| \| |/ p q |
|}
|}
Last question, for now: What is the value of this expression from your current standpoint, that is, evaluated at the point where <math>pq\!</math> is true? Well, substituting <math>1\!</math> for <math>p\!</math> and <math>1\!</math> for <math>q\!</math> in the graph amounts to erasing the labels <math>p\!</math> and <math>q\!,</math> as shown here:
Last question, for now: What is the value of this expression from your current standpoint, that is, evaluated at the point where <math>pq\!</math> is true? Well, substituting <math>1\!</math> for <math>p\!</math> and <math>1\!</math> for <math>q\!</math> in the graph amounts to erasing the labels <math>p\!</math> and <math>q\!,</math> as shown here:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10"
| [[Image:Cactus Graph (( ,dP)( ,dQ), ).jpg|500px]]
| \| |/ |
|}
|}
And this is equivalent to the following graph:
And this is equivalent to the following graph:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10"
| [[Image:Cactus Graph ((dP)(dQ)).jpg|500px]]
| |
|}
|}
We have just met with the fact that the differential of the '''''and''''' is the '''''or''''' of the differentials.
We have just met with the fact that the differential of the '''''and''''' is the '''''or''''' of the differentials.
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10" style="text-align:center; width:90%"
|
<math>p ~\operatorname{and}~ q \quad \xrightarrow{~\operatorname{Diff}~} \quad \operatorname{d}p ~\operatorname{or}~ \operatorname{d}q</math>
<math>\begin{matrix}
|-
p ~\operatorname{and}~ q
| align="center" |
& \quad &
\xrightarrow{\quad\operatorname{Diff}\quad}
& \quad &
\operatorname{d}p ~\operatorname{or}~ \operatorname{d}q
\end{matrix}\!</math>
| o o |
|}
{| align="center" cellpadding="10"
| [[Image:Cactus Graph PQ Diff ((dP)(dQ)).jpg|500px]]
|}
|}
A function like this has an abstract type and a concrete type. The abstract type is what we invoke when we write things like <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}</math> or <math>f : \mathbb{B}^2 \to \mathbb{B}.</math> The concrete type takes into account the qualitative dimensions or the "units" of the case, which can be explained as follows.
A function like this has an abstract type and a concrete type. The abstract type is what we invoke when we write things like <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}</math> or <math>f : \mathbb{B}^2 \to \mathbb{B}.</math> The concrete type takes into account the qualitative dimensions or the "units" of the case, which can be explained as follows.
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10" width="90%"
| Let <math>P\!</math> be the set of values <math>\{ \texttt{(} p \texttt{)},~ p \} ~=~ \{ \operatorname{not}~ p,~ p \} ~\cong~ \mathbb{B}.</math>
| Let <math>P\!</math> be the set of values <math>\{ \texttt{(} p \texttt{)},~ p \} ~=~ \{ \operatorname{not}~ p,~ p \} ~\cong~ \mathbb{B}.</math>
|-
|-
The first couple of operators that we need to consider are logical analogues of the pair that play a founding role in the classical finite difference calculus, namely:
The first couple of operators that we need to consider are logical analogues of the pair that play a founding role in the classical finite difference calculus, namely:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10" width="90%"
| The ''difference operator'' <math>\Delta,\!</math> written here as <math>\operatorname{D}.</math>
| The ''difference operator'' <math>\Delta,\!</math> written here as <math>\operatorname{D}.</math>
|-
|-
In order to describe the universe in which these operators operate, it is necessary to enlarge the original universe of discourse. Starting from the initial space <math>X = P \times Q,</math> its ''(first order) differential extension'' <math>\operatorname{E}X</math> is constructed according to the following specifications:
In order to describe the universe in which these operators operate, it is necessary to enlarge the original universe of discourse. Starting from the initial space <math>X = P \times Q,</math> its ''(first order) differential extension'' <math>\operatorname{E}X</math> is constructed according to the following specifications:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10" width="90%"
|
|
<math>\begin{array}{rcc}
<math>\begin{array}{rcc}
\operatorname{E}X & = & X \times \operatorname{d}X
\operatorname{E}X & = & X \times \operatorname{d}X
\end{array}</math>
\end{array}\!</math>
|}
|}
where:
where:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10" width="90%"
|
|
<math>\begin{array}{rcc}
<math>\begin{array}{rcc}
& = &
& = &
\{ \texttt{(} \operatorname{d}q \texttt{)},~ \operatorname{d}q \}
\{ \texttt{(} \operatorname{d}q \texttt{)},~ \operatorname{d}q \}
\end{array}</math>
\end{array}\!</math>
|}
|}
The interpretations of these new symbols can be diverse, but the easiest
The interpretations of these new symbols can be diverse, but the easiest option for now is just to say that <math>\operatorname{d}p\!</math> means "change <math>p\!</math>" and <math>\operatorname{d}q</math> means "change <math>q\!</math>".
option for now is just to say that <math>\operatorname{d}p</math> means "change <math>p\!</math>" and <math>\operatorname{d}q</math> means "change <math>q\!</math>".
