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

Revision as of 16:38, 29 November 2015

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→‎Cactus Language for Propositional Logic: try alternate table without graphics
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Table 1 collects a sample of basic propositional forms as expressed in terms of cactus language connectives.
 
Table 1 collects a sample of basic propositional forms as expressed in terms of cactus language connectives.
   −
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
+
<br>
−
|+ <math>\text{Table 1.}~~\text{Syntax and Semantics of a Calculus for Propositional Logic}\!</math>
+
 
−
|- style="background:#f0f0ff"
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
−
| <math>\text{Graph}\!</math>
+
|+ style="height:30px" | <math>\text{Table 1.} ~~ \text{Syntax and Semantics of a Calculus for Propositional Logic}\!</math>
 +
|- style="height:40px; background:ghostwhite"
 
| <math>\text{Expression}~\!</math>
 
| <math>\text{Expression}~\!</math>
 
| <math>\text{Interpretation}\!</math>
 
| <math>\text{Interpretation}\!</math>
 
| <math>\text{Other Notations}\!</math>
 
| <math>\text{Other Notations}\!</math>
 
|-
 
|-
−
| height="100px" | [[Image:Rooted Node.jpg|20px]]
+
| &nbsp;
−
| <math>~\!</math>
+
| <math>\text{True}\!</math>
−
| <math>\mathrm{true}\!</math>
   
| <math>1\!</math>
 
| <math>1\!</math>
 
|-
 
|-
−
| height="100px" | [[Image:Rooted Edge.jpg|20px]]
   
| <math>\texttt{(~)}\!</math>
 
| <math>\texttt{(~)}\!</math>
−
| <math>\mathrm{false}\!</math>
+
| <math>\text{False}\!</math>
 
| <math>0\!</math>
 
| <math>0\!</math>
 
|-
 
|-
−
| height="100px" | [[Image:Cactus A Big.jpg|20px]]
+
| <math>x\!</math>
−
| <math>a\!</math>
+
| <math>x\!</math>
−
| <math>a\!</math>
+
| <math>x\!</math>
−
| <math>a\!</math>
   
|-
 
|-
−
| height="120px" | [[Image:Cactus (A) Big.jpg|20px]]
+
| <math>\texttt{(} x \texttt{)}\!</math>
−
| <math>\texttt{(} a \texttt{)}\!</math>
+
| <math>\text{Not}~ x\!</math>
−
| <math>\mathrm{not}~ a\!</math>
+
|
−
| <math>\lnot a \quad \bar{a} \quad \tilde{a} \quad a^\prime~\!</math>
+
<math>\begin{matrix}
 +
x'
 +
\\
 +
\tilde{x}
 +
\\
 +
\lnot x
 +
\end{matrix}\!</math>
 
|-
 
|-
−
| height="100px" | [[Image:Cactus ABC Big.jpg|50px]]
+
| <math>x~y~z\!</math>
−
| <math>a ~ b ~ c\!</math>
+
| <math>x ~\text{and}~ y ~\text{and}~ z\!</math>
−
| <math>a ~\mathrm{and}~ b ~\mathrm{and}~ c\!</math>
+
| <math>x \land y \land z\!</math>
−
| <math>a \land b \land c\!</math>
   
|-
 
|-
−
| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]]
+
| <math>\texttt{((} x \texttt{)(} y \texttt{)(} z \texttt{))}\!</math>
−
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}\!</math>
+
| <math>x ~\text{or}~ y ~\text{or}~ z\!</math>
−
| <math>a ~\mathrm{or}~ b ~\mathrm{or}~ c\!</math>
+
| <math>x \lor y \lor z\!</math>
−
| <math>a \lor b \lor c\!</math>
   
