# User:Jon Awbrey/Figures and Tables

## 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 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

 $S$ $I$ $\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}$ $\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}$ $\begin{matrix} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}$ $\begin{matrix} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}$
 $S$ $I$ $\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}$ $\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}$ $\begin{matrix} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}$ $\begin{matrix} {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}$

## 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

 $\text{Object}$ $\text{Sign}$ $\text{Interpretant}$ $\mathrm{A}$ ${}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}$ ${}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}$ $\mathrm{A}$ ${}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}$ ${}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}$ $\mathrm{A}$ ${}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}$ ${}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}$ $\mathrm{A}$ ${}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}$ ${}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}$ $\mathrm{B}$ ${}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}$ ${}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}$ $\mathrm{B}$ ${}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}$ ${}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}$ $\mathrm{B}$ ${}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}$ ${}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}$ $\mathrm{B}$ ${}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}$ ${}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}$

## 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

 $\text{Figure 16.} ~~ \text{Higher Order Universe of Discourse} ~ \left[ \ell_{00}, \ell_{01}, \ell_{10}, \ell_{11} \right] \subseteq \left[\left[ u, v \right]\right]$

### Example 2

 $\text{Figure 16.} ~~ \text{A Couple of Fourth Gear Orbits}$

### Example 3

 $\text{Figure 1. Conjunction}~ pq$

## 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.

 $\text{Graph}$ $\text{Expression}$ $\text{Interpretation}$ $\text{Other Notations}$ $~$ $\mathrm{true}$ $1$ $\texttt{(}~\texttt{)}$ $\mathrm{false}$ $0$ $a$ $a$ $a$ $\texttt{(} a \texttt{)}$ $\mathrm{not}~ a$ $\lnot a \quad \bar{a} \quad \tilde{a} \quad a^\prime$ $a ~ b ~ c$ $a ~\mathrm{and}~ b ~\mathrm{and}~ c$ $a \land b \land c$ $\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}$ $a ~\mathrm{or}~ b ~\mathrm{or}~ c$ $a \lor b \lor c$ $\texttt{(} a \texttt{(} b \texttt{))}$ $\begin{matrix} a ~\mathrm{implies}~ b \\[6pt] \mathrm{if}~ a ~\mathrm{then}~ b \end{matrix}$ $a \Rightarrow b$ 65px $\texttt{(} a \texttt{,} b \texttt{)}$ $\begin{matrix} a ~\mathrm{not~equal~to}~ b \\[6pt] a ~\mathrm{exclusive~or}~ b \end{matrix}$ $\begin{matrix} a \neq b \\[6pt] a + b \end{matrix}$ $\texttt{((} a \texttt{,} b \texttt{))}$ $\begin{matrix} a ~\mathrm{is~equal~to}~ b \\[6pt] a ~\mathrm{if~and~only~if}~ b \end{matrix}$ $\begin{matrix} a = b \\[6pt] a \Leftrightarrow b \end{matrix}$ $\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}$ $\begin{matrix} \mathrm{just~one~of} \\ a, b, c \\ \mathrm{is~false}. \end{matrix}$ $\begin{matrix} & \bar{a} ~ b ~ c \\ \lor & a ~ \bar{b} ~ c \\ \lor & a ~ b ~ \bar{c} \end{matrix}$ $\texttt{((} a \texttt{),(} b \texttt{),(} c \texttt{))}$ $\begin{matrix} \mathrm{just~one~of} \\ a, b, c \\ \mathrm{is~true}. \\[6pt] \mathrm{partition~all} \\ \mathrm{into}~ a, b, c. \end{matrix}$ $\begin{matrix} & a ~ \bar{b} ~ \bar{c} \\ \lor & \bar{a} ~ b ~ \bar{c} \\ \lor & \bar{a} ~ \bar{b} ~ c \end{matrix}$ $\texttt{(} a \texttt{,(} b \texttt{,} c \texttt{))}$ $\begin{matrix} \mathrm{oddly~many~of} \\ a, b, c \\ \mathrm{are~true}. \end{matrix}$ $a + b + c$ $\begin{matrix} & a ~ b ~ c \\ \lor & a ~ \bar{b} ~ \bar{c} \\ \lor & \bar{a} ~ b ~ \bar{c} \\ \lor & \bar{a} ~ \bar{b} ~ c \end{matrix}$ $\texttt{(} x \texttt{,(} a \texttt{),(} b \texttt{),(} c \texttt{))}$ $\begin{matrix} \mathrm{partition}~ x \\ \mathrm{into}~ a, b, c. \\[6pt] \mathrm{genus}~ x ~\mathrm{comprises} \\ \mathrm{species}~ a, b, c. \end{matrix}$ $\begin{matrix} & \bar{x} ~ \bar{a} ~ \bar{b} ~ \bar{c} \\ \lor & x ~ a ~ \bar{b} ~ \bar{c} \\ \lor & x ~ \bar{a} ~ b ~ \bar{c} \\ \lor & x ~ \bar{a} ~ \bar{b} ~ c \end{matrix}~$

