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May 23, 2019, 10:54:13 AM5/23/19

to Ontolog Forum, Structural Modeling, SysSciWG, Cybernetic Communications, Laws Of Form Group

Survey of Animated Logical Graphs

https://inquiryintoinquiry.com/2019/05/22/survey-of-animated-logical-graphs-%e2%80%a2-2/

The blog post linked above updates my Survey of Resources having to do with

Animated Logical Graphs. There you will find links to basic expositions and

extended discussions of the graphs themselves, deriving from the Alpha Graphs

C.S. Peirce used for propositional logic, more recently revived and augmented

by G. Spencer Brown in his Laws of Form. What I added was the extension from

tree-like forms to what graph theorists know as cacti, and thereby hangs many

a tale yet to be told. I hope to add more proof animations as time goes on.

Regards,

Jon

inquiry into inquiry: https://inquiryintoinquiry.com/

academia: https://independent.academia.edu/JonAwbrey

oeiswiki: https://www.oeis.org/wiki/User:Jon_Awbrey

isw: http://intersci.ss.uci.edu/wiki/index.php/JLA

facebook page: https://www.facebook.com/JonnyCache

https://inquiryintoinquiry.com/2019/05/22/survey-of-animated-logical-graphs-%e2%80%a2-2/

The blog post linked above updates my Survey of Resources having to do with

Animated Logical Graphs. There you will find links to basic expositions and

extended discussions of the graphs themselves, deriving from the Alpha Graphs

C.S. Peirce used for propositional logic, more recently revived and augmented

by G. Spencer Brown in his Laws of Form. What I added was the extension from

tree-like forms to what graph theorists know as cacti, and thereby hangs many

a tale yet to be told. I hope to add more proof animations as time goes on.

Regards,

Jon

inquiry into inquiry: https://inquiryintoinquiry.com/

academia: https://independent.academia.edu/JonAwbrey

oeiswiki: https://www.oeis.org/wiki/User:Jon_Awbrey

isw: http://intersci.ss.uci.edu/wiki/index.php/JLA

facebook page: https://www.facebook.com/JonnyCache

May 23, 2019, 1:20:20 PM5/23/19

to Sys Sci, Ontolog Forum, Structural Modeling, Cybernetic Communications, Laws Of Form Group

Jon:

Thanks for posting this material, it makes some aspects of Theme One program more clear.

Take care and have fun,

Joe

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Joe Simpson

# “Reasonable people adapt themselves to the world.

# Unreasonable people attempt to adapt the world to themselves.

# All progress, therefore, depends on unreasonable people.”

- George Bernard Shaw
- Git Hub link:
- Research Gate link:
- YouTube Channel
- Web Site:

May 28, 2019, 3:00:19 PM5/28/19

to ontolo...@googlegroups.com, joseph simpson, Sys Sci, Structural Modeling

Re: Survey of Animated Logical Graphs ??? 2

At: https://inquiryintoinquiry.com/2019/05/22/survey-of-animated-logical-graphs-%e2%80%a2-2/

Joe, All ???

One of the things I added to the Survey this time around was an

earlier piece of work titled ???Futures Of Logical Graphs??? (FOLG),

which takes up a number of difficult issues in more detail than

I've found the ability or audacity to do since. In particular,

it gives an indication of the steps I took from trees to cacti

in the graph-theoretic representation of logical propositions

and boolean functions, along with the factors that forced me

to make that transition.

??? https://oeis.org/wiki/Futures_Of_Logical_Graphs

A lot of the text goes back to the dusty Ascii days of the old discussion lists

where I last shared it, so I'll be working on converting the figures and tables

and trying to make the presentation more understandable.

Regards,

Jon

At: https://inquiryintoinquiry.com/2019/05/22/survey-of-animated-logical-graphs-%e2%80%a2-2/

Joe, All ???

One of the things I added to the Survey this time around was an

earlier piece of work titled ???Futures Of Logical Graphs??? (FOLG),

which takes up a number of difficult issues in more detail than

I've found the ability or audacity to do since. In particular,

it gives an indication of the steps I took from trees to cacti

in the graph-theoretic representation of logical propositions

and boolean functions, along with the factors that forced me

to make that transition.

??? https://oeis.org/wiki/Futures_Of_Logical_Graphs

A lot of the text goes back to the dusty Ascii days of the old discussion lists

where I last shared it, so I'll be working on converting the figures and tables

and trying to make the presentation more understandable.

Regards,

Jon

May 28, 2019, 11:05:56 PM5/28/19

to Sys Sci, Ontolog Forum @ GG, Structural Modeling

Jon:

Very interesting work.

Many of the patterns are now becoming more clear... just need more time to study

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Contributions to the discussion are licensed by authors under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

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Jul 1, 2019, 1:20:15 PM7/1/19

to Cybernetic Communications, Laws Of Form Group, SysSciWG, Structural Modeling, Ontolog Forum

Cf: Animated Logical Graphs : 15

At: https://inquiryintoinquiry.com/2019/06/30/animated-logical-graphs-%e2%80%a2-15/

In George Spencer Brown's "Laws of Form" the relation between the primary arithmetic

and the primary algebra is founded on the idea that a variable name appearing as an

operand in an algebraic expression indicates the contemplated absence or presence of

any expression in the arithmetic, with the understanding that each appearance of the

same variable name indicates the same state of contemplation with respect to the same

expression of the arithmetic.

For example, consider the following expression:

Figure 1. Cactus Graph (a(a))

https://inquiryintoinquiry.files.wordpress.com/2019/06/box-aa.jpg

We may regard this algebraic expression as a general expression

for an infinite set of arithmetic expressions, starting like so:

Figure 2. Cactus Graph Series (a(a))

https://inquiryintoinquiry.files.wordpress.com/2019/06/box-aa-series.jpg

Now consider what this says about the following algebraic law:

Figure 3. Cactus Graph Equation (a(a)) = <blank>

https://inquiryintoinquiry.files.wordpress.com/2019/06/box-aa-.jpg

It permits us to understand the algebraic law as saying, in effect, that every one

of the arithmetic expressions of the contemplated pattern evaluates to the very same

canonical expression as the upshot of that evaluation. This is, as far as I know,

just about as close as we can come to a conceptually and ontologically minimal way

of understanding the relation between an algebra and its corresponding arithmetic.

To be continued ...

Regards,

Jon

On 5/30/2019 9:16 AM, Jon Awbrey wrote:

> Re: Survey of Animated Logical Graphs : 2

> At: https://inquiryintoinquiry.com/2019/05/22/survey-of-animated-logical-graphs-%e2%80%a2-2/

At: https://inquiryintoinquiry.com/2019/06/30/animated-logical-graphs-%e2%80%a2-15/

In George Spencer Brown's "Laws of Form" the relation between the primary arithmetic

and the primary algebra is founded on the idea that a variable name appearing as an

operand in an algebraic expression indicates the contemplated absence or presence of

any expression in the arithmetic, with the understanding that each appearance of the

same variable name indicates the same state of contemplation with respect to the same

expression of the arithmetic.

For example, consider the following expression:

Figure 1. Cactus Graph (a(a))

https://inquiryintoinquiry.files.wordpress.com/2019/06/box-aa.jpg

We may regard this algebraic expression as a general expression

for an infinite set of arithmetic expressions, starting like so:

Figure 2. Cactus Graph Series (a(a))

https://inquiryintoinquiry.files.wordpress.com/2019/06/box-aa-series.jpg

Now consider what this says about the following algebraic law:

Figure 3. Cactus Graph Equation (a(a)) = <blank>

https://inquiryintoinquiry.files.wordpress.com/2019/06/box-aa-.jpg

It permits us to understand the algebraic law as saying, in effect, that every one

of the arithmetic expressions of the contemplated pattern evaluates to the very same

canonical expression as the upshot of that evaluation. This is, as far as I know,

just about as close as we can come to a conceptually and ontologically minimal way

of understanding the relation between an algebra and its corresponding arithmetic.

To be continued ...

Regards,

Jon

On 5/30/2019 9:16 AM, Jon Awbrey wrote:

> Re: Survey of Animated Logical Graphs : 2

> At: https://inquiryintoinquiry.com/2019/05/22/survey-of-animated-logical-graphs-%e2%80%a2-2/

>

> One of the things I added to the Survey this time around was an

> earlier piece of work titled "Futures Of Logical Graphs" (FOLG),
> One of the things I added to the Survey this time around was an

> which takes up a number of difficult issues in more detail than

> I've found the ability or audacity to do since.?? In particular,
> it gives an indication of the steps I took from trees to cacti

> in the graph-theoretic representation of logical propositions

> and boolean functions, along with the factors that forced me

> to make that transition.

>

> See: https://oeis.org/wiki/Futures_Of_Logical_Graphs
> in the graph-theoretic representation of logical propositions

> and boolean functions, along with the factors that forced me

> to make that transition.

>

>

> A lot of the text goes back to the dusty Ascii days of the

> old discussion lists where I last shared it, so I will be
> A lot of the text goes back to the dusty Ascii days of the

> working on converting the figures and tables and trying

> to make the presentation more understandable.

>

> Regards,

>

> Jon

--
> to make the presentation more understandable.

>

> Regards,

>

> Jon

Jul 8, 2019, 2:56:20 PM7/8/19

to Ontolog Forum, Structural Modeling, SysSciWG

Cf: Animated Logical Graphs ??? 16

At: https://inquiryintoinquiry.com/2019/07/07/animated-logical-graphs-%e2%80%a2-16/

In lieu of a field study requirement for my bachelor's degree I spent

a couple years in a host of state and university libraries reading

everything I could find by and about Peirce, poring most memorably

through the reels of microfilmed Peirce manuscripts Michigan State

had at the time, all in trying to track down some hint of a clue to

a puzzling passage in Peirce's "Simplest Mathematics", most acutely

coming to a head with that bizarre line of type at CP 4.306, which

the editors of the ''Collected Papers'', no doubt compromised by the

typographer's resistance to cutting new symbols, transmogrified into

a script more cryptic than even the manuscript???s original hieroglyphic.

I found one key to the mystery in Peirce's use of "operator variables",

which he and his students Christine Ladd-Franklin and O.H. Mitchell

explored in depth. I will shortly discuss this theme as it affects

logical graphs but it may be useful to give a shorter and sweeter

explanation of how the basic idea typically arises in common

logical practice.

Think of De Morgan???s rules:

: ??(A ??? B) = ??A ??? ??B

: ??(A ??? B) = ??A ??? ??B

We could capture the common form of these two rules in a single formula

by taking "o1" and "o2" as variable names ranging over a set of logical

operators, and then by asking what substitutions for o1 and o2 would

satisfy the following equation:

: ??(A o1 B) = ??A o2 ??B

We already know two solutions to this "operator equation", namely,

(o1, o2) = (???, ???) and (o1, o2) = (???, ???). Wouldn't it be just like

Peirce to ask if there are others?

Having broached the subject of logical operator variables,

I will leave it for now in the same way Peirce himself did:

<QUOTE>

I shall not further enlarge upon this matter at this point,

although the conception mentioned opens a wide field; because

it cannot be set in its proper light without overstepping the

limits of dichotomic mathematics. (Collected Papers, CP 4.306).

</QUOTE>

Further exploration of operator variables and operator invariants

treads on grounds traditionally known as "second intentional logic"

and "opens a wide field", as Peirce says. For now, however, I will

tend to that corner of the field where our garden variety logical

graphs grow, observing the ways operative variations and operative

themes naturally develop on those grounds.

At: https://inquiryintoinquiry.com/2019/07/07/animated-logical-graphs-%e2%80%a2-16/

In lieu of a field study requirement for my bachelor's degree I spent

a couple years in a host of state and university libraries reading

everything I could find by and about Peirce, poring most memorably

through the reels of microfilmed Peirce manuscripts Michigan State

had at the time, all in trying to track down some hint of a clue to

a puzzling passage in Peirce's "Simplest Mathematics", most acutely

coming to a head with that bizarre line of type at CP 4.306, which

the editors of the ''Collected Papers'', no doubt compromised by the

typographer's resistance to cutting new symbols, transmogrified into

a script more cryptic than even the manuscript???s original hieroglyphic.

I found one key to the mystery in Peirce's use of "operator variables",

which he and his students Christine Ladd-Franklin and O.H. Mitchell

explored in depth. I will shortly discuss this theme as it affects

logical graphs but it may be useful to give a shorter and sweeter

explanation of how the basic idea typically arises in common

logical practice.

Think of De Morgan???s rules:

: ??(A ??? B) = ??A ??? ??B

: ??(A ??? B) = ??A ??? ??B

We could capture the common form of these two rules in a single formula

by taking "o1" and "o2" as variable names ranging over a set of logical

operators, and then by asking what substitutions for o1 and o2 would

satisfy the following equation:

: ??(A o1 B) = ??A o2 ??B

We already know two solutions to this "operator equation", namely,

(o1, o2) = (???, ???) and (o1, o2) = (???, ???). Wouldn't it be just like

Peirce to ask if there are others?

Having broached the subject of logical operator variables,

I will leave it for now in the same way Peirce himself did:

<QUOTE>

I shall not further enlarge upon this matter at this point,

although the conception mentioned opens a wide field; because

it cannot be set in its proper light without overstepping the

limits of dichotomic mathematics. (Collected Papers, CP 4.306).

