Genus, Species, Pie Charts, Radio Buttons

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Jon Awbrey

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Nov 10, 2021, 3:00:23 PMNov 10
to Laws of Form, Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Genus, Species, Pie Charts, Radio Buttons • 1
https://inquiryintoinquiry.com/2021/11/10/genus-species-pie-charts-radio-buttons-1/

Re: Minimal Negation Operators
https://inquiryintoinquiry.com/2017/08/27/minimal-negation-operators-1/
https://inquiryintoinquiry.com/2017/08/30/minimal-negation-operators-2/
https://inquiryintoinquiry.com/2017/08/30/minimal-negation-operators-3/
https://inquiryintoinquiry.com/2017/09/01/minimal-negation-operators-4/
Re: Laws of Form ( https://groups.io/g/lawsofform/topic/checkboxes/86874727 )
::: Bruce Schuman ( https://groups.io/g/lawsofform/message/1153 )

<QUOTE BS:>
Leon Conrad's presentation talks about “marked” and “unmarked” states.

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” […]
</QUOTE>

Dear Bruce,

I posted an expanded and better-formatted version of my last message on my blog.

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, suggesting that “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
both efficient and elegant representations in the cactus graph extension of
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.

tabula rasa = true

( ) = false

(x) = not x

x y = x and y

(x (y)) = x ⇒ y

((x)(y)) = x or y

and so on.

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

Minimal Negation Operators
https://oeis.org/wiki/Minimal_negation_operator

The cactus expression (x, y, z) evaluates to true
if and only if exactly one of the variables x, y, z is false.

So the cactus expression ((x),(y),(z)) 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 ((x),(y),(z)) partitions the universe of discourse
into three mutually exclusive and exhaustive regions.

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

Logical Graphs • Appendices
https://oeis.org/wiki/Logical_Graphs#Appendices

Figure 1 shows the cactus graph for ((a),(b),(c)).

Figure 1. Cactus ((a),(b),(c))
https://inquiryintoinquiry.files.wordpress.com/2018/03/cactus-e28691e28691ae28693-e288a5-e28691be28693-e288a5-e28691ce28693e28693-big.jpg

Now consider the expression (x, (a),(b),(c)).

Figure 2 shows the cactus graph for (x, (a),(b),(c)).

Figure 2. Cactus (x, (a),(b),(c))
https://inquiryintoinquiry.files.wordpress.com/2021/11/cactus-e28691x-e288a5-e28691ae28693-e288a5-e28691be28693-e288a5-e28691ce28693e28693-big.jpg

If x is true, i.e. blank, the expression reduces to ((a),(b),(c)),
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 (a) and (b) and (c) 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
Cactus ↑↑A↓ ∥ ↑B↓ ∥ ↑C↓↓ Big.jpg
Cactus ↑X ∥ ↑A↓ ∥ ↑B↓ ∥ ↑C↓↓ Big.jpg

Jon Awbrey

unread,
Nov 18, 2021, 10:20:11 AM (13 days ago) Nov 18
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Genus, Species, Pie Charts, Radio Buttons • Discussion 1
https://inquiryintoinquiry.com/2021/11/18/genus-species-pie-charts-radio-buttons-discussion-1/

Re: Genus, Species, Pie Charts, Radio Buttons • 1
https://inquiryintoinquiry.com/2021/11/10/genus-species-pie-charts-radio-buttons-1/
Re: Laws of Form
https://groups.io/g/lawsofform/topic/genus_species_pie_charts/86943252
::: William Bricken ( https://groups.io/g/lawsofform/message/1191 )

<QUOTE 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)
( https://groups.io/g/lawsofform/attachment/1191/0/symmetry-and-figures.020404.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)
( https://groups.io/g/lawsofform/attachment/1191/1/syntactic-variety.060322.pdf )
</QUOTE>

Dear William,

Thanks for the readings.

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.

• Survey of Animated Logical Graphs
( https://inquiryintoinquiry.com/2021/05/01/survey-of-animated-logical-graphs-4/ )

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.

