Animated Logical Graphs • 25

Re: Animated Logical Graphs • 23

Let’s examine the Formal Operation Table for the third in our series of reflective forms to see if we can elicit the general pattern.

Or, thinking in terms of the corresponding cactus graphs, writing ${}^{\backprime\backprime} \texttt{o} {}^{\prime\prime}$ for a blank node and ${}^{\backprime\backprime} \texttt{|} {}^{\prime\prime}$ for a terminal edge, we get the following Table.

Evidently, the rule is that ${}^{\backprime\backprime} \texttt{(} a \texttt{,} b \texttt{,} c \texttt{)} {}^{\prime\prime}$ denotes the value denoted by ${}^{\backprime\backprime} \texttt{o} {}^{\prime\prime}$ if and only if exactly one of the variables $a, b, c$ has the value denoted by ${}^{\backprime\backprime} \texttt{|} {}^{\prime\prime},$ otherwise ${}^{\backprime\backprime} \texttt{(} a \texttt{,} b \texttt{,} c \texttt{)} {}^{\prime\prime}$ denotes the value denoted by ${}^{\backprime\backprime} \texttt{|} {}^{\prime\prime}.$ Examining the whole series of reflective forms shows this is the general rule.

In the Entitative Interpretation $(\mathrm{En}),$ where $\texttt{o}$ = false and $\texttt{|}$ = true,
${}^{\backprime\backprime} \texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)} {}^{\prime\prime}$ translates as “not just one of the $x_j$ is true”.
In the Existential Interpretation $(\mathrm{Ex}),$ where $\texttt{o}$ = true and $\texttt{|}$ = false,
${}^{\backprime\backprime} \texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)} {}^{\prime\prime}$ translates as “just one of the $x_j$ is not true”.

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cc: Structural Modeling (1) (2) • Systems Science (1) (2)
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