### Futures Of Logical Graphs

#### Reflective Series • Variant 1

 $\text{Form}$ $\text{String}$ $\text{Graph}$ $\texttt{(a)}$ $\texttt{(a,b)}$ $\texttt{(a,b,c)}$

#### Reflective Series • Variant 2

 $\text{Form}$ $\text{String}$ $\text{Graph}$ $\texttt{(} a \texttt{)}$ $\texttt{(} a \texttt{,} b \texttt{)}$ $\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}$

### Higher Order Propositions

#### Higher Order Propositions (n = 1)

##### Version 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

 $x:$ 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

 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

 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

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

##### Version 2

 $u\!:$$v\!:$ 11001010 $f$ $m_0$ $m_1$ $m_2$ $m_3$ $m_4$ $m_5$ $m_6$ $m_7$ $m_8$ $m_9$ $m_{10}$ $m_{11}$ $m_{12}$ $m_{13}$ $m_{14}$ $m_{15}$ $m_{16}$ $m_{17}$ $m_{18}$ $m_{19}$ $m_{20}$ $m_{21}$ $m_{22}$ $m_{23}$ $f_0$ 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 Looks Now

 $\mathcal{L}_1$ $\mathcal{L}_2$ $\mathcal{L}_3$ $\mathcal{L}_4$ $p\colon$ $1~1~1~1~0~0~0~0$ $q\colon$ $1~1~0~0~1~1~0~0$ $r\colon$ $1~0~1~0~1~0~1~0$ $\begin{matrix} f_{104} \\[4pt] f_{148} \\[4pt] f_{146} \\[4pt] f_{97} \\[4pt] f_{134} \\[4pt] f_{73} \\[4pt] f_{41} \\[4pt] f_{22} \end{matrix}$ $\begin{matrix} f_{01101000} \\[4pt] f_{10010100} \\[4pt] f_{10010010} \\[4pt] f_{01100001} \\[4pt] f_{10000110} \\[4pt] f_{01001001} \\[4pt] f_{00101001} \\[4pt] f_{00010110} \end{matrix}$ $\begin{matrix} 0~1~1~0~1~0~0~0 \\[4pt] 1~0~0~1~0~1~0~0 \\[4pt] 1~0~0~1~0~0~1~0 \\[4pt] 0~1~1~0~0~0~0~1 \\[4pt] 1~0~0~0~0~1~1~0 \\[4pt] 0~1~0~0~1~0~0~1 \\[4pt] 0~0~1~0~1~0~0~1 \\[4pt] 0~0~0~1~0~1~1~0 \end{matrix}$ $\begin{matrix} \texttt{(~p~,~q~,~r~)} \\[4pt] \texttt{(~p~,~q~,(r))} \\[4pt] \texttt{(~p~,(q),~r~)} \\[4pt] \texttt{(~p~,(q),(r))} \\[4pt] \texttt{((p),~q~,~r~)} \\[4pt] \texttt{((p),~q~,(r))} \\[4pt] \texttt{((p),(q),~r~)} \\[4pt] \texttt{((p),(q),(r))} \end{matrix}$ $\begin{matrix} f_{233} \\[4pt] f_{214} \\[4pt] f_{182} \\[4pt] f_{121} \\[4pt] f_{158} \\[4pt] f_{109} \\[4pt] f_{107} \\[4pt] f_{151} \end{matrix}$ $\begin{matrix} f_{11101001} \\[4pt] f_{11010110} \\[4pt] f_{10110110} \\[4pt] f_{01111001} \\[4pt] f_{10011110} \\[4pt] f_{01101101} \\[4pt] f_{01101011} \\[4pt] f_{10010111} \end{matrix}$ $\begin{matrix} 1~1~1~0~1~0~0~1 \\[4pt] 1~1~0~1~0~1~1~0 \\[4pt] 1~0~1~1~0~1~1~0 \\[4pt] 0~1~1~1~1~0~0~1 \\[4pt] 1~0~0~1~1~1~1~0 \\[4pt] 0~1~1~0~1~1~0~1 \\[4pt] 0~1~1~0~1~0~1~1 \\[4pt] 1~0~0~1~0~1~1~1 \end{matrix}$ $\begin{matrix} \texttt{(((p),(q),(r)))} \\[4pt] \texttt{(((p),(q),~r~))} \\[4pt] \texttt{(((p),~q~,(r)))} \\[4pt] \texttt{(((p),~q~,~r~))} \\[4pt] \texttt{((~p~,(q),(r)))} \\[4pt] \texttt{((~p~,(q),~r~))} \\[4pt] \texttt{((~p~,~q~,(r)))} \\[4pt] \texttt{((~p~,~q~,~r~))} \end{matrix}$