</QUOTE>

Further exploration of operator variables and operator invariants

treads on grounds traditionally known as "second intentional logic"

and "opens a wide field", as Peirce says. For now, however, I will

tend to that corner of the field where our garden variety logical

graphs grow, observing the ways operative variations and operative

themes naturally develop on those grounds.

Jul 8, 2019, 5:18:35 PM7/8/19

to Ontolog Forum, Structural Modeling, SysSciWG, Cybernetic Communications, Laws Of Form Group

| I'm still waiting to hear from the T-bird Hive Mind why

| everything got so iffy with unicodes a couple months ago,

| and what to do about it. In the meantime here's plaintext

| version of that last post. As always, there's a better

| formatted version at the blog post linked below.

Cf: Animated Logical Graphs : 16

I found one key to the mystery in Peirce's use of "operator variables",

which he and his students Christine Ladd-Franklin and O.H. Mitchell

explored in depth. I will shortly discuss this theme as it affects

logical graphs but it may be useful to give a shorter and sweeter

explanation of how the basic idea typically arises in common

logical practice.

Think of De Morgan's rules:

: not (A and B) = (not A) or (not B)

: not (A or B) = (not A) and (not B)

We could capture the common form of these two rules in a single formula

by taking "o1" and "o2" as variable names ranging over a set of logical

operators, and then by asking what substitutions for o1 and o2 would

satisfy the following equation:

: not (A o1 B) = (not A) o2 (not B)

We already know two solutions to this "operator equation", namely,

(o1, o2) = (and, or) and (o1, o2) = (or, and). Wouldn't it be

| everything got so iffy with unicodes a couple months ago,

| and what to do about it. In the meantime here's plaintext

| version of that last post. As always, there's a better

| formatted version at the blog post linked below.

Cf: Animated Logical Graphs : 16

At: https://inquiryintoinquiry.com/2019/07/07/animated-logical-graphs-%e2%80%a2-16/

In lieu of a field study requirement for my bachelor's degree I spent

a couple years in a host of state and university libraries reading

everything I could find by and about Peirce, poring most memorably

through the reels of microfilmed Peirce manuscripts Michigan State

had at the time, all in trying to track down some hint of a clue to

a puzzling passage in Peirce's "Simplest Mathematics", most acutely

coming to a head with that bizarre line of type at CP 4.306, which

the editors of the ''Collected Papers'', no doubt compromised by the

typographer's resistance to cutting new symbols, transmogrified into

a script more cryptic than even the manuscript's original hieroglyphic.
In lieu of a field study requirement for my bachelor's degree I spent

a couple years in a host of state and university libraries reading

everything I could find by and about Peirce, poring most memorably

through the reels of microfilmed Peirce manuscripts Michigan State

had at the time, all in trying to track down some hint of a clue to

a puzzling passage in Peirce's "Simplest Mathematics", most acutely

coming to a head with that bizarre line of type at CP 4.306, which

the editors of the ''Collected Papers'', no doubt compromised by the

typographer's resistance to cutting new symbols, transmogrified into

I found one key to the mystery in Peirce's use of "operator variables",

which he and his students Christine Ladd-Franklin and O.H. Mitchell

explored in depth. I will shortly discuss this theme as it affects

logical graphs but it may be useful to give a shorter and sweeter

explanation of how the basic idea typically arises in common

logical practice.

: not (A and B) = (not A) or (not B)

: not (A or B) = (not A) and (not B)

We could capture the common form of these two rules in a single formula

by taking "o1" and "o2" as variable names ranging over a set of logical

operators, and then by asking what substitutions for o1 and o2 would

satisfy the following equation:

We already know two solutions to this "operator equation", namely,

just like Peirce to ask if there are others?

Having broached the subject of logical operator variables,

I will leave it for now in the same way Peirce himself did:

<QUOTE>

I shall not further enlarge upon this matter at this point,

although the conception mentioned opens a wide field; because

it cannot be set in its proper light without overstepping the

limits of dichotomic mathematics. (Collected Papers, CP 4.306).

</QUOTE>

Further exploration of operator variables and operator invariants

treads on grounds traditionally known as "second intentional logic"

and "opens a wide field", as Peirce says. For now, however, I will

tend to that corner of the field where our garden variety logical

graphs grow, observing the ways operative variations and operative

themes naturally develop on those grounds.

Having broached the subject of logical operator variables,

I will leave it for now in the same way Peirce himself did:

<QUOTE>

I shall not further enlarge upon this matter at this point,

although the conception mentioned opens a wide field; because

it cannot be set in its proper light without overstepping the

limits of dichotomic mathematics. (Collected Papers, CP 4.306).

</QUOTE>

Further exploration of operator variables and operator invariants

treads on grounds traditionally known as "second intentional logic"

and "opens a wide field", as Peirce says. For now, however, I will

tend to that corner of the field where our garden variety logical

graphs grow, observing the ways operative variations and operative

themes naturally develop on those grounds.

Jul 8, 2019, 7:13:50 PM7/8/19

to Sys Sci, Ontolog Forum, Structural Modeling

Jon:

Excellent material..

I will add it to the list..

Take care, be good to yourself and have fun,

Joe

--

The SysSciWG wiki is at https://sites.google.com/site/syssciwg/.

Contributions to the discussion are licensed by authors under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

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Jul 9, 2019, 8:12:22 AM7/9/19

to syss...@googlegroups.com, joseph simpson, Ontolog Forum, Structural Modeling

Thanks, Joe,

This material is coming from the section of my "Futures Of Logical Graphs" (FOLG)

titled "Themes and Variations" where I explain how I came down from logical trees

and learned to love logical cacti (ouch) --

https://oeis.org/wiki/Futures_Of_Logical_Graphs#Themes_and_variations

I spent the last month upgrading the ancient ascii graphics to jpegs and the text

will hopefully get less rambling and clearer as I serialize it to my inquiry blog.

Previous Installments:

Survey of Animated Logical Graphs

https://inquiryintoinquiry.com/2019/05/22/survey-of-animated-logical-graphs-%e2%80%a2-2/

Animated Logical Graphs

(13) https://inquiryintoinquiry.com/2019/05/24/animated-logical-graphs-%e2%80%a2-13/

(14) https://inquiryintoinquiry.com/2019/05/28/animated-logical-graphs-%e2%80%a2-14/

(15) https://inquiryintoinquiry.com/2019/06/30/animated-logical-graphs-%e2%80%a2-15/

(16) https://inquiryintoinquiry.com/2019/07/07/animated-logical-graphs-%e2%80%a2-16/

Regards,

Jon

This material is coming from the section of my "Futures Of Logical Graphs" (FOLG)

titled "Themes and Variations" where I explain how I came down from logical trees

and learned to love logical cacti (ouch) --

https://oeis.org/wiki/Futures_Of_Logical_Graphs#Themes_and_variations

I spent the last month upgrading the ancient ascii graphics to jpegs and the text

will hopefully get less rambling and clearer as I serialize it to my inquiry blog.

Previous Installments:

Survey of Animated Logical Graphs

https://inquiryintoinquiry.com/2019/05/22/survey-of-animated-logical-graphs-%e2%80%a2-2/

(13) https://inquiryintoinquiry.com/2019/05/24/animated-logical-graphs-%e2%80%a2-13/

(14) https://inquiryintoinquiry.com/2019/05/28/animated-logical-graphs-%e2%80%a2-14/

(15) https://inquiryintoinquiry.com/2019/06/30/animated-logical-graphs-%e2%80%a2-15/

(16) https://inquiryintoinquiry.com/2019/07/07/animated-logical-graphs-%e2%80%a2-16/

Regards,

Jon

Jul 10, 2019, 9:24:52 AM7/10/19

to Jon Awbrey, Sys Sci, Ontolog Forum, Structural Modeling

Jon:

Great, I will program some time to engage this material.

Take care and have fun,

Joe

Jul 10, 2019, 10:00:14 AM7/10/19

to Ontolog Forum, Structural Modeling, SysSciWG

Cf: Animated Logical Graphs : 17

At: https://inquiryintoinquiry.com/2019/07/09/animated-logical-graphs-%e2%80%a2-17/

To get a clearer view of the relation between

primary arithmetic and primary algebra consider

the following extremely simple algebraic expression:

Figure 4. Cactus Graph (a)

https://inquiryintoinquiry.files.wordpress.com/2019/07/box-a.jpg

In this expression the variable name "a" appears as an "operand name".

In functional terms, "a" is called an "argument name", but it's best

to avoid the potentially confusing connotations of the word "argument"

here, since it also refers in logical discussions to a more or less

specific pattern of reasoning.

As we've discussed, the algebraic variable name indicates the

contemplated absence or presence of any arithmetic expression

taking its place in the surrounding template, which expression

is proxied well enough by its logical value, and of which values

we know but two. Thus, the given algebraic expression varies

between these two choices:

Figure 5. Cactus Graph Set (),(())

https://inquiryintoinquiry.files.wordpress.com/2019/07/box-.jpg

The above selection of arithmetic expressions is what it means

to contemplate the absence or presence of the operand "a" in

the algebraic expression "(a)". But what would it mean to

contemplate the absence or presence of the operator "( )"

in the algebraic expression "(a)"?

That is the question I'll take up next.

Regards,

Jon

At: https://inquiryintoinquiry.com/2019/07/09/animated-logical-graphs-%e2%80%a2-17/

To get a clearer view of the relation between

primary arithmetic and primary algebra consider

the following extremely simple algebraic expression:

Figure 4. Cactus Graph (a)

https://inquiryintoinquiry.files.wordpress.com/2019/07/box-a.jpg

In this expression the variable name "a" appears as an "operand name".

In functional terms, "a" is called an "argument name", but it's best

to avoid the potentially confusing connotations of the word "argument"

here, since it also refers in logical discussions to a more or less

specific pattern of reasoning.

As we've discussed, the algebraic variable name indicates the

contemplated absence or presence of any arithmetic expression

taking its place in the surrounding template, which expression

is proxied well enough by its logical value, and of which values

we know but two. Thus, the given algebraic expression varies

between these two choices:

Figure 5. Cactus Graph Set (),(())

https://inquiryintoinquiry.files.wordpress.com/2019/07/box-.jpg

The above selection of arithmetic expressions is what it means

to contemplate the absence or presence of the operand "a" in

the algebraic expression "(a)". But what would it mean to

contemplate the absence or presence of the operator "( )"

in the algebraic expression "(a)"?

That is the question I'll take up next.

Regards,

Jon

Jul 10, 2019, 6:00:22 PM7/10/19

to Ontolog Forum, Structural Modeling, SysSciWG

Cf: Animated Logical Graphs : 18

At: https://inquiryintoinquiry.com/2019/07/10/animated-logical-graphs-%e2%80%a2-18/

We had been contemplating the penultimately simple

algebraic expression "(a)" as a name for a set of

arithmetic expressions, namely, (a) = { () , (()) },

taking the equality sign in the appropriate sense.

Figure 6. Cactus Graph Equation (a) = {(),(())}

https://inquiryintoinquiry.files.wordpress.com/2019/07/box-a-1.jpg

Then we asked the corresponding question about the operator "( )".

the operator "( )" in the algebraic expression "(a)" refers to

a variation between the algebraic expressions "a" and "(a)",

respectively, somewhat as pictured below:

Figure 7. Cactus Graph Equation ?a? = {a,(a)}

https://inquiryintoinquiry.files.wordpress.com/2019/07/box-a-queaa.jpg

But how shall we signify such variations in a coherent calculus?

(end of season cliff-hanger ...)

Regards,

Jon

At: https://inquiryintoinquiry.com/2019/07/10/animated-logical-graphs-%e2%80%a2-18/

We had been contemplating the penultimately simple

algebraic expression "(a)" as a name for a set of

arithmetic expressions, namely, (a) = { () , (()) },

taking the equality sign in the appropriate sense.

Figure 6. Cactus Graph Equation (a) = {(),(())}

https://inquiryintoinquiry.files.wordpress.com/2019/07/box-a-1.jpg

Then we asked the corresponding question about the operator "( )".

The above selection of arithmetic expressions is what it means

to contemplate the absence or presence of the operand "a" in

the algebraic expression "(a)". But what would it mean to

contemplate the absence or presence of the operator "( )"

in the algebraic expression "(a)"?

Clearly, a variation between the absence and the presence of
to contemplate the absence or presence of the operand "a" in

the algebraic expression "(a)". But what would it mean to

contemplate the absence or presence of the operator "( )"

in the algebraic expression "(a)"?

the operator "( )" in the algebraic expression "(a)" refers to

a variation between the algebraic expressions "a" and "(a)",

respectively, somewhat as pictured below:

Figure 7. Cactus Graph Equation ?a? = {a,(a)}

https://inquiryintoinquiry.files.wordpress.com/2019/07/box-a-queaa.jpg

But how shall we signify such variations in a coherent calculus?

(end of season cliff-hanger ...)

Regards,

Jon

Jul 11, 2019, 10:48:54 AM7/11/19

to Ontolog Forum, Structural Modeling, SysSciWG, Cybernetic Communications, Laws Of Form Group

We have encountered the question of how to extend our

formal calculus to take account of operator variables.

In the days when I scribbled these things on the backs of computer punchcards,

the first thing I tried was drawing big loopy script characters, placing some

inside the loops of others. Lower case alphas, betas, gammas, deltas, and

so on worked best. Graphics like these conveyed the idea that a character-

shaped boundary drawn around another space can be viewed as absent or present

depending on whether the formal value of the character is unmarked or marked.

