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May 5, 2024, 4:16:42 PMMay 5

to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG

Transformations of Logical Graphs • 1

• https://inquiryintoinquiry.com/2024/05/05/transformations-of-logical-graphs-1/

Re: Interpretive Duality in Logical Graphs • 1

• https://inquiryintoinquiry.com/2024/04/22/interpretive-duality-in-logical-graphs-1/

Re: Mathematical Duality in Logical Graphs • 1

• https://inquiryintoinquiry.com/2024/05/03/mathematical-duality-in-logical-graphs-1/

All,

Anything called a “duality” is naturally associated with

a transformation group of order 2, say a group G acting on

a set X. Transformation groupies generally refer to X as

a set of “points” even when the elements have additional

structure of their own, as they often do. A group of order

two has the form G = {1, t}, where 1 is the identity element

and the remaining element t satisfies the equation t² = 1,

being on that account self‑inverse.

A first look at the dual interpretation of logical graphs from

a group-theoretic point of view is provided by the Table below.

Interpretive Duality as Group Symmetry

• https://inquiryintoinquiry.files.wordpress.com/2021/02/peirce-duality-as-group-symmetry.png

The sixteen boolean functions f : B × B → B on two variables

are listed in Column 1.

Column 2 lists the elements of the set X, specifically,

the sixteen logical graphs γ giving canonical expression

to the boolean functions in Column 1.

Column 2 shows the graphs in existential order but

the order is arbitrary since only the transformations

of the set X into itself are material in this setting.

Column 3 shows the result 1γ of the group element 1

acting on each graph γ in X, which is of course the

same graph γ back again.

Column 4 shows the result tγ of the group element t

acting on each graph γ in X, which is the entitative

graph dual to the existential graph in Column 2.

The last Row of the Table displays a statistic of considerable

interest to transformation group theorists. It is the total

incidence of “fixed points”, in other words, the number of

points in X left invariant or unchanged by the various

group actions. I'll explain the significance of the

fixed point parameter next time.

Regards,

Jon

cc: https://www.academia.edu/community/l7jBGO

• https://inquiryintoinquiry.com/2024/05/05/transformations-of-logical-graphs-1/

Re: Interpretive Duality in Logical Graphs • 1

• https://inquiryintoinquiry.com/2024/04/22/interpretive-duality-in-logical-graphs-1/

Re: Mathematical Duality in Logical Graphs • 1

• https://inquiryintoinquiry.com/2024/05/03/mathematical-duality-in-logical-graphs-1/

All,

Anything called a “duality” is naturally associated with

a transformation group of order 2, say a group G acting on

a set X. Transformation groupies generally refer to X as

a set of “points” even when the elements have additional

structure of their own, as they often do. A group of order

two has the form G = {1, t}, where 1 is the identity element

and the remaining element t satisfies the equation t² = 1,

being on that account self‑inverse.

A first look at the dual interpretation of logical graphs from

a group-theoretic point of view is provided by the Table below.

Interpretive Duality as Group Symmetry

• https://inquiryintoinquiry.files.wordpress.com/2021/02/peirce-duality-as-group-symmetry.png

The sixteen boolean functions f : B × B → B on two variables

are listed in Column 1.

Column 2 lists the elements of the set X, specifically,

the sixteen logical graphs γ giving canonical expression

to the boolean functions in Column 1.

Column 2 shows the graphs in existential order but

the order is arbitrary since only the transformations

of the set X into itself are material in this setting.

Column 3 shows the result 1γ of the group element 1

acting on each graph γ in X, which is of course the

same graph γ back again.

Column 4 shows the result tγ of the group element t

acting on each graph γ in X, which is the entitative

graph dual to the existential graph in Column 2.

The last Row of the Table displays a statistic of considerable

interest to transformation group theorists. It is the total

incidence of “fixed points”, in other words, the number of

points in X left invariant or unchanged by the various

group actions. I'll explain the significance of the

fixed point parameter next time.