|-
 
|-
−
| height="120px" | [[Image:Cactus (A(B)) Big.jpg|60px]]
+
| <math>\texttt{(} x ~ \texttt{(} y \texttt{))}\!</math>
−
| <math>\texttt{(} a \texttt{(} b \texttt{))}\!</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
a ~\text{implies}~ b
+
x ~\text{implies}~ y
−
\\[6pt]
+
\\
−
\mathrm{if}~ a ~\mathrm{then}~ b
+
\mathrm{If}~ x ~\text{then}~ y
−
\end{matrix}\!</math>
+
\end{matrix}</math>
−
| <math>a \Rightarrow b\!</math>
+
| <math>x \Rightarrow y\!</math>
 
|-
 
|-
−
| height="120px" | [[Image:Cactus (A,B) Big ISW.jpg|65px]]
+
| <math>\texttt{(} x \texttt{,} y \texttt{)}\!</math>
−
| <math>\texttt{(} a \texttt{,} b \texttt{)}\!</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
a ~\mathrm{not~equal~to}~ b
+
x ~\text{not equal to}~ y
−
\\[6pt]
+
\\
−
a ~\mathrm{exclusive~or}~ b
+
x ~\text{exclusive or}~ y
−
\end{matrix}\!</math>
+
\end{matrix}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
a \neq b
+
x \ne y
−
\\[6pt]
+
\\
−
a + b
+
x + y
−
\end{matrix}\!</math>
+
\end{matrix}</math>
 
|-
 
|-
−
| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]]
+
| <math>\texttt{((} x \texttt{,} y \texttt{))}\!</math>
−
| <math>\texttt{((} a \texttt{,} b \texttt{))}\!</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
a ~\mathrm{is~equal~to}~ b
+
x ~\text{is equal to}~ y
−
\\[6pt]
+
\\
−
a ~\mathrm{if~and~only~if}~ b
+
x ~\text{if and only if}~ y
−
\end{matrix}\!</math>
+
\end{matrix}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
a = b
+
x = y
−
\\[6pt]
+
\\
−
a \Leftrightarrow b
+
x \Leftrightarrow y
−
\end{matrix}\!</math>
+
\end{matrix}</math>
 
|-
 
|-
−
| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]]
+
| <math>\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}\!</math>
−
| <math>\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}\!</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\mathrm{just~one~of}
+
\text{Just one of}
 
\\
 
\\
−
a, b, c
+
x, y, z
 
\\
 
\\
−
\mathrm{is~false}.
+
\text{is false}.
−
\end{matrix}\!</math>
+
\end{matrix}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
& \bar{a} ~ b ~ c
+
x'y~z~ & \lor
 
\\
 
\\
−
\lor & a ~ \bar{b} ~ c
+
x~y'z~ & \lor
 
\\
 
\\
−
\lor & a ~ b ~ \bar{c}
+
x~y~z' &
−
\end{matrix}\!</math>
+
\end{matrix}</math>
 
|-
 
|-
−
| height="160px" | [[Image:Cactus ((A),(B),(C)) Big.jpg|65px]]
+
| <math>\texttt{((} x \texttt{),(} y \texttt{),(} z \texttt{))}\!</math>
−
| <math>\texttt{((} a \texttt{),(} b \texttt{),(} c \texttt{))}\!</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\mathrm{just~one~of}
+
\text{Just one of}
 +
\\
 +
x, y, z
 +
\\
 +
\text{is true}.
 +
\\
 +
&
 +
\\
 +
\text{Partition all}
 
\\
 
\\
−
a, b, c
+
\text{into}~ x, y, z.
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
x~y'z' & \lor
 
\\
 
\\
−
\mathrm{is~true}.
+
x'y~z' & \lor
−
\\[6pt]
  −
\mathrm{partition~all}
   
\\
 
\\
−
\mathrm{into}~ a, b, c.
+
x'y'z~ &
−
\end{matrix}\!</math>
+
\end{matrix}</math>
 +
|-
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
& a ~ \bar{b} ~ \bar{c}
+
\texttt{((} x \texttt{,} y \texttt{),} z \texttt{)}
 
\\
 
\\
−
\lor & \bar{a} ~ b ~ \bar{c}
+
&
 
\\
 
\\
−
\lor & \bar{a} ~ \bar{b} ~ c
+
\texttt{(} x \texttt{,(} y \texttt{,} z \texttt{))}
 
