User:Jon Awbrey/Figures and Tables

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Logical Graphs

Old Versions

Example 1

o-----------------------------------------------------------o
|                                                           |
|   o o o             o o               o                   |
|    \| |             | |               |                   |
|     o o o           o o o             o o             o   |
|      \|/             \|/              |/              |   |
|       @       =       @       =       @       =       @   |
|                                                           |
o-----------------------------------------------------------o
|                                                           |
| (()())(())()  =   (())(())()  =     (())()    =      ( )  |
|                                                           |
o-----------------------------------------------------------o

Example 2

o-------------------o-------------------o-------------------o
| Object            | Sign              | Interpretant      |
o-------------------o-------------------o-------------------o
|                   |                   |                   |
| Falsity           | "(()())(())()"    | "(())(())()"      |
|                   |                   |                   |
| Falsity           | "(())(())()"      | "(())()"          |
|                   |                   |                   |
| Falsity           | "(())()"          | "()"              |
|                   |                   |                   |
o-------------------o-------------------o-------------------o

Example 3

o-------------------o-------------------o-------------------o
| Object            | Sign              | Interpretant      |
o-------------------o-------------------o-------------------o
|                   |                   |                   |
| Falsity           | "(()())(())()"    | "(()())(())()"    |
|                   |                   |                   |
| Falsity           | "(()())(())()"    | "(())(())()"      |
|                   |                   |                   |
| Falsity           | "(()())(())()"    | "(())()"          |
|                   |                   |                   |
| Falsity           | "(()())(())()"    | "()"              |
|                   |                   |                   |
o-------------------o-------------------o-------------------o
|                   |                   |                   |
| Falsity           | "(())(())()"      | "(()())(())()"    |
|                   |                   |                   |
| Falsity           | "(())(())()"      | "(())(())()"      |
|                   |                   |                   |
| Falsity           | "(())(())()"      | "(())()"          |
|                   |                   |                   |
| Falsity           | "(())(())()"      | "()"              |
|                   |                   |                   |
o-------------------o-------------------o-------------------o
|                   |                   |                   |
| Falsity           | "(())()"          | "(()())(())()"    |
|                   |                   |                   |
| Falsity           | "(())()"          | "(())(())()"      |
|                   |                   |                   |
| Falsity           | "(())()"          | "(())()"          |
|                   |                   |                   |
| Falsity           | "(())()"          | "()"              |
|                   |                   |                   |
o-------------------o-------------------o-------------------o
|                   |                   |                   |
| Falsity           | "()"              | "(()())(())()"    |
|                   |                   |                   |
| Falsity           | "()"              | "(())(())()"      |
|                   |                   |                   |
| Falsity           | "()"              | "(())()"          |
|                   |                   |                   |
| Falsity           | "()"              | "()"              |
|                   |                   |                   |
o-------------------o-------------------o-------------------o

Example 4

o-------------------o-------------------o-------------------o
|         a         |         b         |      (a , b)      |
o-------------------o-------------------o-------------------o
|                   |                   |                   |
|       blank       |       blank       |       cross       |
|                   |                   |                   |
|       blank       |       cross       |       blank       |
|                   |                   |                   |
|       cross       |       blank       |       blank       |
|                   |                   |                   |
|       cross       |       cross       |       cross       |
|                   |                   |                   |
o-------------------o-------------------o-------------------o

Example 5

o-------o-------o-------o-----------o
|   a   |   b   |   c   | (a, b, c) |
o-------o-------o-------o-----------o
|       |       |       |           |
| blank | blank | blank |   cross   |
|       |       |       |           |
| blank | blank | cross |   blank   |
|       |       |       |           |
| blank | cross | blank |   blank   |
|       |       |       |           |
| blank | cross | cross |   cross   |
|       |       |       |           |
| cross | blank | blank |   blank   |
|       |       |       |           |
| cross | blank | cross |   cross   |
|       |       |       |           |
| cross | cross | blank |   cross   |
|       |       |       |           |
| cross | cross | cross |   cross   |
|       |       |       |           |
o-------o-------o-------o-----------o

Example 6

o-------o-------o-------o-----------o
|   a   |   b   |   c   | (a, b, c) |
o-------o-------o-------o-----------o
|       |       |       |           |
|   o   |   o   |   o   |     |     |
|       |       |       |           |
|   o   |   o   |   |   |     o     |
|       |       |       |           |
|   o   |   |   |   o   |     o     |
|       |       |       |           |
|   o   |   |   |   |   |     |     |
|       |       |       |           |
|   |   |   o   |   o   |     o     |
|       |       |       |           |
|   |   |   o   |   |   |     |     |
|       |       |       |           |
|   |   |   |   |   o   |     |     |
|       |       |       |           |
|   |   |   |   |   |   |     |     |
|       |       |       |           |
o-------o-------o-------o-----------o

New Versions

Example 1

Example 2a

\text{Object} \text{Sign} \text{Interpretant}
\mathrm{Falsity} {}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime} {}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\mathrm{Falsity} {}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime} {}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\mathrm{Falsity} {}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime} {}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}

Example 2b

\text{Object} \text{Sign} \text{Interpretant}

\begin{array}{l}
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\end{array}

\begin{array}{l}
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\end{array}

\begin{array}{l}
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\end{array}

Example 3

\text{Object} \text{Sign} \text{Interpretant}

\begin{array}{l}
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\end{array}

\begin{array}{l}
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\end{array}

\begin{array}{l}
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\end{array}

\begin{array}{l}
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\end{array}

\begin{array}{l}
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\end{array}

\begin{array}{l}
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\end{array}

\begin{array}{l}
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\end{array}

\begin{array}{l}
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\end{array}

\begin{array}{l}
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\end{array}

\begin{array}{l}
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\\[6pt]
\mathrm{Falsity}
\end{array}

\begin{array}{l}
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\end{array}

\begin{array}{l}
{}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}
\end{array}

Example 4

a b \texttt{(} a \texttt{,} b \texttt{)}

\begin{matrix}
\texttt{Blank}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Cross}
\end{matrix}

\begin{matrix}
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\end{matrix}

\begin{matrix}
\texttt{Cross}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\end{matrix}

Example 5

a b c \texttt{(} a \texttt{,} b \texttt{,} \texttt{c} \texttt{)}

\begin{matrix}
\texttt{Blank}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Cross}
\end{matrix}

\begin{matrix}
\texttt{Blank}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Cross}
\end{matrix}

\begin{matrix}
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\end{matrix}

\begin{matrix}
\texttt{Cross}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Blank}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Cross}
\\[6pt]
\texttt{Cross}
\end{matrix}