Drawing a venn diagram for the differential extension <math>\operatorname{E}X = X \times \operatorname{d}X</math> requires four logical dimensions, <math>P, Q, \operatorname{d}P, \operatorname{d}Q,</math> but it is possible to project a suggestion of what the differential features <math>\operatorname{d}p</math> and <math>\operatorname{d}q</math> are about on the 2-dimensional base space <math>X = P \times Q</math> by drawing arrows that cross the boundaries of the basic circles in the venn diagram for <math>X\!,</math> reading an arrow as <math>\operatorname{d}p</math> if it crosses the boundary between <math>p\!</math> and <math>\texttt{(} p \texttt{)}</math> in either direction and reading an arrow as <math>\operatorname{d}q</math> if it crosses the boundary between <math>q\!</math> and <math>\texttt{(} q \texttt{)}</math> in either direction.
Drawing a venn diagram for the differential extension <math>\operatorname{E}X = X \times \operatorname{d}X</math> requires four logical dimensions, <math>P, Q, \operatorname{d}P, \operatorname{d}Q,</math> but it is possible to project a suggestion of what the differential features <math>\operatorname{d}p</math> and <math>\operatorname{d}q</math> are about on the 2-dimensional base space <math>X = P \times Q</math> by drawing arrows that cross the boundaries of the basic circles in the venn diagram for <math>X\!,</math> reading an arrow as <math>\operatorname{d}p</math> if it crosses the boundary between <math>p\!</math> and <math>\texttt{(} p \texttt{)}</math> in either direction and reading an arrow as <math>\operatorname{d}q</math> if it crosses the boundary between <math>q\!</math> and <math>\texttt{(} q \texttt{)}</math> in either direction.
following formula:
following formula:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10" style="text-align:center"
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{E}f(p, q, \operatorname{d}p, \operatorname{d}q)
\operatorname{E}f(p, q, \operatorname{d}p, \operatorname{d}q)
In the example <math>f(p, q) = pq,\!</math> the enlargement <math>\operatorname{E}f</math> is computed as follows:
In the example <math>f(p, q) = pq,\!</math> the enlargement <math>\operatorname{E}f</math> is computed as follows:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10" style="text-align:center"
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{E}f(p, q, \operatorname{d}p, \operatorname{d}q)
\operatorname{E}f(p, q, \operatorname{d}p, \operatorname{d}q)
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| align="center" |
| [[Image:Cactus Graph Ef = (P,dP)(Q,dQ).jpg|500px]]
|}
|}
Given the proposition <math>f(p, q)\!</math> over <math>X = P \times Q,</math> the ''(first order) difference'' of <math>f\!</math> is the proposition <math>\operatorname{D}f</math> over <math>\operatorname{E}X</math> that is defined by the formula <math>\operatorname{D}f = \operatorname{E}f - f,</math> or, written out in full:
Given the proposition <math>f(p, q)\!</math> over <math>X = P \times Q,</math> the ''(first order) difference'' of <math>f\!</math> is the proposition <math>\operatorname{D}f</math> over <math>\operatorname{E}X</math> that is defined by the formula <math>\operatorname{D}f = \operatorname{E}f - f,</math> or, written out in full:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10" style="text-align:center"
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{D}f(p, q, \operatorname{d}p, \operatorname{d}q)
\operatorname{D}f(p, q, \operatorname{d}p, \operatorname{d}q)
In the example <math>f(p, q) = pq,\!</math> the difference <math>\operatorname{D}f</math> is computed as follows:
In the example <math>f(p, q) = pq,\!</math> the difference <math>\operatorname{D}f</math> is computed as follows:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10" style="text-align:center"
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{D}f(p, q, \operatorname{d}p, \operatorname{d}q)
\operatorname{D}f(p, q, \operatorname{d}p, \operatorname{d}q)
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| align="center" |
| [[Image:Cactus Graph Df = ((P,dP)(Q,dQ),PQ).jpg|500px]]
|}
|}
We did not yet go through the trouble to interpret this (first order) ''difference of conjunction'' fully, but were happy simply to evaluate it with respect to a single location in the universe of discourse, namely, at the point picked out by the singular proposition <math>pq,\!</math> that is, at the place where <math>p = 1\!</math> and <math>q = 1.\!</math> This evaluation is written in the form <math>\operatorname{D}f|_{pq}</math> or <math>\operatorname{D}f|_{(1, 1)},</math> and we arrived at the locally applicable law that is stated and illustrated as follows:
