Theme One • A Program Of Inquiry : 17

Re: Ontolog Forum • (1)
Re: Systems Science • (1)
Re: Laws Of Form • (1)(2)(3)(4)

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}
This entry was posted in Algorithms, Animata, Artificial Intelligence, Boolean Functions, C.S. Peirce, Cactus Graphs, Computation, Computational Complexity, Cybernetics, Data Structures, Differential Logic, Form, Formal Languages, Graph Theory, Inquiry, Inquiry Driven Systems, Intelligent Systems, Laws of Form, Learning, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Pragmatics, Programming, Propositional Calculus, Propositional Equation Reasoning Systems, Reasoning, Semantics, Semiotics, Spencer Brown, Syntax, Theorem Proving and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

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