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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 30, 2019, 9:16:16 AM5/30/19

to Cybernetic Communications, Laws Of Form Group

Re: Survey of Animated Logical Graphs : 2

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

Dear CybCom, LoF Groups,

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.

See: 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 will be

working on converting the figures and tables and trying

to make the presentation more understandable.

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

Dear CybCom, LoF Groups,

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.

See: 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 will be

working on converting the figures and tables and trying

to make the presentation more understandable.

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),

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

> I've found the ability or audacity to do since.?? In particular,
> earlier piece of work titled "Futures Of Logical Graphs" (FOLG),

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

> 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

>

> 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

> working on converting the figures and tables and trying

> to make the presentation more understandable.

>

> Regards,

>

> Jon

--
> 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

>

> 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

> working on converting the figures and tables and trying

> to make the presentation more understandable.

>

> Regards,

>

> Jon

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

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:

: 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

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.

| 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.

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

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 9, 2019, 9:10:57 AM7/9/19

to Cybernetic Communications, Laws Of Form Group

Dear CybCom and LoF groups,

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, 10:10:09 AM7/10/19

to Cybernetic Communications, Laws Of Form Group

Cf: Animated Logical Graphs : 17

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

Dear CybCom and LoF groups,

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/

Dear CybCom and LoF groups,

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 11, 2019, 7:52:18 AM7/11/19

to Cybernetic Communications, Laws Of Form Group

Cf: Animated Logical Graphs : 18

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

Dear CybCom and LoF groups,

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?

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

Dear CybCom and LoF groups,

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?

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, 5:30:09 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 12, 2019, 2:00:30 PM7/12/19

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

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

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

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 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 26, 2019, 11:45:56 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, 10:20:32 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 6, 2019, 4:34:04 PM8/6/19

to cyb...@googlegroups.com

Hi all,

I wonder if anyone here has found the model useful and if so, where was it applied and what were the learnings that you could share with us?

Thank you so much for your kind help and guidance.

Yeu Wen

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Aug 6, 2019, 4:54:11 PM8/6/19

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Hi Yue Wen,

I suggest you using "4U Test": Is it Useful? Does it Work? Is it Used?

Many academics fall into traps of "misplaced concerns"... we need to guard ourselves not fall victim of that. However, identifying the potential usefulness of any given theory is itself a big challenge...

*---------------------------------***Jason Jixuan Hu, Ph.D.***Independent Research Scholar &** Discussant, Club of REMY*

Intro-English: https://en.wikipedia.org/wiki/Jason_Jixuan_Hu

office: j...@wintopgroup.com

mobile: jasonth...@gmail.com

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Aug 6, 2019, 5:49:32 PM8/6/19

to cyb...@googlegroups.com

To put the two conceptualizers on one page is a sign of ignorance.

Korzybski was committed to a representational theory of language which presumes objective knowledge of the world of references as distinct from signs. He had s dualist conception of the world, now largely dismissed.

Bateson pointed out the evolutionary nature of Ashby’s cybernetics: One can never know whether one’s conceptions of the world are correct. One can only experience their failures when enacting them.

Klaus

Sent from my iPhone

Aug 6, 2019, 6:11:43 PM8/6/19

to cyb...@googlegroups.com

Dear Klaus, could you share a source URL of that Bateson point you just quoted? Many thanks - Jason

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Aug 6, 2019, 10:50:31 PM8/6/19

to cyb...@googlegroups.com

Hi Jason,

One source is Bateson’s “Cybernetic explanation” pages 405-515 in his Steps to an ecology of mind. You would gain further insight in some of Ross Ashby’s writing.

I recently contributed a paper for Tomas Fischer’s design cybernetics book, titled “the cybernetics of design and the design of cybernetics.”

Klaus

Sent from my iPhone

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Aug 7, 2019, 1:21:53 AM8/7/19

to cyb...@googlegroups.com

Thanks, Klaus. I'll read.

Best,

Loet

Loet
Leydesdorff

Professor emeritus,
University of Amsterdam

Amsterdam School of Communication Research (ASCoR)

lo...@leydesdorff.net ; http://www.leydesdorff.net/

Associate Faculty, SPRU, University of Sussex;

Guest Professor Zhejiang Univ., Hangzhou; Visiting Professor, ISTIC, Beijing;

Visiting Fellow, Birkbeck,
University of London;

To view this discussion on the web visit https://groups.google.com/d/msgid/cybcom/BE6DB8F1-66D0-4001-B75F-65CE549D102B%40asc.upenn.edu.

Aug 7, 2019, 2:10:01 PM8/7/19

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Thank you Klaus! Is your paper available online? If yes please share the link. Best regards - Jason

To view this discussion on the web visit https://groups.google.com/d/msgid/cybcom/BE6DB8F1-66D0-4001-B75F-65CE549D102B%40asc.upenn.edu.

Aug 7, 2019, 6:53:18 PM8/7/19

to cyb...@googlegroups.com

Available as an e-book:

Fischer, T., & Herr, C. M. (Eds.). (2019). Design Cybernetics: Navigating the new. Cham: Springer.

To view this discussion on the web visit https://groups.google.com/d/msgid/cybcom/CA%2BSRckuMzurfS136zPYFUUhKuCMDzftH1VGM1GcRjkCeFXygdA%40mail.gmail.com.

Aug 8, 2019, 6:18:02 AM8/8/19

to cyb...@googlegroups.com

Thank you all for your guidance!

I will take a while to go through all the wonderful references.

Please bear with me. I am sure I am going to come back with more questions to seek more help.

Thank you all once again!

Yeu Wen

To view this discussion on the web visit https://groups.google.com/d/msgid/cybcom/CAKvNPyWtkbq0gYHrf9ZE1YU4_-HPQjRf0Ar7TAXOtutbXN5KwA%40mail.gmail.com.

Aug 13, 2019, 5:15:36 PM8/13/19

to Cybernetic Communications, Laws Of Form Group

Cf: Animated Logical Graphs : 29

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

Re: Animated Logical Graphs : 21

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

I invoked the general concepts of equivalence and distinction at this point

in order to keep the wider backdrop of ideas 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 links 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.

Then B^k is the set of k-tuples of elements of B.

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

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

The correspondence between boolean functions

and boolean relations may be articulated as follows:

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.

The set L_f is called the "fiber of 1" or the "pre-image of 1" in B^k

and is commonly notated as [f^(-1)](1).

Regards,

Jon

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

Re: Animated Logical Graphs : 21

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

I invoked the general concepts of equivalence and distinction at this point

in order to keep the wider backdrop of ideas 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 links 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.

Then B^k is the set of k-tuples of elements of B.

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

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

The correspondence between boolean functions

and boolean relations may be articulated as follows:

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.

The set L_f is called the "fiber of 1" or the "pre-image of 1" in B^k

and is commonly notated as [f^(-1)](1).

Regards,

Jon

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