\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}
 
<math>\begin{matrix}
−
\mathrm{oddly~many~of}
+
\text{Oddly many of}
 
\\
 
\\
−
a, b, c
+
x, y, z
 
\\
 
\\
−
\mathrm{are~true}.
+
\text{are true}.
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
 
|
 
|
−
<p><math>a + b + c\!</math></p>
+
<p><math>x + y + z\!</math></p>
 
<br>
 
<br>
 
<p><math>\begin{matrix}
 
<p><math>\begin{matrix}
−
& a ~ b ~ c
+
x~y~z~ & \lor
 
\\
 
\\
−
\lor & a ~ \bar{b} ~ \bar{c}
+
x~y'z' & \lor
 
\\
 
\\
−
\lor & \bar{a} ~ b ~ \bar{c}
+
x'y~z' & \lor
 
\\
 
\\
−
\lor & \bar{a} ~ \bar{b} ~ c
+
x'y'z~ &
 
\end{matrix}\!</math></p>
 
\end{matrix}\!</math></p>
 
|-
 
|-
−
| height="160px" | [[Image:Cactus (X,(A),(B),(C)) Big.jpg|90px]]
+
| <math>\texttt{(} w \texttt{,(} x \texttt{),(} y \texttt{),(} z \texttt{))}\!</math>
−
| <math>\texttt{(} x \texttt{,(} a \texttt{),(} b \texttt{),(} c \texttt{))}\!</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\mathrm{partition}~ x
+
\text{Partition}~ w
 
\\
 
\\
−
\mathrm{into}~ a, b, c.
+
\text{into}~ x, y, z.
−
\\[6pt]
  −
\mathrm{genus}~ x ~\mathrm{comprises}
   
\\
 
\\
−
\mathrm{species}~ a, b, c.
+
&
−
\end{matrix}\!</math>
+
\\
 +
\text{Genus}~ w ~\text{comprises}
 +
\\
 +
\text{species}~ x, y, z.
 +
\end{matrix}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
& \bar{x} ~ \bar{a} ~ \bar{b} ~ \bar{c}
+
w'x'y'z' & \lor
 
\\
 
\\
−
\lor & x ~ a ~ \bar{b} ~ \bar{c}
+
w~x~y'z' & \lor
 
\\
 
\\
−
\lor & x ~ \bar{a} ~ b ~ \bar{c}
+
w~x'y~z' & \lor
 
\\
 
\\
−
\lor & x ~ \bar{a} ~ \bar{b} ~ c
+
w~x'y'z~ &
−
\end{matrix}~\!</math>
+
\end{matrix}</math>
 
|}
 
|}
 +
 +
<br>
    
The simplest expression for logical truth is the empty word, usually denoted by <math>\boldsymbol\varepsilon\!</math> or <math>\lambda\!</math> in formal languages, where it forms the identity element for concatenation.  To make it visible in context, it may be denoted by the equivalent expression <math>{}^{\backprime\backprime} \texttt{((~))} {}^{\prime\prime},\!</math> or, especially if operating in an algebraic context, by a simple <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}.\!</math>  Also when working in an algebraic mode, the plus sign <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> may be used for [[exclusive disjunction]].  For example, we have the following paraphrases of algebraic expressions by means of parenthesized expressions:
 
The simplest expression for logical truth is the empty word, usually denoted by <math>\boldsymbol\varepsilon\!</math> or <math>\lambda\!</math> in formal languages, where it forms the identity element for concatenation.  To make it visible in context, it may be denoted by the equivalent expression <math>{}^{\backprime\backprime} \texttt{((~))} {}^{\prime\prime},\!</math> or, especially if operating in an algebraic context, by a simple <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}.\!</math>  Also when working in an algebraic mode, the plus sign <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> may be used for [[exclusive disjunction]].  For example, we have the following paraphrases of algebraic expressions by means of parenthesized expressions:
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