Animated Logical Graphs • 21

A funny thing just happened.  Let’s see if we can tell where.  We started with the algebraic expression {}^{\backprime\backprime} \texttt{(} a \texttt{)} {}^{\prime\prime}, in which the operand {}^{\backprime\backprime} a {}^{\prime\prime} suggests the contemplated absence or presence of any arithmetic expression or its value, then we contemplated the absence or presence of the operator {}^{\backprime\backprime} \texttt{(} ~ \texttt{)} {}^{\prime\prime} in {}^{\backprime\backprime} \texttt{(} a \texttt{)} {}^{\prime\prime} to be indicated by a cross or a space, respectively, for the value of a newly introduced variable, {}^{\backprime\backprime} b {}^{\prime\prime}, placed in a new slot of a newly extended operator form, as suggested by this picture:

Control Form (a)_b

What happened here is this.  Our contemplation of a constant operator as being potentially variable gave rise to the contemplation of a newly introduced but otherwise quite ordinary operand variable, albeit in a newly-fashioned formula.  In its interpretation for logic the newly formed operation may be viewed as an extension of ordinary negation, one in which the negation of the first variable is controlled by the value of the second variable.  Thus, we may regard this development as marking a form of controlled reflection, or a form of reflective control.  From here on out we will use the inline syntax {}^{\backprime\backprime} \texttt{(} a \texttt{,} b \texttt{)} {}^{\prime\prime} for the corresponding operation on two variables, whose operation table is given below:

\begin{array}{|c|c|c|}  \hline  a & b & \texttt{(} a \texttt{,} b \texttt{)} \\  \hline\hline  \texttt{Space} & \texttt{Space} & \texttt{Cross} \\  \texttt{Space} & \texttt{Cross} & \texttt{Space} \\  \texttt{Cross} & \texttt{Space} & \texttt{Space} \\  \texttt{Cross} & \texttt{Cross} & \texttt{Cross} \\  \hline  \end{array}

  • The Entitative Interpretation (\mathrm{En}), for which Space = False and Cross = True, calls this operation equivalence.
  • The Existential Interpretation (\mathrm{Ex}), for which Space = True and Cross = False, calls this operation distinction.

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.

3 Responses to Animated Logical Graphs • 21

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

  2. Pingback: Animated Logical Graphs • 29 | Inquiry Into Inquiry

  3. Pingback: Animated Logical Graphs • 31 | Inquiry Into Inquiry

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