#### Test 1

 $\mathcal{L}_1$ $\mathcal{L}_2$ $\mathcal{L}_3$ $\mathcal{L}_4$ $p\colon$ $1~1~1~1~0~0~0~0$ $q\colon$ $1~1~0~0~1~1~0~0$ $r\colon$ $1~0~1~0~1~0~1~0$ $\begin{matrix} f_{104} \\[4pt] f_{148} \\[4pt] f_{146} \\[4pt] f_{97} \\[4pt] f_{134} \\[4pt] f_{73} \\[4pt] f_{41} \\[4pt] f_{22} \end{matrix}$ $\begin{matrix} f_{01101000} \\[4pt] f_{10010100} \\[4pt] f_{10010010} \\[4pt] f_{01100001} \\[4pt] f_{10000110} \\[4pt] f_{01001001} \\[4pt] f_{00101001} \\[4pt] f_{00010110} \end{matrix}$ $\begin{matrix} 0~1~1~0~1~0~0~0 \\[4pt] 1~0~0~1~0~1~0~0 \\[4pt] 1~0~0~1~0~0~1~0 \\[4pt] 0~1~1~0~0~0~0~1 \\[4pt] 1~0~0~0~0~1~1~0 \\[4pt] 0~1~0~0~1~0~0~1 \\[4pt] 0~0~1~0~1~0~0~1 \\[4pt] 0~0~0~1~0~1~1~0 \end{matrix}$ $\begin{matrix} \texttt{( p , q , r )} \\[4pt] \texttt{( p , q ,(r))} \\[4pt] \texttt{( p ,(q), r )} \\[4pt] \texttt{( p ,(q),(r))} \\[4pt] \texttt{((p), q , r )} \\[4pt] \texttt{((p), q ,(r))} \\[4pt] \texttt{((p),(q), r )} \\[4pt] \texttt{((p),(q),(r))} \end{matrix}$ $\begin{matrix} f_{233} \\[4pt] f_{214} \\[4pt] f_{182} \\[4pt] f_{121} \\[4pt] f_{158} \\[4pt] f_{109} \\[4pt] f_{107} \\[4pt] f_{151} \end{matrix}$ $\begin{matrix} f_{11101001} \\[4pt] f_{11010110} \\[4pt] f_{10110110} \\[4pt] f_{01111001} \\[4pt] f_{10011110} \\[4pt] f_{01101101} \\[4pt] f_{01101011} \\[4pt] f_{10010111} \end{matrix}$ $\begin{matrix} 1~1~1~0~1~0~0~1 \\[4pt] 1~1~0~1~0~1~1~0 \\[4pt] 1~0~1~1~0~1~1~0 \\[4pt] 0~1~1~1~1~0~0~1 \\[4pt] 1~0~0~1~1~1~1~0 \\[4pt] 0~1~1~0~1~1~0~1 \\[4pt] 0~1~1~0~1~0~1~1 \\[4pt] 1~0~0~1~0~1~1~1 \end{matrix}$ $\begin{matrix} \texttt{(((p),(q),(r)))} \\[4pt] \texttt{(((p),(q), r ))} \\[4pt] \texttt{(((p), q ,(r)))} \\[4pt] \texttt{(((p), q , r ))} \\[4pt] \texttt{(( p ,(q),(r)))} \\[4pt] \texttt{(( p ,(q), r ))} \\[4pt] \texttt{(( p , q ,(r)))} \\[4pt] \texttt{(( p , q , r ))} \end{matrix}$