We did not yet go through the trouble to interpret this (first order) ''difference of conjunction'' fully, but were happy simply to evaluate it with respect to a single location in the universe of discourse, namely, at the point picked out by the singular proposition <math>pq,\!</math> that is, at the place where <math>p = 1\!</math> and <math>q = 1.\!</math> This evaluation is written in the form <math>\operatorname{D}f|_{pq}</math> or <math>\operatorname{D}f|_{(1, 1)},</math> and we arrived at the locally applicable law that is stated and illustrated as follows:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10" style="text-align:center"
|
<math>f(p, q) ~=~ pq ~=~ p ~\operatorname{and}~ q \quad \Rightarrow \quad \operatorname{D}f|_{pq} ~=~ \texttt{((} \operatorname{dp} \texttt{)(} \operatorname{d}q \texttt{))} ~=~ \operatorname{d}p ~\operatorname{or}~ \operatorname{d}q</math>
<math>f(p, q) ~=~ pq ~=~ p ~\operatorname{and}~ q \quad \Rightarrow \quad \operatorname{D}f|_{pq} ~=~ \texttt{((} \operatorname{dp} \texttt{)(} \operatorname{d}q \texttt{))} ~=~ \operatorname{d}p ~\operatorname{or}~ \operatorname{d}q</math>
|-
|-
| align="center" |
| [[Image:Venn Diagram PQ Difference Conj At Conj.jpg|500px]]
[[Image:Venn Diagram PQ Difference Conj At Conj.jpg|500px]]
|-
|-
| align="center" |
| [[Image:Cactus Graph PQ Difference Conj At Conj.jpg|500px]]
[[Image:Cactus Graph PQ Difference Conj At Conj.jpg|500px]]
|}
|}
The picture shows the analysis of the inclusive disjunction <math>\texttt{((} \operatorname{d}p \texttt{)(} \operatorname{d}q \texttt{))}</math> into the following exclusive disjunction:
The picture shows the analysis of the inclusive disjunction <math>\texttt{((} \operatorname{d}p \texttt{)(} \operatorname{d}q \texttt{))}</math> into the following exclusive disjunction:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10" style="text-align:center"
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{d}p ~\texttt{(} \operatorname{d}q \texttt{)}
\operatorname{d}p ~\texttt{(} \operatorname{d}q \texttt{)}
In the example <math>f(p, q) = pq,\!</math> the value of the difference proposition <math>\operatorname{D}f_x</math> at each of the four points in <math>x \in X\!</math> may be computed in graphical fashion as shown below:
In the example <math>f(p, q) = pq,\!</math> the value of the difference proposition <math>\operatorname{D}f_x</math> at each of the four points in <math>x \in X\!</math> may be computed in graphical fashion as shown below:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10"
| [[Image:Cactus Graph Df = ((P,dP)(Q,dQ),PQ).jpg|500px]]
| Df = ((p, dp)(q, dq), pq) |
|-
|-
| align="center" |
| [[Image:Cactus Graph Df@PQ = ((dP)(dQ)).jpg|500px]]
|-
|-
| align="center" |
| [[Image:Cactus Graph Df@P(Q) = (dP)dQ.jpg|500px]]
|-
|-
| align="center" |
| [[Image:Cactus Graph Df@(P)Q = dP(dQ).jpg|500px]]
|-
|-
| align="center" |
| [[Image:Cactus Graph Df@(P)(Q) = dP dQ.jpg|500px]]
|}
|}
The easy way to visualize the values of these graphical expressions is just to notice the following equivalents:
The easy way to visualize the values of these graphical expressions is just to notice the following equivalents:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10"
| [[Image:Cactus Graph Lobe Rule.jpg|500px]]
| o-o-o-...-o-o-o |
|-
|-
| align="center" |
| [[Image:Cactus Graph Spike Rule.jpg|500px]]
|}
|}
Laying out the arrows on the augmented venn diagram, one gets a picture of a ''differential vector field''.
Laying out the arrows on the augmented venn diagram, one gets a picture of a ''differential vector field''.
The Figure shows the points of the extended universe <math>\operatorname{E}X = P \times Q \times \operatorname{d}P \times \operatorname{d}Q</math> that are indicated by the difference map <math>\operatorname{D}f : \operatorname{E}X \to \mathbb{B},</math> namely, the following six points or singular propositions::
The Figure shows the points of the extended universe <math>\operatorname{E}X = P \times Q \times \operatorname{d}P \times \operatorname{d}Q</math> that are indicated by the difference map <math>\operatorname{D}f : \operatorname{E}X \to \mathbb{B},</math> namely, the following six points or singular propositions::
{| align="center" cellpadding="6"
{| align="center" cellpadding="10"
|
|
<math>\begin{array}{rcccc}
<math>\begin{array}{rcccc}
Abstracting from the augmented venn diagram that shows how the ''models'' or ''satisfying interpretations'' of <math>\operatorname{D}f</math> distribute over the extended universe of discourse <math>\operatorname{E}X = P \times Q \times \operatorname{d}P \times \operatorname{d}Q,</math> the difference map <math>\operatorname{D}f</math> can be represented in the form of a ''digraph'' or ''directed graph'', one whose points are labeled with the elements of <math>X = P \times Q</math> and whose arrows are labeled with the elements of <math>\operatorname{d}X = \operatorname{d}P \times \operatorname{d}Q,</math> as shown in the following Figure.