Example 6

a b c \texttt{(} a \texttt{,} b \texttt{,} \texttt{c} \texttt{)}

\begin{matrix}
\texttt{o}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\end{matrix}

\begin{matrix}
\texttt{o}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\end{matrix}

\begin{matrix}
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\end{matrix}

\begin{matrix}
\texttt{|}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{o}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\\[6pt]
\texttt{|}
\end{matrix}

Semiotic Equivalence Relations


\text{Table 6a.} ~~ \mathrm{Con}(L_\mathrm{A}) = \mathrm{proj}_{SI}(L_\mathrm{A})
S I

\begin{matrix}
{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}
\end{matrix}

\begin{matrix}
{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}
\end{matrix}

\begin{matrix}
{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}
\end{matrix}

\begin{matrix}
{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}
\end{matrix}

\text{Table 6b.} ~~ \mathrm{Con}(L_\mathrm{B}) = \mathrm{proj}_{SI}(L_\mathrm{B})
S I

\begin{matrix}
{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}
\end{matrix}

\begin{matrix}
{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}
\end{matrix}

\begin{matrix}
{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}
\end{matrix}

\begin{matrix}
{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}
\end{matrix}


Fourier Analysis


\begin{array}{|c||*{4}{c}|}
\hline
g & f_{8} & f_{4} & f_{2} & f_{1} \\
&
\texttt{ } u \texttt{  } v \texttt{ } &
\texttt{ } u \texttt{ (} v \texttt{)} &
\texttt{(} u \texttt{) } v \texttt{ } &
\texttt{(} u \texttt{)(} v \texttt{)} \\
\hline\hline
f_{7}  & 0 & 1 & 1 & 1 \\
f_{11} & 1 & 0 & 1 & 1 \\
f_{13} & 1 & 1 & 0 & 1 \\
f_{14} & 1 & 1 & 1 & 0 \\
\hline
\end{array}


Sign Relations

\text{Table 1a.} ~ {L_\mathrm{A}} = \text{Sign Relation of Interpreter A}
\text{Object} \text{Sign} \text{Interpretant}
\mathrm{A} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}
\mathrm{A} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}
\mathrm{A} {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}
\mathrm{A} {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}
\mathrm{B} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}
\mathrm{B} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}
\mathrm{B} {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}
\mathrm{B} {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}

Combinators

Composer

\begin{array}{c}
((x \overset{ }{\underset{A}{\Downarrow}} ~
  y \overset{A}{\underset{B}{\Downarrow}}
  ) \overset{ }{\underset{B}{\Downarrow}} ~
  z \overset{B}{\underset{C}{\Downarrow}}
  ) \overset{ }{\underset{C}{\Downarrow}}
\\[6pt]
=
\\[6pt]
(x \overset{ }{\underset{A}{\Downarrow}} ~
(y \overset{A}{\underset{B}{\Downarrow}} ~
(z \overset{B}{\underset{C}{\Downarrow}} ~
 P \overset{B \Rightarrow C}{\underset{(A \Rightarrow B) \Rightarrow (A \Rightarrow C)}{\Downarrow}}
 ) \overset{A \Rightarrow B}{\underset{A \Rightarrow C}{\Downarrow}}
 ) \overset{A}{\underset{C}{\Downarrow}}
 ) \overset{ }{\underset{C}{\Downarrow}}
\end{array}

Transposer

\begin{array}{c}
(y \overset{ }{\underset{B}{\Downarrow}} ~
(x \overset{ }{\underset{A}{\Downarrow}} ~
 z \overset{A}{\underset{B \Rightarrow C}{\Downarrow}}
 ) \overset{B}{\underset{C}{\Downarrow}}
 ) \overset{ }{\underset{C}{\Downarrow}}
\\[6pt]
=
\\[6pt]
((x \overset{ }{\underset{A}{\Downarrow}} ~
 (y \overset{ }{\underset{B}{\Downarrow}} ~
  K \overset{B}{\underset{A \Rightarrow B}{\Downarrow}}
  ) \overset{A}{\underset{B}{\Downarrow}}
  ) \overset{ }{\underset{B}{\Downarrow}} ~
 (x \overset{ }{\underset{A}{\Downarrow}} ~
  z \overset{A}{\underset{B \Rightarrow C}{\Downarrow}}
  ) \overset{B}{\underset{C}{\Downarrow}}
  ) \overset{ }{\underset{C}{\Downarrow}}
\\[6pt]
=
\\[6pt]
 (x \overset{ }{\underset{A}{\Downarrow}} ~
((y \overset{ }{\underset{B}{\Downarrow}} ~
  K \overset{B}{\underset{A \Rightarrow B}{\Downarrow}} 
  ) \overset{A}{\underset{B}{\Downarrow}} ~
 (z \overset{A}{\underset{B \Rightarrow C}{\Downarrow}} ~
  S \overset{A \Rightarrow (B \Rightarrow C)}{\underset{(A \Rightarrow B) \Rightarrow (A \Rightarrow C)}{\Downarrow}}
  ) \overset{A \Rightarrow B}{\underset{A \Rightarrow C}{\Downarrow}}
  ) \overset{A}{\underset{C}{\Downarrow}}
  ) \overset{ }{\underset{C}{\Downarrow}}
\end{array}

Figures

Example 1

Venn Diagram 4 Dimensions UV Cacti 8 Inch.jpg
\text{Figure 16.} ~~ \text{Higher Order Universe of Discourse} ~ \left[ \ell_{00}, \ell_{01}, \ell_{10}, \ell_{11} \right] \subseteq \left[\left[ u, v \right]\right]

Example 2

Diff Log Dyn Sys -- Figure 16 -- A Couple of Fourth Gear Orbits.gif
\text{Figure 16.} ~~ \text{A Couple of Fourth Gear Orbits}

Example 3

Venn Diagram P And Q.jpg
\text{Figure 1. Conjunction}~ pq

Tables

Appendices

Syntax and Semantics of a Calculus for Propositional Logic

Table 1 collects a sample of basic propositional forms as expressed in terms of cactus language connectives.