#### Test 2

 $\mathcal{L}_1$ $\mathcal{L}_2$ $\mathcal{L}_3$ $\mathcal{L}_4$ $p\colon$ $1~1~1~1~0~0~0~0$ $q\colon$ $1~1~0~0~1~1~0~0$ $r\colon$ $1~0~1~0~1~0~1~0$ $\begin{matrix} f_{104} \\[4pt] f_{148} \\[4pt] f_{146} \\[4pt] f_{97} \\[4pt] f_{134} \\[4pt] f_{73} \\[4pt] f_{41} \\[4pt] f_{22} \end{matrix}$ $\begin{matrix} f_{01101000} \\[4pt] f_{10010100} \\[4pt] f_{10010010} \\[4pt] f_{01100001} \\[4pt] f_{10000110} \\[4pt] f_{01001001} \\[4pt] f_{00101001} \\[4pt] f_{00010110} \end{matrix}$ $\begin{matrix} 0~1~1~0~1~0~0~0 \\[4pt] 1~0~0~1~0~1~0~0 \\[4pt] 1~0~0~1~0~0~1~0 \\[4pt] 0~1~1~0~0~0~0~1 \\[4pt] 1~0~0~0~0~1~1~0 \\[4pt] 0~1~0~0~1~0~0~1 \\[4pt] 0~0~1~0~1~0~0~1 \\[4pt] 0~0~0~1~0~1~1~0 \end{matrix}$ $\begin{matrix} \texttt{(}~p~\texttt{,}~q~\texttt{,}~r~\texttt{)} \\[4pt] \texttt{(}~p~\texttt{,}~q~\texttt{,}(r\texttt{))} \\[4pt] \texttt{(}~p~\texttt{,(}q\texttt{),}~r~\texttt{)} \\[4pt] \texttt{(}~p~\texttt{,(}q\texttt{),(}r\texttt{))} \\[4pt] \texttt{((}p\texttt{),}~q~\texttt{,}~r~\texttt{)} \\[4pt] \texttt{((}p\texttt{),}~q~\texttt{,(}r\texttt{))} \\[4pt] \texttt{((}p\texttt{),(}q\texttt{),}~r~\texttt{)} \\[4pt] \texttt{((}p\texttt{),(}q\texttt{),(}r\texttt{))} \end{matrix}$ $\begin{matrix} f_{233} \\[4pt] f_{214} \\[4pt] f_{182} \\[4pt] f_{121} \\[4pt] f_{158} \\[4pt] f_{109} \\[4pt] f_{107} \\[4pt] f_{151} \end{matrix}$ $\begin{matrix} f_{11101001} \\[4pt] f_{11010110} \\[4pt] f_{10110110} \\[4pt] f_{01111001} \\[4pt] f_{10011110} \\[4pt] f_{01101101} \\[4pt] f_{01101011} \\[4pt] f_{10010111} \end{matrix}$ $\begin{matrix} 1~1~1~0~1~0~0~1 \\[4pt] 1~1~0~1~0~1~1~0 \\[4pt] 1~0~1~1~0~1~1~0 \\[4pt] 0~1~1~1~1~0~0~1 \\[4pt] 1~0~0~1~1~1~1~0 \\[4pt] 0~1~1~0~1~1~0~1 \\[4pt] 0~1~1~0~1~0~1~1 \\[4pt] 1~0~0~1~0~1~1~1 \end{matrix}$ $\begin{matrix} \texttt{(((}p\texttt{),(}q\texttt{),(}r\texttt{)))} \\[4pt] \texttt{(((}p\texttt{),(}q\texttt{),}~r~\texttt{))} \\[4pt] \texttt{(((}p\texttt{),}~q~\texttt{,(}r\texttt{)))} \\[4pt] \texttt{(((}p\texttt{),}~q~\texttt{,}~r~\texttt{))} \\[4pt] \texttt{((}~p~\texttt{,(}q\texttt{),(}r\texttt{)))} \\[4pt] \texttt{((}~p~\texttt{,(}q\texttt{),}~r~\texttt{))} \\[4pt] \texttt{((}~p~\texttt{,}~q~\texttt{,(}r\texttt{)))} \\[4pt] \texttt{((}~p~\texttt{,}~q~\texttt{,}~r~\texttt{))} \end{matrix}$