Abstracting from the augmented venn diagram that shows how the ''models'' or ''satisfying interpretations'' of <math>\operatorname{D}f</math> distribute over the extended universe of discourse <math>\operatorname{E}X = P \times Q \times \operatorname{d}P \times \operatorname{d}Q,</math> the difference map <math>\operatorname{D}f</math> can be represented in the form of a ''digraph'' or ''directed graph'', one whose points are labeled with the elements of <math>X = P \times Q</math> and whose arrows are labeled with the elements of <math>\operatorname{d}X = \operatorname{d}P \times \operatorname{d}Q,</math> as shown in the following Figure.
{| align="center" cellpadding="10"
{| align="center" cellpadding="10" style="text-align:center"
| [[Image:Directed Graph PQ Difference Conj.jpg|500px]]
| [[Image:Directed Graph PQ Difference Conj.jpg|500px]]
|}
|-
|
|
<math>\begin{array}{rcccccc}
<math>\begin{array}{rcccccc}
A suitably generic definition of the extended universe of discourse is afforded by the following set-up:
A suitably generic definition of the extended universe of discourse is afforded by the following set-up:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10" width="90%"
|
|
<math>\begin{array}{lccl}
<math>\begin{array}{lccl}
For a proposition of the form <math>f : X_1 \times \ldots \times X_k \to \mathbb{B},</math> the ''(first order) enlargement'' of <math>f\!</math> is the proposition <math>\operatorname{E}f : \operatorname{E}X \to \mathbb{B}</math> that is defined by the following equation:
For a proposition of the form <math>f : X_1 \times \ldots \times X_k \to \mathbb{B},</math> the ''(first order) enlargement'' of <math>f\!</math> is the proposition <math>\operatorname{E}f : \operatorname{E}X \to \mathbb{B}</math> that is defined by the following equation:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10" width="90%"
|
|
<math>\begin{array}{l}
<math>\begin{array}{l}
In the example of logical conjunction, <math>f(p, q) = pq,\!</math> the enlargement <math>\operatorname{E}f</math> is formulated as follows:
In the example of logical conjunction, <math>f(p, q) = pq,\!</math> the enlargement <math>\operatorname{E}f</math> is formulated as follows:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10" width="90%"
|
|
<math>\begin{array}{l}
<math>\begin{array}{l}
Given that this expression uses nothing more than the boolean ring operations of addition and multiplication, it is permissible to "multiply things out" in the usual manner to arrive at the following result:
Given that this expression uses nothing more than the boolean ring operations of addition and multiplication, it is permissible to "multiply things out" in the usual manner to arrive at the following result:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10" width="90%"
|
|
<math>\begin{matrix}
<math>\begin{matrix}
To understand what the ''enlarged'' or ''shifted'' proposition means in logical terms, it serves to go back and analyze the above expression for <math>\operatorname{E}f</math> in the same way that we did for <math>\operatorname{D}f.</math> Toward that end, the value of <math>\operatorname{E}f_x</math> at each <math>x \in X</math> may be computed in graphical fashion as shown below:
To understand what the ''enlarged'' or ''shifted'' proposition means in logical terms, it serves to go back and analyze the above expression for <math>\operatorname{E}f</math> in the same way that we did for <math>\operatorname{D}f.</math> Toward that end, the value of <math>\operatorname{E}f_x</math> at each <math>x \in X</math> may be computed in graphical fashion as shown below:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10" style="text-align:center"
| [[Image:Cactus Graph Ef = (P,dP)(Q,dQ).jpg|500px]]
| Ef = (p, dp) (q, dq) |
|-
|-
| align="center" |
| [[Image:Cactus Graph Ef@PQ = (dP)(dQ).jpg|500px]]
|-
|-
| align="center" |
| [[Image:Cactus Graph Ef@P(Q) = (dP)dQ.jpg|500px]]
|-
|-
| align="center" |
| [[Image:Cactus Graph Ef@(P)Q = dP(dQ).jpg|500px]]
|-
|-
| align="center" |
| [[Image:Cactus Graph Ef@(P)(Q) = dP dQ.jpg|500px]]
|}
|}
Given the data that develops in this form of analysis, the disjoined ingredients can now be folded back into a boolean expansion or a disjunctive normal form (DNF) that is equivalent to the enlarged proposition <math>\operatorname{E}f.</math>