\text{Table 1.} ~~ \text{Syntax and Semantics of a Calculus for Propositional Logic}
\text{Graph} \text{Expression} \text{Interpretation} \text{Other Notations}
Rooted Node.jpg ~ \mathrm{true} 1
Rooted Edge.jpg \texttt{(}~\texttt{)} \mathrm{false} 0
Cactus A Big.jpg a a a
Cactus (A) Big.jpg \texttt{(} a \texttt{)} \mathrm{not}~ a \lnot a \quad \bar{a} \quad \tilde{a} \quad a^\prime
Cactus ABC Big.jpg a ~ b ~ c a ~\mathrm{and}~ b ~\mathrm{and}~ c a \land b \land c
Cactus ((A)(B)(C)) Big.jpg \texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))} a ~\mathrm{or}~ b ~\mathrm{or}~ c a \lor b \lor c
Cactus (A(B)) Big.jpg \texttt{(} a \texttt{(} b \texttt{))}

\begin{matrix}
a ~\mathrm{implies}~ b
\\[6pt]
\mathrm{if}~ a ~\mathrm{then}~ b
\end{matrix}

a \Rightarrow b
65px \texttt{(} a \texttt{,} b \texttt{)}

\begin{matrix}
a ~\mathrm{not~equal~to}~ b
\\[6pt]
a ~\mathrm{exclusive~or}~ b
\end{matrix}

\begin{matrix}
a \neq b
\\[6pt]
a + b
\end{matrix}

Cactus ((A,B)) Big.jpg \texttt{((} a \texttt{,} b \texttt{))}

\begin{matrix}
a ~\mathrm{is~equal~to}~ b
\\[6pt]
a ~\mathrm{if~and~only~if}~ b
\end{matrix}

\begin{matrix}
a = b
\\[6pt]
a \Leftrightarrow b
\end{matrix}

Cactus (A,B,C) Big.jpg \texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}

\begin{matrix}
\mathrm{just~one~of}
\\
a, b, c
\\
\mathrm{is~false}.
\end{matrix}

\begin{matrix}
& \bar{a} ~ b ~ c
\\
\lor & a ~ \bar{b} ~ c
\\
\lor & a ~ b ~ \bar{c}
\end{matrix}

Cactus ((A),(B),(C)) Big.jpg \texttt{((} a \texttt{),(} b \texttt{),(} c \texttt{))}

\begin{matrix}
\mathrm{just~one~of}
\\
a, b, c
\\
\mathrm{is~true}.
\\[6pt]
\mathrm{partition~all}
\\
\mathrm{into}~ a, b, c.
\end{matrix}

\begin{matrix}
& a ~ \bar{b} ~ \bar{c}
\\
\lor & \bar{a} ~ b ~ \bar{c}
\\
\lor & \bar{a} ~ \bar{b} ~ c
\end{matrix}

Cactus (A,(B,C)) Big.jpg \texttt{(} a \texttt{,(} b \texttt{,} c \texttt{))}

\begin{matrix}
\mathrm{oddly~many~of}
\\
a, b, c
\\
\mathrm{are~true}.
\end{matrix}

a + b + c


\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}

Cactus (X,(A),(B),(C)) Big.jpg \texttt{(} x \texttt{,(} a \texttt{),(} b \texttt{),(} c \texttt{))}

\begin{matrix}
\mathrm{partition}~ x
\\
\mathrm{into}~ a, b, c.
\\[6pt]
\mathrm{genus}~ x ~\mathrm{comprises}
\\
\mathrm{species}~ a, b, c.
\end{matrix}

\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}~

Futures Of Logical Graphs

Reflective Series • Variant 1

\text{Form} \text{String} \text{Graph}
Form (A) Plain.jpg \texttt{(a)} Cactus (A) Big Plain.jpg
Form (A,B) Plain.jpg \texttt{(a,b)} Cactus (A,B) Big Plain.jpg
Form (A,B,C) Plain.jpg \texttt{(a,b,c)} Cactus (A,B,C) Big Plain.jpg

Reflective Series • Variant 2

\text{Form} \text{String} \text{Graph}
Form (A) Italic.jpg \texttt{(} a \texttt{)} Cactus (A) Big Italic.jpg
Form (A,B) Italic.jpg \texttt{(} a \texttt{,} b \texttt{)} Cactus (A,B) Big Italic.jpg
Form (A,B,C) Italic.jpg \texttt{(} a \texttt{,} b \texttt{,} c \texttt{)} Cactus (A,B,C) Big Italic.jpg

Higher Order Propositions

Higher Order Propositions (n = 1)

Version 1


\text{Table 1.} ~~ \text{Higher Order Propositions} ~ (n = 1)
x: 1 ~ 0 f m_{0} m_{1} m_{2} m_{3} m_{4} m_{5} m_{6} m_{7} m_{8} m_{9} m_{10} m_{11} m_{12} m_{13} m_{14} m_{15}
f_{0} 0 ~ 0 \texttt{(~)} 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
f_{1} 0 ~ 1 \texttt{(} x \texttt{)} 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
f_{2} 1 ~ 0 x 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
f_{3} 1 ~ 1 \texttt{((~))} 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1


Version 2


\text{Table 1.} ~~ \text{Higher Order Propositions} ~~ (n = 1)
x: 1 0 f m_0 m_1 m_2 m_3 m_4 m_5 m_6 m_7 m_8 m_9 m_{10} m_{11} m_{12} m_{13} m_{14} m_{15}
f_0 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
f_1 0 1 (x) 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
f_2 1 0 x 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
f_3 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1


Interpretive Categories (n = 1)

Version 1


\text{Table 2.} ~~ \text{Interpretive Categories for Higher Order Propositions} ~ (n = 1)
Measure Happening Exactness Existence Linearity Uniformity Information
m_{0} Nothing happens          
m_{1}   Just false Nothing exists      
m_{2}   Just not x        
m_{3}     Nothing is x      
m_{4}   Just x        
m_{5}     Everything is x f is linear    
m_{6}         f is not uniform f is informed
m_{7}   Not just true        
m_{8}   Just true        
m_{9}         f is uniform f is not informed
m_{10}     Something is not x f is not linear    
m_{11}   Not just x        
m_{12}     Something is x      
m_{13}   Not just not x        
m_{14}   Not just false Something exists      
m_{15} Anything happens          


Version 2


\text{Table 2.} ~~ \text{Interpretive Categories for Higher Order Propositions} ~~ (n = 1)
Measure Happening Exactness Existence Linearity Uniformity Information
m_0 Nothing happens          
m_1   Just false Nothing exists      
m_2   Just not x        
m_3     Nothing is x      
m_4   Just x        
m_5     Everything is x f is linear    
m_6         f is not uniform f is informed
m_7   Not just true        
m_8   Just true        
m_9         f is uniform f is not informed
m_{10}     Something is not x f is not linear    
m_{11}   Not just x        
m_{12}     Something is x      
m_{13}   Not just not x        
m_{14}   Not just false Something exists      
m_{15} Anything happens          


Higher Order Propositions (n = 2)

