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Directory talk:Jon Awbrey/Papers/Differential Logic : Introduction (view source)

Revision as of 18:38, 2 December 2015

4,107 bytes added ,  18:38, 2 December 2015
→‎Logical Cacti
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==Discussion Area==
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==Work Area • Logical Cacti==
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==Logical Cacti==
      
; Theme One Program — Logical Cacti
 
; Theme One Program — Logical Cacti
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\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
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| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|70px]]
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| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]]
 
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
 
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
 
|
 
|
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\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
−
| height="120px" | [[Image:Cactus (A,B) Big.jpg|70px]]
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| height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]]
−
| <math>\texttt{(} a, b \texttt{)}</math>
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| <math>\texttt{(} a \texttt{,} b \texttt{)}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
−
| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|70px]]
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| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]]
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| <math>\texttt{((} a, b \texttt{))}</math>
+
| <math>\texttt{((} a \texttt{,} b \texttt{))}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
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| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|70px]]
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| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]]
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| <math>\texttt{(} a, b, c \texttt{)}</math>
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| <math>\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
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| height="160px" | [[Image:Cactus ((A),(B),(C)) Big.jpg|70px]]
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| height="160px" | [[Image:Cactus ((A),(B),(C)) Big.jpg|65px]]
−
| <math>\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
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| <math>\texttt{((} a \texttt{),(} b \texttt{),(} c \texttt{))}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
−
| height="160px" | [[Image:Cactus (A,(B),(C)) Big.jpg|70px]]
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| height="160px" | [[Image:Cactus (A,(B),(C)) Big.jpg|65px]]
−
| <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
+
| <math>\texttt{(} a \texttt{,(} b \texttt{),(} c \texttt{))}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
−
| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|70px]]
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| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]]
 
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
 
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
 
|
 
|
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\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
−
| height="120px" | [[Image:Cactus (A,B) Big.jpg|70px]]
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| height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]]
−
| <math>\texttt{(} a, b \texttt{)}</math>
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| <math>\texttt{(} a \texttt{,} b \texttt{)}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
−
| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|70px]]
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| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]]
−
| <math>\texttt{((} a, b \texttt{))}</math>
+
| <math>\texttt{((} a \texttt{,} b \texttt{))}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
−
| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|70px]]
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| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]]
−
| <math>\texttt{(} a, b, c \texttt{)}</math>
+
| <math>\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
−
| height="160px" | [[Image:Cactus ((A,B,C)) Big.jpg|70px]]
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| height="160px" | [[Image:Cactus ((A,B,C)) Big.jpg|65px]]
−
| <math>\texttt{((} a, b, c \texttt{))}</math>
+
| <math>\texttt{((} a \texttt{,} b \texttt{,} c \texttt{))}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
−
| height="200px" | [[Image:Cactus (((A),B,C)) Big.jpg|70px]]
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| height="200px" | [[Image:Cactus (((A),B,C)) Big.jpg|65px]]
−
| <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math>
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| <math>\texttt{(((} a \texttt{),} b \texttt{,} c \texttt{))}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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===Tables===
 
===Tables===
−
  −
<br>
      
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
 
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
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| <math>a \land b \land c</math>
 
| <math>a \land b \land c</math>
 
|-
 
|-
−
| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|70px]]
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| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]]
 
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
 
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
 
| <math>a ~\operatorname{or}~ b ~\operatorname{or}~ c</math>
 
| <math>a ~\operatorname{or}~ b ~\operatorname{or}~ c</math>
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| <math>a \Rightarrow b</math>
 
| <math>a \Rightarrow b</math>
 
|-
 
|-
−
| height="120px" | [[Image:Cactus (A,B) Big.jpg|70px]]
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| height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]]
−
| <math>\texttt{(} a, b \texttt{)}</math>
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| <math>\texttt{(} a \texttt{,} b \texttt{)}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
−
| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|70px]]
+
| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]]
−
| <math>\texttt{((} a, b \texttt{))}</math>
+
| <math>\texttt{((} a \texttt{,} b \texttt{))}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
−
| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|70px]]
+
| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]]
−
| <math>\texttt{(} a, b, c \texttt{)}</math>
+
| <math>\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
−
| height="160px" | [[Image:Cactus ((A),(B),(C)) Big.jpg|70px]]
+
| height="160px" | [[Image:Cactus ((A),(B),(C)) Big.jpg|65px]]
 
| <math>\texttt{((} a \texttt{),(} b \texttt{),(} c \texttt{))}</math>
 
| <math>\texttt{((} a \texttt{),(} b \texttt{),(} c \texttt{))}</math>
 
|
 
|
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\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
−
|
+
| height="160px" | [[Image:Cactus (A,(B,C)) Big.jpg|90px]]
−
|
+
| <math>\texttt{(} a \texttt{,(} b \texttt{,} c \texttt{))}</math>
−
<math>\begin{matrix}
  −
\texttt{((} a, b \texttt{)}, c \texttt{)}
  −
\\[6pt]
  −
\texttt{(} a, \texttt{(} b, c \texttt{))}
  −
\end{matrix}</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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\end{matrix}</math></p>
 
\end{matrix}</math></p>
 
|-
 
|-
−
|
+
| height="160px" | [[Image:Cactus (X,(A),(B),(C)) Big.jpg|90px]]
−
| <math>\texttt{(} x, \texttt{(} a \texttt{),(} b \texttt{),(} c \texttt{))}</math>
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| <math>\texttt{(} x \texttt{,(} a \texttt{),(} b \texttt{),(} c \texttt{))}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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\\
 