The same idea can be conveyed by attaching characters directly to the edges

of graphs.

Here is how we might suggest an algebraic expression of the form "(q)"

where the absence or presence of the operator "( )" depends on the value

of the algebraic expression "p", the operator "( )" being absent whenever

p is unmarked and present whenever p is marked.

Figure 8. Cactus Graph (q)_p = {q,(q)}

https://inquiryintoinquiry.files.wordpress.com/2019/07/box-q-que-pqq.jpg

It was obvious to me from the outset that this sort of tactic would need

a lot of work to become a usable calculus, especially when it came time

to feed those punchcards back into the computer.

Regards,

Jon

formal calculus to take account of operator variables.

In the days when I scribbled these things on the backs of computer punchcards,

the first thing I tried was drawing big loopy script characters, placing some

inside the loops of others. Lower case alphas, betas, gammas, deltas, and

so on worked best. Graphics like these conveyed the idea that a character-

shaped boundary drawn around another space can be viewed as absent or present

depending on whether the formal value of the character is unmarked or marked.

The same idea can be conveyed by attaching characters directly to the edges

of graphs.

Here is how we might suggest an algebraic expression of the form "(q)"

where the absence or presence of the operator "( )" depends on the value

of the algebraic expression "p", the operator "( )" being absent whenever

p is unmarked and present whenever p is marked.

Figure 8. Cactus Graph (q)_p = {q,(q)}

https://inquiryintoinquiry.files.wordpress.com/2019/07/box-q-que-pqq.jpg

It was obvious to me from the outset that this sort of tactic would need

a lot of work to become a usable calculus, especially when it came time

to feed those punchcards back into the computer.

Regards,

Jon

Jul 11, 2019, 5:30:08 PM7/11/19

to Ontolog Forum, Structural Modeling, SysSciWG, Cybernetic Communications, Laws Of Form Group

Cf: Animated Logical Graphs

19: https://inquiryintoinquiry.com/2019/07/11/animated-logical-graphs-%e2%80%a2-19/

20: https://inquiryintoinquiry.com/2019/07/11/animated-logical-graphs-%e2%80%a2-20/

Another tactic I tried by way of porting operator variables into logical graphs and

laws of form was to hollow out a leg of Spencer-Brown's crosses, gnomons, markers,

whatever you wish to call them, as shown below:

Figure 9. Transitional (q)_p = {q,(q)}

https://inquiryintoinquiry.files.wordpress.com/2019/07/box-q-qua-p.jpg

The initial idea I had in mind was the same as before, that the operator

over q would be counted as absent when p evaluates to a space and present

when p evaluates to a cross.

However, much in the same way that operators with a shade of negativity

tend to be more generative than the purely positive brand, it turned out

more useful to reverse this initial polarity of operation, letting the

operator over q be counted as absent when p evaluates to a cross and

present when p evaluates to a space.

So that is the convention I'll adopt from here on.

Regards,

Jon

19: https://inquiryintoinquiry.com/2019/07/11/animated-logical-graphs-%e2%80%a2-19/

20: https://inquiryintoinquiry.com/2019/07/11/animated-logical-graphs-%e2%80%a2-20/

Another tactic I tried by way of porting operator variables into logical graphs and

laws of form was to hollow out a leg of Spencer-Brown's crosses, gnomons, markers,

whatever you wish to call them, as shown below:

Figure 9. Transitional (q)_p = {q,(q)}

https://inquiryintoinquiry.files.wordpress.com/2019/07/box-q-qua-p.jpg

The initial idea I had in mind was the same as before, that the operator

over q would be counted as absent when p evaluates to a space and present

when p evaluates to a cross.

However, much in the same way that operators with a shade of negativity

tend to be more generative than the purely positive brand, it turned out

more useful to reverse this initial polarity of operation, letting the

operator over q be counted as absent when p evaluates to a cross and

present when p evaluates to a space.

So that is the convention I'll adopt from here on.

Regards,

Jon

Jul 11, 2019, 10:58:19 PM7/11/19

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Jon:

This section sent me back to the 'Laws of Form' for a quick review.

I think I see the connection, time will tell..

Have fun,

Joe

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Jul 12, 2019, 2:00:31 PM7/12/19

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Cf: Animated Logical Graphs : 21

At: https://inquiryintoinquiry.com/2019/07/12/animated-logical-graphs-%e2%80%a2-21/

A funny thing just happened. Let's see if we can tell where.

We started with the algebraic expression "(a)", in which the

operand "a" suggests the contemplated absence or presence of

any arithmetic expression or its value, then we contemplated

the absence or presence of the operator "( )" in "(a)" to be

indicated by a cross or a space, respectively, for the value

of a newly introduced variable, "b", placed in a new slot of

a newly extended operator form, as suggested by this picture:

Figure 10. Control Form (a)_b

https://inquiryintoinquiry.files.wordpress.com/2019/07/box-a-quo-b.jpg

What happened here is this. Our contemplation of an operator variable

just as quickly transformed into the contemplation of a newly introduced

but otherwise quite ordinary operand variable, fitting into a new form of

formula. In its interpretation for logic the newly formed operation may be

viewed as an extension of ordinary negation, one in which the negation of the

first variable is "controlled" by the value of the second variable. Thus, we

may regard this development as marking a form of "controlled reflection", or a

form of "reflective control". From this point on we will use the inline syntax

"(a , b)" for the associated operation on two variables, whose operation table

is given below:

Operation Table for (a , b)

https://inquiryintoinquiry.files.wordpress.com/2019/07/table-ab-space-cross.png

: The Entitative Interpretation (En), for which Space = False and Cross = True,

calls this operation "equivalence".

: The Existential Interpretation (Ex), for which Space = True and Cross = False,

calls this operation "distinction".

Regards,

Jon

At: https://inquiryintoinquiry.com/2019/07/12/animated-logical-graphs-%e2%80%a2-21/

A funny thing just happened. Let's see if we can tell where.

We started with the algebraic expression "(a)", in which the

operand "a" suggests the contemplated absence or presence of

any arithmetic expression or its value, then we contemplated

the absence or presence of the operator "( )" in "(a)" to be

indicated by a cross or a space, respectively, for the value

of a newly introduced variable, "b", placed in a new slot of

a newly extended operator form, as suggested by this picture:

Figure 10. Control Form (a)_b

https://inquiryintoinquiry.files.wordpress.com/2019/07/box-a-quo-b.jpg

What happened here is this. Our contemplation of an operator variable

just as quickly transformed into the contemplation of a newly introduced

but otherwise quite ordinary operand variable, fitting into a new form of

formula. In its interpretation for logic the newly formed operation may be

viewed as an extension of ordinary negation, one in which the negation of the

first variable is "controlled" by the value of the second variable. Thus, we

may regard this development as marking a form of "controlled reflection", or a

form of "reflective control". From this point on we will use the inline syntax

"(a , b)" for the associated operation on two variables, whose operation table

is given below:

Operation Table for (a , b)

https://inquiryintoinquiry.files.wordpress.com/2019/07/table-ab-space-cross.png

: The Entitative Interpretation (En), for which Space = False and Cross = True,

calls this operation "equivalence".

: The Existential Interpretation (Ex), for which Space = True and Cross = False,

calls this operation "distinction".

Regards,

Jon

Jul 13, 2019, 5:33:11 AM7/13/19

to Sys Sci, Ontolog Forum, Structural Modeling, Cybernetic Communications, Laws Of Form Group

Each step on its own, as far as I can follow them, makes sense. You are, if I understand it correctly, trying to figure out something fundamental, the rock bottom reality. When can we expect that results of such a research to become "applicable to more than one of the traditional departments of knowledge" (http://isss.org/world/about-the-isss)? What kinds of tragedy, disaster, misunderstanding, mismanagement, or failure would/will be preventable by your approach?

Aleksandar

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Jul 13, 2019, 11:45:05 PM7/13/19

to structura...@googlegroups.com, Sys Sci, Ontolog Forum, Cybernetic Communications, Laws Of Form Group

Aleksander:

I am commenting on my view of the potential value associated with Jon's work.

Most, if not all, of the benefits associated with the "Laws of Form," appear to be associated with Jon's work on animated logical graphs. These benefits include:

-- complexity reduction (both cognitive and computational)

-- increased communication precision

-- interface between informal and formal languages.

Many large scale human activities could be improved using the above listed benefits.

However, another more interesting (to me) aspect of this work is the dynamic, responsive form of the logical graphs.

In any case, this is interesting material that could be applied in many areas.

I have started to map the relationships to the augmented model-exchange isomorphism (AMEI) logical groups. Later, I plan of creating a collection of abstract relation types (ART) to document and communicate the application of this type of logical analysis.

Take care, be good to yourself and have fun,

Joe

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Jul 15, 2019, 7:23:36 PM7/15/19

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Bruce:

Interesting observations and comments.

Much of the material contained in the "Laws Of Form," relates a formal language to natural language.

The last statement of Chapter 12, on page 76, states:

"We now see that the first distinction, the mark, and the observer are not only interchangeable, but, in the form, identical."

The statement above, to me, indicates that the formal language representations associated with the "Laws Of Form," are highly restricted.

No one would say that a mark on a sheet of paper is an observer, in real life.

The challenge is to find a suitable natural language that properly applies the Laws Of Form to a range of real situations.

The space, state or contents associated with the distinction is a key consideration in the proper application of the Laws Of Form.

The relationships associated with the real space, state or contents are not as restricted as the relationships associated with the Laws Of Form.

Anyway, interesting material.

Take care and have fun,

Joe

On Sun, Jul 14, 2019 at 9:58 AM <bruces...@cox.net> wrote:

Thanks for these comments, Joe – thanks to Jon for exploring the connection with Laws of Form. Thanks for linking to a few other conversations.

I just want to mention that G. Spencer Brown and LOF had a significant influence on me as well. This book was popular in the late 1960’s or early 1970’s, and it was very up my line. I was exploring topological approaches to logic, and Brown’s book was very striking and influential.

I did not really understand his graphic symbolism – the notion of “crossing”, etc. – and I think a lot of people were mystified by it, but yet intrigued. It just so happened that in 1973, I met a couple of times with the famous anthropologist/epistemologist Gregory Bateson (author of the very influential Steps to an Ecology of Mind, which introduced the theme of “The Pattern That Connects” into the new-thought conversation). He was in residence at UC Santa Cruz at the time, and one of the things we talked about was Laws of Form. Like me, Bateson was intrigued – but asked me if I thought the book was a trick, a mystifying joke being played on a gullible public. He found some citation in the index that he thought was a clue. https://link.springer.com/chapter/10.1007/978-1-4020-6706-8_14

But for me, this question was just a distraction. The idea that stuck with me – and shaped everything I’ve been doing since, including a lot of messages posted to Ontolog – has to do with the concept of “distinction”, which LOF first put in my head.

“The first command – draw a distinction”

That’s something I’ve been feeling for a long time.

In what I am doing right now, this idea is at the core –

“Draw a distinction in a distinction”

There’s a lot we can say about this.

In Dr. Susan Carey’s “The Origin of Concepts”, she talks about “Quinian Bootstrapping” (W.V.O. Quine) – which I think relates to this kind of mysterious coalescence of concept and form from a mysterious figure/ground tension. She is talking about concept formation in children. Something organic drives an emerging fuzzy notion that becomes explicitly codified.

What I want to suggest is that there is a organic drive dynamic that pushes conceptual form out of the continuum (Tao, real number line, unit interval, etc.) , under the force of some kind of “local” motivation. A distinction gets drawn in a distinction. Maybe this relates to Helen Keller’s powerful experience with the concept/word W-A-T-E-R as drawn on her hand….

In any case, thank you.

It feels to me like there is a powerful transcendental theorem in the air right now. We are figuring this out. Something wants to explode through the keyhole.

For those who want to decode this mystery, there’s a lot to consider right here in this excerpt from Wikipedia.

https://en.wikipedia.org/wiki/Laws_of_Form

Bruce Schuman

Santa Barbara CA USA

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Jul 22, 2019, 8:12:16 PM7/22/19

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Cf: Animated Logical Graphs : 22

At: https://inquiryintoinquiry.com/2019/07/22/animated-logical-graphs-%e2%80%a2-22/

The step of controlled reflection we just took can be iterated as far

as we wish to take it, as suggested by the following series of forms:

Figure 11. Reflective Series (a) to (a, b, c, d)

https://inquiryintoinquiry.files.wordpress.com/2019/07/reflective-series-a-to-abcd.jpg

Written inline, we have the series "(a)", "(a, b)", "(a, b, c)", "(a, b, c, d)",

and so on, whose general form is "(x_1, x_2, ..., x_k)". With this move we have

passed beyond the graph-theoretical form of rooted trees to what graph theorists

know as "rooted cacti".

I will discuss this "cactus language" and its logical interpretations next.

At: https://inquiryintoinquiry.com/2019/07/22/animated-logical-graphs-%e2%80%a2-22/

The step of controlled reflection we just took can be iterated as far

as we wish to take it, as suggested by the following series of forms:

Figure 11. Reflective Series (a) to (a, b, c, d)

https://inquiryintoinquiry.files.wordpress.com/2019/07/reflective-series-a-to-abcd.jpg

Written inline, we have the series "(a)", "(a, b)", "(a, b, c)", "(a, b, c, d)",

and so on, whose general form is "(x_1, x_2, ..., x_k)". With this move we have

passed beyond the graph-theoretical form of rooted trees to what graph theorists

know as "rooted cacti".

