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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

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

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

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

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/

Re: Laws of Form

https://groups.io/g/lawsofform/topic/genus_species_pie_charts/86943252

::: William Bricken ( https://groups.io/g/lawsofform/message/1191 )

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

https://inquiryintoinquiry.com/2021/11/19/genus-species-pie-charts-radio-buttons-discussion-2/

Re: Laws of Form

https://groups.io/g/lawsofform/topic/genus_species_pie_charts/86943252

::: William Bricken ( https://groups.io/g/lawsofform/message/1191 )

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

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

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

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

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

Regards,

Jon

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

https://inquiryintoinquiry.com/2021/11/25/genus-species-pie-charts-radio-buttons-discussion-5/

Re: 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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