Given the data that develops in this form of analysis, the disjoined ingredients can now be folded back into a boolean expansion or a disjunctive normal form (DNF) that is equivalent to the enlarged proposition <math>\operatorname{E}f.</math>
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10" width="90%"
|
|
<math>\begin{matrix}
<math>\begin{matrix}
Here is a summary of the result, illustrated by means of a digraph picture, where the "no change" element <math>(\operatorname{d}p)(\operatorname{d}q)</math> is drawn as a loop at the point <math>p~q.</math>
Here is a summary of the result, illustrated by means of a digraph picture, where the "no change" element <math>(\operatorname{d}p)(\operatorname{d}q)</math> is drawn as a loop at the point <math>p~q.</math>
{| align="center" cellpadding="10"
{| align="center" cellpadding="10" style="text-align:center"
| [[Image:Directed Graph PQ Enlargement Conj.jpg|500px]]
| [[Image:Directed Graph PQ Enlargement Conj.jpg|500px]]
|}
|-
|
|
<math>\begin{array}{rcccccc}
<math>\begin{array}{rcccccc}
f
f
(p)~q~
(p)~q~
\\[4pt]
\\[4pt]
(p)~~~
(p)[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]])
\\[4pt]
\\[4pt]
~p~(q)
~p~(q)
\\[4pt]
\\[4pt]
[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]])(q)
\\[4pt]
\\[4pt]
(p,~q)
(p,~q)
((p,~q))
((p,~q))
\\[4pt]
\\[4pt]
17:54, 5 December 2014 (UTC)q~~
\\[4pt]
\\[4pt]
~(p~(q))
~(p~(q))
\\[4pt]
\\[4pt]
~~p~~~~~
~~p17:54, 5 December 2014 (UTC)
\\[4pt]
\\[4pt]
((p)~q)~
((p)~q)~
|}
|}
For example, given the set <math>X = \{ a, b, c \},\!</math> suppose that we have the 2-adic relative term <math>\mathit{m} = {}^{\backprime\backprime}\, \text{marker for}\, \underline{~~~~}\, {}^{\prime\prime}</math> and
For example, given the set <math>X = \{ a, b, c \},\!</math> suppose that we have the 2-adic relative term <math>\mathit{m} = {}^{\backprime\backprime}\, \text{marker for}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 17:54, 5 December 2014 (UTC)}\, {}^{\prime\prime}</math> and
the associated 2-adic relation <math>M \subseteq X \times X,</math> the general pattern of whose common structure is represented by the following matrix:
the associated 2-adic relation <math>M \subseteq X \times X,</math> the general pattern of whose common structure is represented by the following matrix:
|}
|}
Recognizing that <math>a\!:\!a + b\!:\!b + c\!:\!c</math> is the identity transformation otherwise known as <math>\mathit{1},\!</math> the 2-adic relative term <math>m = {}^{\backprime\backprime}\, \text{marker for}\, \underline{~~~~}\, {}^{\prime\prime}</math> can be parsed as an element <math>\mathit{1} + a\!:\!b + b\!:\!c + c\!:\!a</math> of the so-called ''group ring'', all of which makes this element just a special sort of linear transformation.
Recognizing that <math>a\!:\!a + b\!:\!b + c\!:\!c</math> is the identity transformation otherwise known as <math>\mathit{1},\!</math> the 2-adic relative term <math>m = {}^{\backprime\backprime}\, \text{marker for}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 17:54, 5 December 2014 (UTC)}\, {}^{\prime\prime}</math> can be parsed as an element <math>\mathit{1} + a\!:\!b + b\!:\!c + c\!:\!a</math> of the so-called ''group ring'', all of which makes this element just a special sort of linear transformation.
Up to this point, we are still reading the elementary relatives of the form <math>i\!:\!j</math> in the way that Peirce read them in logical contexts: <math>i\!</math> is the relate, <math>j\!</math> is the correlate, and in our current example <math>i\!:\!j,</math> or more exactly, <math>m_{ij} = 1,\!</math> is taken to say that <math>i\!</math> is a marker for <math>j.\!</math> This is the mode of reading that we call "multiplying on the left".
Up to this point, we are still reading the elementary relatives of the form <math>i\!:\!j</math> in the way that Peirce read them in logical contexts: <math>i\!</math> is the relate, <math>j\!</math> is the correlate, and in our current example <math>i\!:\!j,</math> or more exactly, <math>m_{ij} = 1,\!</math> is taken to say that <math>i\!</math> is a marker for <math>j.\!</math> This is the mode of reading that we call "multiplying on the left".
