Genus, Species, Pie Charts, Radio Buttons


 

Bruce & All,

All the forms mentioned in the subject line have
nice (both efficient and pretty) representations
in the cactus graph extension of Peirce's logical
graphs and Spencer Brown's calculus of indications.

Let's keep to the existential interpretation for now,
in other words, where we have the following readings:

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

Thus the expression ((x),(y),(z)), when it's true,
says exactly one of the variables x, y, z is true.
Push x “on” and the other two go “off”, just like
radio buttons.

Viewing the truth conditions as a venn diagram,
they partition the universe of discourse into
three mutually exclusive and exhaustive regions.

Now look at Table 1 at the end of the following article.

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

The cactus graph for ((a),(b),(c)) is linked below.

Figure 1. Cactus ((a),(b),(c))
https://oeis.org/w/images/4/48/Cactus_%28%28A%29%2C%28B%29%2C%28C%29%29_Big.jpg

But wait, there's more ...

Consider the expression (x, (a),(b),(c)).

Figure 2. Cactus (x, (a),(b),(c))
https://oeis.org/w/images/5/54/Cactus_%28X%2C%28A%29%2C%28B%29%2C%28C%29%29_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 x-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 the 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


 

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


bruceschuman@...
 

Thanks, Jon.

 

It would be interesting to compile a comprehensive list of abstract categories that can be defined by a common graphical logic -- or show ways that all abstract classes like these can be defined in a common language.  And then to really push -- show that some selected common language or method (cactus graph, hierarchy?) is actually "best" (or "optimal, all things considered") for some reason.   I am interested in all those you mention, and no doubt many others.  Can they all be usefully fitted together under a common logic and method?

 

This would involve a task rather like the work taken on by practical working semantic ontologists -- the kind of people who subscribe to the ontolog mailing list -- because their work involves negotiating best common solutions for logic and database issues that extend across nations or cultures or sectors or industries or scientific disciplines.  We need one general-purpose solution, maybe with "patches" for various special cases that don't quite fit.

 

And if possible, we might also want to combine the supposedly "mystical" elements in LOF -- issues with the Tao, for example.  This might be something very powerful and meaningful in a globalized world.

 

I was just looking at this series of graphics on Wikipedia that compare Euler circles with Venn Diagrams.  Pretty interesting.

 

There's a whole series of related diagrams (16 of them) linked from this URL:

 

https://en.wikipedia.org/wiki/Euler_diagram#/media/File:Veitch_and_Karnaugh_truth_table_4.jpg

 

Bruce Schuman

Santa Barbara CA USA

bruceschuman@... / 805-705-9174

www.origin.org / www.integralontology.net / www.newcongress.net

 

 

 

 

 

-----Original Message-----
From: lawsofform@groups.io <lawsofform@groups.io> On Behalf Of Jon Awbrey
Sent: Tuesday, November 9, 2021 1:56 PM
To: Laws of Form <lawsofform@groups.io>
Subject: [lawsofform] Genus, Species, Pie Charts, Radio Buttons

 

Bruce & All,

 

All the forms mentioned in the subject line have nice (both efficient and pretty) representations in the cactus graph extension of Peirce's logical graphs and Spencer Brown's calculus of indications.

 

Let's keep to the existential interpretation for now, in other words, where we have the following readings:

 

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

 

Thus the expression ((x),(y),(z)), when it's true, says exactly one of the variables x, y, z is true.

Push x “on” and the other two go “off”, just like radio buttons.

 

Viewing the truth conditions as a venn diagram, they partition the universe of discourse into three mutually exclusive and exhaustive regions.

 

Now look at Table 1 at the end of the following article.

 

Logical Graphs • Appendices

https://oeis.org/wiki/Logical_Graphs#Appendices

 

The cactus graph for ((a),(b),(c)) is linked below.

 

Figure 1.  Cactus ((a),(b),(c))

https://oeis.org/w/images/4/48/Cactus_%28%28A%29%2C%28B%29%2C%28C%29%29_Big.jpg

 

But wait, there's more ...

 

Consider the expression (x, (a),(b),(c)).

 

Figure 2.  Cactus (x, (a),(b),(c))

https://oeis.org/w/images/5/54/Cactus_%28X%2C%28A%29%2C%28B%29%2C%28C%29%29_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 x-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 the 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

 

 

 


william bricken
 

Hi Folks,

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 LoF distinction structure.