• Survey of Differential Logic
( https://inquiryintoinquiry.com/2021/05/15/survey-of-differential-logic-3/ )

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

• Survey of Inquiry Driven Systems
( https://inquiryintoinquiry.com/2020/12/27/survey-of-inquiry-driven-systems-3/ )

Regards,

Jon

Jon Awbrey

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Nov 19, 2021, 12:00:14 PM (12 days ago) Nov 19
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Genus, Species, Pie Charts, Radio Buttons • Discussion 2
https://inquiryintoinquiry.com/2021/11/19/genus-species-pie-charts-radio-buttons-discussion-2/
All,

For the sake of comparison, a couple of Tables I drew up may be useful
at this point and also for future reference. They present two arrangements
of the 16 boolean functions on 2 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
===================================================
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Appendices#Appendix_1._Propositional_Forms_and_Differential_Expansions


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

Differential Logic • 8
======================
https://inquiryintoinquiry.com/2020/04/08/differential-logic-8/

Table A1. Propositional Forms on Two Variables • Index Order
https://inquiryintoinquiry.files.wordpress.com/2020/04/table-a1.-propositional-forms-on-two-variables.png

Differential Logic • 9
======================
https://inquiryintoinquiry.com/2020/04/11/differential-logic-9/

Table A2. Propositional Forms on Two Variables • Orbit Order
https://inquiryintoinquiry.files.wordpress.com/2020/04/table-a2.-propositional-forms-on-two-variables-1.png

Regards,

Jon
Table A1. Propositional Forms on Two Variables.png
Table A2. Propositional Forms on Two Variables.png

Jon Awbrey

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Nov 21, 2021, 3:06:35 PM (10 days ago) Nov 21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Genus, Species, Pie Charts, Radio Buttons • Discussion 3
https://inquiryintoinquiry.com/2021/11/21/genus-species-pie-charts-radio-buttons-discussion-3/

All,

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.

Table 1. Boolean Functions and Logical Graphs on Two Variables • Index Order
https://inquiryintoinquiry.files.wordpress.com/2021/10/boolean-functions-and-logical-graphs-on-two-variables.png

Table 2. Boolean Functions and Logical Graphs on Two Variables • Orbit Order
https://inquiryintoinquiry.files.wordpress.com/2021/10/boolean-functions-and-logical-graphs-on-two-variables-e280a2-orbit-order.png

Regards,

Jon
Boolean Functions and Logical Graphs on Two Variables.png
Boolean Functions and Logical Graphs on Two Variables • Orbit Order.png

Jon Awbrey

unread,
Nov 23, 2021, 11:55:18 AM (8 days ago) Nov 23
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG, Conceptual Graphs
Cf: Genus, Species, Pie Charts, Radio Buttons • Discussion 4
https://inquiryintoinquiry.com/2021/11/23/genus-species-pie-charts-radio-buttons-discussion-4/

Re: Genus, Species, Pie Charts, Radio Buttons • 1
https://inquiryintoinquiry.com/2021/11/10/genus-species-pie-charts-radio-buttons-1/
::: John Mingers ( https://groups.io/g/lawsofform/message/1239 )

<QUOTE 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.
</QUOTE>

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 Google's
now-defunct Knol. People can find a version of that on the following page of my blog.

Logical Graphs • Introduction
https://inquiryintoinquiry.com/2008/07/29/logical-graphs-1/

Resources
=========

Minimal Negations Operators
https://oeis.org/wiki/Minimal_negation_operator

Survey of Animated Logical Graphs
https://inquiryintoinquiry.com/2021/05/01/survey-of-animated-logical-graphs-4/

Regards,

Jon

Jon Awbrey

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Nov 25, 2021, 12:18:16 PM (6 days ago) Nov 25
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG, Conceptual Graphs
Cf: Genus, Species, Pie Charts, Radio Buttons • Discussion 5
https://inquiryintoinquiry.com/2021/11/25/genus-species-pie-charts-radio-buttons-discussion-5/

Re: Genus, Species, Pie Charts, Radio Buttons • 1
( https://inquiryintoinquiry.com/2021/11/10/genus-species-pie-charts-radio-buttons-1/ )
Re: Laws of Form
( https://groups.io/g/lawsofform/topic/genus_species_pie_charts/86943252 )
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 your “what's 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)
( https://inquiryintoinquiry.files.wordpress.com/2021/11/theme-one-guide-e280a2-mollys-world-2.0.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
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