Work H

Node Connective

$\text{Table 1.} ~~ \text{Existential Interpretation}$
$\text{Graph}$	$\text{Expression}$	$\text{Interpretation}$
	$~$	$\mathrm{true}$
	$\texttt{(} ~ \texttt{)}$	$\mathrm{false}$
	$a$	$a$
	$\texttt{(} a \texttt{)}$	$\begin{matrix} \tilde{a} \\[2pt] a^\prime \\[2pt] \lnot a \\[2pt] \mathrm{not}~ a \end{matrix}$
	$a~b~c$	$\begin{matrix} a \land b \land c \\[6pt] a ~\mathrm{and}~ b ~\mathrm{and}~ c \end{matrix}$
	$\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}$	$\begin{matrix} a \lor b \lor c \\[6pt] a ~\mathrm{or}~ b ~\mathrm{or}~ c \end{matrix}$
	$\texttt{(} a \texttt{(} b \texttt{))}$	$\begin{matrix} a \Rightarrow b \\[2pt] a ~\mathrm{implies}~ b \\[2pt] \mathrm{if}~ a ~\mathrm{then}~ b \\[2pt] \mathrm{not}~ a ~\mathrm{without}~ b \end{matrix}$
	$\texttt{(} a, b \texttt{)}$	$\begin{matrix} a + b \\[2pt] a \neq b \\[2pt] a ~\mathrm{exclusive~or}~ b \\[2pt] a ~\mathrm{not~equal~to}~ b \end{matrix}$
	$\texttt{((} a, b \texttt{))}$	$\begin{matrix} a = b \\[2pt] a \iff b \\[2pt] a ~\mathrm{equals}~ b \\[2pt] a ~\mathrm{if~and~only~if}~ b \end{matrix}$
	$\texttt{(} a, b, c \texttt{)}$	$\begin{matrix} \mathrm{just~one~of} \\ a, b, c \\ \mathrm{is~false} \end{matrix}$
	$\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}$	$\begin{matrix} \mathrm{just~one~of} \\ a, b, c \\ \mathrm{is~true} \end{matrix}$
	$\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}$	$\begin{matrix} \mathrm{genus}~ a ~\mathrm{of~species}~ b, c \\[6pt] \mathrm{partition}~ a ~\mathrm{into}~ b, c \\[6pt] \mathrm{pie}~ a ~\mathrm{of~slices}~ b, c \end{matrix}$

$\text{Table 2.} ~~ \text{Entitative Interpretation}$
$\text{Graph}$	$\text{Expression}$	$\text{Interpretation}$
	$~$	$\mathrm{false}$
	$\texttt{(} ~ \texttt{)}$	$\mathrm{true}$
	$a$	$a$
	$\texttt{(} a \texttt{)}$	$\begin{matrix} \tilde{a} \\[2pt] a^\prime \\[2pt] \lnot a \\[2pt] \mathrm{not}~ a \end{matrix}$
	$a~b~c$	$\begin{matrix} a \lor b \lor c \\[6pt] a ~\mathrm{or}~ b ~\mathrm{or}~ c \end{matrix}$
	$\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}$	$\begin{matrix} a \land b \land c \\[6pt] a ~\mathrm{and}~ b ~\mathrm{and}~ c \end{matrix}$
	$\texttt{(} a \texttt{)} b$	$\begin{matrix} a \Rightarrow b \\[2pt] a ~\mathrm{implies}~ b \\[2pt] \mathrm{if}~ a ~\mathrm{then}~ b \\[2pt] \mathrm{not}~ a, \mathrm{or}~ b \end{matrix}$
	$\texttt{(} a, b \texttt{)}$	$\begin{matrix} a = b \\[2pt] a \iff b \\[2pt] a ~\mathrm{equals}~ b \\[2pt] a ~\mathrm{if~and~only~if}~ b \end{matrix}$
	$\texttt{((} a, b \texttt{))}$	$\begin{matrix} a + b \\[2pt] a \neq b \\[2pt] a ~\mathrm{exclusive~or}~ b \\[2pt] a ~\mathrm{not~equal~to}~ b \end{matrix}$
	$\texttt{(} a, b, c \texttt{)}$	$\begin{matrix} \mathrm{not~just~one~of} \\ a, b, c \\ \mathrm{is~true} \end{matrix}$
	$\texttt{((} a, b, c \texttt{))}$	$\begin{matrix} \mathrm{just~one~of} \\ a, b, c \\ \mathrm{is~true} \end{matrix}$
	$\texttt{(((} a \texttt{)}, b, c \texttt{))}$	$\begin{matrix} \mathrm{genus}~ a ~\mathrm{of~species}~ b, c \\[6pt] \mathrm{partition}~ a ~\mathrm{into}~ b, c \\[6pt] \mathrm{pie}~ a ~\mathrm{of~slices}~ b, c \end{matrix}$

$\text{Table 3.} ~~ \text{Logical Interpretations of Cactus Structures}$
$\text{Expression}$	$\begin{matrix} \text{Existential} \\ \text{Interpretation} \end{matrix}$	$\begin{matrix} \text{Entitative} \\ \text{Interpretation} \end{matrix}$
$~$	$\mathrm{true}$	$\mathrm{false}$
$\texttt{(} ~ \texttt{)}$	$\mathrm{false}$	$\mathrm{true}$
$C_1 \ldots C_k$	$C_1 \land \ldots \land C_k$	$C_1 \lor \ldots \lor C_k$
$\texttt{(} C_1 \texttt{,} \ldots \texttt{,} C_k \texttt{)}$	$\begin{matrix} \text{just one of} \\[6px] C_1, \ldots, C_k \\[6px] \text{is false} \end{matrix}$	$\begin{matrix} \text{not just one of} \\[6px] C_1, \ldots, C_k \\[6px] \text{is true} \end{matrix}$