Here is the operation table for <math>S_3,\!</math> given in abstract fashion:
Here is the operation table for <math>S_3,\!</math> given in abstract fashion:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10" style="text-align:center"
| <math>\text{Symmetric Group}~ S_3</math>
<pre>
|-
Symmetric Group S_3
| [[Image:Symmetric Group S(3).jpg|500px]]
| / \ |
| / e \ |
|}
|}
To construct the regular representations of <math>S_3,\!</math> we begin with the data of its operation table:
To construct the regular representations of <math>S_3,\!</math> we begin with the data of its operation table:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10" style="text-align:center"
| <math>\text{Symmetric Group}~ S_3</math>
<pre>
|-
Symmetric Group S_3
| [[Image:Symmetric Group S(3).jpg|500px]]
| / \ |
| / e \ |
|}
|}
Since we have a function of the type <math>L : G \times G \to G,</math> we can define a couple of substitution operators:
Since we have a function of the type <math>L : G \times G \to G,</math> we can define a couple of substitution operators:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10" width="90%"
| valign="top" | 1.
| valign="top" | 1.
| <math>\operatorname{Sub}(x, (\underline{~~}, y))</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(\underline{~~}, y),</math> with the effect of producing the saturated rheme <math>(x, y)\!</math> that evaluates to <math>xy.\!</math>
| <math>\operatorname{Sub}(x, (\underline{~~}, y))</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(\underline{~~}, y),</math> with the effect of producing the saturated rheme <math>(x, y)\!</math> that evaluates to <math>xy.\!</math>
In (1) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(\underline{~~}, y),</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(\underline{~~}, y)</math> into <math>xy,\!</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : xy) ~|~ y \in G \}.</math> The pairs <math>(y : xy)\!</math> can be found by picking an <math>x\!</math> from the left margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run along the right margin. This produces the ''regular ante-representation'' of <math>S_3,\!</math> like so:
In (1) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(\underline{~~}, y),</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(\underline{~~}, y)</math> into <math>xy,\!</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : xy) ~|~ y \in G \}.</math> The pairs <math>(y : xy)\!</math> can be found by picking an <math>x\!</math> from the left margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run along the right margin. This produces the ''regular ante-representation'' of <math>S_3,\!</math> like so:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10" style="text-align:center"
|
<math>\begin{array}{*{13}{c}}
<math>\begin{array}{*{13}{c}}
\operatorname{e}
\operatorname{e}
In (2) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(y, \underline{~~}),</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(y, \underline{~~})</math> into <math>yx,\!</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : yx) ~|~ y \in G \}.</math> The pairs <math>(y : yx)\!</math> can be found by picking an <math>x\!</math> on the right margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run along the left margin. This produces the ''regular post-representation'' of <math>S_3,\!</math> like so:
In (2) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(y, \underline{~~}),</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(y, \underline{~~})</math> into <math>yx,\!</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : yx) ~|~ y \in G \}.</math> The pairs <math>(y : yx)\!</math> can be found by picking an <math>x\!</math> on the right margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run along the left margin. This produces the ''regular post-representation'' of <math>S_3,\!</math> like so:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10" style="text-align:center"
|
<math>\begin{array}{*{13}{c}}
<math>\begin{array}{*{13}{c}}
\operatorname{e}
\operatorname{e}
Let us take a moment to view an old proposition in this new light, for example, the logical conjunction <math>pq : X \to \mathbb{B}</math> pictured in Figure 22-a.
Let us take a moment to view an old proposition in this new light, for example, the logical conjunction <math>pq : X \to \mathbb{B}</math> pictured in Figure 22-a.
{| align="center" cellspacing="10" style="text-align:center"
{| align="center" cellpadding="10" style="text-align:center"
| [[Image:Field Picture PQ Conjunction.jpg|500px]]
| [[Image:Field Picture PQ Conjunction.jpg|500px]]
|-
|-
The field of changes produced by <math>\operatorname{E}</math> on <math>pq\!</math> is shown in Figure 22-b.
The field of changes produced by <math>\operatorname{E}</math> on <math>pq\!</math> is shown in Figure 22-b.
{| align="center" cellspacing="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 Enlargement Conjunction.jpg|500px]]
|-
|-
The field of changes produced by <math>\operatorname{D}\!</math> on <math>pq\!</math> is shown in Figure 22-c.
The field of changes produced by <math>\operatorname{D}\!</math> on <math>pq\!</math> is shown in Figure 22-c.
{| align="center" cellspacing="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 Difference Conjunction.jpg|500px]]
|-
|-
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.
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" cellspacing="10" style="text-align:center; width:90%"
{| align="center" cellpadding="10" style="text-align:center; width:90%"
|
|
<pre>
<pre>
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| o G o |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
| | G | |
| | o<---------T---------o |
| | | |
| | | |
| | | |
| | | |
| \ / |
| \ / |
| \ / |
| \ / |
| \ T / |
| \ / |
| \ o<------------/-------------o |
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
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.
