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Differential Logic
ASCII Graphics
o-------------------------------------------------o
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| o-------------o o-------------o |
| / \ / \ |
| / o \ |
| / /%\ \ |
| / /%%%\ \ |
| o o%%%%%o o |
| | |%%%%%| | |
| | P |%%%%%| Q | |
| | |%%%%%| | |
| o o%%%%%o o |
| \ \%%%/ / |
| \ \%/ / |
| \ o / |
| \ / \ / |
| o-------------o o-------------o |
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o-------------------------------------------------o
| f = p q |
o-------------------------------------------------o
Figure 22-a. Conjunction pq : X -> B
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o-------------------------------------------------o
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| o-------------o o-------------o |
| / \ / \ |
| / P o Q \ |
| / /%\ \ |
| / /%%%\ \ |
| o o.->-.o o |
| | p(q)(dp)dq |%\%/%| (p)q dp(dq) | |
| | o---------------|->o<-|---------------o | |
| | |%%^%%| | |
| o o%%|%%o o |
| \ \%|%/ / |
| \ \|/ / |
| \ o / |
| \ /|\ / |
| o-------------o | o-------------o |
| | |
| | |
| | |
| o |
| (p)(q) dp dq |
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o-------------------------------------------------o
| f = p q |
o-------------------------------------------------o
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| Ef = p q (dp)(dq) |
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| + p (q) (dp) dq |
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| + (p) q dp (dq) |
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| + (p)(q) dp dq |
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o-------------------------------------------------o
Figure 22-b. Enlargement E[pq] : EX -> B
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o-------------------------------------------------o
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| o-------------o o-------------o |
| / \ / \ |
| / P o Q \ |
| / /%\ \ |
| / /%%%\ \ |
| o o%%%%%o o |
| | (dp)dq |%%%%%| dp(dq) | |
| | o<--------------|->o<-|-------------->o | |
| | |%%^%%| | |
| o o%%|%%o o |
| \ \%|%/ / |
| \ \|/ / |
| \ o / |
| \ /|\ / |
| o-------------o | o-------------o |
| | |
| | |
| v |
| o |
| dp dq |
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o-------------------------------------------------o
| f = p q |
o-------------------------------------------------o
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| Df = p q ((dp)(dq)) |
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| + p (q) (dp) dq |
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| + (p) q dp (dq) |
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| + (p)(q) dp dq |
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o-------------------------------------------------o
Figure 22-c. Difference D[pq] : EX -> B
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o---------------------------------------------------------------------o
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| X |
| o-------------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| | | |
| | | |
| | | |
| | G | |
| | | |
| | | |
| | | |
| o o |
| \ / |
| \ / |
| \ T / |
| \ o<------------/-------------o |
| \ / |
| \ / |
| \ / |
| o-------------------o |
| |
| |
o---------------------------------------------------------------------o
Figure 23. Elements of a Cybernetic System
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o---------------------------------------------------------------------o
| |
| X |
| o-------------------o o-------------------o |
| / \ / \ |
| / o \ |
| / /%\ \ |
| / /%%%\ \ |
| / /%%%%%\ \ |
| / /%%%%%%%\ \ |
| / /%%%%%%%%%\ \ |
| o o%%%%%%%%%%%o o |
| | |%%%%%%%%%%%| | |
| | |%%%%%%%%%%%| | |
| | |%%%%%%%%%%%| | |
| | P |%%%%%%%%%%%| Q | |