Regards,

Jon

cc: https://www.academia.edu/community/l7jBGO

May 6, 2024, 2:15:30 PMMay 6

to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG

Transformations of Logical Graphs • 2

• https://inquiryintoinquiry.com/2024/05/06/transformations-of-logical-graphs-2/

Re: Transformations of Logical Graphs • 1

• https://inquiryintoinquiry.com/2024/05/05/transformations-of-logical-graphs-1/

All,

Another way of looking at the dual interpretation of logical graphs

from a group-theoretic point of view is provided by the following Table.

Interpretive Duality as Group Symmetry • Orbit Order

• https://inquiryintoinquiry.files.wordpress.com/2021/03/peirce-duality-as-group-symmetry-e280a2-orbit-order.png

In this arrangement we have sorted the rows of the previous Table to

bring together similar graphs γ belonging to the set X, the similarity

being determined by the action of the group G = {1, t}. Transformation

group theorists refer to the corresponding similarity classes as “orbits”

of the group action under consideration. The orbits are defined by the

group acting “transitively” on them, meaning elements of the same orbit

can always be transformed into one another by some group operation while

elements of different orbits cannot.

Scanning the Table we observe the 16 points of X fall into 10 orbits

total, divided into 4 orbits of 1 point each and 6 orbits of 2 points

each. The points in singleton orbits are called “fixed points” of the

transformation group since they are not moved, or mapped into themselves,

by all group actions. The bottom row of the Table tabulates the total

number of fixed points for the group operations 1 and t respectively.

The group identity 1 always fixes all points, so its total is 16.

The group action t fixes only the four points in singleton orbits,

giving a total of 4.

I leave it as an exercise for the reader to investigate the

relationship between the group order |G| = 2, the number of

orbits 10, and the total number of fixed points 16 + 4 = 20.

Resources —

Logic Syllabus

• https://inquiryintoinquiry.com/logic-syllabus/

Logical Graphs • First Impressions

• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Logical Graphs • Formal Development

• https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-a/

Regards,

Jon

cc: https://www.academia.edu/community/LxnPx3

• https://inquiryintoinquiry.com/2024/05/06/transformations-of-logical-graphs-2/

Re: Transformations of Logical Graphs • 1

• https://inquiryintoinquiry.com/2024/05/05/transformations-of-logical-graphs-1/

All,

Another way of looking at the dual interpretation of logical graphs

from a group-theoretic point of view is provided by the following Table.

Interpretive Duality as Group Symmetry • Orbit Order

• https://inquiryintoinquiry.files.wordpress.com/2021/03/peirce-duality-as-group-symmetry-e280a2-orbit-order.png

In this arrangement we have sorted the rows of the previous Table to

bring together similar graphs γ belonging to the set X, the similarity

being determined by the action of the group G = {1, t}. Transformation

group theorists refer to the corresponding similarity classes as “orbits”

of the group action under consideration. The orbits are defined by the

group acting “transitively” on them, meaning elements of the same orbit

can always be transformed into one another by some group operation while

elements of different orbits cannot.

Scanning the Table we observe the 16 points of X fall into 10 orbits

total, divided into 4 orbits of 1 point each and 6 orbits of 2 points

each. The points in singleton orbits are called “fixed points” of the

transformation group since they are not moved, or mapped into themselves,

by all group actions. The bottom row of the Table tabulates the total

number of fixed points for the group operations 1 and t respectively.

The group identity 1 always fixes all points, so its total is 16.

The group action t fixes only the four points in singleton orbits,

giving a total of 4.

I leave it as an exercise for the reader to investigate the

relationship between the group order |G| = 2, the number of

orbits 10, and the total number of fixed points 16 + 4 = 20.