#### Test 3

 $\mathcal{L}_1$ $\mathcal{L}_2$ $\mathcal{L}_3$ $\mathcal{L}_4$ $p\colon$ $1~1~1~1~0~0~0~0$ $q\colon$ $1~1~0~0~1~1~0~0$ $r\colon$ $1~0~1~0~1~0~1~0$ $\begin{matrix} f_{104} \\[4pt] f_{148} \\[4pt] f_{146} \\[4pt] f_{97} \\[4pt] f_{134} \\[4pt] f_{73} \\[4pt] f_{41} \\[4pt] f_{22} \end{matrix}$ $\begin{matrix} f_{01101000} \\[4pt] f_{10010100} \\[4pt] f_{10010010} \\[4pt] f_{01100001} \\[4pt] f_{10000110} \\[4pt] f_{01001001} \\[4pt] f_{00101001} \\[4pt] f_{00010110} \end{matrix}$ $\begin{matrix} 0~1~1~0~1~0~0~0 \\[4pt] 1~0~0~1~0~1~0~0 \\[4pt] 1~0~0~1~0~0~1~0 \\[4pt] 0~1~1~0~0~0~0~1 \\[4pt] 1~0~0~0~0~1~1~0 \\[4pt] 0~1~0~0~1~0~0~1 \\[4pt] 0~0~1~0~1~0~0~1 \\[4pt] 0~0~0~1~0~1~1~0 \end{matrix}$ $\begin{matrix} \texttt{(} & \texttt{p} & \texttt{,} & \texttt{q} & \texttt{,} & \texttt{r} & \texttt{)} \\[4pt] \texttt{(} & \texttt{p} & \texttt{,} & \texttt{q} & \texttt{,} & \texttt{(r)} & \texttt{)} \\[4pt] \texttt{(} & \texttt{p} & \texttt{,} & \texttt{(q)} & \texttt{,} & \texttt{r} & \texttt{)} \\[4pt] \texttt{(} & \texttt{p} & \texttt{,} & \texttt{(q)} & \texttt{,} & \texttt{(r)} & \texttt{)} \\[4pt] \texttt{(} & \texttt{(p)} & \texttt{,} & \texttt{q} & \texttt{,} & \texttt{r} & \texttt{)} \\[4pt] \texttt{(} & \texttt{(p)} & \texttt{,} & \texttt{q} & \texttt{,} & \texttt{(r)} & \texttt{)} \\[4pt] \texttt{(} & \texttt{(p)} & \texttt{,} & \texttt{(q)} & \texttt{,} & \texttt{r} & \texttt{)} \\[4pt] \texttt{(} & \texttt{(p)} & \texttt{,} & \texttt{(q)} & \texttt{,} & \texttt{(r)} & \texttt{)} \end{matrix}$ $\begin{matrix} f_{233} \\[4pt] f_{214} \\[4pt] f_{182} \\[4pt] f_{121} \\[4pt] f_{158} \\[4pt] f_{109} \\[4pt] f_{107} \\[4pt] f_{151} \end{matrix}$ $\begin{matrix} f_{11101001} \\[4pt] f_{11010110} \\[4pt] f_{10110110} \\[4pt] f_{01111001} \\[4pt] f_{10011110} \\[4pt] f_{01101101} \\[4pt] f_{01101011} \\[4pt] f_{10010111} \end{matrix}$ $\begin{matrix} 1~1~1~0~1~0~0~1 \\[4pt] 1~1~0~1~0~1~1~0 \\[4pt] 1~0~1~1~0~1~1~0 \\[4pt] 0~1~1~1~1~0~0~1 \\[4pt] 1~0~0~1~1~1~1~0 \\[4pt] 0~1~1~0~1~1~0~1 \\[4pt] 0~1~1~0~1~0~1~1 \\[4pt] 1~0~0~1~0~1~1~1 \end{matrix}$ $\begin{matrix} \texttt{((} & \texttt{(p)} & \texttt{,} & \texttt{(q)} & \texttt{,} & \texttt{(r)} & \texttt{))} \\[4pt] \texttt{((} & \texttt{(p)} & \texttt{,} & \texttt{(q)} & \texttt{,} & \texttt{r} & \texttt{))} \\[4pt] \texttt{((} & \texttt{(p)} & \texttt{,} & \texttt{q} & \texttt{,} & \texttt{(r)} & \texttt{))} \\[4pt] \texttt{((} & \texttt{(p)} & \texttt{,} & \texttt{q} & \texttt{,} & \texttt{r} & \texttt{))} \\[4pt] \texttt{((} & \texttt{p} & \texttt{,} & \texttt{(q)} & \texttt{,} & \texttt{(r)} & \texttt{))} \\[4pt] \texttt{((} & \texttt{p} & \texttt{,} & \texttt{(q)} & \texttt{,} & \texttt{r} & \texttt{))} \\[4pt] \texttt{((} & \texttt{p} & \texttt{,} & \texttt{q} & \texttt{,} & \texttt{(r)} & \texttt{))} \\[4pt] \texttt{((} & \texttt{p} & \texttt{,} & \texttt{q} & \texttt{,} & \texttt{r} & \texttt{))} \end{matrix}$