I will discuss this "cactus language" and its logical interpretations next.

Jul 23, 2019, 7:55:37 PM7/23/19

to Sys Sci, Ontolog Forum, Structural Modeling, Cybernetic Communications, Laws Of Form Group

Jon:

Very interesting work and ideas.

I went back to this section to address an area where I have some specific questions.

The area is associated with the operations of "equivalence" and "distinction".

I tend to view equivalence and distinction as relationships as opposed to operations. I do not know if this makes any significant difference in this context.

Given the relationship of equivalence, the logical properties of the relationship are given as:

--- reflexive

--- symmetric

--- transitive

Object A is equivalent to Object A. (reflexive property)

Object B is equivalent to Object A and Object A is equivalent to Object B. (symmetric property)

If Object B is equivalent to Object A and Object A is equivalent to Object C, then Object B is equivalent to Object C. (transitive property)

Given the relationship of distinction, the logical properties of the relationship are given as:

--- irreflexive

--- symmetric

--- transitive

Object A is not distinct from Object A. (irreflexive property)

Object B is distinct from Object A and Object A is distinct from Object B. (symmetric property)

If
Object B is distinct from Object A and Object A is distinct from Object
C, then Object B is distinct from Object C. (transitive property)

The questions relate to the two different types of reflexive logical attributes; reflexive and irreflexive.

In the operations of controlled reflection are both types of reflexive logical properties allowed?

Does the term "controlled reflection" relate to a collection of operations or logical properties or both?

This is my first attempt to start associating your work to the augmented model-exchange isomorphism (AMEI) logical groups.

The AMEI logical groups may then be used to help associate these logical graphic terms to a natural language implementation.

Thanks for continuing to share and explain your work.

Take care, be good to yourself and have fun,

Joe

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Jul 24, 2019, 9:40:54 AM7/24/19

to ontolo...@googlegroups.com, Sys Sci, Structural Modeling

Joe, Thanks, those are very good questions. We just got back from

our annual "dramatic immersion" at the Stratford Festival in Canada

and I learned a lot from the plays this year that I'm still working

to metabolize and integrate into my everyday outlook on life. I'll

hold off just a bit on replying to your questions and the intervening

discussions until I get my next blog post out, as it has a table that

should help things along.

Regards,

Jon

>> ...

our annual "dramatic immersion" at the Stratford Festival in Canada

and I learned a lot from the plays this year that I'm still working

to metabolize and integrate into my everyday outlook on life. I'll

hold off just a bit on replying to your questions and the intervening

discussions until I get my next blog post out, as it has a table that

should help things along.

Regards,

Jon

Jul 24, 2019, 2:09:01 PM7/24/19

to Ontolog Forum, Structural Modeling, SysSciWG, Cybernetic Communications, Laws Of Form Group

Cf: Animated Logical Graphs : 23

At: https://inquiryintoinquiry.com/2019/07/23/animated-logical-graphs-%e2%80%a2-23/

The following Table will suffice to show how the "streamer-cross" forms

C.S. Peirce used in his essay on "Qualitative Logic" and Spencer Brown

used in his "Laws of Form", as they get extended by successive steps

of controlled reflection, translate into syntactic strings and

rooted cactus graphs:

Table. Syntactic Correspondences

https://inquiryintoinquiry.com/syntactic-correspondences/

At: https://inquiryintoinquiry.com/2019/07/23/animated-logical-graphs-%e2%80%a2-23/

The following Table will suffice to show how the "streamer-cross" forms

C.S. Peirce used in his essay on "Qualitative Logic" and Spencer Brown

used in his "Laws of Form", as they get extended by successive steps

of controlled reflection, translate into syntactic strings and

rooted cactus graphs:

Table. Syntactic Correspondences

https://inquiryintoinquiry.com/syntactic-correspondences/

Jul 25, 2019, 8:18:18 PM7/25/19

to Sys Sci, Ontolog Forum, Structural Modeling, Cybernetic Communications, Laws Of Form Group

Jon:

Today I found an interesting publication that might relate to the current discussion of Animated Logical Graphs (ALG).

Please see:

The “sensitivity” conjecture may be a topic that could be explored using ALG.

There appear to be many interesting connections between ALG and the sensitivity conjecture.

I am looking for an area where ALG application examples may be created and discussed.

My first attempt at an example, using the augmented model-exchange isomorphism (AMEI), raised a number of conceptual and structural application issues. Which can be addressed as we move forward.

My plan is to continue the search for specific application areas.

I believe that finding domain specific applications will help me better understand the ALG material.

Take care, be good to yourself and have fun,

Joe

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Jul 26, 2019, 11:45:57 AM7/26/19

to Ontolog Forum, Structural Modeling, SysSciWG, Cybernetic Communications, Laws Of Form Group

Cf: Animated Logical Graphs : 24

At: https://inquiryintoinquiry.com/2019/07/26/animated-logical-graphs-%e2%80%a2-24/

Re: Joseph Simpson

At: https://groups.google.com/d/msg/ontolog-forum/wF03K5KG1vQ/ZRlQdavJBgAJ

Boolean functions f : B^k -> B and different ways of contemplating their

complexity are definitely the right ballpark, or at least the right planet,

for field-testing logical graphs.

I don't know much about the Boolean Sensitivity Conjecture but I did run across

an enlightening article about it just yesterday and I did once begin an exploration

of what appears to be a related question, Peter Frankl's "Union-Closed Sets Conjecture".

See the resource pages linked below.

At any rate, now that we've entered the ballpark, or standard orbit, of boolean functions,

I can skip a bit of dancing around and jump to the next blog post I have on deck.

Resources

=========

* Frankl Conjecture

( https://inquiryintoinquiry.com/category/frankl-conjecture/ )

* R.J. Lipton and K.W. Regan : Discrepancy Games and Sensitivity

( https://rjlipton.wordpress.com/2019/07/25/discrepancy-games-and-sensitivity/ )

At: https://inquiryintoinquiry.com/2019/07/26/animated-logical-graphs-%e2%80%a2-24/

Re: Joseph Simpson

At: https://groups.google.com/d/msg/ontolog-forum/wF03K5KG1vQ/ZRlQdavJBgAJ

Boolean functions f : B^k -> B and different ways of contemplating their

complexity are definitely the right ballpark, or at least the right planet,

for field-testing logical graphs.

I don't know much about the Boolean Sensitivity Conjecture but I did run across

an enlightening article about it just yesterday and I did once begin an exploration

of what appears to be a related question, Peter Frankl's "Union-Closed Sets Conjecture".

See the resource pages linked below.

At any rate, now that we've entered the ballpark, or standard orbit, of boolean functions,

I can skip a bit of dancing around and jump to the next blog post I have on deck.

Resources

=========

* Frankl Conjecture

( https://inquiryintoinquiry.com/category/frankl-conjecture/ )

* R.J. Lipton and K.W. Regan : Discrepancy Games and Sensitivity

( https://rjlipton.wordpress.com/2019/07/25/discrepancy-games-and-sensitivity/ )

Jul 27, 2019, 10:20:31 AM7/27/19

to Ontolog Forum, Structural Modeling, SysSciWG, Cybernetic Communications, Laws Of Form Group

Cf: Animated Logical Graphs : 25

At: https://inquiryintoinquiry.com/2019/07/26/animated-logical-graphs-%e2%80%a2-25/

Let's examine the formal operation table for the third in our series

of reflective forms to see if we can elicit the general pattern:

Formal Operation Table (a, b, c) Variant 1

https://inquiryintoinquiry.com/formal-operation-table-a-b-c-variant-1/

Or, thinking in terms of the corresponding cactus graphs,

writing "o" for a blank node and "|" for a terminal edge,

we get the following Table:

Formal Operation Table (a, b, c) Variant 2

https://inquiryintoinquiry.com/formal-operation-table-a-b-c-variant-2/

Evidently, the rule is that "(a, b, c)" denotes the value denoted by "o"

if and only if exactly one of the variables a, b, c has the value denoted

by "|", otherwise "(a, b, c)" denotes the value denoted by "|". Examining

the whole series of reflective forms shows this is the general rule.

* In the Entitative Interpretation (En), where o = false and | = true,

"(x_1, ..., x_k)" translates as "not just one of the x_j is true".

* In the Existential Interpretation (Ex), where o = true and | = false,

"(x_1, ..., x_k)" translates as "just one of the x_j is not true".

Regards,

Jon

At: https://inquiryintoinquiry.com/2019/07/26/animated-logical-graphs-%e2%80%a2-25/

Let's examine the formal operation table for the third in our series

of reflective forms to see if we can elicit the general pattern:

Formal Operation Table (a, b, c) Variant 1

https://inquiryintoinquiry.com/formal-operation-table-a-b-c-variant-1/

Or, thinking in terms of the corresponding cactus graphs,

writing "o" for a blank node and "|" for a terminal edge,

we get the following Table:

Formal Operation Table (a, b, c) Variant 2

https://inquiryintoinquiry.com/formal-operation-table-a-b-c-variant-2/

Evidently, the rule is that "(a, b, c)" denotes the value denoted by "o"

if and only if exactly one of the variables a, b, c has the value denoted

by "|", otherwise "(a, b, c)" denotes the value denoted by "|". Examining

the whole series of reflective forms shows this is the general rule.

* In the Entitative Interpretation (En), where o = false and | = true,

"(x_1, ..., x_k)" translates as "not just one of the x_j is true".

* In the Existential Interpretation (Ex), where o = true and | = false,

"(x_1, ..., x_k)" translates as "just one of the x_j is not true".

Regards,

Jon

Jul 29, 2019, 3:24:18 PM7/29/19

to Ontolog Forum, Structural Modeling, SysSciWG, Cybernetic Communications, Laws Of Form Group

Cf: Animated Logical Graphs : 26

At: https://inquiryintoinquiry.com/2019/07/28/animated-logical-graphs-%e2%80%a2-26/

This post and the next wrap up the Themes and Variations section

of my speculation on Futures Of Logical Graphs. I made an effort

to "show my work", reviewing the steps I took to arrive at the

present perspective on logical graphs, whistling past the least

productive of the blind alleys, cul-de-sacs, detours, and forking

paths I explored along the way. It can be useful to tell the story

that way, partly because others may find things I missed down those

roads, but it does call for a recap of the main ideas I would like

readers to take away.

Partly through my reflections on Peirce's use of operator variables I was led

to what I called the "reflective extension of logical graphs", or what I now

call the "cactus language", after its principal graph-theoretic data structure.

This graphical formal language arises from generalizing the negation operator "( )"

in a particular direction, treating "( )" as the "controlled", "moderated", or

"reflective" negation operator of order 1, and adding another operator for each

integer parameter greater than 1. This family of operators is symbolized by

bracketed argument lists of the forms "( )", "( , )", "( , , )", and so on,

where the number of places is the order of the reflective negation operator

in question.

Two rules suffice for evaluating cactus graphs:

* The rule for evaluating a k-node operator,

corresponding to an expression of the form

"x_1 x_2 ... x_{k-1} x_k", is as follows:

Figure 12. Node Evaluation Rule

https://inquiryintoinquiry.files.wordpress.com/2019/07/box-xj-node-evaluation-rule.jpg

* The rule for evaluating a k-lobe operator,

corresponding to an expression of the form

"(x_1, x_2, ..., x_{k-1}, x_k)", is as follows:

Figure 13. Lobe Evaluation Rule

https://inquiryintoinquiry.files.wordpress.com/2019/07/box-xj-lobe-evaluation-rule.jpg

References

==========

Futures Of Logical Graphs

https://oeis.org/wiki/Futures_Of_Logical_Graphs

Themes and Variations

https://oeis.org/wiki/Futures_Of_Logical_Graphs#Themes_and_variations

Regards,

Jon

At: https://inquiryintoinquiry.com/2019/07/28/animated-logical-graphs-%e2%80%a2-26/

This post and the next wrap up the Themes and Variations section

of my speculation on Futures Of Logical Graphs. I made an effort

to "show my work", reviewing the steps I took to arrive at the

present perspective on logical graphs, whistling past the least

productive of the blind alleys, cul-de-sacs, detours, and forking

paths I explored along the way. It can be useful to tell the story

that way, partly because others may find things I missed down those

roads, but it does call for a recap of the main ideas I would like

readers to take away.

Partly through my reflections on Peirce's use of operator variables I was led

to what I called the "reflective extension of logical graphs", or what I now

call the "cactus language", after its principal graph-theoretic data structure.

This graphical formal language arises from generalizing the negation operator "( )"

in a particular direction, treating "( )" as the "controlled", "moderated", or

"reflective" negation operator of order 1, and adding another operator for each

integer parameter greater than 1. This family of operators is symbolized by

bracketed argument lists of the forms "( )", "( , )", "( , , )", and so on,

where the number of places is the order of the reflective negation operator

in question.