Resources —

Logic Syllabus

• https://inquiryintoinquiry.com/logic-syllabus/

Logical Graphs • First Impressions

• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Logical Graphs • Formal Development

• https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-a/

Regards,

Jon

cc: https://www.academia.edu/community/LxnPx3

May 7, 2024, 3:40:40 PMMay 7

to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG

Transformations of Logical Graphs • 3

• https://inquiryintoinquiry.com/2024/05/07/transformations-of-logical-graphs-3/

Re: Transformations of Logical Graphs • 1 • 2

• https://inquiryintoinquiry.com/2024/05/05/transformations-of-logical-graphs-1/

• https://inquiryintoinquiry.com/2024/05/06/transformations-of-logical-graphs-2/

We've been using the duality between entitative and

existential interpretations of logical graphs to get a

handle on the mathematical forms pervading logical laws.

A few posts ago we took up the tools of groups and symmetries

and transformations to study the duality and we looked to the

space of 2‑variable boolean functions as a basic training grounds.

On those grounds the translation between interpretations presents

as a group G of order 2 acting on a set X of sixteen logical graphs

denoting boolean functions.

Last time we arrived at a Table showing how the group G

partitions the set X into 10 orbits of logical graphs.

Here again is that Table.

Interpretive Duality as Group Symmetry • Orbit Order

• https://inquiryintoinquiry.files.wordpress.com/2021/03/peirce-duality-as-group-symmetry-e280a2-orbit-order.png

I invited the reader to investigate the relationship between

product of the group order (2) and the number of orbits (10)

is equal to the sum of the fixed points (20). Is that just

a fluke? If not, why so? And does it reflect a general rule?

We can make a beginning toward answering those questions

by inspecting the “incidence relation” of fixed points and

orbits in the Table above. Each singleton orbit accumulates

two hits, one from the group identity and one from the other

group operation.

But each doubleton orbit also accumulates two hits, since

the group identity fixes both of its two points. Thus all

the orbits are double‑counted by counting the incidence of

fixed points and orbits.

In sum, dividing the total number of fixed points by the order

of the group brings us back to the exact number of orbits.

Resources —

Logic Syllabus

• https://inquiryintoinquiry.com/logic-syllabus/

Logical Graphs • First Impressions

• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Logical Graphs • Formal Development

• https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-a/

Regards,

Jon

cc: https://www.academia.edu/community/5M7moJ

• https://inquiryintoinquiry.com/2024/05/07/transformations-of-logical-graphs-3/

Re: Transformations of Logical Graphs • 1 • 2

• https://inquiryintoinquiry.com/2024/05/05/transformations-of-logical-graphs-1/

• https://inquiryintoinquiry.com/2024/05/06/transformations-of-logical-graphs-2/

We've been using the duality between entitative and

existential interpretations of logical graphs to get a

handle on the mathematical forms pervading logical laws.

A few posts ago we took up the tools of groups and symmetries

and transformations to study the duality and we looked to the

space of 2‑variable boolean functions as a basic training grounds.

On those grounds the translation between interpretations presents

as a group G of order 2 acting on a set X of sixteen logical graphs

denoting boolean functions.

Last time we arrived at a Table showing how the group G

partitions the set X into 10 orbits of logical graphs.

Here again is that Table.

Interpretive Duality as Group Symmetry • Orbit Order

• https://inquiryintoinquiry.files.wordpress.com/2021/03/peirce-duality-as-group-symmetry-e280a2-orbit-order.png

the group order |G| = 2, the number of orbits 10, and the total

number of fixed points 16 + 4 = 20. In the present case the
product of the group order (2) and the number of orbits (10)

is equal to the sum of the fixed points (20). Is that just

a fluke? If not, why so? And does it reflect a general rule?

We can make a beginning toward answering those questions

by inspecting the “incidence relation” of fixed points and

orbits in the Table above. Each singleton orbit accumulates

two hits, one from the group identity and one from the other

group operation.

But each doubleton orbit also accumulates two hits, since

the group identity fixes both of its two points. Thus all

the orbits are double‑counted by counting the incidence of

fixed points and orbits.

In sum, dividing the total number of fixed points by the order

of the group brings us back to the exact number of orbits.

