Work J

Operation Tables

Formal Operation Table (a,b) • Variant 1

LaTeX

$\begin{array}{|c|c|c|} \hline a & b & \texttt{(} a \texttt{,} b \texttt{)} \\ \hline\hline \texttt{Space} & \texttt{Space} & \texttt{Cross} \\ \texttt{Space} & \texttt{Cross} & \texttt{Space} \\ \texttt{Cross} & \texttt{Space} & \texttt{Space} \\ \texttt{Cross} & \texttt{Cross} & \texttt{Cross} \\ \hline \end{array}$

PNG

Formal Operation Table (a,b) • Variant 2

LaTeX

$\begin{array}{|c|c|c|} \multicolumn{3}{c}{\text{Formal Operation Table} ~ \texttt{(} a \texttt{,} b \texttt{)}} \\[4pt] \hline a & b & \texttt{(} a \texttt{,} b \texttt{)} \\ \hline\hline \texttt{Space} & \texttt{Space} & \texttt{Cross} \\ \texttt{Space} & \texttt{Cross} & \texttt{Space} \\ \texttt{Cross} & \texttt{Space} & \texttt{Space} \\ \texttt{Cross} & \texttt{Cross} & \texttt{Cross} \\ \hline \end{array}$

PNG

Formal Operation Table (a,b,c) • Variant 1

LaTeX

$\begin{array}{|*{3}{c}||c|} \multicolumn{4}{c}{\text{Formal Operation Table} ~ \texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}} \\[4pt] \hline a & b & c & \texttt{(}a\texttt{,}b\texttt{,}c\texttt{)} \\ \hline\hline \texttt{Space}&\texttt{Space}&\texttt{Space}&\texttt{Cross} \\ \texttt{Space}&\texttt{Space}&\texttt{Cross}&\texttt{Space} \\ \texttt{Space}&\texttt{Cross}&\texttt{Space}&\texttt{Space} \\ \texttt{Space}&\texttt{Cross}&\texttt{Cross}&\texttt{Cross} \\ \hline \texttt{Cross}&\texttt{Space}&\texttt{Space}&\texttt{Space} \\ \texttt{Cross}&\texttt{Space}&\texttt{Cross}&\texttt{Cross} \\ \texttt{Cross}&\texttt{Cross}&\texttt{Space}&\texttt{Cross} \\ \texttt{Cross}&\texttt{Cross}&\texttt{Cross}&\texttt{Cross} \\ \hline \end{array}$

PNG

Formal Operation Table (a,b,c) • Variant 2

LaTeX

$\begin{array}{|*{3}{c}||c|} \multicolumn{4}{c}{\text{Formal Operation Table} ~ \texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}} \\[4pt] \hline \quad a \quad & \quad b \quad & \quad c \quad & \texttt{(}a\texttt{,}b\texttt{,}c\texttt{)} \\ \hline\hline \texttt{o} & \texttt{o} & \texttt{o} & \texttt{|} \\ \texttt{o} & \texttt{o} & \texttt{|} & \texttt{o} \\ \texttt{o} & \texttt{|} & \texttt{o} & \texttt{o} \\ \texttt{o} & \texttt{|} & \texttt{|} & \texttt{|} \\ \hline \texttt{|} & \texttt{o} & \texttt{o} & \texttt{o} \\ \texttt{|} & \texttt{o} & \texttt{|} & \texttt{|} \\ \texttt{|} & \texttt{|} & \texttt{o} & \texttt{|} \\ \texttt{|} & \texttt{|} & \texttt{|} & \texttt{|} \\ \hline \end{array}$

PNG

Truth Tables

LaTeX

The entitative interpretation of $\texttt{(} a \texttt{,} b \texttt{)}$ produces the truth table for logical equality.

$\begin{array}{|c|c|c|} \multicolumn{3}{c}{\text{En} ~ \texttt{(} a \texttt{,} b \texttt{)}} \\[2pt] \hline a & b & \texttt{(} a \texttt{,} b \texttt{)} \\ \hline\hline \texttt{F} & \texttt{F} & \texttt{T} \\ \texttt{F} & \texttt{T} & \texttt{F} \\ \texttt{T} & \texttt{F} & \texttt{F} \\ \texttt{T} & \texttt{T} & \texttt{T} \\ \hline \end{array}$

The existential interpretation of $\texttt{(} a \texttt{,} b \texttt{)}$ produces the truth table for logical inequality, also known as exclusive disjunction.

$\begin{array}{|c|c|c|} \multicolumn{3}{c}{\text{Ex} ~ \texttt{(} a \texttt{,} b \texttt{)}} \\[2pt] \hline a & b & \texttt{(} a \texttt{,} b \texttt{)} \\ \hline\hline \texttt{T} & \texttt{T} & \texttt{F} \\ \texttt{T} & \texttt{F} & \texttt{T} \\ \texttt{F} & \texttt{T} & \texttt{T} \\ \texttt{F} & \texttt{F} & \texttt{F} \\ \hline \end{array}$

PNG

Syntactic Correspondences

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

Evaluation Rules

Interpretation Rules

$\text{Logical Interpretations of Cactus Structures}$
$\text{Expression}$	$\begin{matrix} \text{Entitative} \\ \text{Interpretation} \end{matrix}$	$\begin{matrix} \text{Existential} \\ \text{Interpretation} \end{matrix}$
$~$	$\mathrm{false}$	$\mathrm{true}$
$\texttt{(} ~ \texttt{)}$	$\mathrm{true}$	$\mathrm{false}$
$x_1 \ldots x_k$	$x_1 \lor \ldots \lor x_k$	$x_1 \land \ldots \land x_k$
$\texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)}$	$\begin{matrix} \text{not just one of} \\[6px] x_1, \ldots, x_k \\[6px] \text{is true} \end{matrix}$	$\begin{matrix} \text{just one of} \\[6px] x_1, \ldots, x_k \\[6px] \text{is false} \end{matrix}$