Two rules suffice for evaluating cactus graphs:

* The rule for evaluating a k-node operator,

corresponding to an expression of the form

"x_1 x_2 ... x_{k-1} x_k", is as follows:

Figure 12. Node Evaluation Rule

https://inquiryintoinquiry.files.wordpress.com/2019/07/box-xj-node-evaluation-rule.jpg

* The rule for evaluating a k-lobe operator,

corresponding to an expression of the form

"(x_1, x_2, ..., x_{k-1}, x_k)", is as follows:

Figure 13. Lobe Evaluation Rule

https://inquiryintoinquiry.files.wordpress.com/2019/07/box-xj-lobe-evaluation-rule.jpg

References

==========

Futures Of Logical Graphs

https://oeis.org/wiki/Futures_Of_Logical_Graphs

Themes and Variations

https://oeis.org/wiki/Futures_Of_Logical_Graphs#Themes_and_variations

Regards,

Jon

Aug 1, 2019, 10:48:17 AM8/1/19

to Ontolog Forum, Structural Modeling, SysSciWG, Cybernetic Communications, Laws Of Form Group

Cf: Animated Logical Graphs : 27

At: https://inquiryintoinquiry.com/2019/07/31/animated-logical-graphs-%e2%80%a2-27/

The rules given in the previous post for evaluating cactus graphs were

given in purely formal terms, that is, by referring to the mathematical

forms of cacti without mentioning their potential for logical meaning.

As it turns out, two ways of mapping cactus graphs to logical meanings

are commonly found in practice. These two mappings of mathematical

structure to logical meaning are formally dual to each other and known

as the Entitative and Existential interpretations respectively. The

following Table compares the entitative and existential interpretations

of the primary cactus structures, from which the rest of their semantics

can be derived.

Table. Logical Interpretations of Cactus Structures

https://inquiryintoinquiry.files.wordpress.com/2019/07/logical-interpretations-of-cactus-structures-en-ex.jpg

This concludes the "New Cacti for Old Trees" episode

of my prospectus on Futures Of Logical Graphs. I've

got to take care of some procrastinations about home

and garden and then try to catch up with the comments

and questions that have accumulated in the meantime.

Regards,

Jon

At: https://inquiryintoinquiry.com/2019/07/31/animated-logical-graphs-%e2%80%a2-27/

The rules given in the previous post for evaluating cactus graphs were

given in purely formal terms, that is, by referring to the mathematical

forms of cacti without mentioning their potential for logical meaning.

As it turns out, two ways of mapping cactus graphs to logical meanings

are commonly found in practice. These two mappings of mathematical

structure to logical meaning are formally dual to each other and known

as the Entitative and Existential interpretations respectively. The

following Table compares the entitative and existential interpretations

of the primary cactus structures, from which the rest of their semantics

can be derived.

Table. Logical Interpretations of Cactus Structures

https://inquiryintoinquiry.files.wordpress.com/2019/07/logical-interpretations-of-cactus-structures-en-ex.jpg

This concludes the "New Cacti for Old Trees" episode

of my prospectus on Futures Of Logical Graphs. I've

got to take care of some procrastinations about home

and garden and then try to catch up with the comments

and questions that have accumulated in the meantime.

Regards,

Jon

Aug 3, 2019, 11:15:42 AM8/3/19

to Ontolog Forum, Structural Modeling, SysSciWG, Cybernetic Communications, Laws Of Form Group

Cf: Animated Logical Graphs : 28

At: https://inquiryintoinquiry.com/2019/08/03/animated-logical-graphs-%e2%80%a2-28/

I will have to focus on other business for a couple of weeks ?

so just by way of reminding myself what we were talking about

at this juncture where logical graphs and differential logic

intersect, here's my comment on R.J. Lipton and K.W. Regan's

blog post about Discrepancy Games and Sensitivity.

At: https://rjlipton.wordpress.com/2019/07/25/discrepancy-games-and-sensitivity/

<QUOTE>

Just by way of a general observation, concepts like discrepancy,

influence, sensitivity, etc. are differential in character, so

I tend to think the proper grounds for approaching them more

systematically will come from developing the logical analogue

of differential geometry.

I took a few steps in this direction some years ago in connection

with an effort to understand a certain class of intelligent systems

as dynamical systems. There's a motley assortment of links here:

* Survey of Differential Logic

https://inquiryintoinquiry.com/2015/05/11/survey-of-differential-logic-%E2%80%A2-1/

</QUOTE>

Resources

=========

* Logical Graphs

https://oeis.org/wiki/Logical_Graphs

* Differential Logic

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Introduction

Regards,

Jon

At: https://inquiryintoinquiry.com/2019/08/03/animated-logical-graphs-%e2%80%a2-28/

I will have to focus on other business for a couple of weeks ?

so just by way of reminding myself what we were talking about

at this juncture where logical graphs and differential logic

intersect, here's my comment on R.J. Lipton and K.W. Regan's

blog post about Discrepancy Games and Sensitivity.

At: https://rjlipton.wordpress.com/2019/07/25/discrepancy-games-and-sensitivity/

<QUOTE>

Just by way of a general observation, concepts like discrepancy,

influence, sensitivity, etc. are differential in character, so

I tend to think the proper grounds for approaching them more

systematically will come from developing the logical analogue

of differential geometry.

I took a few steps in this direction some years ago in connection

with an effort to understand a certain class of intelligent systems

as dynamical systems. There's a motley assortment of links here:

* Survey of Differential Logic

https://inquiryintoinquiry.com/2015/05/11/survey-of-differential-logic-%E2%80%A2-1/

</QUOTE>

Resources

=========

* Logical Graphs

https://oeis.org/wiki/Logical_Graphs

* Differential Logic

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Introduction

Regards,

Jon

Aug 11, 2019, 1:10:36 PM8/11/19

to ontolo...@googlegroups.com, joseph simpson, Sys Sci, Structural Modeling

Joe & All,

Cf: Animated Logical Graphs : 21

At: https://inquiryintoinquiry.com/2019/07/12/animated-logical-graphs-%e2%80%a2-21/

Re: JS

to connect with a wider backdrop of ideas we need to keep in mind

but since we've been focusing on boolean functions to coordinate

the semantics of propositional calculi we can get a sense of the

connection between operations and relations by looking at their

relationship in a boolean frame of reference.

Let B = {0, 1} and let k be a positive integer.

A k-variable boolean function is a mapping B^k -> B.

A k-place boolean relation is a subset of B^k.

Any k-place relation L, as a subset of B^k, has a corresponding

"indicator function", also called a "characteristic function",

f_L : B^k -> B defined such that f_L (x) = 1 if x is in L and

f_L = 0 if x is not in L.

Any k-variable function f : B^k -> B is the indicator function of

a k-place relation L_f consisting of all the x in B^k where f(x) = 1.

This called the "fiber of 1" or the "pre-image of 1" in B^k and is

commonly notated rather obtusely as [f^(-1)](1).

I'll post a better formatted copy on my blog when I get a chance.

Regards,

Jon

On 7/23/2019 7:55 PM, joseph simpson wrote:

Cf: Animated Logical Graphs : 21

At: https://inquiryintoinquiry.com/2019/07/12/animated-logical-graphs-%e2%80%a2-21/

> The area is associated with the operations

> of "equivalence" and "distinction".

>

> I tend to view equivalence and distinction

> as relationships as opposed to operations.

> I do not know if this makes any significant

> difference in this context.

I invoked the general concepts of equivalence and distinction here
> of "equivalence" and "distinction".

>

> I tend to view equivalence and distinction

> as relationships as opposed to operations.

> I do not know if this makes any significant

> difference in this context.

to connect with a wider backdrop of ideas we need to keep in mind

but since we've been focusing on boolean functions to coordinate

the semantics of propositional calculi we can get a sense of the

connection between operations and relations by looking at their

relationship in a boolean frame of reference.

Let B = {0, 1} and let k be a positive integer.

A k-variable boolean function is a mapping B^k -> B.

A k-place boolean relation is a subset of B^k.

Any k-place relation L, as a subset of B^k, has a corresponding

"indicator function", also called a "characteristic function",

f_L : B^k -> B defined such that f_L (x) = 1 if x is in L and

f_L = 0 if x is not in L.

Any k-variable function f : B^k -> B is the indicator function of

a k-place relation L_f consisting of all the x in B^k where f(x) = 1.

This called the "fiber of 1" or the "pre-image of 1" in B^k and is

commonly notated rather obtusely as [f^(-1)](1).

I'll post a better formatted copy on my blog when I get a chance.

Regards,

Jon

On 7/23/2019 7:55 PM, joseph simpson wrote:

Aug 12, 2019, 9:04:15 AM8/12/19

to ontolo...@googlegroups.com, joseph simpson, Sys Sci, Structural Modeling

Joe & All,

Here's a link to the blog edition of my last post:

Animated Logical Graphs : 29

https://inquiryintoinquiry.com/2019/08/11/animated-logical-graphs-%e2%80%a2-29/

Should be easier on the eyes, and I added a few bits for clarity's sake.

Regards,

Jon

> This is called the "fiber of 1" or the "pre-image of 1" in B^k and is

Here's a link to the blog edition of my last post:

Animated Logical Graphs : 29

https://inquiryintoinquiry.com/2019/08/11/animated-logical-graphs-%e2%80%a2-29/

Should be easier on the eyes, and I added a few bits for clarity's sake.

Regards,

Jon

> commonly notated rather obtusely as [f^(-1)](1).

>

> I'll post a better formatted copy on my blog when I get a chance.

>

> Regards,

>

> Jon

>

> On 7/23/2019 7:55 PM, joseph simpson wrote:

>> Jon:

>>

>> Very interesting work and ideas.

>>

>> I went back to this section to address an area where I have some specific

>> questions.

>>

>> The area is associated with the operations of "equivalence" and

>> "distinction".

>>

>> I tend to view equivalence and distinction as relationships as opposed to

>> operations.? I do not know if this makes any significant difference in this
>

> I'll post a better formatted copy on my blog when I get a chance.

>

> Regards,

>

> Jon

>

> On 7/23/2019 7:55 PM, joseph simpson wrote:

>> Jon:

>>

>> Very interesting work and ideas.

>>

>> I went back to this section to address an area where I have some specific

>> questions.

>>

>> The area is associated with the operations of "equivalence" and

>> "distinction".

>>

>> I tend to view equivalence and distinction as relationships as opposed to

Aug 25, 2019, 3:54:32 PM8/25/19

to Ontolog Forum, Structural Modeling, SysSciWG, Cybernetic Communications, Laws Of Form Group

Cf: Animated Logical Graphs : 30

At: https://inquiryintoinquiry.com/2019/08/25/animated-logical-graphs-%e2%80%a2-30/

The duality between Entitative and Existential interpretations

of logical graphs is a basic example of a mathematical symmetry,

in this case a symmetry of order 2. Symmetries of this and

higher orders give us conceptual handles on excess complexities

in the manifold of sensuous impressions, making it well worth

our trouble to seek them out and grasp them where we find them.

In that vein, here's a Rosetta Stone to give us a grounding in

the relationship between boolean functions and our two readings

of logical graphs:

| Table too big to get a good screen shot.

| Please see the blog post linked above.

Regards,

Jon

At: https://inquiryintoinquiry.com/2019/08/25/animated-logical-graphs-%e2%80%a2-30/

The duality between Entitative and Existential interpretations

of logical graphs is a basic example of a mathematical symmetry,

in this case a symmetry of order 2. Symmetries of this and

higher orders give us conceptual handles on excess complexities

in the manifold of sensuous impressions, making it well worth

our trouble to seek them out and grasp them where we find them.

In that vein, here's a Rosetta Stone to give us a grounding in

the relationship between boolean functions and our two readings

of logical graphs:

| Table too big to get a good screen shot.

| Please see the blog post linked above.

Regards,

Jon

Sep 2, 2019, 8:56:23 AM9/2/19

to syss...@googlegroups.com, Aleksandar Malečić, Ontolog Forum, Structural Modeling, Cybernetic Communications, Laws Of Form Group

Cf: Animated Logical Graphs : 31

At: https://inquiryintoinquiry.com/2019/09/01/animated-logical-graphs-%e2%80%a2-31/

On 7/13/2019 5:33 AM, Aleksandar Male??i?? wrote:

> Each step on its own, as far as I can follow them, makes sense. You are,

> if I understand it correctly, trying to figure out something fundamental,

> the rock bottom reality. When can we expect that results of such a research

> to become "applicable to more than one of the traditional departments of

> knowledge" (http://isss.org/world/about-the-isss)? What kinds of tragedy,

> disaster, misunderstanding, mismanagement, or failure would/will be

> preventable by your approach?

Aleksandar, All,

The larger questions asked above -- interdisciplinary inquiry, the

interest in integration, the synthesis of ideas across isolated silos

of specialization, and what it might mean for the future -- are issues

Susan Awbrey and I addressed from a pragmatic semiotic perspective:

* Awbrey, S.M., and Awbrey, J.L. (2001),

"Conceptual Barriers to Creating Integrative Universities", Organization :

The Interdisciplinary Journal of Organization, Theory, and Society 8(2),

Sage Publications, London, UK, 269-284.

Abstract ( https://journals.sagepub.com/doi/abs/10.1177/1350508401082013 )

Online ( https://www.academia.edu/1266492/Conceptual_Barriers_to_Creating_Integrative_Universities )

* Awbrey, S.M., and Awbrey, J.L. (1999),

"Organizations of Learning or Learning Organizations : The Challenge

of Creating Integrative Universities for the Next Century", Second

International Conference of the Journal 'Organization', Re-Organizing

Knowledge, Trans-Forming Institutions : Knowing, Knowledge, and the

University in the 21st Century, University of Massachusetts, Amherst, MA.