Resources —

Logic Syllabus

• https://inquiryintoinquiry.com/logic-syllabus/

Logical Graphs • First Impressions

• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Logical Graphs • Formal Development

• https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-a/

Regards,

Jon

May 8, 2024, 12:00:34 PMMay 8

to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG

Transformations of Logical Graphs • 4

• https://inquiryintoinquiry.com/2024/05/08/transformations-of-logical-graphs-4/

Semiotic Transformations —

Once we bring the dual interpretations of logical graphs to

the same Table and relate their parleys to the same objects,

it is clear we are dealing with a triadic sign relation of the

sort taken up in C.S. Peirce's “semiotics” or theory of signs.

A “sign relation” L ⊆ O × S × I, as a set L embedded in a cartesian

product O × S × I, tells how the “signs” in S and the “interpretant

signs” in I correlate with the “objects” or objective situations in O.

There are many ways of using sign relations to model various types

of sign‑theoretic situations and processes. The following cases

are often seen.

• Some sign relations model co‑referring signs or transitions

between signs within a single language or symbol system.

In that event L ⊆ O × S × I has S = I.

• Other sign relations model translations between different languages

or different interpretations of the same language, in other words,

different ways of referring the same set of signs to a shared

object domain.

The next Table extracts the sign relation L ⊆ O × S × I

involved in switching between existential and entitative

interpretations of logical graphs.

Interpretive Duality as Sign Relation

• https://inquiryintoinquiry.files.wordpress.com/2021/03/peirce-duality-as-sign-relation.png

• Column 1 shows the object domain O as the

set of 16 boolean functions on 2 variables.

• Column 2 shows the sign domain S as a representative set

of logical graphs denoting the objects in O according to

the existential interpretation.

• Column 3 shows the interpretant domain I as the same set

of logical graphs denoting the objects in O according to

the entitative interpretation.

Resources —

C.S. Peirce • On the Definition of Logic

• https://inquiryintoinquiry.com/2012/06/01/c-s-peirce-on-the-definition-of-logic/

C.S. Peirce • Logic as Formal Semiotic

• https://inquiryintoinquiry.com/2012/06/04/c-s-peirce-logic-as-semiotic/

Semeiotic • Sign Relations • Triadic Relations

• https://oeis.org/wiki/Semeiotic

• https://oeis.org/wiki/Sign_relation

• https://oeis.org/wiki/Triadic_relation

Survey of Semiotics, Semiosis, Sign Relations

• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

Regards,

Jon

cc: https://www.academia.edu/community/laaO6E

• https://inquiryintoinquiry.com/2024/05/08/transformations-of-logical-graphs-4/

Semiotic Transformations —

Once we bring the dual interpretations of logical graphs to

the same Table and relate their parleys to the same objects,

it is clear we are dealing with a triadic sign relation of the

sort taken up in C.S. Peirce's “semiotics” or theory of signs.

A “sign relation” L ⊆ O × S × I, as a set L embedded in a cartesian

product O × S × I, tells how the “signs” in S and the “interpretant

signs” in I correlate with the “objects” or objective situations in O.

There are many ways of using sign relations to model various types

of sign‑theoretic situations and processes. The following cases

are often seen.

• Some sign relations model co‑referring signs or transitions

between signs within a single language or symbol system.

In that event L ⊆ O × S × I has S = I.

• Other sign relations model translations between different languages

or different interpretations of the same language, in other words,

different ways of referring the same set of signs to a shared

object domain.

The next Table extracts the sign relation L ⊆ O × S × I

involved in switching between existential and entitative

interpretations of logical graphs.

Interpretive Duality as Sign Relation

• https://inquiryintoinquiry.files.wordpress.com/2021/03/peirce-duality-as-sign-relation.png

• Column 1 shows the object domain O as the

set of 16 boolean functions on 2 variables.

• Column 2 shows the sign domain S as a representative set

of logical graphs denoting the objects in O according to

the existential interpretation.

