Theme One • A Program Of Inquiry 17

The move is all over but the unpacking, and the time looks ripe to pick up this thread from last spring.  Here, by way of a quick refresher, are a few Tables from earlier discussions.

  • Theme One • A Program Of Inquiry 11
    • Tables 1 and 2 illustrate the existential and entitative interpretations of cactus graphs and cactus expressions by means of English translations for a few of the most basic forms.

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 a

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

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

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

(a(b))
\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}

(a, b)
\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}

((a, b))
\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}

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

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

(a, (b),(c))
\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 a

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

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

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

(a)b
\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}

(a, b)
\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}

((a, b))
\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}

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

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

(((a), b, c))
\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}

cc: Ontolog ForumSystems Science

This entry was posted in Algorithms, Animata, Artificial Intelligence, Boolean Functions, C.S. Peirce, Cactus Graphs, Cognition, Computation, Constraint Satisfaction Problems, Data Structures, Differential Logic, Equational Inference, Formal Languages, Graph Theory, Inquiry Driven Systems, Laws of Form, Learning Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Semiotics, Spencer Brown, Visualization and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

2 Responses to Theme One • A Program Of Inquiry 17

  1. Pingback: Survey of Theme One Program • 2 | Inquiry Into Inquiry

  2. Pingback: Survey of Theme One Program • 3 | Inquiry Into Inquiry

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