## Genus, Species, Pie Charts, Radio Buttons • 1

Re: Minimal Negation Operators • (1)(2)(3)(4)
Re: Laws of FormBruce Schuman

BS:

He uses checkboxes to illustrate this choice, which seem to be “either/or” (and not, for example, “both”).

Just strictly in terms of programming and web forms, if Leon does mean “either/or” — maybe he should use “radio buttons” and not “checkboxes” […]

Dear Bruce,

What programmers call radio button logic is related to what physicists call exclusion principles, both of which fall under a theme from the first-linked post above.  As I wrote there, “taking minimal negations as primitive operators enables efficient expressions for many natural constructs and affords a bridge between boolean domains of two values and domains with finite numbers of values, for example, finite sets of individuals”.

To illustrate, let’s look at how the forms mentioned in the subject line have efficient and elegant representations in the cactus graph extension of C.S. Peirce’s logical graphs and Spencer Brown’s calculus of indications.

Keeping to the existential interpretation for now, we have the following readings of our formal expressions.

$\begin{matrix} \textit{tabula rasa} & = & \mathrm{true} \\ \texttt{( )} & = & \mathrm{false} \\ \texttt{(} x \texttt{)} & = & \lnot x \\ x y & = & x \land y \\ \texttt{(} x \texttt{(} y \texttt{))} & = & x \Rightarrow y \\ \texttt{((} x \texttt{)(} y \texttt{))} & = & x \lor y \\ \textit{and so on} & \ldots & \ldots \end{matrix}$

Take a look at the following article on minimal negation operators.

The cactus expression $\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}$ evaluates to true if and only if exactly one of the variables $x, y, z$ is false.  So the cactus expression $\texttt{((} x \texttt{),(} y \texttt{),(} z \texttt{))}$ says exactly one of the variables $x, y, z$ is true.  Push one variable “on” and the other two go “off”, just like radio buttons.  Drawn as a venn diagram, the proposition $\texttt{((} x \texttt{),(} y \texttt{),(} z \texttt{))}$ partitions the universe of discourse into three mutually exclusive and exhaustive regions.

Refer now to Table 1 at the end of the following article.

Figure 1 shows the cactus graph for $\texttt{((} a \texttt{),(} b \texttt{),(} c \texttt{))}.$

Now consider the expression $\texttt{(} x \texttt{,(} a \texttt{),(} b \texttt{),(} c \texttt{))}.$

Figure 2 shows the cactus graph for $\texttt{(} x \texttt{,(} a \texttt{),(} b \texttt{),(} c \texttt{))}.$

If $x$ is true, i.e. blank, the expression $\texttt{(} x \texttt{,(} a \texttt{),(} b \texttt{),(} c \texttt{))}$ reduces to $\texttt{((} a \texttt{),(} b \texttt{),(} c \texttt{))},$ so we have a partition of the region where $x$ is true into three mutually exclusive and exhaustive regions where $a, b, c,$ respectively, are true.

If $x$ is false, it is the unique false variable, meaning $\texttt{(} a \texttt{)}$ and $\texttt{(} b \texttt{)}$ and $\texttt{(} c \texttt{)}$ are all true, so none of $a, b, c$ are true.

We can picture this as a pie chart where a pie $x$ is divided into exactly three slices $a, b, c.$

It is the same thing as having a genus $x$ with exactly three species $a, b, c.$

Regards,

Jon

### 6 Responses to Genus, Species, Pie Charts, Radio Buttons • 1

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