• Column 3 shows the interpretant domain I as the same set

of logical graphs denoting the objects in O according to

the entitative interpretation.

Resources —

C.S. Peirce • On the Definition of Logic

• https://inquiryintoinquiry.com/2012/06/01/c-s-peirce-on-the-definition-of-logic/

C.S. Peirce • Logic as Formal Semiotic

• https://inquiryintoinquiry.com/2012/06/04/c-s-peirce-logic-as-semiotic/

Semeiotic • Sign Relations • Triadic Relations

• https://oeis.org/wiki/Semeiotic

• https://oeis.org/wiki/Sign_relation

• https://oeis.org/wiki/Triadic_relation

Survey of Semiotics, Semiosis, Sign Relations

• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

Regards,

Jon

cc: https://www.academia.edu/community/laaO6E

May 9, 2024, 4:30:34 PMMay 9

to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG

Transformations of Logical Graphs • 5

• https://inquiryintoinquiry.com/2024/05/09/transformations-of-logical-graphs-5/

Semiotic Transformations —

“I know what you mean but I say it another way” — it's a thing

I find myself saying often enough, if only under my breath, to

rate an acronym for it ☞ IKWYMBISIAW ☜ and not too coincidentally

it's a rubric of relevance to many situations in semiotics where

sundry manners of speaking and thinking converge, more or less,

on the same patch of pragmata.

We encountered just such a situation in our exploration of the duality

between entitative and existential interpretations of logical graphs.

The two interpretations afford distinct but equally adequate ways of

reasoning about a shared objective domain.

To cut our teeth on a simple but substantial example of an object domain,

we picked the space of boolean functions or propositional forms on two

variables. That brought us to the following Table, highlighting the

• https://inquiryintoinquiry.com/2024/05/09/transformations-of-logical-graphs-5/

Semiotic Transformations —

“I know what you mean but I say it another way” — it's a thing

I find myself saying often enough, if only under my breath, to

rate an acronym for it ☞ IKWYMBISIAW ☜ and not too coincidentally

it's a rubric of relevance to many situations in semiotics where

sundry manners of speaking and thinking converge, more or less,

on the same patch of pragmata.

We encountered just such a situation in our exploration of the duality

between entitative and existential interpretations of logical graphs.

The two interpretations afford distinct but equally adequate ways of

reasoning about a shared objective domain.

To cut our teeth on a simple but substantial example of an object domain,

we picked the space of boolean functions or propositional forms on two

variables. That brought us to the following Table, highlighting the

sign relation L ⊆ O × S × I involved in switching between existential

and entitative interpretations of logical graphs.

Interpretive Duality as Sign Relation

• https://inquiryintoinquiry.files.wordpress.com/2021/03/peirce-duality-as-sign-relation.png

• Column 1 shows the object domain O as the set of 16 boolean functions on 2 variables.

• Column 2 shows the sign domain S as a representative set of logical graphs

denoting the objects in O according to the existential interpretation.

• Column 3 shows the interpretant domain I as the same set of logical graphs

denoting the objects in O according to the entitative interpretation.

Resources —

C.S. Peirce • On the Definition of Logic

• https://inquiryintoinquiry.com/2012/06/01/c-s-peirce-on-the-definition-of-logic/

C.S. Peirce • Logic as Formal Semiotic

• https://inquiryintoinquiry.com/2012/06/04/c-s-peirce-logic-as-semiotic/

Semeiotic • Sign Relations • Triadic Relations

• https://oeis.org/wiki/Semeiotic

• https://oeis.org/wiki/Sign_relation

• https://oeis.org/wiki/Triadic_relation

Survey of Semiotics, Semiosis, Sign Relations

• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

Regards,

Jon

cc: https://www.academia.edu/community/Lxn3dRand entitative interpretations of logical graphs.

Interpretive Duality as Sign Relation

• https://inquiryintoinquiry.files.wordpress.com/2021/03/peirce-duality-as-sign-relation.png

• Column 1 shows the object domain O as the set of 16 boolean functions on 2 variables.