Version 1


\text{Table 3.} ~~ \text{Higher Order Propositions} ~ (n = 2)
\begin{matrix}u\!:\\v\!:\end{matrix} \begin{matrix}1100\\1010\end{matrix} f {\underset{0}{m}} {\underset{1}{m}} {\underset{2}{m}} {\underset{3}{m}} {\underset{4}{m}} {\underset{5}{m}} {\underset{6}{m}} {\underset{7}{m}} {\underset{8}{m}} {\underset{9}{m}} {\underset{10}{m}} {\underset{11}{m}} {\underset{12}{m}} {\underset{13}{m}} {\underset{14}{m}} {\underset{15}{m}} {\underset{16}{m}} {\underset{17}{m}} {\underset{18}{m}} {\underset{19}{m}} {\underset{20}{m}} {\underset{21}{m}} {\underset{22}{m}} {\underset{23}{m}}
f_{0} 0000 \texttt{(~)} 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
f_{1} 0001 \texttt{(} u \texttt{)(} v \texttt{)} 00 1 1 00 1 1 00 1 1 00 1 1 00 1 1 00 1 1
f_{2} 0010 \texttt{(} u\texttt{)} ~ v 0000 1 1 1 1 0000 1 1 1 1 0000 1 1 1 1
f_{3} 0011 \texttt{(} u \texttt{)} 0000 0000 1 1 1 1 1 1 1 1 0000 0000
f_{4} 0100 u ~ \texttt{(} v \texttt{)} 0000 0000 0000 0000 1 1 1 1 1 1 1 1
f_{5} 0101 \texttt{(} v \texttt{)} 0000 0000 0000 0000 0000 0000
f_{6} 0110 \texttt{(} u \texttt{,} v \texttt{)} 0000 0000 0000 0000 0000 0000
f_{7} 0111 \texttt{(} u ~ v \texttt{)} 0000 0000 0000 0000 0000 0000
f_{8} 1000 u ~ v 0000 0000 0000 0000 0000 0000
f_{9} 1001 \texttt{((} u \texttt{,} v \texttt{))} 0000 0000 0000 0000 0000 0000
f_{10} 1010 v 0000 0000 0000 0000 0000 0000
f_{11} 1011 \texttt{(} u ~ \texttt{(} v \texttt{))} 0000 0000 0000 0000 0000 0000
f_{12} 1100 u 0000 0000 0000 0000 0000 0000
f_{13} 1101 \texttt{((} u \texttt{)} ~ v \texttt{)} 0000 0000 0000 0000 0000 0000
f_{14} 1110 \texttt{((} u \texttt{)(} v \texttt{))} 0000 0000 0000 0000 0000 0000
f_{15} 1111 \texttt{((~))} 0000 0000 0000 0000 0000 0000


Version 2


\text{Table 3.} ~~ \text{Higher Order Propositions} ~~ (n = 2)
u\!:
v\!:
1100
1010
f m_0 m_1 m_2 m_3 m_4 m_5 m_6 m_7 m_8 m_9 m_{10} m_{11} m_{12} m_{13} m_{14} m_{15} m_{16} m_{17} m_{18} m_{19} m_{20} m_{21} m_{22} m_{23}
f_0 0000 (~) 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
f_1 0001 (u)(v) 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
f_2 0010 (u) v 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
f_3 0011 (u) 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
f_4 0100 u (v) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
f_5 0101 (v) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
f_6 0110 (u, v) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
f_7 0111 (u v) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
f_8 1000 u v 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
f_9 1001 ((u, v)) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
f_{10} 1010 v 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
f_{11} 1011 (u (v)) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
f_{12} 1100 u 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
f_{13} 1101 ((u) v) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
f_{14} 1110 ((u)(v)) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
f_{15} 1111 ((~)) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0


Minimal Negation Operators

Table 3 is a truth table for the sixteen boolean functions of type f : \mathbb{B}^3 \to \mathbb{B} whose fibers of 1 are either the boundaries of points in \mathbb{B}^3 or the complements of those boundaries.

How It Used To Look

How It Looks Now


\text{Table 3.} ~~ \text{Logical Boundaries and Their Complements}
\mathcal{L}_1 \mathcal{L}_2 \mathcal{L}_3 \mathcal{L}_4
  p\colon 1~1~1~1~0~0~0~0  
  q\colon 1~1~0~0~1~1~0~0  
  r\colon 1~0~1~0~1~0~1~0  

\begin{matrix}
f_{104}
\\[4pt]
f_{148}
\\[4pt]
f_{146}
\\[4pt]
f_{97}
\\[4pt]
f_{134}
\\[4pt]
f_{73}
\\[4pt]
f_{41}
\\[4pt]
f_{22}
\end{matrix}

\begin{matrix}
f_{01101000}
\\[4pt]
f_{10010100}
\\[4pt]
f_{10010010}
\\[4pt]
f_{01100001}
\\[4pt]
f_{10000110}
\\[4pt]
f_{01001001}
\\[4pt]
f_{00101001}
\\[4pt]
f_{00010110}
\end{matrix}

\begin{matrix}
0~1~1~0~1~0~0~0
\\[4pt]
1~0~0~1~0~1~0~0
\\[4pt]
1~0~0~1~0~0~1~0
\\[4pt]
0~1~1~0~0~0~0~1
\\[4pt]
1~0~0~0~0~1~1~0
\\[4pt]
0~1~0~0~1~0~0~1
\\[4pt]
0~0~1~0~1~0~0~1
\\[4pt]
0~0~0~1~0~1~1~0
\end{matrix}

\begin{matrix}
\texttt{(~p~,~q~,~r~)}
\\[4pt]
\texttt{(~p~,~q~,(r))}
\\[4pt]
\texttt{(~p~,(q),~r~)}
\\[4pt]
\texttt{(~p~,(q),(r))}
\\[4pt]
\texttt{((p),~q~,~r~)}
\\[4pt]
\texttt{((p),~q~,(r))}
\\[4pt]
\texttt{((p),(q),~r~)}
\\[4pt]
\texttt{((p),(q),(r))}
\end{matrix}

\begin{matrix}
f_{233}
\\[4pt]
f_{214}
\\[4pt]
f_{182}
\\[4pt]
f_{121}
\\[4pt]
f_{158}
\\[4pt]
f_{109}
\\[4pt]
f_{107}
\\[4pt]
f_{151}
\end{matrix}

\begin{matrix}
f_{11101001}
\\[4pt]
f_{11010110}
\\[4pt]
f_{10110110}
\\[4pt]
f_{01111001}
\\[4pt]
f_{10011110}
\\[4pt]
f_{01101101}
\\[4pt]
f_{01101011}
\\[4pt]
f_{10010111}
\end{matrix}

\begin{matrix}
1~1~1~0~1~0~0~1
\\[4pt]
1~1~0~1~0~1~1~0
\\[4pt]
1~0~1~1~0~1~1~0
\\[4pt]
0~1~1~1~1~0~0~1
\\[4pt]
1~0~0~1~1~1~1~0
\\[4pt]
0~1~1~0~1~1~0~1
\\[4pt]
0~1~1~0~1~0~1~1
\\[4pt]
1~0~0~1~0~1~1~1
\end{matrix}

\begin{matrix}
\texttt{(((p),(q),(r)))}
\\[4pt]
\texttt{(((p),(q),~r~))}
\\[4pt]
\texttt{(((p),~q~,(r)))}
\\[4pt]
\texttt{(((p),~q~,~r~))}
\\[4pt]
\texttt{((~p~,(q),(r)))}
\\[4pt]
\texttt{((~p~,(q),~r~))}
\\[4pt]
\texttt{((~p~,~q~,(r)))}
\\[4pt]
\texttt{((~p~,~q~,~r~))}
\end{matrix}


