## Genus, Species, Pie Charts, Radio Buttons • Discussion 5

Dear John,

Once we grasp the utility of minimal negation operators for partitioning a universe of discourse into several regions and any region into further parts, there are quite a few directions we might explore as far as our next steps go.

One thing I always did when I reached a new level of understanding about any logical issue was to see if I could actualize the insight in whatever programming projects I was working on at the time.  Conversely and recursively the trials of doing that would often force me to modify my initial understanding in the direction of what works in brass tacks practice.

The use of cactus graphs to implement minimal negation operators made its way into the Theme One Program I worked on all through the 1980s and the applications I made of it went into the work I did for a master’s in psych.  At any rate, I can finally answer the “what next” question by pointing to one of the exercises I set for the logical reasoning module of that program, as described in the following excerpt from its User Guide.

• Theme One Guide • Molly’s World (pdf)

The writing there is a little rough by my current standards, so I’ll work on revising it over the next few days.

Regards,

Jon

## Genus, Species, Pie Charts, Radio Buttons • Discussion 4

JM:
I feel as though you have posted these same diagrams many times, and it is always portrayed as clearing the ground for something else.  But the something else never arrives!  I would be really interested to know what the next step is in your ideas.

Dear John,

Thanks for the question.  Bruce Schuman mentioned radio button logic and I jumped on it “like a duck on a June bug” — as they say in several southern States I know — because that very thing marks an important first step in the application of minimal negation operators to represent finite domains of values, contextual individuals, genus and species, partitions, and so on.  But some of the comments I got next gave me pause and made me feel I should go back and clarify a few points.

I wasn’t sure, but I got the sense Bruce was reading the cactus graphs I posted as an order of hierarchical, ontological, or taxonomic diagrams.  What they really amount to are the abstract, human-viewable renditions of linked data structures or “pointer” data structures in computer memory.  I explained the transformation from planar forms of enclosure to their topological dual trees to the pointer structures in one of the articles on logical graphs I wrote for Wikipedia and later for Google’s now-defunct Knol.  People can find a version of that on the following page of my blog.

## Genus, Species, Pie Charts, Radio Buttons • Discussion 3

Last time I alluded to the general problem of relating a variety of formal languages to a shared domain of formal objects, taking six notations for the boolean functions on two variables as a simple but critical illustration of the larger task.  This time we’ll take up a subtler example of cross-calculus communication, where the same syntactic forms bear different logical interpretations.

In each of the Tables below —

• Column 1 shows a conventional name $f_{i}$ and a venn diagram for each of the sixteen boolean functions on two variables.
• Column 2 shows the logical graph canonically representing the boolean function in Column 1 under the entitative interpretation.  This is the interpretation C.S. Peirce used in his earlier work on entitative graphs and the one Spencer Brown used in his book Laws of Form.
• Column 3 shows the logical graph canonically representing the boolean function in Column 1 under the existential interpretation.  This is the interpretation C.S. Peirce used in his later work on existential graphs.

$\text{Boolean Functions and Logical Graphs on Two Variables} \stackrel{_\bullet}{} \text{Index Order}$

$\text{Boolean Functions and Logical Graphs on Two Variables} \stackrel{_\bullet}{} \text{Orbit Order}$

### Resources

• Logical Graphs, Iconicity, Interpretation • (1)(2)
• Minimal Negation Operators • (1)(2)(3)(4)

## Genus, Species, Pie Charts, Radio Buttons • Discussion 2

A problem we often encounter is the need to relate a variety of formal languages to the same domain of formal objects.  In our present engagement we are using languages not only to describe but further to compute with the objects in question and so we call our languages so many diverse calculi.

When it comes to propositional calculi, a couple of Tables may be useful at this point and also for future reference.  They present two arrangements of the sixteen boolean functions on two variables, collating their truth tables with their expressions in several systems of notation, including the parenthetical versions of cactus expressions, here read under the existential interpretation.  They appear as the first two Tables on the following page.

### Differential Logic and Dynamic Systems • Appendices

The copies I posted to my blog will probably load faster.