• Column 2 shows the sign domain S as a representative set of logical graphs

denoting the objects in O according to the existential interpretation.

• Column 3 shows the interpretant domain I as the same set of logical graphs

denoting the objects in O according to the entitative interpretation.

Resources —

C.S. Peirce • On the Definition of Logic

• https://inquiryintoinquiry.com/2012/06/01/c-s-peirce-on-the-definition-of-logic/

C.S. Peirce • Logic as Formal Semiotic

• https://inquiryintoinquiry.com/2012/06/04/c-s-peirce-logic-as-semiotic/

Semeiotic • Sign Relations • Triadic Relations

• https://oeis.org/wiki/Semeiotic

• https://oeis.org/wiki/Sign_relation

• https://oeis.org/wiki/Triadic_relation

Survey of Semiotics, Semiosis, Sign Relations

• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

Regards,

Jon

May 11, 2024, 7:04:28 AMMay 11

to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG

Transformations of Logical Graphs • 6

• https://inquiryintoinquiry.com/2024/05/10/transformations-of-logical-graphs-6/

Semiotic Transformations —

Our study of the duality between entitative and existential

interpretations of logical graphs has brought to light its

fully sign-relational character, casting the interpretive

duality as a transformation of signs revolving about a

common object domain. The overall picture is a triadic

relation linking an object domain with two sign domains,

whose signs denote the objects in two distinct ways.

By way of constructing a concrete example, we let our object domain

consist of the 16 boolean functions on 2 variables and we let our

sign domains consist of representative logical graphs for those

16 functions. Thus we arrived at the Table in the previous post,

linked by its title below.

Interpretive Duality as Sign Relation

• https://inquiryintoinquiry.files.wordpress.com/2021/03/peirce-duality-as-sign-relation.png

• Column 1 shows the object domain O as the

set of 16 boolean functions on 2 variables.

• Column 2 shows the sign domain S as a representative set

of logical graphs denoting the objects in O according to

the existential interpretation.

• Column 3 shows the interpretant domain I as the same set

of logical graphs denoting the objects in O according to

the entitative interpretation.

Additional aspects of the sign relation's structure can be

brought out by sorting the Table in accord with the orbits

induced on the object domain by the group action inherent

in the interpretive duality. Performing that sort produces

the following Table.

Interpretive Duality as Sign Relation • Orbit Order

• https://inquiryintoinquiry.files.wordpress.com/2021/03/peirce-duality-as-sign-relation-e280a2-orbit-order.png

Resources —

Logic Syllabus

• https://inquiryintoinquiry.com/logic-syllabus/

Logical Graphs • First Impressions

• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Logical Graphs • Formal Development

• https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-a/

Jon

cc: https://www.academia.edu/community/5REZPa

• https://inquiryintoinquiry.com/2024/05/10/transformations-of-logical-graphs-6/

Semiotic Transformations —

Our study of the duality between entitative and existential

interpretations of logical graphs has brought to light its

fully sign-relational character, casting the interpretive

duality as a transformation of signs revolving about a

common object domain. The overall picture is a triadic

relation linking an object domain with two sign domains,

whose signs denote the objects in two distinct ways.

By way of constructing a concrete example, we let our object domain

consist of the 16 boolean functions on 2 variables and we let our

sign domains consist of representative logical graphs for those

16 functions. Thus we arrived at the Table in the previous post,

linked by its title below.

Interpretive Duality as Sign Relation

• https://inquiryintoinquiry.files.wordpress.com/2021/03/peirce-duality-as-sign-relation.png

• Column 1 shows the object domain O as the

set of 16 boolean functions on 2 variables.

• Column 2 shows the sign domain S as a representative set

of logical graphs denoting the objects in O according to

the existential interpretation.

• Column 3 shows the interpretant domain I as the same set

of logical graphs denoting the objects in O according to

the entitative interpretation.

brought out by sorting the Table in accord with the orbits

induced on the object domain by the group action inherent

in the interpretive duality. Performing that sort produces

the following Table.