Online ( http://www.iupui.edu/~arisbe/menu/library/aboutcsp/awbrey/integrat.htm )

From that vantage point, what I'm about here is just a subgoal of a subgoal,

panning what bits of elemental substrate can be found ever nearer that elusive

"rock bottom reality".

Regards,

Jon

At: https://inquiryintoinquiry.com/2019/09/01/animated-logical-graphs-%e2%80%a2-31/

On 7/13/2019 5:33 AM, Aleksandar Male??i?? wrote:

> Each step on its own, as far as I can follow them, makes sense. You are,

> if I understand it correctly, trying to figure out something fundamental,

> the rock bottom reality. When can we expect that results of such a research

> to become "applicable to more than one of the traditional departments of

> knowledge" (http://isss.org/world/about-the-isss)? What kinds of tragedy,

> disaster, misunderstanding, mismanagement, or failure would/will be

> preventable by your approach?

The larger questions asked above -- interdisciplinary inquiry, the

interest in integration, the synthesis of ideas across isolated silos

of specialization, and what it might mean for the future -- are issues

Susan Awbrey and I addressed from a pragmatic semiotic perspective:

* Awbrey, S.M., and Awbrey, J.L. (2001),

"Conceptual Barriers to Creating Integrative Universities", Organization :

The Interdisciplinary Journal of Organization, Theory, and Society 8(2),

Sage Publications, London, UK, 269-284.

Abstract ( https://journals.sagepub.com/doi/abs/10.1177/1350508401082013 )

Online ( https://www.academia.edu/1266492/Conceptual_Barriers_to_Creating_Integrative_Universities )

* Awbrey, S.M., and Awbrey, J.L. (1999),

"Organizations of Learning or Learning Organizations : The Challenge

of Creating Integrative Universities for the Next Century", Second

International Conference of the Journal 'Organization', Re-Organizing

Knowledge, Trans-Forming Institutions : Knowing, Knowledge, and the

University in the 21st Century, University of Massachusetts, Amherst, MA.

Online ( http://www.iupui.edu/~arisbe/menu/library/aboutcsp/awbrey/integrat.htm )

From that vantage point, what I'm about here is just a subgoal of a subgoal,

panning what bits of elemental substrate can be found ever nearer that elusive

"rock bottom reality".

Regards,

Jon

Jun 14, 2020, 10:25:13 AM6/14/20

to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Cf: Animated Logical Graphs • 32

At: http://inquiryintoinquiry.com/2020/06/14/animated-logical-graphs-%e2%80%a2-32/

Re: R.J. Lipton and K.W. Regan

https://rjlipton.wordpress.com/about-me/

::: Proof Checking

https://rjlipton.wordpress.com/2020/06/13/proof-checking-not-line-by-line/

Here's a place where I explore several different shapes of proofs within propositional calculus deriving from the

graphical systems of Charles S. Peirce and G. Spencer Brown.

• Propositional Equation Reasoning Systems • Analysis of Contingent Propositions

https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems#Analysis_of_contingent_propositions

I don't know whether that helps any with P ≟ NP but it does supply a lot of nice pictures to contemplate.

Resources

=========

• Logical Graphs

https://oeis.org/wiki/Logical_Graphs

• Propositional Equation Reasoning Systems

https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems

Regards,

Jon

inquiry into inquiry: https://inquiryintoinquiry.com/

academia: https://independent.academia.edu/JonAwbrey

oeiswiki: https://www.oeis.org/wiki/User:Jon_Awbrey

facebook page: https://www.facebook.com/JonnyCache

At: http://inquiryintoinquiry.com/2020/06/14/animated-logical-graphs-%e2%80%a2-32/

Re: R.J. Lipton and K.W. Regan

https://rjlipton.wordpress.com/about-me/

::: Proof Checking

https://rjlipton.wordpress.com/2020/06/13/proof-checking-not-line-by-line/

Here's a place where I explore several different shapes of proofs within propositional calculus deriving from the

graphical systems of Charles S. Peirce and G. Spencer Brown.

• Propositional Equation Reasoning Systems • Analysis of Contingent Propositions

https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems#Analysis_of_contingent_propositions

I don't know whether that helps any with P ≟ NP but it does supply a lot of nice pictures to contemplate.

Resources

=========

• Logical Graphs

https://oeis.org/wiki/Logical_Graphs

• Propositional Equation Reasoning Systems

https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems

Regards,

Jon

inquiry into inquiry: https://inquiryintoinquiry.com/

academia: https://independent.academia.edu/JonAwbrey

oeiswiki: https://www.oeis.org/wiki/User:Jon_Awbrey

Jun 18, 2020, 3:12:06 PM6/18/20

to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Cf: Animated Logical Graphs • 33

At: http://inquiryintoinquiry.com/2020/06/18/animated-logical-graphs-%e2%80%a2-33/

A reader's request for more examples of animated logical graphs

prompted me to look again at the User Guide for my Theme One Program,

whose exposition develops a series of logical graphs increasing in

complexity from extremely simple to more complex and interesting

than any I've posted online so far.

• Theme One Program • User Guide

https://www.academia.edu/5211369/Theme_One_Program_User_Guide

I'm thinking now it may be worthwhile to look at those examples again

and see if they're suitable for recycling as a series of blog posts.

• Proof Animations

https://oeis.org/wiki/User:Jon_Awbrey/ANIMATION#Proof_Animations

• Survey of Theme One Program

https://inquiryintoinquiry.com/2018/02/25/survey-of-theme-one-program-%e2%80%a2-2/

• Survey of Animated Logical Graphs

https://inquiryintoinquiry.com/2019/05/22/survey-of-animated-logical-graphs-%e2%80%a2-2/

At: http://inquiryintoinquiry.com/2020/06/18/animated-logical-graphs-%e2%80%a2-33/

A reader's request for more examples of animated logical graphs

prompted me to look again at the User Guide for my Theme One Program,

whose exposition develops a series of logical graphs increasing in

complexity from extremely simple to more complex and interesting

than any I've posted online so far.

• Theme One Program • User Guide

https://www.academia.edu/5211369/Theme_One_Program_User_Guide

I'm thinking now it may be worthwhile to look at those examples again

and see if they're suitable for recycling as a series of blog posts.

• Proof Animations

https://oeis.org/wiki/User:Jon_Awbrey/ANIMATION#Proof_Animations

• Survey of Theme One Program

https://inquiryintoinquiry.com/2018/02/25/survey-of-theme-one-program-%e2%80%a2-2/

• Survey of Animated Logical Graphs

https://inquiryintoinquiry.com/2019/05/22/survey-of-animated-logical-graphs-%e2%80%a2-2/

Aug 11, 2020, 11:20:19 AM8/11/20

to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Cf: Animated Logical Graphs • 34

http://inquiryintoinquiry.com/2020/08/11/animated-logical-graphs-34/

Re: Ontolog Forum ( https://groups.google.com/d/topic/ontolog-forum/7iwyulzDpFA/overview )

::: John Sowa ( https://groups.google.com/d/msg/ontolog-forum/7iwyulzDpFA/dwlvIiMaBwAJ )

Re: Peirce List ( https://list.iupui.edu/sympa/arc/peirce-l/2020-08/thrd2.html#00051 )

::: John Sowa ( https://list.iupui.edu/sympa/arc/peirce-l/2020-08/msg00051.html )

All,

This adds a few resource links to an earlier reply on

the Ontolog Forum and the Peirce List. I added it to

my blog series on Animated Logical Graphs mostly just

by way of reminding myself to get back to that.

Dear John,

I can't imagine why anyone would bother with Peirce's logic

if it's just Frege and Russell in a different syntax, which

has been the opinion I usually get from FOL fans. But the fact

is Peirce's 1870 "Logic of Relatives" is already far in advance

of anything we'd see again for a century, in principle in most

places, in practice in many others, chock full of revolutionary

ideas, not all of which he developed fully in subsequent work.

Although I studied the 1870 Logic from early on I did not realize

how far ahead of its time it was until I began reading approaches

to logic from category-theoretic and computation-theoretic angles

in the 1970s and 1980s. An indication of Peirce's innovations can

be found in the series of selections and commentary I started on

the 1870 Logic of Relatives.

Here's the work in progress so far on the OEIS Wiki.

* Peirce's 1870 Logic Of Relatives

https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Overview

* Part 1

https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1

* Part 2

https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2

* Part 3

https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_3

Here's the overview for a parallel series of blog posts.

* Peirce's 1870 Logic Of Relatives • Overview

https://inquiryintoinquiry.com/2019/09/24/peirces-1870-logic-of-relatives-%e2%80%a2-overview/

* Proof Animations

https://oeis.org/wiki/User:Jon_Awbrey/ANIMATION#Proof_Animations

* Survey of Theme One Program

https://inquiryintoinquiry.com/2018/02/25/survey-of-theme-one-program-%e2%80%a2-2/

* Survey of Animated Logical Graphs

https://inquiryintoinquiry.com/2019/05/22/survey-of-animated-logical-graphs-%e2%80%a2-2/

* Propositional Equation Reasoning Systems

https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems

Regards,

Jon

http://inquiryintoinquiry.com/2020/08/11/animated-logical-graphs-34/

Re: Ontolog Forum ( https://groups.google.com/d/topic/ontolog-forum/7iwyulzDpFA/overview )

::: John Sowa ( https://groups.google.com/d/msg/ontolog-forum/7iwyulzDpFA/dwlvIiMaBwAJ )

Re: Peirce List ( https://list.iupui.edu/sympa/arc/peirce-l/2020-08/thrd2.html#00051 )

::: John Sowa ( https://list.iupui.edu/sympa/arc/peirce-l/2020-08/msg00051.html )

All,

This adds a few resource links to an earlier reply on

the Ontolog Forum and the Peirce List. I added it to

my blog series on Animated Logical Graphs mostly just

by way of reminding myself to get back to that.

Dear John,

I can't imagine why anyone would bother with Peirce's logic

if it's just Frege and Russell in a different syntax, which

has been the opinion I usually get from FOL fans. But the fact

is Peirce's 1870 "Logic of Relatives" is already far in advance

of anything we'd see again for a century, in principle in most

places, in practice in many others, chock full of revolutionary

ideas, not all of which he developed fully in subsequent work.

Although I studied the 1870 Logic from early on I did not realize

how far ahead of its time it was until I began reading approaches

to logic from category-theoretic and computation-theoretic angles

in the 1970s and 1980s. An indication of Peirce's innovations can

be found in the series of selections and commentary I started on

the 1870 Logic of Relatives.

Here's the work in progress so far on the OEIS Wiki.

* Peirce's 1870 Logic Of Relatives

https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Overview

* Part 1

https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1

* Part 2

https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2

* Part 3

https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_3

Here's the overview for a parallel series of blog posts.

* Peirce's 1870 Logic Of Relatives • Overview

https://inquiryintoinquiry.com/2019/09/24/peirces-1870-logic-of-relatives-%e2%80%a2-overview/

* Proof Animations

https://oeis.org/wiki/User:Jon_Awbrey/ANIMATION#Proof_Animations

* Survey of Theme One Program

https://inquiryintoinquiry.com/2018/02/25/survey-of-theme-one-program-%e2%80%a2-2/

* Survey of Animated Logical Graphs

https://inquiryintoinquiry.com/2019/05/22/survey-of-animated-logical-graphs-%e2%80%a2-2/

* Propositional Equation Reasoning Systems

https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems

Regards,

Jon

Aug 19, 2020, 10:40:09 AM8/19/20

to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Cf: Animated Logical Graphs ??? 35

http://inquiryintoinquiry.com/2020/08/19/animated-logical-graphs-35/

Re: Richard J. Lipton

https://rjlipton.wordpress.com/about-me/

::: Logical Complexity Of Proofs

https://rjlipton.wordpress.com/2020/08/19/logical-complexity-of-proofs/

The smoothest way I know to do propositional calculus

is by using minimal negation operators as primitives,

parsing propositional formulas into (painted and rooted)

cactus graphs, and using the appropriate extension of

the axiom set from Charles S. Peirce's logical graphs

and G. Spencer Brown's laws of form.

There's a quick link here:

* Cactus Language for Propositional Logic

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Cactus_Language_for_Propositional_Logic

Resources

=========

* Logic Syllabus

https://oeis.org/wiki/Logic_Syllabus

* Futures Of Logical Graphs

https://oeis.org/wiki/Futures_Of_Logical_Graphs

* Minimal Negation Operators

https://oeis.org/wiki/Minimal_negation_operator

http://inquiryintoinquiry.com/2020/08/19/animated-logical-graphs-35/

Re: Richard J. Lipton

https://rjlipton.wordpress.com/about-me/

::: Logical Complexity Of Proofs

https://rjlipton.wordpress.com/2020/08/19/logical-complexity-of-proofs/

The smoothest way I know to do propositional calculus

is by using minimal negation operators as primitives,

parsing propositional formulas into (painted and rooted)

cactus graphs, and using the appropriate extension of

the axiom set from Charles S. Peirce's logical graphs

and G. Spencer Brown's laws of form.