### Differential Logic • 8

Table A1.  Propositional Forms on Two Variables • Index Order

### Differential Logic • 9

Table A2.  Propositional Forms on Two Variables • Orbit Order

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

WB:
Here’s an analysis of “Boolean” structure.  It’s actually a classification of the structure of distinctions containing 2 and 3 variables.  The work was originally done within the context of optimization of combinational silicon circuits, so I used “boolean” for that community, but we all know that “boolean” is just one interpretation of Laws of Form distinction structure.

• Bricken, W. (1997/2002), “Symmetry in Boolean Functions
with Examples for Two and Three Variables” (pdf).

And here’s some different visualizations of distinction structures in general.  Section 4 is relevant to us, the rest is just too many words for an academic community.

• Bricken, W. (n.d.), “Syntactic Variety in Boundary Logic” (pdf).

Dear William,

Here’s a few resources on the angle I’ve been taking, greatly impacted from the beginning by reading Peirce and Spencer Brown in parallel and by implementing their forms as list and pointer data structures, first in Lisp and later in Pascal.

One thing my computational work taught me early on is that planar representations are an efficiency death trap on numerous grounds.  For one thing we don’t want to be computing on bitmap images and for another the representations of logical equality and exclusive disjunction, whether they require two occurrences of each variable or whether they introduce a new symbol like “=” requiring separate handling, lead to combinatorially explosive branching.  A decade of wrangling with that and other issues eventually led me to generalize trees to cacti, and this had the serendipitous benefit of leading to differential logic.

Not too coincidentally, differential logic is one of the very tools I needed to analyze and model Inquiry Driven Systems.

## Functional Logic • Inquiry and Analogy • Preliminaries

This report discusses C.S. Peirce’s treatment of analogy, placing it in relation to his overall theory of inquiry.  We begin by introducing three basic types of reasoning Peirce adopted from classical logic.  In Peirce’s analysis both inquiry and analogy are complex programs of logical inference which develop through stages of these three types, though normally in different orders.

Note on notation.  The discussion to follow uses logical conjunctions, expressed in the form of concatenated tuples $e_1 ~\ldots~ e_k,$ and minimal negation operations, expressed in the form of bracketed tuples $\texttt{(} e_1 \texttt{,} \ldots \texttt{,} e_k \texttt{)},$ as the principal expression-forming operations of a calculus for boolean-valued functions, that is, for propositions.  The expressions of this calculus parse into data structures whose underlying graphs are called cacti by graph theorists.  Hence the name cactus language for this dialect of propositional calculus.

## 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

## Survey of Relation Theory • 5

In this Survey of blog and wiki posts on Relation Theory, relations are viewed from the perspective of combinatorics, in other words, as a topic in discrete mathematics, with special attention to finite structures and concrete set-theoretic constructions, many of which arise quite naturally in applications.  This approach to relation theory is distinct from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other.

### Triadic Relations • Examples from Semiotics

The study of signs — the full variety of significant forms of expression — in relation to all the affairs signs are significant of, and in relation to all the beings signs are significant to, is known as semiotics or the theory of signs.  As described, semiotics treats of a $3$-place relation among signs, their objects, and their interpreters.

The term semiosis refers to any activity or process involving signs.  Studies of semiosis focusing on its abstract form are not concerned with every concrete detail of the entities acting as signs, as objects, or as agents of semiosis, but only with the most salient patterns of relationship among those three roles.  In particular, the formal theory of signs does not consider all the properties of the interpretive agent but only the more striking features of the impressions signs make on a representative interpreter.  From a formal point of view this impact or influence may be treated as just another sign, called the interpretant sign, or the interpretant for short.  A triadic relation of this type, among objects, signs, and interpretants, is called a sign relation.

For example, consider the aspects of sign use involved when two people, say Ann and Bob, use their own proper names, “Ann” and “Bob”, along with the pronouns, “I” and “you”, to refer to themselves and each other.  For brevity, these four signs may be abbreviated to the set $\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.$  The abstract consideration of how $\mathrm{A}$ and $\mathrm{B}$ use this set of signs leads to the contemplation of a pair of triadic relations, the sign relations $L_\mathrm{A}$ and $L_\mathrm{B},$ reflecting the differential use of these signs by $\mathrm{A}$ and $\mathrm{B},$ respectively.