| | |%%%%%%%%%%%| | |
| | |%%%%%%%%%%%| | |
| | |%%%%%%%%%%%| | |
| o o%%%%%%%%%%%o o |
| \ \%%%%%%%%%/ / |
| \ \%%%%%%%/ / |
| \ \%%%%%/ / |
| \ \%%%/ / |
| \ \%/ / |
| \ o / |
| \ / \ / |
| o-------------------o o-------------------o |
| |
| |
o---------------------------------------------------------------------o
Figure 24-1. Proposition pq : X -> B
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o---------------------------------------------------------------------o
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| X |
| o-------------------o o-------------------o |
| / \ / \ |
| / P o Q \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| o o (dp) (dq) o o |
| | | o-->--o | | |
| | | \ / | | |
| | (dp) dq | \ / | dp (dq) | |
| | o<-----------------o----------------->o | |
| | | | | | |
| | | | | | |
| | | | | | |
| o o | o o |
| \ \ | / / |
| \ \ | / / |
| \ \ | / / |
| \ \ | / / |
| \ \|/ / |
| \ | / |
| \ /|\ / |
| o-------------------o | o-------------------o |
| | |
| dp | dq |
| | |
| v |
| o |
| |
o---------------------------------------------------------------------o
Figure 24-2. Tacit Extension !e![pq] : EX -> B
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o---------------------------------------------------------------------o
| |
| X |
| o-------------------o o-------------------o |
| / \ / \ |
| / P o Q \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| o o (dp) (dq) o o |
| | | o-->--o | | |
| | | \ / | | |
| | (dp) dq | \ / | dp (dq) | |
| | o----------------->o<-----------------o | |
| | | ^ | | |
| | | | | | |
| | | | | | |
| o o | o o |
| \ \ | / / |
| \ \ | / / |
| \ \ | / / |
| \ \ | / / |
| \ \|/ / |
| \ | / |
| \ /|\ / |
| o-------------------o | o-------------------o |
| | |
| dp | dq |
| | |
| | |
| o |
| |
o---------------------------------------------------------------------o
Figure 25-1. Enlargement E[pq] : EX -> B
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o---------------------------------------------------------------------o
| |
| X |
| o-------------------o o-------------------o |
| / \ / \ |
| / P o Q \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| o o o o |
| | | | | |
| | | | | |
| | (dp) dq | | dp (dq) | |
| | o<---------------->o<---------------->o | |
| | | ^ | | |
| | | | | | |
| | | | | | |
| o o | o o |
| \ \ | / / |
| \ \ | / / |
| \ \ | / / |
| \ \ | / / |
| \ \|/ / |
| \ | / |
| \ /|\ / |
| o-------------------o | o-------------------o |
| | |
| dp | dq |
| | |
| v |
| o |
| |
o---------------------------------------------------------------------o
Figure 25-2. Difference Map D[pq] : EX -> B
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o---------------------------------------------------------------------o
| |
| X |
| o-------------------o o-------------------o |
| / \ / \ |
| / P o Q \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / o \ \ |
| / / ^ ^ \ \ |
| o o / \ o o |
| | | / \ | | |
| | | / \ | | |
| | |/ \| | |
| | (dp)/ dq dp \(dq) | |
| | /| |\ | |
| | / | | \ | |
| | / | | \ | |
| o / o o \ o |
| \ v \ dp dq / v / |
| \ o<--------------------->o / |
| \ \ / / |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| o-------------------o o-------------------o |
| |
| |
o---------------------------------------------------------------------o
Figure 26-1. Differential or Tangent d[pq] : EX -> B
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o---------------------------------------------------------------------o
| |
| X |
| o-------------------o o-------------------o |
| / \ / \ |
| / P o Q \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| o o o o |
| | | | | |
| | | | | |
| | | dp dq | | |
| | o<------------------------------->o | |
| | | | | |
| | | | | |
| | | o | | |
| o o ^ o o |
| \ \ | / / |
| \ \ | / / |
| \ \ | / / |
| \ \ | / / |