There's a quick link here:

* Cactus Language for Propositional Logic

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Cactus_Language_for_Propositional_Logic

Resources

=========

* Logic Syllabus

https://oeis.org/wiki/Logic_Syllabus

* Futures Of Logical Graphs

https://oeis.org/wiki/Futures_Of_Logical_Graphs

* Minimal Negation Operators

https://oeis.org/wiki/Minimal_negation_operator

Aug 21, 2020, 11:00:24 AM8/21/20

to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Cf: Animated Logical Graphs ??? 36

http://inquiryintoinquiry.com/2020/08/21/animated-logical-graphs-36/

Re: Richard J. Lipton

https://rjlipton.wordpress.com/about-me/

::: Logical Complexity Of Proofs

https://rjlipton.wordpress.com/2020/08/19/logical-complexity-of-proofs/

Dear Dick,

You asked, "Is this measure, the logical flow of a proof, of any interest?"

I wasn't quite clear how you define the measure of flow in a proof --

it seemed to have something to do with the number of implication arrows

in the argument structure?

But this does bring up interesting issues of "proof style" ...

Propositional calculus as a formal language and boolean functions

as an object domain form an instructive microcosm for many issues

of logic writ large. The relation between proof theory and model

theory is one of those issues, despite, or maybe in virtue of,

propositional logic's status as a special case.

Folks who pursue the CSP-GSB line of development

in graphical syntax for propositional calculus are

especially likely to notice the following dimensions

of proof style.

Formal Duality

==============

This goes back to Peirce's discovery of the "amphecks"

( https://oeis.org/wiki/Ampheck ) and the duality between

Not Both (nand ( https://oeis.org/wiki/Logical_NAND ) ) and

Both Not (nnor ( https://oeis.org/wiki/Logical_NNOR ) ).

The same duality is present in Peirce's graphical systems

for propositional calculus. It is analogous to the duality

in projective geometry and it means we are always proving

two theorems for the price of one. That's a reduction in

complexity -- it raises the question of how many such

group-theoretic reductions we can find.

To be continued ...

Resources

=========

* Cactus Language

https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview

Applications

============

* Applications of a Propositional Calculator ??? Constraint Satisfaction Problems

https://www.academia.edu/4727842/Applications_of_a_Propositional_Calculator_Constraint_Satisfaction_Problems

* Exploratory Qualitative Analysis of Sequential Observation Data

http://web.archive.org/web/20180828161616/http://intersci.ss.uci.edu/wiki/index.php/Exploratory_Qualitative_Analysis_of_Sequential_Observation_Data

Regards,

Jon

http://inquiryintoinquiry.com/2020/08/21/animated-logical-graphs-36/

Re: Richard J. Lipton

https://rjlipton.wordpress.com/about-me/

::: Logical Complexity Of Proofs

https://rjlipton.wordpress.com/2020/08/19/logical-complexity-of-proofs/

You asked, "Is this measure, the logical flow of a proof, of any interest?"

I wasn't quite clear how you define the measure of flow in a proof --

it seemed to have something to do with the number of implication arrows

in the argument structure?

But this does bring up interesting issues of "proof style" ...

Propositional calculus as a formal language and boolean functions

as an object domain form an instructive microcosm for many issues

of logic writ large. The relation between proof theory and model

theory is one of those issues, despite, or maybe in virtue of,

propositional logic's status as a special case.

Folks who pursue the CSP-GSB line of development

in graphical syntax for propositional calculus are

especially likely to notice the following dimensions

of proof style.

Formal Duality

==============

This goes back to Peirce's discovery of the "amphecks"

( https://oeis.org/wiki/Ampheck ) and the duality between

Not Both (nand ( https://oeis.org/wiki/Logical_NAND ) ) and

Both Not (nnor ( https://oeis.org/wiki/Logical_NNOR ) ).

The same duality is present in Peirce's graphical systems

for propositional calculus. It is analogous to the duality

in projective geometry and it means we are always proving

two theorems for the price of one. That's a reduction in

complexity -- it raises the question of how many such

group-theoretic reductions we can find.

To be continued ...

Resources

=========

* Cactus Language

https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview

Applications

============

* Applications of a Propositional Calculator ??? Constraint Satisfaction Problems

https://www.academia.edu/4727842/Applications_of_a_Propositional_Calculator_Constraint_Satisfaction_Problems

* Exploratory Qualitative Analysis of Sequential Observation Data

http://web.archive.org/web/20180828161616/http://intersci.ss.uci.edu/wiki/index.php/Exploratory_Qualitative_Analysis_of_Sequential_Observation_Data

Regards,

Jon

Aug 22, 2020, 2:20:22 PM8/22/20

to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Cf: Animated Logical Graphs • 37

http://inquiryintoinquiry.com/2020/08/22/animated-logical-graphs-37/

Re: Richard J. Lipton

https://rjlipton.wordpress.com/about-me/

::: Logical Complexity Of Proofs

https://rjlipton.wordpress.com/2020/08/19/logical-complexity-of-proofs/

Another dimension of proof style has to do with how much information

is kept or lost as the argument develops. For the moment let's focus

on classical deductive reasoning at the propositional level. Then we

can distinguish between "equational inferences", which keep all the

information represented by the input propositions, and "implicational

inferences", which permit information to be lost as the proof proceeds.

Information-Preserving vs. Information-Reducing Inferences

==========================================================

Implicit in Peirce's systems of logical graphs is the ability to use

equational inferences. Spencer Brown drew this out and turned it to

great advantage in his revival of Peirce's graphical forms. As it

affects "logical flow" this allows for bi-directional or reversible

flows, you might even say a "logical equilibrium" between two states

of information.

It is probably obvious when we stop to think about it, but

seldom remarked, that all the more familiar inference rules,

like modus ponens and resolution or transitivity, entail in

general a loss of information as we traverse their arrows or

turnstiles.

For example, the usual form of modus ponens takes us from knowing

p and p => q to knowing q but in fact we know more, we actually know

p and q. With that in mind we can formulate two variants of modus ponens,

one reducing and one preserving the actual state of information, as shown

in the following figure.

Modus Ponens Variants

https://inquiryintoinquiry.files.wordpress.com/2020/08/modus-ponens-variants.png

There's more discussion of this topic at the following location.

* Propositional Equation Reasoning Systems : Computation and Inference as Semiosis

https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems#Computation_and_inference_as_semiosis

To be continued ...

Regards,

Jon

http://inquiryintoinquiry.com/2020/08/22/animated-logical-graphs-37/

Re: Richard J. Lipton

https://rjlipton.wordpress.com/about-me/

::: Logical Complexity Of Proofs

https://rjlipton.wordpress.com/2020/08/19/logical-complexity-of-proofs/

is kept or lost as the argument develops. For the moment let's focus

on classical deductive reasoning at the propositional level. Then we

can distinguish between "equational inferences", which keep all the

information represented by the input propositions, and "implicational

inferences", which permit information to be lost as the proof proceeds.

Information-Preserving vs. Information-Reducing Inferences

==========================================================

Implicit in Peirce's systems of logical graphs is the ability to use

equational inferences. Spencer Brown drew this out and turned it to

great advantage in his revival of Peirce's graphical forms. As it

affects "logical flow" this allows for bi-directional or reversible

flows, you might even say a "logical equilibrium" between two states

of information.

It is probably obvious when we stop to think about it, but

seldom remarked, that all the more familiar inference rules,

like modus ponens and resolution or transitivity, entail in

general a loss of information as we traverse their arrows or

turnstiles.

For example, the usual form of modus ponens takes us from knowing

p and p => q to knowing q but in fact we know more, we actually know

p and q. With that in mind we can formulate two variants of modus ponens,

one reducing and one preserving the actual state of information, as shown

in the following figure.

Modus Ponens Variants

https://inquiryintoinquiry.files.wordpress.com/2020/08/modus-ponens-variants.png

There's more discussion of this topic at the following location.

* Propositional Equation Reasoning Systems : Computation and Inference as Semiosis

https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems#Computation_and_inference_as_semiosis

To be continued ...

Regards,

Jon

Aug 24, 2020, 10:04:07 AM8/24/20

to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Cf: Survey of Animated Logical Graphs • 3

http://inquiryintoinquiry.com/2020/08/23/survey-of-animated-logical-graphs-3/

All,

I just updated my Survey of blog and wiki posts relating to

Animated Logical Graphs. A great many links went missing

when my old worksite, the InterSciWiki, went offline so

I've been repairing those as I run across them.

Beginnings

==========

Logical Graphs : Introduction

https://inquiryintoinquiry.com/2008/07/29/logical-graphs-1/

Logical Graphs : Formal Development

https://inquiryintoinquiry.com/2008/09/19/logical-graphs-2/

Elements

========

Logic Syllabus

https://oeis.org/wiki/Logic_Syllabus

Logical Graphs

https://oeis.org/wiki/Logical_Graphs

========

Peirce's Law

* This Blog

https://inquiryintoinquiry.com/2008/10/06/peirce-s-law/

* OEIS Wiki

https://oeis.org/wiki/Peirce%27s_law

Praeclarum Theorema

* This Blog

https://inquiryintoinquiry.com/2008/10/05/praeclarum-theorema/

* OEIS Wiki

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

Excursions

==========

Cactus Language

https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview

Applications

============

Applications of a Propositional Calculator :

https://oeis.org/wiki/Differential_Analytic_Turing_Automata

Survey of Theme One Program

https://inquiryintoinquiry.com/2018/02/25/survey-of-theme-one-program-2/

Blog Dialogs

============

Animated Logical Graphs

https://inquiryintoinquiry.com/2015/01/08/animated-logical-graphs-1/

https://inquiryintoinquiry.com/2015/01/14/animated-logical-graphs-2/

https://inquiryintoinquiry.com/2015/01/26/animated-logical-graphs-3/

...

https://inquiryintoinquiry.com/2020/08/11/animated-logical-graphs-34/

https://inquiryintoinquiry.com/2020/08/19/animated-logical-graphs-35/

https://inquiryintoinquiry.com/2020/08/21/animated-logical-graphs-36/

https://inquiryintoinquiry.com/2020/08/22/animated-logical-graphs-37/

Regards,

Jon

http://inquiryintoinquiry.com/2020/08/23/survey-of-animated-logical-graphs-3/

All,

I just updated my Survey of blog and wiki posts relating to

Animated Logical Graphs. A great many links went missing

when my old worksite, the InterSciWiki, went offline so

I've been repairing those as I run across them.

Beginnings

==========

Logical Graphs : Introduction

https://inquiryintoinquiry.com/2008/07/29/logical-graphs-1/

Logical Graphs : Formal Development

https://inquiryintoinquiry.com/2008/09/19/logical-graphs-2/

Elements

========

Logic Syllabus

https://oeis.org/wiki/Logic_Syllabus

Logical Graphs

https://oeis.org/wiki/Logical_Graphs

Propositional Equation Reasoning Systems

https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems

Examples
https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems

========

Peirce's Law

* This Blog

https://inquiryintoinquiry.com/2008/10/06/peirce-s-law/

* OEIS Wiki

https://oeis.org/wiki/Peirce%27s_law

Praeclarum Theorema

* This Blog

https://inquiryintoinquiry.com/2008/10/05/praeclarum-theorema/

* OEIS Wiki

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

Excursions

==========

Cactus Language

https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview

Applications

============

Applications of a Propositional Calculator :

Constraint Satisfaction Problems

https://www.academia.edu/4727842/Applications_of_a_Propositional_Calculator_Constraint_Satisfaction_Problems

https://www.academia.edu/4727842/Applications_of_a_Propositional_Calculator_Constraint_Satisfaction_Problems

Exploratory Qualitative Analysis of Sequential Observation Data

http://web.archive.org/web/20180828161616/http://intersci.ss.uci.edu/wiki/index.php/Exploratory_Qualitative_Analysis_of_Sequential_Observation_Data

Differential Analytic Turing Automata
http://web.archive.org/web/20180828161616/http://intersci.ss.uci.edu/wiki/index.php/Exploratory_Qualitative_Analysis_of_Sequential_Observation_Data

https://oeis.org/wiki/Differential_Analytic_Turing_Automata

Survey of Theme One Program

Blog Dialogs

============

Animated Logical Graphs

https://inquiryintoinquiry.com/2015/01/08/animated-logical-graphs-1/

https://inquiryintoinquiry.com/2015/01/14/animated-logical-graphs-2/

https://inquiryintoinquiry.com/2015/01/26/animated-logical-graphs-3/

...

https://inquiryintoinquiry.com/2020/08/11/animated-logical-graphs-34/

https://inquiryintoinquiry.com/2020/08/19/animated-logical-graphs-35/

https://inquiryintoinquiry.com/2020/08/21/animated-logical-graphs-36/

https://inquiryintoinquiry.com/2020/08/22/animated-logical-graphs-37/

Regards,

Jon

Aug 25, 2020, 4:01:07 PM8/25/20

to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Cf: Animated Logical Graphs • 38

http://inquiryintoinquiry.com/2020/08/25/animated-logical-graphs-38/

Re: Richard J. Lipton

https://rjlipton.wordpress.com/about-me/

::: Logical Complexity Of Proofs

https://rjlipton.wordpress.com/2020/08/19/logical-complexity-of-proofs/

Three examples of propositional proofs in logical graphs using

equational inference rules can be found at the following location.

* Propositional Equation Reasoning Systems • Exemplary Proofs

https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems#Exemplary_proofs

By way of a quick overview, here are the animated proofs of

these examples, along with links to their detailed descriptions.