Test 1


\text{Table 3.} ~~ \text{Logical Boundaries and Their Complements}
\mathcal{L}_1 \mathcal{L}_2 \mathcal{L}_3 \mathcal{L}_4
  p\colon 1~1~1~1~0~0~0~0  
  q\colon 1~1~0~0~1~1~0~0  
  r\colon 1~0~1~0~1~0~1~0  

\begin{matrix}
f_{104}
\\[4pt]
f_{148}
\\[4pt]
f_{146}
\\[4pt]
f_{97}
\\[4pt]
f_{134}
\\[4pt]
f_{73}
\\[4pt]
f_{41}
\\[4pt]
f_{22}
\end{matrix}

\begin{matrix}
f_{01101000}
\\[4pt]
f_{10010100}
\\[4pt]
f_{10010010}
\\[4pt]
f_{01100001}
\\[4pt]
f_{10000110}
\\[4pt]
f_{01001001}
\\[4pt]
f_{00101001}
\\[4pt]
f_{00010110}
\end{matrix}

\begin{matrix}
0~1~1~0~1~0~0~0
\\[4pt]
1~0~0~1~0~1~0~0
\\[4pt]
1~0~0~1~0~0~1~0
\\[4pt]
0~1~1~0~0~0~0~1
\\[4pt]
1~0~0~0~0~1~1~0
\\[4pt]
0~1~0~0~1~0~0~1
\\[4pt]
0~0~1~0~1~0~0~1
\\[4pt]
0~0~0~1~0~1~1~0
\end{matrix}

\begin{matrix}
\texttt{( p , q , r )}
\\[4pt]
\texttt{( p , q ,(r))}
\\[4pt]
\texttt{( p ,(q), r )}
\\[4pt]
\texttt{( p ,(q),(r))}
\\[4pt]
\texttt{((p), q , r )}
\\[4pt]
\texttt{((p), q ,(r))}
\\[4pt]
\texttt{((p),(q), r )}
\\[4pt]
\texttt{((p),(q),(r))}
\end{matrix}

\begin{matrix}
f_{233}
\\[4pt]
f_{214}
\\[4pt]
f_{182}
\\[4pt]
f_{121}
\\[4pt]
f_{158}
\\[4pt]
f_{109}
\\[4pt]
f_{107}
\\[4pt]
f_{151}
\end{matrix}

\begin{matrix}
f_{11101001}
\\[4pt]
f_{11010110}
\\[4pt]
f_{10110110}
\\[4pt]
f_{01111001}
\\[4pt]
f_{10011110}
\\[4pt]
f_{01101101}
\\[4pt]
f_{01101011}
\\[4pt]
f_{10010111}
\end{matrix}

\begin{matrix}
1~1~1~0~1~0~0~1
\\[4pt]
1~1~0~1~0~1~1~0
\\[4pt]
1~0~1~1~0~1~1~0
\\[4pt]
0~1~1~1~1~0~0~1
\\[4pt]
1~0~0~1~1~1~1~0
\\[4pt]
0~1~1~0~1~1~0~1
\\[4pt]
0~1~1~0~1~0~1~1
\\[4pt]
1~0~0~1~0~1~1~1
\end{matrix}

\begin{matrix}
\texttt{(((p),(q),(r)))}
\\[4pt]
\texttt{(((p),(q), r ))}
\\[4pt]
\texttt{(((p), q ,(r)))}
\\[4pt]
\texttt{(((p), q , r ))}
\\[4pt]
\texttt{(( p ,(q),(r)))}
\\[4pt]
\texttt{(( p ,(q), r ))}
\\[4pt]
\texttt{(( p , q ,(r)))}
\\[4pt]
\texttt{(( p , q , r ))}
\end{matrix}


Test 2


\text{Table 3.} ~~ \text{Logical Boundaries and Their Complements}
\mathcal{L}_1 \mathcal{L}_2 \mathcal{L}_3 \mathcal{L}_4
  p\colon 1~1~1~1~0~0~0~0  
  q\colon 1~1~0~0~1~1~0~0  
  r\colon 1~0~1~0~1~0~1~0  

\begin{matrix}
f_{104}
\\[4pt]
f_{148}
\\[4pt]
f_{146}
\\[4pt]
f_{97}
\\[4pt]
f_{134}
\\[4pt]
f_{73}
\\[4pt]
f_{41}
\\[4pt]
f_{22}
\end{matrix}

\begin{matrix}
f_{01101000}
\\[4pt]
f_{10010100}
\\[4pt]
f_{10010010}
\\[4pt]
f_{01100001}
\\[4pt]
f_{10000110}
\\[4pt]
f_{01001001}
\\[4pt]
f_{00101001}
\\[4pt]
f_{00010110}
\end{matrix}

\begin{matrix}
0~1~1~0~1~0~0~0
\\[4pt]
1~0~0~1~0~1~0~0
\\[4pt]
1~0~0~1~0~0~1~0
\\[4pt]
0~1~1~0~0~0~0~1
\\[4pt]
1~0~0~0~0~1~1~0
\\[4pt]
0~1~0~0~1~0~0~1
\\[4pt]
0~0~1~0~1~0~0~1
\\[4pt]
0~0~0~1~0~1~1~0
\end{matrix}

\begin{matrix}
\texttt{(}~p~\texttt{,}~q~\texttt{,}~r~\texttt{)}
\\[4pt]
\texttt{(}~p~\texttt{,}~q~\texttt{,}(r\texttt{))}
\\[4pt]
\texttt{(}~p~\texttt{,(}q\texttt{),}~r~\texttt{)}
\\[4pt]
\texttt{(}~p~\texttt{,(}q\texttt{),(}r\texttt{))}
\\[4pt]
\texttt{((}p\texttt{),}~q~\texttt{,}~r~\texttt{)}
\\[4pt]
\texttt{((}p\texttt{),}~q~\texttt{,(}r\texttt{))}
\\[4pt]
\texttt{((}p\texttt{),(}q\texttt{),}~r~\texttt{)}
\\[4pt]
\texttt{((}p\texttt{),(}q\texttt{),(}r\texttt{))}
\end{matrix}

