Animated Logical Graphs • 25

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:

\begin{array}{|*{3}{c}||c|}  \multicolumn{4}{c}{\text{Formal Operation Table} ~ \texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}} \\[4pt]  \hline  a & b & c & \texttt{(}a\texttt{,}b\texttt{,}c\texttt{)} \\  \hline\hline  \texttt{Space}&\texttt{Space}&\texttt{Space}&\texttt{Cross} \\  \texttt{Space}&\texttt{Space}&\texttt{Cross}&\texttt{Space} \\  \texttt{Space}&\texttt{Cross}&\texttt{Space}&\texttt{Space} \\  \texttt{Space}&\texttt{Cross}&\texttt{Cross}&\texttt{Cross} \\  \hline  \texttt{Cross}&\texttt{Space}&\texttt{Space}&\texttt{Space} \\  \texttt{Cross}&\texttt{Space}&\texttt{Cross}&\texttt{Cross} \\  \texttt{Cross}&\texttt{Cross}&\texttt{Space}&\texttt{Cross} \\  \texttt{Cross}&\texttt{Cross}&\texttt{Cross}&\texttt{Cross} \\  \hline  \end{array}

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:

\begin{array}{|*{3}{c}||c|}  \multicolumn{4}{c}{\text{Formal Operation Table} ~ \texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}} \\[4pt]  \hline  \quad a \quad & \quad b \quad & \quad c \quad & \texttt{(}a\texttt{,}b\texttt{,}c\texttt{)} \\  \hline\hline  \texttt{o} & \texttt{o} & \texttt{o} & \texttt{|} \\  \texttt{o} & \texttt{o} & \texttt{|} & \texttt{o} \\  \texttt{o} & \texttt{|} & \texttt{o} & \texttt{o} \\  \texttt{o} & \texttt{|} & \texttt{|} & \texttt{|} \\  \hline  \texttt{|} & \texttt{o} & \texttt{o} & \texttt{o} \\  \texttt{|} & \texttt{o} & \texttt{|} & \texttt{|} \\  \texttt{|} & \texttt{|} & \texttt{o} & \texttt{|} \\  \texttt{|} & \texttt{|} & \texttt{|} & \texttt{|} \\  \hline  \end{array}

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”.

cc: Systems ScienceStructural ModelingOntolog ForumLaws of FormCybernetics

This entry was posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Constraint Satisfaction Problems, Deduction, Diagrammatic Reasoning, Duality, Equational Inference, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Painted Cacti, Peirce, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Theorem Proving, Visualization and tagged , , , , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

1 Response to Animated Logical Graphs • 25

  1. Pingback: Survey of Animated Logical Graphs • 2 | Inquiry Into Inquiry

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