Work H

Existential Interpretation

$\text{Table 1.} ~~ \text{Existential Interpretation}$
$\text{Graph}$ $\text{Expression}$ $\text{Interpretation}$

$~$ $\mathrm{true}$

$\texttt{(} ~ \texttt{)}$ $\mathrm{false}$

$a$ $a$

$\texttt{(} a \texttt{)}$ $\begin{matrix} \tilde{a} \\[2pt] a^\prime \\[2pt] \lnot a \\[2pt] \mathrm{not}~ a \end{matrix}$

$a~b~c$ $\begin{matrix} a \land b \land c \\[6pt] a ~\mathrm{and}~ b ~\mathrm{and}~ c \end{matrix}$

$\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}$ $\begin{matrix} a \lor b \lor c \\[6pt] a ~\mathrm{or}~ b ~\mathrm{or}~ c \end{matrix}$

$\texttt{(} a \texttt{(} b \texttt{))}$ $\begin{matrix} a \Rightarrow b \\[2pt] a ~\mathrm{implies}~ b \\[2pt] \mathrm{if}~ a ~\mathrm{then}~ b \\[2pt] \mathrm{not}~ a ~\mathrm{without}~ b \end{matrix}$

$\texttt{(} a, b \texttt{)}$ $\begin{matrix} a + b \\[2pt] a \neq b \\[2pt] a ~\mathrm{exclusive~or}~ b \\[2pt] a ~\mathrm{not~equal~to}~ b \end{matrix}$

$\texttt{((} a, b \texttt{))}$ $\begin{matrix} a = b \\[2pt] a \iff b \\[2pt] a ~\mathrm{equals}~ b \\[2pt] a ~\mathrm{if~and~only~if}~ b \end{matrix}$

$\texttt{(} a, b, c \texttt{)}$ $\begin{matrix} \mathrm{just~one~of} \\ a, b, c \\ \mathrm{is~false} \end{matrix}$

$\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}$ $\begin{matrix} \mathrm{just~one~of} \\ a, b, c \\ \mathrm{is~true} \end{matrix}$

$\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}$ $\begin{matrix} \mathrm{genus}~ a ~\mathrm{of~species}~ b, c \\[6pt] \mathrm{partition}~ a ~\mathrm{into}~ b, c \\[6pt] \mathrm{pie}~ a ~\mathrm{of~slices}~ b, c \end{matrix}$

Entitative Interpretation

$\text{Table 2.} ~~ \text{Entitative Interpretation}$
$\text{Graph}$ $\text{Expression}$ $\text{Interpretation}$

$~$ $\mathrm{false}$

$\texttt{(} ~ \texttt{)}$ $\mathrm{true}$

$a$ $a$

$\texttt{(} a \texttt{)}$ $\begin{matrix} \tilde{a} \\[2pt] a^\prime \\[2pt] \lnot a \\[2pt] \mathrm{not}~ a \end{matrix}$

$a~b~c$ $\begin{matrix} a \lor b \lor c \\[6pt] a ~\mathrm{or}~ b ~\mathrm{or}~ c \end{matrix}$

$\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}$ $\begin{matrix} a \land b \land c \\[6pt] a ~\mathrm{and}~ b ~\mathrm{and}~ c \end{matrix}$

$\texttt{(} a \texttt{)} b$ $\begin{matrix} a \Rightarrow b \\[2pt] a ~\mathrm{implies}~ b \\[2pt] \mathrm{if}~ a ~\mathrm{then}~ b \\[2pt] \mathrm{not}~ a, \mathrm{or}~ b \end{matrix}$

$\texttt{(} a, b \texttt{)}$ $\begin{matrix} a = b \\[2pt] a \iff b \\[2pt] a ~\mathrm{equals}~ b \\[2pt] a ~\mathrm{if~and~only~if}~ b \end{matrix}$

$\texttt{((} a, b \texttt{))}$ $\begin{matrix} a + b \\[2pt] a \neq b \\[2pt] a ~\mathrm{exclusive~or}~ b \\[2pt] a ~\mathrm{not~equal~to}~ b \end{matrix}$

$\texttt{(} a, b, c \texttt{)}$ $\begin{matrix} \mathrm{not~just~one~of} \\ a, b, c \\ \mathrm{is~true} \end{matrix}$

$\texttt{((} a, b, c \texttt{))}$ $\begin{matrix} \mathrm{just~one~of} \\ a, b, c \\ \mathrm{is~true} \end{matrix}$

$\texttt{(((} a \texttt{)}, b, c \texttt{))}$ $\begin{matrix} \mathrm{genus}~ a ~\mathrm{of~species}~ b, c \\[6pt] \mathrm{partition}~ a ~\mathrm{into}~ b, c \\[6pt] \mathrm{pie}~ a ~\mathrm{of~slices}~ b, c \end{matrix}$

Logical Interpretations of Cactus Structures

$\text{Table 3.} ~~ \text{Logical Interpretations of Cactus Structures}$
$\text{Graph}$ $\text{Expression}$ $\begin{matrix} \text{Existential} \\ \text{Interpretation} \end{matrix}$ $\begin{matrix} \text{Entitative} \\ \text{Interpretation} \end{matrix}$

$~$ $\mathrm{true}$ $\mathrm{false}$

$\texttt{(} ~ \texttt{)}$ $\mathrm{false}$ $\mathrm{true}$

$C_1 \ldots C_k$ $C_1 \land \ldots \land C_k$ $C_1 \lor \ldots \lor C_k$

$\texttt{(} C_1 \texttt{,} \ldots \texttt{,} C_k \texttt{)}$ $\begin{matrix} \text{just one of} \\[6px] C_1, \ldots, C_k \\[6px] \text{is false} \end{matrix}$ $\begin{matrix} \text{not just one of} \\[6px] C_1, \ldots, C_k \\[6px] \text{is true} \end{matrix}$