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" cellspacing="10" style="text-align:center"
{| align="center" cellpadding="10" style="text-align:center"
| [[Image:Field Picture PQ Conjunction.jpg|500px]]
| [[Image:Field Picture PQ Conjunction.jpg|500px]]
|-
|-
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>
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" cellspacing="10" style="text-align:center"
{| align="center" cellpadding="10" style="text-align:center"
| [[Image:Field Picture PQ Tacit Extension Conjunction.jpg|500px]]
| [[Image:Field Picture PQ Tacit Extension Conjunction.jpg|500px]]
|-
|-
<math>\begin{array}{rcccccc}
<math>\begin{array}{rcccccc}
\varepsilon (pq)
\varepsilon (pq)
& = & p & \cdot & q & \cdot & (\operatorname{d}p)(\operatorname{d}q)
& = &
p & \cdot & q & \cdot &
\texttt{(} \operatorname{d}p \texttt{)}
\texttt{(} \operatorname{d}q \texttt{)}
\\[4pt]
\\[4pt]
& + & p & \cdot & q & \cdot & (\operatorname{d}p)~\operatorname{d}q~
& + &
p & \cdot & q & \cdot &
\texttt{(} \operatorname{d}p \texttt{)}
\texttt{~} \operatorname{d}q \texttt{~}
\\[4pt]
\\[4pt]
& + & p & \cdot & q & \cdot & ~\operatorname{d}p~(\operatorname{d}q)
& + &
p & \cdot & q & \cdot &
\texttt{~} \operatorname{d}p \texttt{~}
\texttt{(} \operatorname{d}q \texttt{)}
\\[4pt]
\\[4pt]
& + & p & \cdot & q & \cdot & ~\operatorname{d}p~~\operatorname{d}q~
& + &
p & \cdot & q & \cdot &
\texttt{~} \operatorname{d}p \texttt{~}
\texttt{~} \operatorname{d}q \texttt{~}
\end{array}</math>
\end{array}</math>
|}
|}
==Note 25==
==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>
the enlargement or shift map E[pq] : EX -> B in the same
style of differential field picture that we drew for the
tacit extension !e![pq] : EX -> B.
{| align="center" cellpadding="10" style="text-align:center"
| |
| [[Image:Field Picture PQ Enlargement Conjunction.jpg|500px]]
| X |
|-
| o-------------------o o-------------------o |
| <math>\text{Figure 25-1. Enlargement Map}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}</math>
|-
| / P o Q \ |
|
<math>\begin{array}{rcccccc}
\operatorname{E}(pq)
& = &
p
& \cdot &
q
& \cdot &
\texttt{(} \operatorname{d}p \texttt{)}
\texttt{(} \operatorname{d}q \texttt{)}
\\[4pt]
& + &
p
& \cdot &
\texttt{(} q \texttt{)}
& \cdot &
\texttt{(} \operatorname{d}p \texttt{)}
\texttt{~} \operatorname{d}q \texttt{~}
\\[4pt]
& + &
\texttt{(} p \texttt{)}
& \cdot &
q
& \cdot &
\texttt{~} \operatorname{d}p \texttt{~}
\texttt{(} \operatorname{d}q \texttt{)}
\\[4pt]
& + &
| |
\texttt{(} p \texttt{)}
& \cdot &
\texttt{(} q \texttt{)}
& \cdot &
\texttt{~} \operatorname{d}p \texttt{~}
\texttt{~} \operatorname{d}q \texttt{~}
\end{array}</math>
|}
A very important conceptual transition has just occurred here,
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.
almost tacitly, as it were. Generally speaking, having a set
of mathematical objects of compatible types, in this case the
two differential fields !e!f and Ef, both of the type EX -> B,
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 !e!f and the
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.
enlargement Ef are in a certain sense dual to each other, with
true, and with Ef indicating all of the arrows into the region
where f is true. The only arc that they have in common is the
no-change loop (dp)(dq) at pq. If we add the two sets of arcs
mod 2, then the common loop drops out, leaving the 6 arrows of
D[pq] = !e![pq] + E[pq] that are illustrated in Figure 25-2.