\begin{matrix}
f_{233}
\\[4pt]
f_{214}
\\[4pt]
f_{182}
\\[4pt]
f_{121}
\\[4pt]
f_{158}
\\[4pt]
f_{109}
\\[4pt]
f_{107}
\\[4pt]
f_{151}
\end{matrix}

\begin{matrix}
f_{11101001}
\\[4pt]
f_{11010110}
\\[4pt]
f_{10110110}
\\[4pt]
f_{01111001}
\\[4pt]
f_{10011110}
\\[4pt]
f_{01101101}
\\[4pt]
f_{01101011}
\\[4pt]
f_{10010111}
\end{matrix}

\begin{matrix}
1~1~1~0~1~0~0~1
\\[4pt]
1~1~0~1~0~1~1~0
\\[4pt]
1~0~1~1~0~1~1~0
\\[4pt]
0~1~1~1~1~0~0~1
\\[4pt]
1~0~0~1~1~1~1~0
\\[4pt]
0~1~1~0~1~1~0~1
\\[4pt]
0~1~1~0~1~0~1~1
\\[4pt]
1~0~0~1~0~1~1~1
\end{matrix}

\begin{matrix}
\texttt{(((}p\texttt{),(}q\texttt{),(}r\texttt{)))}
\\[4pt]
\texttt{(((}p\texttt{),(}q\texttt{),}~r~\texttt{))}
\\[4pt]
\texttt{(((}p\texttt{),}~q~\texttt{,(}r\texttt{)))}
\\[4pt]
\texttt{(((}p\texttt{),}~q~\texttt{,}~r~\texttt{))}
\\[4pt]
\texttt{((}~p~\texttt{,(}q\texttt{),(}r\texttt{)))}
\\[4pt]
\texttt{((}~p~\texttt{,(}q\texttt{),}~r~\texttt{))}
\\[4pt]
\texttt{((}~p~\texttt{,}~q~\texttt{,(}r\texttt{)))}
\\[4pt]
\texttt{((}~p~\texttt{,}~q~\texttt{,}~r~\texttt{))}
\end{matrix}


Test 3


\text{Table 3.} ~~ \text{Logical Boundaries and Their Complements}
\mathcal{L}_1 \mathcal{L}_2 \mathcal{L}_3 \mathcal{L}_4
  p\colon 1~1~1~1~0~0~0~0  
  q\colon 1~1~0~0~1~1~0~0  
  r\colon 1~0~1~0~1~0~1~0  

\begin{matrix}
f_{104}
\\[4pt]
f_{148}
\\[4pt]
f_{146}
\\[4pt]
f_{97}
\\[4pt]
f_{134}
\\[4pt]
f_{73}
\\[4pt]
f_{41}
\\[4pt]
f_{22}
\end{matrix}

\begin{matrix}
f_{01101000}
\\[4pt]
f_{10010100}
\\[4pt]
f_{10010010}
\\[4pt]
f_{01100001}
\\[4pt]
f_{10000110}
\\[4pt]
f_{01001001}
\\[4pt]
f_{00101001}
\\[4pt]
f_{00010110}
\end{matrix}

\begin{matrix}
0~1~1~0~1~0~0~0
\\[4pt]
1~0~0~1~0~1~0~0
\\[4pt]
1~0~0~1~0~0~1~0
\\[4pt]
0~1~1~0~0~0~0~1
\\[4pt]
1~0~0~0~0~1~1~0
\\[4pt]
0~1~0~0~1~0~0~1
\\[4pt]
0~0~1~0~1~0~0~1
\\[4pt]
0~0~0~1~0~1~1~0
\end{matrix}

\begin{matrix}
\texttt{(} & \texttt{p}   & \texttt{,} & \texttt{q}   & \texttt{,} & \texttt{r}   & \texttt{)}
\\[4pt]
\texttt{(} & \texttt{p}   & \texttt{,} & \texttt{q}   & \texttt{,} & \texttt{(r)} & \texttt{)}
\\[4pt]
\texttt{(} & \texttt{p}   & \texttt{,} & \texttt{(q)} & \texttt{,} & \texttt{r}   & \texttt{)}
\\[4pt]
\texttt{(} & \texttt{p}   & \texttt{,} & \texttt{(q)} & \texttt{,} & \texttt{(r)} & \texttt{)}
\\[4pt]
\texttt{(} & \texttt{(p)} & \texttt{,} & \texttt{q}   & \texttt{,} & \texttt{r}   & \texttt{)}
\\[4pt]
\texttt{(} & \texttt{(p)} & \texttt{,} & \texttt{q}   & \texttt{,} & \texttt{(r)} & \texttt{)}
\\[4pt]
\texttt{(} & \texttt{(p)} & \texttt{,} & \texttt{(q)} & \texttt{,} & \texttt{r}   & \texttt{)}
\\[4pt]
\texttt{(} & \texttt{(p)} & \texttt{,} & \texttt{(q)} & \texttt{,} & \texttt{(r)} & \texttt{)}
\end{matrix}

\begin{matrix}
f_{233}
\\[4pt]
f_{214}
\\[4pt]
f_{182}
\\[4pt]
f_{121}
\\[4pt]
f_{158}
\\[4pt]
f_{109}
\\[4pt]
f_{107}
\\[4pt]
f_{151}
\end{matrix}

\begin{matrix}
f_{11101001}
\\[4pt]
f_{11010110}
\\[4pt]
f_{10110110}
\\[4pt]
f_{01111001}
\\[4pt]
f_{10011110}
\\[4pt]
f_{01101101}
\\[4pt]
f_{01101011}
\\[4pt]
f_{10010111}
\end{matrix}

\begin{matrix}
1~1~1~0~1~0~0~1
\\[4pt]
1~1~0~1~0~1~1~0
\\[4pt]
1~0~1~1~0~1~1~0
\\[4pt]
0~1~1~1~1~0~0~1
\\[4pt]
1~0~0~1~1~1~1~0
\\[4pt]
0~1~1~0~1~1~0~1
\\[4pt]
0~1~1~0~1~0~1~1
\\[4pt]
1~0~0~1~0~1~1~1
\end{matrix}

\begin{matrix}
\texttt{((} & \texttt{(p)} & \texttt{,} & \texttt{(q)} & \texttt{,} & \texttt{(r)} & \texttt{))}
\\[4pt]
\texttt{((} & \texttt{(p)} & \texttt{,} & \texttt{(q)} & \texttt{,} & \texttt{r}   & \texttt{))}
\\[4pt]
\texttt{((} & \texttt{(p)} & \texttt{,} & \texttt{q}   & \texttt{,} & \texttt{(r)} & \texttt{))}
\\[4pt]
\texttt{((} & \texttt{(p)} & \texttt{,} & \texttt{q}   & \texttt{,} & \texttt{r}   & \texttt{))}
\\[4pt]
\texttt{((} & \texttt{p}   & \texttt{,} & \texttt{(q)} & \texttt{,} & \texttt{(r)} & \texttt{))}
\\[4pt]
\texttt{((} & \texttt{p}   & \texttt{,} & \texttt{(q)} & \texttt{,} & \texttt{r}   & \texttt{))}
\\[4pt]
\texttt{((} & \texttt{p}   & \texttt{,} & \texttt{q}   & \texttt{,} & \texttt{(r)} & \texttt{))}
\\[4pt]
\texttt{((} & \texttt{p}   & \texttt{,} & \texttt{q}   & \texttt{,} & \texttt{r}   & \texttt{))}
\end{matrix}


Templates for Logical Operators

Exclusive disjunction

Exclusive disjunction, also known as logical inequality or symmetric difference, is an operation on two logical values, typically the values of two propositions, that produces a value of true just in case exactly one of its operands is true.

