## Theme One • A Program Of Inquiry : 11

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

The portions of exposition just skipped over covered with the use of cactus graphs in the program’s learning module to learn sequences of characters called “words” or “strings” and sequences of words called “sentences” or “strands”.  Leaving the matter of grammar to another time we turn to the use of cactus graphs in the program’s reasoning module to represent logical propositions on the order of Peirce’s alpha graphs and Spencer Brown’s calculus of indications.

### Logical Cacti

Up till now we’ve been working to hammer out a two-edged sword of syntax, honing the syntax of cactus graphs and cactus expressions and turning it to use in taming the syntax of two-level formal languages.

But the purpose of a logical syntax is to support a logical semantics, which means, for starters, to bear interpretation as sentential signs capable of denoting objective propositions about a universe of objects.

One of the difficulties we face is that the words interpretation, meaning, semantics, and their ilk take on so many different meanings from one moment to the next of their use.  A dedicated neologician might be able to think up distinctive names for all the aspects of meaning and all the approaches to them that concern us, but I will do the best I can with the common lot of ambiguous terms, leaving it to context and intelligent interpreters to sort it out as much as possible.

The formal language of cacti is formed at such a high level of abstraction that its graphs can bear at least two distinct interpretations as logical propositions.  The two interpretations that concern us here are descended from the ones C.S. Peirce called the entitative and the existential interpretations of his systems of graphical logics.

### Existential Interpretation

Table 1 illustrates the existential interpretation of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.

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

$~$ $\mathrm{true}$

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

$a$ $a$

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

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

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

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

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

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

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

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

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

### Entitative Interpretation

Table 2 illustrates the entitative interpretation of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.

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

$~$ $\mathrm{false}$

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

$a$ $a$

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

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

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

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

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

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

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

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

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

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