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.


take care
wm

On Nov 11, 2021, at 12:16 PM, bruceschuman@... wrote:

Thanks, Jon.
 
It would be interesting to compile a comprehensive list of abstract categories that can be defined by a common graphical logic -- or show ways that all abstract classes like these can be defined in a common language.  And then to really push -- show that some selected common language or method (cactus graph, hierarchy?) is actually "best" (or "optimal, all things considered") for some reason.   I am interested in all those you mention, and no doubt many others.  Can they all be usefully fitted together under a common logic and method?
 
This would involve a task rather like the work taken on by practical working semantic ontologists -- the kind of people who subscribe to the ontolog mailing list -- because their work involves negotiating best common solutions for logic and database issues that extend across nations or cultures or sectors or industries or scientific disciplines.  We need one general-purpose solution, maybe with "patches" for various special cases that don't quite fit.
 
And if possible, we might also want to combine the supposedly "mystical" elements in LOF -- issues with the Tao, for example.  This might be something very powerful and meaningful in a globalized world.
 
I was just looking at this series of graphics on Wikipedia that compare Euler circles with Venn Diagrams.  Pretty interesting.
 
There's a whole series of related diagrams (16 of them) linked from this URL:
 
 
Bruce Schuman
Santa Barbara CA USA
 
 
 
 
 
-----Original Message-----
From: lawsofform@groups.io <lawsofform@groups.io> On Behalf Of Jon Awbrey
Sent: Tuesday, November 9, 2021 1:56 PM
To: Laws of Form <lawsofform@groups.io>
Subject: [lawsofform] Genus, Species, Pie Charts, Radio Buttons
 
Bruce & All,
 
All the forms mentioned in the subject line have nice (both efficient and pretty) representations in the cactus graph extension of Peirce's logical graphs and Spencer Brown's calculus of indications.
 
Let's keep to the existential interpretation for now, in other words, where we have the following readings:
 
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
 
The expression (x, y, z) evaluates to true if and only if exactly one of the variables x, y, z is false.
 
Thus the expression ((x),(y),(z)), when it's true, says exactly one of the variables x, y, z is true.
Push x “on” and the other two go “off”, just like radio buttons.
 
Viewing the truth conditions as a venn diagram, they partition the universe of discourse into three mutually exclusive and exhaustive regions.
 
Now look at Table 1 at the end of the following article.
 
Logical Graphs • Appendices
 
The cactus graph for ((a),(b),(c)) is linked below.
 
Figure 1.  Cactus ((a),(b),(c))
 
But wait, there's more ...
 
Consider the expression (x, (a),(b),(c)).
 
Figure 2.  Cactus (x, (a),(b),(c))
 
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 x-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 the 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
 
 
 


 

Thanks, 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.

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

On 11/15/2021 4:58 AM, william bricken wrote:
Hi Folks,
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 LoF distinction structure.
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.
take care
wm

On Nov 11, 2021, at 12:16 PM, bruceschuman@cox.net wrote:

Thanks, Jon.
It would be interesting to compile a comprehensive list of abstract categories that can be defined by a common graphical logic -- or show ways that all abstract classes like these can be defined in a common language. And then to really push -- show that some selected common language or method (cactus graph, hierarchy?) is actually "best" (or "optimal, all things considered") for some reason. I am interested in all those you mention, and no doubt many others. Can they all be usefully fitted together under a common logic and method?
This would involve a task rather like the work taken on by practical working semantic ontologists -- the kind of people who subscribe to the ontolog mailing list -- because their work involves negotiating best common solutions for logic and database issues that extend across nations or cultures or sectors or industries or scientific disciplines. We need one general-purpose solution, maybe with "patches" for various special cases that don't quite fit.
And if possible, we might also want to combine the supposedly "mystical" elements in LOF -- issues with the Tao, for example. This might be something very powerful and meaningful in a globalized world.
I was just looking at this series of graphics on Wikipedia that compare Euler circles with Venn Diagrams. Pretty interesting.
There's a whole series of related diagrams (16 of them) linked from this URL:
https://en.wikipedia.org/wiki/Euler_diagram#/media/File:Veitch_and_Karnaugh_truth_table_4.jpg <https://en.wikipedia.org/wiki/Euler_diagram#/media/File:Veitch_and_Karnaugh_truth_table_4.jpg>
Bruce Schuman
Santa Barbara CA USA
bruceschuman@cox.net <mailto:bruceschuman@cox.net> / 805-705-9174 <tel:805-705-9174>
www.origin.org <http://www.origin.org/> / www.integralontology.net <http://www.integralontology.net/> / www.newcongress.net <http://www.newcongress.net/>
-----Original Message-----
From: lawsofform@groups.io <mailto:lawsofform@groups.io> <lawsofform@groups.io <mailto:lawsofform@groups.io>> On Behalf Of Jon Awbrey
Sent: Tuesday, November 9, 2021 1:56 PM
To: Laws of Form <lawsofform@groups.io <mailto:lawsofform@groups.io>>
Subject: [lawsofform] Genus, Species, Pie Charts, Radio Buttons
Bruce & All,
All the forms mentioned in the subject line have nice (both efficient and pretty) representations in the cactus graph extension of Peirce's logical graphs and Spencer Brown's calculus of indications.
Let's keep to the existential interpretation for now, in other words, where we have the following readings:
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 <https://oeis.org/wiki/Minimal_negation_operator>
The expression (x, y, z) evaluates to true if and only if exactly one of the variables x, y, z is false.
Thus the expression ((x),(y),(z)), when it's true, says exactly one of the variables x, y, z is true.
Push x “on” and the other two go “off”, just like radio buttons.
Viewing the truth conditions as a venn diagram, they partition the universe of discourse into three mutually exclusive and exhaustive regions.
Now look at Table 1 at the end of the following article.
Logical Graphs • Appendices
https://oeis.org/wiki/Logical_Graphs#Appendices <https://oeis.org/wiki/Logical_Graphs#Appendices>
The cactus graph for ((a),(b),(c)) is linked below.
Figure 1. Cactus ((a),(b),(c))
https://oeis.org/w/images/4/48/Cactus_%28%28A%29%2C%28B%29%2C%28C%29%29_Big.jpg <https://oeis.org/w/images/4/48/Cactus_%28%28A%29%2C%28B%29%2C%28C%29%29_Big.jpg>
But wait, there's more ...
Consider the expression (x, (a),(b),(c)).
Figure 2. Cactus (x, (a),(b),(c))
https://oeis.org/w/images/5/54/Cactus_%28X%2C%28A%29%2C%28B%29%2C%28C%29%29_Big.jpg <https://oeis.org/w/images/5/54/Cactus_%28X%2C%28A%29%2C%28B%29%2C%28C%29%29_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 x-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 the 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


 

William, all ...

For the sake of comparison, as far as the 16 functions on 2 variables go,
here's a pair of tables I posted last month comparing the entitative and
existential interpretations of cactus graphs.

Logical Graphs, Iconicity, Interpretation
https://groups.io/g/lawsofform/topic/86051464

Cf: Logical Graphs, Iconicity, Interpretation • 1
https://inquiryintoinquiry.com/2021/10/03/logical-graphs-iconicity-interpretation-1/

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

Cf: Logical Graphs, Iconicity, Interpretation • 2
https://inquiryintoinquiry.com/2021/10/04/logical-graphs-iconicity-interpretation-2/

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


 

William, all ...

A couple of Tables I drew up may be useful at this point and
also for future reference. They show 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, this time read
under the existential interpretation. They can be found as the
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#Appendices

The copies I posted to my blog and the group 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


 

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


 

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


 

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


johncm22
 

Hi John

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

John

On Sun, 21 Nov 2021 at 20:06, Jon Awbrey <jawbrey@...> wrote:
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






Mauro Bertani
 

Hi all, 
This is a work about minimal negation operator and pie chart operator. I believe this change the way to think the problem. 
https://docs.google.com/document/d/1gFzbMDW_zbqq452-VjWIlJ02dWtyxrNhWgpvMYBcM2I/edit?usp=drivesdk
Thanks in advance
Mauro


 

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/
Re: Laws of Form
https://groups.io/g/lawsofform/topic/genus_species_pie_charts/86943252
::: 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


 

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 )
::: John Mingers ( https://groups.io/g/lawsofform/message/1239 )

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