{| align="center" cellpadding="10" style="text-align:center"
| |
| [[Image:Field Picture PQ Difference Conjunction.jpg|500px]]
| X |
|-
| <math>\text{Figure 25-2. Difference Map}~ \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}</math>
| / \ / \ |
|-
|
<math>\begin{array}{rcccccc}
| / / \ \ |
\operatorname{D}(pq)
& = &
p
& \cdot &
q
& \cdot &
\texttt{(}
\texttt{(} \operatorname{d}p \texttt{)}
\texttt{(} \operatorname{d}q \texttt{)}
\texttt{)}
\\[4pt]
& + &
p
& \cdot &
\texttt{(} q \texttt{)}
& \cdot &
\texttt{~}
\texttt{(} \operatorname{d}p \texttt{)}
\texttt{~} \operatorname{d}q \texttt{~}
\texttt{~}
\\[4pt]
& + &
\texttt{(} p \texttt{)}
& \cdot &
q
& \cdot &
\texttt{~}
\texttt{~} \operatorname{d}p \texttt{~}
\texttt{(} \operatorname{d}q \texttt{)}
\texttt{~}
\\[4pt]
& + &
\texttt{(} p \texttt{)}
& \cdot &
\texttt{(}q \texttt{)}
& \cdot &
\texttt{~}
\texttt{~} \operatorname{d}p \texttt{~}
\texttt{~} \operatorname{d}q \texttt{~}
\texttt{~}
\end{array}</math>
|}
<pre>
==Note 26==
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"
| [[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}
\operatorname{d}(pq)
& = &
p & \cdot & q & \cdot &
\texttt{(} \operatorname{d}p \texttt{,} \operatorname{d}q \texttt{)}
\\[4pt]
& + &
p & \cdot & \texttt{(} q \texttt{)} & \cdot &
\operatorname{d}q
\\[4pt]
& + &
\texttt{(} p \texttt{)} & \cdot & q & \cdot &
\operatorname{d}p
\\[4pt]
& + &
\texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & 0
\end{array}</math>
|}
To understand the extended interpretations, that is, the conjunctions of basic and differential features that are being indicated here, it may help to note the following equivalences:
{| align="center" cellpadding="10" style="text-align:center"
|
<math>\begin{matrix}
\texttt{(}
\operatorname{d}p
\texttt{,}
\operatorname{d}q
\texttt{)}
& = &
\texttt{~} \operatorname{d}p \texttt{~}
\texttt{(} \operatorname{d}q \texttt{)}
& + &
\texttt{(} \operatorname{d}p \texttt{)}
\texttt{~} \operatorname{d}q \texttt{~}
\\[4pt]
dp
& = &
\texttt{~} \operatorname{d}p \texttt{~}
\texttt{~} \operatorname{d}q \texttt{~}
& + &
\texttt{~} \operatorname{d}p \texttt{~}
\texttt{(} \operatorname{d}q \texttt{)}
\\[4pt]
\operatorname{d}q
& = &
\texttt{~} \operatorname{d}p \texttt{~}
\texttt{~} \operatorname{d}q \texttt{~}
& + &
\texttt{(} \operatorname{d}p \texttt{)}
\texttt{~} \operatorname{d}q \texttt{~}
\end{matrix}</math>
|}
Capping the series that analyzes the proposition <math>pq\!</math> in terms of succeeding orders of linear propositions, Figure 26-2 shows the remainder map <math>\operatorname{r}(pq) : \operatorname{E}X \to \mathbb{B},</math> that happens to be linear in pairs of variables.
{| align="center" cellpadding="10" style="text-align:center"
| [[Image:Field Picture PQ Remainder Conjunction.jpg|500px]]
|-
| <math>\text{Figure 26-2. Remainder Map}~ \operatorname{r}(pq) : \operatorname{E}X \to \mathbb{B}</math>
|}
| o-------------------o o-------------------o |
| / / \ \ |
Figure 26-2. Remainder r[pq] : EX -> B
Reading the arrows off the map produces the following data:
Reading the arrows off the map produces the following data:
{| align="center" cellpadding="10" style="text-align:center"
|
<math>\begin{array}{rcccccc}
\operatorname{r}(pq)
& = &
p & \cdot & q & \cdot &
\operatorname{d}p ~ \operatorname{d}q
\\[4pt]
& + &
p & \cdot & \texttt{(} q \texttt{)} & \cdot &
\operatorname{d}p ~ \operatorname{d}q
\\[4pt]
& + &
\texttt{(} p \texttt{)} & \cdot & q & \cdot &
\operatorname{d}p ~ \operatorname{d}q
\\[4pt]
& + &
\texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot &
\operatorname{d}p ~ \operatorname{d}q
\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.
==Further Reading==
A more detailed presentation of Differential Logic can be found here:
A more detailed presentation of Differential Logic can be found here:
:* [[Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0|Differential Logic and Dynamic Systems]]
==Document History==
==Document History==
# http://stderr.org/pipermail/inquiry/2004-May/001418.html
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===NKS Forum (May & Jul 2004)===
===NKS Forum (May & Jul 2004)===
* http://forum.wolframscience.com/archive/topic/420-1.html
* http://forum.wolframscience.com/printthread.php?threadid=420
* http://forum.wolframscience.com/showthread.php?threadid=420
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[[Category:Artificial Intelligence]]
[[Category:Boolean Algebra]]
[[Category:Boolean Functions]]
[[Category:Charles Sanders Peirce]]
[[Category:Combinatorics]]
[[Category:Computational Complexity]]
[[Category:Computer Science]]
[[Category:Cybernetics]]
[[Category:Differential Logic]]
[[Category:Equational Reasoning]]
[[Category:Formal Languages]]
[[Category:Formal Systems]]
[[Category:Graph Theory]]
[[Category:Inquiry]]
[[Category:Inquiry Driven Systems]]
[[Category:Knowledge Representation]]
[[Category:Logic]]
[[Category:Logical Graphs]]
[[Category:Mathematics]]
[[Category:Philosophy]]
[[Category:Propositional Calculus]]
[[Category:Semiotics]]
[[Category:Visualization]]