## 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:

 $p$ $q$ $p + q$ $\mathrm{F}$ $\mathrm{F}$ $\mathrm{F}$ $\mathrm{F}$ $\mathrm{T}$ $\mathrm{T}$ $\mathrm{T}$ $\mathrm{F}$ $\mathrm{T}$ $\mathrm{T}$ $\mathrm{T}$ $\mathrm{F}$

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:

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:

### 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:

 $p$ $q$ $p \land q$ $\mathrm{F}$ $\mathrm{F}$ $\mathrm{F}$ $\mathrm{F}$ $\mathrm{T}$ $\mathrm{F}$ $\mathrm{T}$ $\mathrm{F}$ $\mathrm{F}$ $\mathrm{T}$ $\mathrm{T}$ $\mathrm{T}$

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

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:

### 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:

 $p$ $q$ $p \lor q$ $\mathrm{F}$ $\mathrm{F}$ $\mathrm{F}$ $\mathrm{F}$ $\mathrm{T}$ $\mathrm{T}$ $\mathrm{T}$ $\mathrm{F}$ $\mathrm{T}$ $\mathrm{T}$ $\mathrm{T}$ $\mathrm{T}$

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

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:

### 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:

 $p$ $q$ $p = q$ $\mathrm{F}$ $\mathrm{F}$ $\mathrm{T}$ $\mathrm{F}$ $\mathrm{T}$ $\mathrm{F}$ $\mathrm{T}$ $\mathrm{F}$ $\mathrm{F}$ $\mathrm{T}$ $\mathrm{T}$ $\mathrm{T}$

A logical graph for $p = q$ is shown below:

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:

### 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:

 $p$ $q$ $p \Rightarrow q$ $0$ $0$ $1$ $0$ $1$ $1$ $1$ $0$ $0$ $1$ $1$ $1$

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

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:

### 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:

 $p$ $q$ $p ~\bar\curlywedge~ q$ $\mathrm{F}$ $\mathrm{F}$ $\mathrm{T}$ $\mathrm{F}$ $\mathrm{T}$ $\mathrm{T}$ $\mathrm{T}$ $\mathrm{F}$ $\mathrm{T}$ $\mathrm{T}$ $\mathrm{T}$ $\mathrm{F}$

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

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:

### 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:

 $p$ $q$ $p \curlywedge q$ $\mathrm{F}$ $\mathrm{F}$ $\mathrm{T}$ $\mathrm{F}$ $\mathrm{T}$ $\mathrm{F}$ $\mathrm{T}$ $\mathrm{F}$ $\mathrm{F}$ $\mathrm{T}$ $\mathrm{T}$ $\mathrm{F}$

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

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:

### 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:

 $p$ $\lnot p$ $\mathrm{F}$ $\mathrm{T}$ $\mathrm{T}$ $\mathrm{F}$

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:

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

A logical graph for $\lnot p$ is shown below:

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:

## 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 $⊽$ \text unicode ⊽ $p ~\bar\vee q$ \bar\vee $p \mathop {\bar \lor} q$ \mathop \bar \lor $p \mathop {\bar \vee} q$ \mathop \bar \vee