Minimal Negation Operators • 2

Re: Laws Of Form DiscussionMinimal Negation Operators

The brief description of minimal negation operators given in the previous post is enough to convey the rule of their construction.  For future reference, a slightly more formal definition is given below.

Initial Definition

The minimal negation operator \nu is a multigrade operator (\nu_k)_{k \in \mathbb{N}} where each \nu_k is a k-ary boolean function defined by the rule that \nu_k (x_1, \ldots , x_k) = 1 if and only if exactly one of the arguments x_j is 0.

In contexts where the initial letter \nu is understood, the minimal negation operators can be indicated by argument lists in parentheses.  In the discussion that follows a distinctive typeface will be used for logical expressions based on minimal negation operators, for example, \texttt{(x, y, z)} = \nu (x, y, z).

The first four members of this family of operators are shown below.  The third and fourth columns give paraphrases in two other notations, where tildes and primes, respectively, indicate logical negation.

\begin{matrix}  \texttt{()} & = & \nu_0 & = & 0 & = & \mathrm{false}  \\[6pt]  \texttt{(x)} & = & \nu_1 (x) & = & \tilde{x} & = & x^\prime  \\[6pt]  \texttt{(x, y)} & = & \nu_2 (x, y) & = & \tilde{x}y \lor x\tilde{y} & = & x^\prime y \lor x y^\prime  \\[6pt] \texttt{(x, y, z)} & = & \nu_3 (x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x^\prime y z \lor x y^\prime z \lor x y z^\prime  \end{matrix}


This entry was posted in Amphecks, 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, Zeroth Order Logic and tagged , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

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