\\
 
\lor & x ~ \bar{a} ~ \bar{b} ~ c
 
\lor & x ~ \bar{a} ~ \bar{b} ~ c
 +
\end{matrix}</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
 +
|+ <math>\text{Table C.}~~\text{Dualing Interpretations}</math>
 +
|- style="background:#f0f0ff"
 +
| <math>\text{Graph}\!</math>
 +
| <math>\text{String}\!</math>
 +
| <math>\text{Existential}\!</math>
 +
| <math>\text{Entitative}\!</math>
 +
|-
 +
| height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]]
 +
| <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math>
 +
| <math>\operatorname{true}.</math>
 +
| <math>\operatorname{false}.</math>
 +
|-
 +
| height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]]
 +
| <math>\texttt{(~)}</math>
 +
| <math>\operatorname{false}.</math>
 +
| <math>\operatorname{true}.</math>
 +
|-
 +
| height="100px" | [[Image:Cactus A Big.jpg|20px]]
 +
| <math>a\!</math>
 +
| <math>a.\!</math>
 +
| <math>a.\!</math>
 +
|-
 +
| height="120px" | [[Image:Cactus (A) Big.jpg|20px]]
 +
| <math>\texttt{(} a \texttt{)}</math>
 +
| <math>\lnot a</math>
 +
| <math>\lnot a</math>
 +
|-
 +
| height="100px" | [[Image:Cactus ABC Big.jpg|50px]]
 +
| <math>a~b~c</math>
 +
| <math>a \land b \land c</math>
 +
| <math>a \lor  b \lor  c</math>
 +
|-
 +
| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]]
 +
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
 +
| <math>a \lor  b \lor  c</math>
 +
| <math>a \land b \land c</math>
 +
|-
 +
| height="120px" | [[Image:Cactus (A(B)) Big.jpg|60px]]
 +
| <math>\texttt{(} a \texttt{(} b \texttt{))}</math>
 +
| <math>a \Rightarrow b</math>
 +
| &nbsp;
 +
|-
 +
| height="120px" | [[Image:Cactus (A)B Big.jpg|35px]]
 +
| <math>\texttt{(} a \texttt{)} b</math>
 +
| &nbsp;
 +
| <math>a \Rightarrow b</math>
 +
|-
 +
| height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]]
 +
| <math>\texttt{(} a \texttt{,} b \texttt{)}</math>
 +
| <math>a \neq b</math>
 +
| <math>a = b\!</math>
 +
|-
 +
| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]]
 +
| <math>\texttt{((} a \texttt{,} b \texttt{))}</math>
 +
| <math>a = b\!</math>
 +
| <math>a \neq b\!</math>
 +
|-
 +
| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]]
 +
| <math>\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{just~one}
 +
\\
 +
\operatorname{of}~ a, b, c
 +
\\
 +
\operatorname{is~false}.
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{not~just~one}
 +
\\
 +
\operatorname{of}~ a, b, c
 +
\\
 +
\operatorname{is~true}.
 +
\end{matrix}</math>
 +
|-
 +
| height="160px" | [[Image:Cactus ((A),(B),(C)) Big.jpg|65px]]
 +
| <math>\texttt{((} a \texttt{),(} b \texttt{),(} c \texttt{))}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{just~one}
 +
\\
 +
\operatorname{of}~ a, b, c
 +
\\
 +
\operatorname{is~true}.
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{not~just~one}
 +
\\
 +
\operatorname{of}~ a, b, c
 +
\\
 +
\operatorname{is~false}.
 +
\end{matrix}</math>
 +
|-
 +
| height="160px" | [[Image:Cactus ((A,B,C)) Big.jpg|65px]]
 +
| <math>\texttt{((} a \texttt{,} b \texttt{,} c \texttt{))}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{not~just~one}
 +
\\
 +
\operatorname{of}~ a, b, c
 +
\\
 +
\operatorname{is~false}.
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{just~one}
 +
\\
 +
\operatorname{of}~ a, b, c
 +
\\
 +
\operatorname{is~true}.
 +
\end{matrix}</math>
 +
|-
 +
| height="200px" | [[Image:Cactus (((A),(B),(C))) Big.jpg|65px]]
 +
| <math>\texttt{(((} a \texttt{),(} b \texttt{),(} c \texttt{)))}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{not~just~one}
 +
\\
 +
\operatorname{of}~ a, b, c
 +
\\
 +
\operatorname{is~true}.
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{just~one}
 +
\\
 +
\operatorname{of}~ a, b, c
 +
\\
 +
\operatorname{is~false}.
 +
\end{matrix}</math>
 +
|-
 +
| height="160px" | [[Image:Cactus (A,(B),(C)) Big.jpg|65px]]
 +
| <math>\texttt{(} a \texttt{,(} b \texttt{),(} c \texttt{))}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{partition}~ a
 +
\\
 +
\operatorname{into}~ b, c.
 +
\end{matrix}</math>
 +
| &nbsp;
 +
|-
 +
| height="200px" | [[Image:Cactus (((A),B,C)) Big.jpg|65px]]
 +
| <math>\texttt{(((} a \texttt{),} b \texttt{,} c \texttt{))}</math>
 +
| &nbsp;
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{partition}~ a
 +
\\
 +
\operatorname{into}~ b, c.
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|}
 
|}
    
<br>
 
<br>
Jon Awbrey
12,122

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