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Directory talk:Jon Awbrey/Papers/Differential Logic : Introduction (view source)
Revision as of 18:38, 2 December 2015
, 18:38, 2 December 2015→Logical Cacti
==Discussion Area==
==Work Area • Logical Cacti==
…
; Theme One Program — Logical Cacti
; Theme One Program — Logical Cacti
\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>
|
|
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| height="120px" | [[Image:Cactus (A,B) Big.jpg|70px]]
| height="120px" | [[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}
\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}
\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}
\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{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math>
| <math>\texttt{((} a \texttt{),(} b \texttt{),(} c \texttt{))}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\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{)}, \texttt{(} c \texttt{))}</math>
| <math>\texttt{(} a \texttt{,(} b \texttt{),(} c \texttt{))}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\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>
|
|
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| height="120px" | [[Image:Cactus (A,B) Big.jpg|70px]]
| height="120px" | [[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}
\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}
\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}
\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, b, c \texttt{))}</math>
| <math>\texttt{((} a \texttt{,} b \texttt{,} c \texttt{))}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| height="200px" | [[Image:Cactus (((A),B,C)) Big.jpg|70px]]
| height="200px" | [[Image:Cactus (((A),B,C)) Big.jpg|65px]]
| <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math>
| <math>\texttt{(((} a \texttt{),} b \texttt{,} c \texttt{))}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
| Not just one of the k arguments is true.
| Not just one of the k arguments is true.
|}
|}
===Tables===
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
|+ <math>\text{Table 1.}~~\text{Syntax and Semantics of a Calculus for Propositional Logic}</math>
|- style="background:#f0f0ff"
| <math>\text{Graph}\!</math>
| <math>\text{Expression}\!</math>
| <math>\text{Interpretation}\!</math>
| <math>\text{Other Notations}\!</math>
|-
| height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]]
| <math>~</math>
| <math>\operatorname{true}</math>
| <math>1\!</math>
|-
| height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]]
| <math>\texttt{(~)}</math>
| <math>\operatorname{false}</math>
| <math>0\!</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>\operatorname{not}~ a</math>
| <math>\lnot a \quad \bar{a} \quad \tilde{a} \quad a^\prime</math>
|-
| height="100px" | [[Image:Cactus ABC Big.jpg|50px]]
| <math>a ~ b ~ c</math>
| <math>a ~\operatorname{and}~ b ~\operatorname{and}~ c</math>
| <math>a \land b \land c</math>
|-
| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]]
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>
| <math>a ~\operatorname{or}~ b ~\operatorname{or}~ c</math>
| <math>a \lor b \lor c</math>
|-
| height="120px" | [[Image:Cactus (A(B)) Big.jpg|60px]]
| <math>\texttt{(} a \texttt{(} b \texttt{))}</math>
|
<math>\begin{matrix}
a ~\operatorname{implies}~ b
\\[6pt]
\operatorname{if}~ a ~\operatorname{then}~ b
\end{matrix}</math>
| <math>a \Rightarrow b</math>
|-
| height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]]
| <math>\texttt{(} a \texttt{,} b \texttt{)}</math>
|
<math>\begin{matrix}
a ~\operatorname{not~equal~to}~ b
\\[6pt]
a ~\operatorname{exclusive~or}~ b
\end{matrix}</math>
|
<math>\begin{matrix}
a \neq b
\\[6pt]
a + b
\end{matrix}</math>
|-
| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]]
| <math>\texttt{((} a \texttt{,} b \texttt{))}</math>
|
<math>\begin{matrix}
a ~\operatorname{is~equal~to}~ b
\\[6pt]
a ~\operatorname{if~and~only~if}~ b
\end{matrix}</math>
|
<math>\begin{matrix}
a = b
\\[6pt]
a \Leftrightarrow b
\end{matrix}</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~of}
\\
a, b, c
\\
\operatorname{is~false}.
\end{matrix}</math>
|
<math>\begin{matrix}
& \bar{a} ~ b ~ c
\\
\lor & a ~ \bar{b} ~ c
\\
\lor & a ~ b ~ \bar{c}
\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~of}
\\
a, b, c
\\
\operatorname{is~true}.
\\[6pt]
\operatorname{partition~all}
\\
\operatorname{into}~ a, b, c.
\end{matrix}</math>
|
<math>\begin{matrix}
& a ~ \bar{b} ~ \bar{c}
\\
\lor & \bar{a} ~ b ~ \bar{c}
\\
\lor & \bar{a} ~ \bar{b} ~ c
\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}
\operatorname{oddly~many~of}
\\
a, b, c
\\
\operatorname{are~true}.
\end{matrix}</math>
|
<p><math>a + b + c\!</math></p>
<br>
<p><math>\begin{matrix}
& a ~ b ~ c
\\
\lor & a ~ \bar{b} ~ \bar{c}
\\
\lor & \bar{a} ~ b ~ \bar{c}
\\
\lor & \bar{a} ~ \bar{b} ~ c
\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>
|
<math>\begin{matrix}
\operatorname{partition}~ x
\\
\operatorname{into}~ a, b, c.
\\[6pt]
\operatorname{genus}~ x ~\operatorname{comprises}
\\
\operatorname{species}~ a, b, c.
\end{matrix}</math>
|
<math>\begin{matrix}
& \bar{x} ~ \bar{a} ~ \bar{b} ~ \bar{c}
\\
\lor & x ~ a ~ \bar{b} ~ \bar{c}
\\
\lor & x ~ \bar{a} ~ b ~ \bar{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>
|
|-
| height="120px" | [[Image:Cactus (A)B Big.jpg|35px]]
| <math>\texttt{(} a \texttt{)} b</math>
|
| <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>
|
|-
| height="200px" | [[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>
|}
<br>