| \ \|/ / |
| \ dp | dq / |
| \ /|\ / |
| o-------------------o | o-------------------o |
| | |
| | |
| | |
| v |
| o |
| |
o---------------------------------------------------------------------o
Figure 26-2. Remainder r[pq] : EX -> B
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JPEG Graphics
Series 1
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| \(\text{Figure 22-a. Conjunction}~ pq : X \to \mathbb{B}\)
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| \(\text{Figure 22-b. Enlargement}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}\)
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\(\begin{array}{rcccccc}
\operatorname{E}(pq)
& = &
p
& \cdot &
q
& \cdot &
\texttt{(} \operatorname{d}p \texttt{)}
\texttt{(} \operatorname{d}q \texttt{)}
\\[4pt]
& + &
p
& \cdot &
\texttt{(} q \texttt{)}
& \cdot &
\texttt{(} \operatorname{d}p \texttt{)}
\texttt{~} \operatorname{d}q \texttt{~}
\\[4pt]
& + &
\texttt{(} p \texttt{)}
& \cdot &
q
& \cdot &
\texttt{~} \operatorname{d}p \texttt{~}
\texttt{(} \operatorname{d}q \texttt{)}
\\[4pt]
& + &
\texttt{(} p \texttt{)}
& \cdot &
\texttt{(} q \texttt{)}
& \cdot &
\texttt{~} \operatorname{d}p \texttt{~}
\texttt{~} \operatorname{d}q \texttt{~}
\end{array}\)
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|
| \(\text{Figure 22-c. Difference}~ \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}\)
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|
\(\begin{array}{rcccccc}
\operatorname{D}(pq)
& = & p & \cdot & q & \cdot & ((\operatorname{d}p)(\operatorname{d}q))
\\[4pt]
& + & p & \cdot & (q) & \cdot & ~(\operatorname{d}p)~\operatorname{d}q~~
\\[4pt]
& + & (p) & \cdot & q & \cdot & ~~\operatorname{d}p~(\operatorname{d}q)~
\\[4pt]
& + & (p) & \cdot & (q) & \cdot & ~~\operatorname{d}p~~\operatorname{d}q~~
\end{array}\)
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Series 2
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| \(\text{Figure 24-1. Proposition}~ pq : X \to \mathbb{B}\)
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| \(\text{Figure 24-2. Tacit Extension}~ \varepsilon (pq) : \operatorname{E}X \to \mathbb{B}\)
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\(\begin{array}{rcccccc}
\varepsilon (pq)
& = & p & \cdot & q & \cdot & (\operatorname{d}p)(\operatorname{d}q)
\\[4pt]
& + & p & \cdot & q & \cdot & (\operatorname{d}p)~\operatorname{d}q~
\\[4pt]
& + & p & \cdot & q & \cdot & ~\operatorname{d}p~(\operatorname{d}q)
\\[4pt]
& + & p & \cdot & q & \cdot & ~\operatorname{d}p~~\operatorname{d}q~
\end{array}\)
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|
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| \(\text{Figure 25-1. Enlargement Map}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}\)
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|
\(\begin{array}{rcccccc}
\operatorname{E}(pq)
& = & p & \cdot & q & \cdot & (\operatorname{d}p)(\operatorname{d}q)
\\[4pt]
& + & p & \cdot & (q) & \cdot & (\operatorname{d}p)~\operatorname{d}q~
\\[4pt]
& + & (p) & \cdot & q & \cdot & ~\operatorname{d}p~(\operatorname{d}q)
\\[4pt]
& + & (p) & \cdot & (q) & \cdot & ~\operatorname{d}p~~\operatorname{d}q~
\end{array}\)
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|
|
| \(\text{Figure 25-2. Difference Map}~ \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}\)
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|
\(\begin{array}{rcccccc}
\operatorname{D}(pq)
& = & p & \cdot & q & \cdot & ((\operatorname{d}p)(\operatorname{d}q))
\\[4pt]
& + & p & \cdot & (q) & \cdot & ~(\operatorname{d}p)~\operatorname{d}q~~
\\[4pt]
& + & (p) & \cdot & q & \cdot & ~~\operatorname{d}p~(\operatorname{d}q)~
\\[4pt]
& + & (p) & \cdot & (q) & \cdot & ~~\operatorname{d}p~~\operatorname{d}q~~
\end{array}\)
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