Peirce's Law

============

https://inquiryintoinquiry.com/2008/10/06/peirces-law/

https://oeis.org/wiki/Peirce%27s_law

https://inquiryintoinquiry.files.wordpress.com/2012/01/peirces-law-2-0-animation.gif

Praeclarum Theorema

===================

https://inquiryintoinquiry.com/2008/10/05/praeclarum-theorema/

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

https://inquiryintoinquiry.files.wordpress.com/2012/01/praeclarum-theorema-2-0-animation.gif

Two-Thirds Majority Function

============================

https://oeis.org/wiki/Logical_Graphs#Two-thirds_majority_function

https://oeis.org/wiki/Futures_Of_Logical_Graphs#Two-thirds_majority_function

https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems#Two-thirds_majority_function

https://inquiryintoinquiry.files.wordpress.com/2020/08/two-thirds-majority-function-500-x-250-animation.gif

Regards,

Jon

http://inquiryintoinquiry.com/2020/08/25/animated-logical-graphs-38/

Re: Richard J. Lipton

https://rjlipton.wordpress.com/about-me/

::: Logical Complexity Of Proofs

https://rjlipton.wordpress.com/2020/08/19/logical-complexity-of-proofs/

equational inference rules can be found at the following location.

* Propositional Equation Reasoning Systems • Exemplary Proofs

https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems#Exemplary_proofs

By way of a quick overview, here are the animated proofs of

these examples, along with links to their detailed descriptions.

Peirce's Law

============

https://inquiryintoinquiry.com/2008/10/06/peirces-law/

https://oeis.org/wiki/Peirce%27s_law

https://inquiryintoinquiry.files.wordpress.com/2012/01/peirces-law-2-0-animation.gif

Praeclarum Theorema

===================

https://inquiryintoinquiry.com/2008/10/05/praeclarum-theorema/

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

https://inquiryintoinquiry.files.wordpress.com/2012/01/praeclarum-theorema-2-0-animation.gif

Two-Thirds Majority Function

============================

https://oeis.org/wiki/Logical_Graphs#Two-thirds_majority_function

https://oeis.org/wiki/Futures_Of_Logical_Graphs#Two-thirds_majority_function

https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems#Two-thirds_majority_function

https://inquiryintoinquiry.files.wordpress.com/2020/08/two-thirds-majority-function-500-x-250-animation.gif

Regards,

Jon

Sep 10, 2020, 3:12:45 PM9/10/20

to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Cf: Animated Logical Graphs • 39

http://inquiryintoinquiry.com/2020/09/10/animated-logical-graphs-39/

Happy Peirce's Birthday, Everyone !!!

Re: Richard J. Lipton

https://rjlipton.wordpress.com/about-me/

::: Logical Complexity Of Proofs

https://rjlipton.wordpress.com/2020/08/19/logical-complexity-of-proofs/

We've been discussing aspects of proof style arising in connection with

the complexity of proofs. In previous posts we took up (1) the aspect of

formal duality, reflecting in passing on the prospect of higher symmetries,

and (2) the spectrum ranging from information-reducing to information-preserving

inference rules. Here's a quick recap --

* Animated Logical Graphs

=========================

A third aspect of proof style arising in this connection is the degree

of insight demanded and demonstrated in the performance of a proof.

Generally speaking, the same endpoint can be reached in many different

ways from given starting points, by paths ranging from those exhibiting

appreciable insight to those exercising little more than persistence in

sticking to a set routine.

A modicum of insight suffices to suggest the quality of "insight" resists

pinning down in a succinct definition but we do tend to recognize it when

we see it, so let me inch forward by highlighting its salient features in

a graded series of examples.

http://inquiryintoinquiry.com/2020/09/10/animated-logical-graphs-39/

Happy Peirce's Birthday, Everyone !!!

Re: Richard J. Lipton

https://rjlipton.wordpress.com/about-me/

::: Logical Complexity Of Proofs

https://rjlipton.wordpress.com/2020/08/19/logical-complexity-of-proofs/

the complexity of proofs. In previous posts we took up (1) the aspect of

formal duality, reflecting in passing on the prospect of higher symmetries,

and (2) the spectrum ranging from information-reducing to information-preserving

inference rules. Here's a quick recap --

* Animated Logical Graphs

=========================

https://inquiryintoinquiry.com/2020/08/19/animated-logical-graphs-35/

https://inquiryintoinquiry.com/2020/08/21/animated-logical-graphs-36/

https://inquiryintoinquiry.com/2020/08/22/animated-logical-graphs-37/

https://inquiryintoinquiry.com/2020/08/25/animated-logical-graphs-38/
https://inquiryintoinquiry.com/2020/08/21/animated-logical-graphs-36/

https://inquiryintoinquiry.com/2020/08/22/animated-logical-graphs-37/

A third aspect of proof style arising in this connection is the degree

of insight demanded and demonstrated in the performance of a proof.

Generally speaking, the same endpoint can be reached in many different

ways from given starting points, by paths ranging from those exhibiting

appreciable insight to those exercising little more than persistence in

sticking to a set routine.

A modicum of insight suffices to suggest the quality of "insight" resists

pinning down in a succinct definition but we do tend to recognize it when

we see it, so let me inch forward by highlighting its salient features in

a graded series of examples.

Sep 26, 2020, 7:48:15 PM9/26/20

to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Cf: Animated Logical Graphs • 40

http://inquiryintoinquiry.com/2020/09/26/animated-logical-graphs-40/

| Note: as always, follow the above link for better formatting,

| especially if the unicode math symbols fail to get through.

One way to see the difference between insight proofs and routine proofs

is to pick a single example of a theorem in propositional calculus and

prove it two ways, one more insightful and one more routine.

The praeclarum theorema, or splendid theorem, is a theorem

of propositional calculus noted and named by G.W. Leibniz,

who stated and proved it in the following manner.

<QUOTE>

If a is b and d is c, then ad will be bc.

This is a fine theorem, which is proved in this way:

a is b, therefore ad is bd (by what precedes),

d is c, therefore bd is bc (again by what precedes),

ad is bd, and bd is bc, therefore ad is bc. Q.E.D.

— Leibniz • Logical Papers, p. 41.

</QUOTE>

Expressed in contemporary logical notation,

the theorem may be written as follows.

Expressed in contemporary logical notation, the theorem may be written as follows.

((a ⇒ b) ∧ (d ⇒ c)) ⇒ ((a ∧ d) ⇒ (b ∧ c))

Using teletype parentheses ( ... ) for the logical negation (p)

of a proposition p and simple concatenation pq for the logical

conjunction of propositions p, q permits the theorem to be

written in the following in-line and lispish ways.

Inline Syntax

=============

( (a (b)) (d (c)) ( (ad (bc)) ))

Lispish ("pretty-printed")

==========================

( (a (b)) (d (c))

( (ad (bc))

))

http://inquiryintoinquiry.com/2020/09/26/animated-logical-graphs-40/

| Note: as always, follow the above link for better formatting,

| especially if the unicode math symbols fail to get through.

One way to see the difference between insight proofs and routine proofs

is to pick a single example of a theorem in propositional calculus and

prove it two ways, one more insightful and one more routine.

The praeclarum theorema, or splendid theorem, is a theorem

of propositional calculus noted and named by G.W. Leibniz,

who stated and proved it in the following manner.

<QUOTE>

If a is b and d is c, then ad will be bc.

This is a fine theorem, which is proved in this way:

a is b, therefore ad is bd (by what precedes),

d is c, therefore bd is bc (again by what precedes),

ad is bd, and bd is bc, therefore ad is bc. Q.E.D.

— Leibniz • Logical Papers, p. 41.

</QUOTE>

Expressed in contemporary logical notation,

the theorem may be written as follows.

Expressed in contemporary logical notation, the theorem may be written as follows.

((a ⇒ b) ∧ (d ⇒ c)) ⇒ ((a ∧ d) ⇒ (b ∧ c))

Using teletype parentheses ( ... ) for the logical negation (p)

of a proposition p and simple concatenation pq for the logical

conjunction of propositions p, q permits the theorem to be

written in the following in-line and lispish ways.

Inline Syntax

=============

( (a (b)) (d (c)) ( (ad (bc)) ))

Lispish ("pretty-printed")

==========================

( (a (b)) (d (c))

( (ad (bc))

))

Sep 29, 2020, 11:40:13 AM9/29/20

to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Cf: Animated Logical Graphs • 41

http://inquiryintoinquiry.com/2020/09/29/animated-logical-graphs-41/

All,

Last time we looked at a formula of propositional logic Leibniz

called a Praeclarum Theorema (PT). We don't concur it's a theorem,

of course, until there's a proof it's identically true and Leibniz

gave an argument to demonstrate that. Written out in one of our

more current formalisms, PT takes the following form.

((a ⇒ b) ∧ (d ⇒ c)) ⇒ ((a ∧ d) ⇒ (b ∧ c))

Somewhat in the spirit of Reduced Instruction Set Computing,

we reformulated PT in a propositional calculus using just

two primitive operations, writing the logical negation of

a proposition p as (p) and the logical conjunction of two

propositions p, q as pq. That gave us a text string in

teletype parentheses and proposition letters, formatted

two ways below.

Figure 1. Praeclarum Theorema Text Strings

https://inquiryintoinquiry.files.wordpress.com/2020/09/praeclarum-theorema-text-strings.png

Our next transformation of the theorem’s expression exploits a

standard correspondence in combinatorics and computer science

between parenthesized symbol strings and trees with symbols

attached to the nodes.

Figure 2. Praeclarum Theorema Parse Graph

https://inquiryintoinquiry.files.wordpress.com/2020/09/praeclarum-theorema-parse-graph-2.0.png

We can see the correspondence between text and tree in the case of PT

by starting at the root of the tree and reading off the characters of

the text string as we traverse the edges and nodes of the tree in the

following manner. The initial "(" tells us to ascend the first edge,

the next "(" tells us to ascend the next edge on the left, where we

find the letter "a" from the string checks with the letter "a" attached

to the node of the tree where we are. Another "(" takes us up another

edge, where we find the letter "b" from the string checks with the

letter "b" on the current tree node. Reading the first ")" on the

string entitles us to descend an edge and reading another ")" gives us

licence to descend another. The way of things is most likely clear by

this point — at any rate, I leave the exercise to the reader.

On the scene of the general correspondence between formulas and graphs

the action may be summed up as follows. The tree, called a "parse tree"

or "parse graph", is constructed in the process of checking whether the

text string is syntactically well-formed, in other words, whether it

satisfies the prescriptions of the associated formal grammar and is

therefore a member in good standing of the prescribed formal language.

If the text string checks out, grammatically speaking, we call it a

"traversal string" of the corresponding parse graph, because it can be

reconstructed from the graph by a process like that illustrated above

called "traversing" the graph.

To be continued …

Regards,

Jon

http://inquiryintoinquiry.com/2020/09/29/animated-logical-graphs-41/

All,

Last time we looked at a formula of propositional logic Leibniz

called a Praeclarum Theorema (PT). We don't concur it's a theorem,

of course, until there's a proof it's identically true and Leibniz

gave an argument to demonstrate that. Written out in one of our

more current formalisms, PT takes the following form.

((a ⇒ b) ∧ (d ⇒ c)) ⇒ ((a ∧ d) ⇒ (b ∧ c))

we reformulated PT in a propositional calculus using just

two primitive operations, writing the logical negation of

a proposition p as (p) and the logical conjunction of two

propositions p, q as pq. That gave us a text string in

teletype parentheses and proposition letters, formatted

two ways below.

Figure 1. Praeclarum Theorema Text Strings

https://inquiryintoinquiry.files.wordpress.com/2020/09/praeclarum-theorema-text-strings.png

Our next transformation of the theorem’s expression exploits a

standard correspondence in combinatorics and computer science

between parenthesized symbol strings and trees with symbols

attached to the nodes.

Figure 2. Praeclarum Theorema Parse Graph

https://inquiryintoinquiry.files.wordpress.com/2020/09/praeclarum-theorema-parse-graph-2.0.png

We can see the correspondence between text and tree in the case of PT

by starting at the root of the tree and reading off the characters of

the text string as we traverse the edges and nodes of the tree in the

following manner. The initial "(" tells us to ascend the first edge,

the next "(" tells us to ascend the next edge on the left, where we

find the letter "a" from the string checks with the letter "a" attached

to the node of the tree where we are. Another "(" takes us up another

edge, where we find the letter "b" from the string checks with the

letter "b" on the current tree node. Reading the first ")" on the

string entitles us to descend an edge and reading another ")" gives us

licence to descend another. The way of things is most likely clear by

this point — at any rate, I leave the exercise to the reader.

On the scene of the general correspondence between formulas and graphs

the action may be summed up as follows. The tree, called a "parse tree"

or "parse graph", is constructed in the process of checking whether the

text string is syntactically well-formed, in other words, whether it

satisfies the prescriptions of the associated formal grammar and is

therefore a member in good standing of the prescribed formal language.

If the text string checks out, grammatically speaking, we call it a

"traversal string" of the corresponding parse graph, because it can be

reconstructed from the graph by a process like that illustrated above

called "traversing" the graph.

To be continued …

Regards,

Jon

Oct 3, 2020, 4:16:11 PM10/3/20