Minimal Negation Operators • 3

It will take a few more rounds of stage-setting before we are able to entertain concrete examples of applications but the following may indicate the direction of generalization embodied in minimal negation operators.

To begin, let’s observe two ways of generalizing the logical operation commonly known as exclusive disjunction (\textsc{xor}) or symmetric difference (\Delta).

Let \mathbb{B} = the boolean domain \{ 0, 1 \}.

Exclusive disjunction is a boolean function \Delta : \mathbb{B} \times \mathbb{B} \to \mathbb{B} isomorphic to the algebraic field addition + : \mathbb{B} \times \mathbb{B} \to \mathbb{B}, also known as addition mod 2.  Adding the language of minimal negation operators to the mix we have the following equivalent expressions.

\begin{matrix}  \textsc{xor}(p, q)  & = &  \Delta (p, q)  & = &  p + q  & = &  \nu (p, q)  & = &  \texttt{(} p \texttt{,} q \texttt{)}  \end{matrix}

Minimal Negation \texttt{(} p \texttt{,} q \texttt{)} as Parity Indicator

Generalizing the function p + q of two variables to more variables extends the sequence of functions in the fashion p\!+\!q\!+\!r,  p\!+\!q\!+\!r\!+\!s,  p\!+\!q\!+\!r\!+\!s\!+\!t,  and so on.  These are known as parity sums, returning a value of 0 when there are an even number of 1’s in the sum and returning a value of 1 when there are an odd number of 1’s in the sum.

Minimal Negation \texttt{(} p \texttt{,} q \texttt{)} as Border Indicator

The equivalent expressions \texttt{(} p \texttt{,} q \texttt{)} = \nu(p, q) = p + q = p \,\Delta\, q = p ~\textsc{xor}~ q may be read with a different connotation, indicating the venn diagram cells adjacent to the conjunction p \land q.  Generalizing the function \texttt{(} p \texttt{,} q \texttt{)} of two variables to more variables extends the sequence of functions in the fashion \texttt{(} p \texttt{,} q \texttt{,} r \texttt{)},  \texttt{(} p \texttt{,} q \texttt{,} r \texttt{,} s \texttt{)},  \texttt{(} p \texttt{,} q \texttt{,} r \texttt{,} s \texttt{,} t \texttt{)},  and so on.  That sequence of operators differs from the sequence of parity sums once it passes the 2-variable case.

The triple sum may be written in terms of 2-place minimal negations as follows.

\begin{matrix}  p + q + r  & = &  \texttt{((} p \texttt{,} q \texttt{)}\!\texttt{,} r \texttt{)}  & = &  \texttt{(} p \texttt{,} \texttt{(} q \texttt{,} r \texttt{))}  \end{matrix}

It is important to recognize the triple sum expressions and the 3-place minimal negation \texttt{(} p \texttt{,} q \texttt{,} r \texttt{)} have very different meanings.

Resources

cc: CyberneticsOntolog ForumPeirce ListStructural ModelingSystems Science
cc: FB | Minimal Negation OperatorsLaws of Form

This entry was posted in Amphecks, Animata, Boolean Functions, C.S. Peirce, Cactus Graphs, Differential Logic, Functional Logic, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Propositional Equation Reasoning Systems, Semiotics, Venn Diagrams, Visualization and tagged , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

4 Responses to Minimal Negation Operators • 3

  1. Pingback: Minimal Negation Operators • 5 | Inquiry Into Inquiry

  2. Pingback: Minimal Negation Operators • Discussion 1 | Inquiry Into Inquiry

  3. Pingback: Survey of Animated Logical Graphs • 4 | Inquiry Into Inquiry

  4. Pingback: Minimal Negation Operators • Discussion 2 | Inquiry Into Inquiry

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.