An exclusive disjunction of propositions p and q may be written in various ways.  Among the most common are these:

  • p ~\mathrm{xor}~ q
  • p ~\Delta~ q
  • p \ne q
  • p + q

A truth table for p + q appears below:


\text{Exclusive Disjunction}
p q p + q
\mathrm{F} \mathrm{F} \mathrm{F}
\mathrm{F} \mathrm{T} \mathrm{T}
\mathrm{T} \mathrm{F} \mathrm{T}
\mathrm{T} \mathrm{T} \mathrm{F}


The exclusive disjunction of two variables belongs to the family of minimal negation operators.  Thus, we have the following equivalents:

  • p + q
  • \nu(p, q)
  • \texttt{(} p \texttt{,} q \texttt{)}

A logical graph for p + q is shown below:

Logical Graph (P,Q).jpg

The traversal string of this graph is \texttt{(} p \texttt{,} q \texttt{)}.  The proposition p + q may be taken as a Boolean function f(p, q) having the abstract type f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}, where \mathbb{B} = \{ 0, 1 \} is interpreted in such a way that 0 means \mathrm{false} and 1 means \mathrm{true}.

A Venn diagram for p + q indicates the region where p + q is true by means of a distinctive color or shading.  In this case the region is two single cells, as shown below:

Venn Diagram P + Q 2.0.jpg

Logical conjunction

Logical conjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both of its operands are true.

A logical conjunction of propositions p and q may be written in various ways.  Among the most common are these:

  • p ~\mathrm{and}~ q
  • p \land q
  • p \cdot q
  • p~q
  • pq

A truth table for p \land q appears below:

\text{Logical Conjunction}
p q p \land q
\mathrm{F} \mathrm{F} \mathrm{F}
\mathrm{F} \mathrm{T} \mathrm{F}
\mathrm{T} \mathrm{F} \mathrm{F}
\mathrm{T} \mathrm{T} \mathrm{T}


A logical graph for p \land q is drawn as two letters attached to a root node:

Logical Graph PQ.jpg

Written as a string, this is just the concatenation pq.  The proposition pq may be taken as a Boolean function f(p, q) having the abstract type f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}, where \mathbb{B} = \{ 0, 1 \} is interpreted in such a way that 0 means \mathrm{false} and 1 means \mathrm{true}.

A Venn diagram for p \land q indicates the region where pq is true by means of a distinctive color or shading.  In this case the region is a single cell, as shown below:

Venn Diagram P ∧ Q 2.0.jpg

Logical disjunction

Logical disjunction, also called logical alternation, is an operation on two logical values, typically the values of two propositions, that produces a value of false if and only if both of its operands are false.

A logical disjunction of propositions p and q may be written in various ways.  Among the most common are these:

  • p ~\mathrm{or}~ q
  • p \lor q

A truth table for p \lor q appears below:


\text{Logical Disjunction}
p q p \lor q
\mathrm{F} \mathrm{F} \mathrm{F}
\mathrm{F} \mathrm{T} \mathrm{T}
\mathrm{T} \mathrm{F} \mathrm{T}
\mathrm{T} \mathrm{T} \mathrm{T}


A logical graph for p \lor q is shown below:

Logical Graph ((P)(Q)).jpg

The traversal string of this graph is \texttt{((} p \texttt{)(} q \texttt{))}.  The proposition p \lor q may be taken as a Boolean function f(p, q) having the abstract type f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}, where \mathbb{B} = \{ 0, 1 \} is interpreted in such a way that 0 means \mathrm{false} and 1 means \mathrm{true}.

A Venn diagram for p \lor q indicates the region where p \lor q is true by means of a distinctive color or shading.  In this case the region consists of three adjacent cells, as shown below:

Venn Diagram P ∨ Q 2.0.jpg

Logical equality

Logical equality is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both operands are false or both operands are true.

A logical equality of propositions p and q may be written in various ways.  Among the most common are these:

  • p \Leftrightarrow q
  • p \equiv q
  • p = q

A truth table for p = q appears below:


\text{Logical Equality}
p q p = q
\mathrm{F} \mathrm{F} \mathrm{T}
\mathrm{F} \mathrm{T} \mathrm{F}
\mathrm{T} \mathrm{F} \mathrm{F}
\mathrm{T} \mathrm{T} \mathrm{T}


A logical graph for p = q is shown below:

Logical Graph ((P,Q)).jpg

The traversal string of this graph is \texttt{((} p \texttt{,} q \texttt{))}.  The proposition p = q may be taken as a Boolean function f(p, q) having the abstract type f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}, where \mathbb{B} = \{ 0, 1 \} is interpreted in such a way that 0 means \mathrm{false} and 1 means \mathrm{true}.

A Venn diagram for p = q indicates the region where p = q is true by means of a distinctive color or shading.  In this case the region consists of two single cells, as shown below:

Venn Diagram P = Q 2.0.jpg

Logical implication

The concept of logical implication encompasses a specific logical function, a specific logical relation, and the various symbols that are used to denote this function and this relation. In order to define the specific function, relation, and symbols in question it is first necessary to establish a few ideas about the connections among them.

Close approximations to the concept of logical implication are expressed in ordinary language by means of linguistic forms like the following:

\begin{array}{l}
p ~\text{implies}~ q.
\\[6pt]
\text{if}~ p ~\text{then}~ q.
\end{array}

Here \text{“} p \text{”} and \text{“} q \text{”} are propositional variables that stand for any propositions in a given language. In a statement of the form \text{“} \text{if}~ p ~\text{then}~ q \text{”}, the first term, p, is called the antecedent and the second term, q, is called the consequent, while the statement as a whole is called either the conditional or the consequence. Assuming that the conditional statement is true, then the truth of the antecedent is a sufficient condition for the truth of the consequent, while the truth of the consequent is a necessary condition for the truth of the antecedent.