Each of the sign relations $L_\mathrm{A}$ and $L_\mathrm{B}$ consists of eight triples of the form $(x, y, z),$ where the object $x$ belongs to the object domain $O = \{ \mathrm{A}, \mathrm{B} \},$ the sign $y$ belongs to the sign domain $S,$ the interpretant sign $z$ belongs to the interpretant domain $I,$ and where it happens in this case that $S = I = \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.$  The union $S \cup I$ is often referred to as the syntactic domain, but in this case $S = I = S \cup I.$

The set-up so far is summarized as follows:

$\begin{array}{ccc} L_\mathrm{A}, L_\mathrm{B} & \subseteq & O \times S \times I \\[8pt] O & = & \{ \mathrm{A}, \mathrm{B} \} \\[8pt] S & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \} \\[8pt] I & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \} \end{array}$

The relation $L_\mathrm{A}$ is the following set of eight triples in $O \times S \times I.$

$\begin{array}{cccccc} \{ & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), & \\ & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}) & \}. \end{array}$

The triples in $L_\mathrm{A}$ represent the way interpreter $\mathrm{A}$ uses signs.  For example, the presence of $( \mathrm{B}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} )$ in $L_\mathrm{A}$ says $\mathrm{A}$ uses ${}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}$ to mean the same thing $\mathrm{A}$ uses ${}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}$ to mean, namely, $\mathrm{B}.$

The relation $L_\mathrm{B}$ is the following set of eight triples in $O \times S \times I.$

$\begin{array}{cccccc} \{ & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), & \\ & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}) & \}. \end{array}$

The triples in $L_\mathrm{B}$ represent the way interpreter $\mathrm{B}$ uses signs.  For example, the presence of $( \mathrm{B}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} )$ in $L_\mathrm{B}$ says $\mathrm{B}$ uses ${}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}$ to mean the same thing $\mathrm{B}$ uses ${}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}$ to mean, namely, $\mathrm{B}.$

The triples in the relations $L_\mathrm{A}$ and $L_\mathrm{B}$ are conveniently arranged in the form of relational data tables, as shown below.

### Document History

Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.

cc: Category Theory • Cybernetics (1) (2) • Ontolog (1) (2)
cc: Structural Modeling (1) (2) • Systems Science (1) (2)
cc: FB | Relation TheoryLaws of Form • Peirce List (1) (2)

### Triadic Relations • Examples from Mathematics

For the sake of topics to be taken up later, it is useful to examine a pair of triadic relations in tandem.  We will construct two triadic relations, $L_0$ and $L_1,$ each of which is a subset of the same cartesian product $X \times Y \times Z.$  The structures of $L_0$ and $L_1$ can be described in the following way.

Each space $X, Y, Z$ is isomorphic to the boolean domain $\mathbb{B} = \{ 0, 1 \}$ so $L_0$ and $L_1$ are subsets of the cartesian power $\mathbb{B} \times \mathbb{B} \times \mathbb{B}$ or the boolean cube $\mathbb{B}^3.$

The operation of boolean addition, $+ : \mathbb{B} \times \mathbb{B} \to \mathbb{B},$ is equivalent to addition modulo 2, where $0$ acts in the usual manner but $1 + 1 = 0.$  In its logical interpretation, the plus sign can be used to indicate either the boolean operation of exclusive disjunction or the boolean relation of logical inequality.

The relation $L_0$ is defined by the following formula.

$L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}.$

The relation $L_0$ is the following set of four triples in $\mathbb{B}^3.$

$L_0 ~=~ \{ ~ (0, 0, 0), ~ (0, 1, 1), ~ (1, 0, 1), ~ (1, 1, 0) ~ \}.$

The relation $L_1$ is defined by the following formula.

$L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}.$

The relation $L_1$ is the following set of four triples in $\mathbb{B}^3.$

$L_1 ~=~ \{ ~ (0, 0, 1), ~ (0, 1, 0), ~ (1, 0, 0), ~ (1, 1, 1) ~ \}.$

The triples in the relations $L_0$ and $L_1$ are conveniently arranged in the form of relational data tables, as shown below.

### Document History

Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.

cc: Category Theory • Cybernetics (1) (2) • Ontolog (1) (2)
cc: Structural Modeling (1) (2) • Systems Science (1) (2)
cc: FB | Relation TheoryLaws of Form • Peirce List (1) (2)