## Theme One • A Program Of Inquiry : 12

Re: Laws Of Form Discussions • (1)(2)(3)
Re: Peirce List Discussions • (1)(2)(3)

### Logical Cacti (cont.)

The main things to take away from the previous post are the following two ideas, one syntactic and one semantic:

• The compositional structures of cactus graphs and cactus expressions are constructed from two kinds of connective operations.
• There are two ways of mapping these compositional structures into the compositional structures of propositional sentences.

The two kinds of connective operations are described as follows:

• The node connective joins a number of component cacti $C_1, \ldots, C_k$ to a node:

• The lobe connective joins a number of component cacti $C_1, \ldots, C_k$ to a lobe:

The two ways of mapping cactus structures to logical meanings are summarized in Table 3, which compares the existential and entitative interpretations of the basic cactus structures, in effect, the graphical constants and connectives.

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

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

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

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

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