Note. Many writers draw a technical distinction between the form \text{“} p ~\text{implies}~ q \text{”} and the form \text{“} \text{if}~ p ~\text{then}~ q \text{”}. In this usage, writing \text{“} p ~\text{implies}~ q \text{”} asserts the existence of a certain relation between the logical value of p and the logical value of q, whereas writing \text{“} \text{if}~ p ~\text{then}~ q \text{”} merely forms a compound statement whose logical value is a function of the logical values of p and q. This will be discussed in detail below.

The concept of logical implication is associated with an operation on two logical values, typically the values of two propositions, that produces a value of false just in case the first operand is true and the second operand is false.

In the interpretation where 0 = \mathrm{false} and 1 = \mathrm{true}, the truth table associated with the statement \text{“} p ~\text{implies}~ q \text{”}, symbolized as \text{“} p \Rightarrow q \text{”}, appears below:


\text{Logical Implication}
p q p \Rightarrow q
0 0 1
0 1 1
1 0 0
1 1 1


A logical graph for p \Rightarrow q is shown below:

Logical Graph (P(Q)).jpg

The traversal string of this graph is \texttt{(} p \texttt{(} q \texttt{))}.  The proposition p \Rightarrow q may be taken as a Boolean function f(p, q) having the abstract type f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}, where \mathbb{B} = \{ 0, 1 \} is interpreted in such a way that 0 means \mathrm{false} and 1 means \mathrm{true}.

A Venn diagram for p \Rightarrow q indicates the region where p \Rightarrow q is true by means of a distinctive color or shading.  In this case the region is three adjacent cells, as shown below:

Venn Diagram P ⇒ Q 2.0.jpg

Logical NAND

Logical NAND (“Not And”) is an operation on two logical values, typically the values of two propositions, that produces a value of false if and only if both of its operands are true.  In other words, it produces a value of true if and only if at least one of its operands is false.

A logical NAND of propositions p and q may be written in various ways.  Among the most common are these:

  • p ~\bar\curlywedge~ q
  • p \barwedge q
  • p \uparrow q

A truth table for p ~\bar\curlywedge~ q appears below:


\text{Logical NAND}
p q p ~\bar\curlywedge~ q
\mathrm{F} \mathrm{F} \mathrm{T}
\mathrm{F} \mathrm{T} \mathrm{T}
\mathrm{T} \mathrm{F} \mathrm{T}
\mathrm{T} \mathrm{T} \mathrm{F}


A logical graph for p ~\bar\curlywedge~ q is drawn as two letters attached to the free node of a rooted edge:

Logical Graph (PQ).jpg

The traversal string of this graph is \texttt{(} pq \texttt{)}.  The proposition p ~\bar\curlywedge~ q may be taken as a Boolean function f(p, q) having the abstract type f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}, where \mathbb{B} = \{ 0, 1 \} is interpreted in such a way that 0 means \mathrm{false} and 1 means \mathrm{true}.

A Venn diagram for p ~\bar\curlywedge~ q indicates the region where p ~\bar\curlywedge~ q is true by means of a distinctive color or shading.  In this case the region consists of three adjacent cells, as shown below:

Venn Diagram P ⊼ Q 2.0.jpg

Logical NNOR

Logical NNOR (“Neither Nor”) is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both of its operands are false.  In other words, it produces a value of false if and only if at least one of its operands is true.

A logical NNOR of propositions p and q may be written in various ways.  Among the most common are these:

  • p \curlywedge q
  • p ~\bar\lor~ q
  • p \downarrow q

A truth table for p \curlywedge q appears below:


\text{Logical NNOR}
p q p \curlywedge q
\mathrm{F} \mathrm{F} \mathrm{T}
\mathrm{F} \mathrm{T} \mathrm{F}
\mathrm{T} \mathrm{F} \mathrm{F}
\mathrm{T} \mathrm{T} \mathrm{F}


A logical graph for p \curlywedge q is shown below:

Logical Graph (P)(Q).jpg

The traversal string of this graph is \texttt{(} p  \texttt{)(}q \texttt{)}.  The proposition p \curlywedge q may be taken as a Boolean function f(p, q) having the abstract type f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}, where \mathbb{B} = \{ 0, 1 \} is interpreted in such a way that 0 means \mathrm{false} and 1 means \mathrm{true}.

A Venn diagram for p \curlywedge q indicates the region where p \curlywedge q is true by means of a distinctive color or shading.  In this case the region is a single cell, as shown below:

Venn Diagram P ⊽ Q 2.0.jpg

Logical negation

Logical negation is an operation on one logical value, typically the value of a proposition, that produces a value of true when its operand is false and a value of false when its operand is true.

A truth table for \mathrm{NOT}~ p, also written \lnot p, appears below:


\text{Logical Negation}
p \lnot p
\mathrm{F} \mathrm{T}
\mathrm{T} \mathrm{F}


The negation of a proposition p may be found notated in various ways in various contexts of application, often merely for typographical convenience.  Among these variants are the following:


\text{Variant Notations}
\text{Notation} \text{Vocalization}
\bar{p} p bar
\tilde{p} p tilde
p' p prime
p complement
!p bang p


A logical graph for \lnot p is shown below:

Logical Graph (P).jpg

The traversal string of this graph is \texttt{(} p \texttt{)}.  The proposition \lnot p may be taken as a Boolean function f(p) having the abstract type f : \mathbb{B} \to \mathbb{B}, where \mathbb{B} = \{ 0, 1 \} is interpreted in such a way that 0 means \mathrm{false} and 1 means \mathrm{true}.

A Venn diagram for \lnot p indicates the region where \lnot p is true by means of a distinctive color or shading.  In this case the region is a single cell, as shown below:

Venn Diagram ¬P 2.0.jpg

Symbol Renderings

NAND

  • p ~\bar\curlywedge~ q \bar\curlywedge
  • p \barwedge q \barwedge
  • p \uparrow q \uparrow

  • p ~\text{⊼}~ q \text unicode ⊼
  • p ~\stackrel{\circ}{\curlywedge}~ q \stackrel{\circ}{\curlywedge}

  • p ~\bar\land~ q \bar\land
  • p ~\bar{\wedge}~ q \bar\wedge
  • p \mathop {\bar \land} q \mathop \bar \land
  • p \mathop {\bar \land} q \mathop \bar \wedge
  • p \mathop {\bar \curlywedge} q \mathop \bar \curlywedge

NNOR

  • p \curlywedge q \curlywedge
  • p ~\bar\lor~ q \bar\lor
  • p \downarrow q \downarrow

  • Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): p ~\text{⊽}~ q \text unicode ⊽

  • p ~\bar\vee q \bar\vee
  • p \mathop {\bar \lor} q \mathop \bar \lor
  • p \mathop {\bar \vee} q \mathop \bar \vee