Interpretive Duality as Sign Relation • Orbit Order

• https://inquiryintoinquiry.files.wordpress.com/2021/03/peirce-duality-as-sign-relation-e280a2-orbit-order.png

Resources —

Logic Syllabus

• https://inquiryintoinquiry.com/logic-syllabus/

Logical Graphs • First Impressions

• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Logical Graphs • Formal Development

• https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-a/

Semeiotic • Sign Relations • Triadic Relations

• https://oeis.org/wiki/Semeiotic

• https://oeis.org/wiki/Sign_relation

• https://oeis.org/wiki/Triadic_relation

Regards,
• https://oeis.org/wiki/Semeiotic

• https://oeis.org/wiki/Sign_relation

• https://oeis.org/wiki/Triadic_relation

Jon

cc: https://www.academia.edu/community/5REZPa

May 12, 2024, 12:16:46 AMMay 12

to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG

Transformations of Logical Graphs • 7

• https://inquiryintoinquiry.com/2024/05/11/transformations-of-logical-graphs-7/

Semiotic Transformations —

Our investigation has brought us to the point of seeing

both a transformation group and a triadic sign relation in

Given the level of the foregoing abstractions it helps to anchor

them in concrete structural experience. In that spirit we've been

pursuing the case of a group action T : X → X and a sign relation

L ⊆ O × X × X where O is the set of boolean functions on two variables

and X is a set of logical graphs denoting those functions. We drew up

a Table combining the aspects of both structures and sorted it according

to the orbits T induces on X and consequently on O.

Interpretive Duality as Sign Relation • Orbit Order

• https://inquiryintoinquiry.files.wordpress.com/2021/03/peirce-duality-as-sign-relation-e280a2-orbit-order.png

In the next few posts we'll take up the orbits of logical graphs

one by one, comparing and contrasting their syntax and semantics.

Survey of Animated Logical Graphs

• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/

Survey of Semiotics, Semiosis, Sign Relations

• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

Regards,

Jon

cc: https://www.academia.edu/community/lKZmwg

• https://inquiryintoinquiry.com/2024/05/11/transformations-of-logical-graphs-7/

Semiotic Transformations —

Our investigation has brought us to the point of seeing

both a transformation group and a triadic sign relation in

the duality between entitative and existential interpretations

of logical graphs.
Given the level of the foregoing abstractions it helps to anchor

them in concrete structural experience. In that spirit we've been

pursuing the case of a group action T : X → X and a sign relation

L ⊆ O × X × X where O is the set of boolean functions on two variables

and X is a set of logical graphs denoting those functions. We drew up

a Table combining the aspects of both structures and sorted it according

to the orbits T induces on X and consequently on O.

Interpretive Duality as Sign Relation • Orbit Order

• https://inquiryintoinquiry.files.wordpress.com/2021/03/peirce-duality-as-sign-relation-e280a2-orbit-order.png

one by one, comparing and contrasting their syntax and semantics.

Survey of Animated Logical Graphs

• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/

Survey of Semiotics, Semiosis, Sign Relations

• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

Regards,

Jon

May 12, 2024, 8:45:27 PMMay 12

to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG

Transformations of Logical Graphs • 8

• https://inquiryintoinquiry.com/2024/05/12/transformations-of-logical-graphs-8/

Semiotic Transformations —

Re: Transformations of Logical Graphs • 2

• https://inquiryintoinquiry.com/2024/05/06/transformations-of-logical-graphs-2/

Turning again to our Table of Orbits let's see

what we can learn about the structure of the

sign relational system in view.

As we saw in Episode 2, the transformation group T = {1, t}

partitions the set X of 16 logical graphs and also the set O

of 16 boolean functions into 10 orbits, all together amounting

to 4 singleton orbits and 6 doubleton orbits.

Points in singleton orbits are called “fixed points” of the

transformation group T : X → X since they are left unchanged,

or changed into themselves, by all group actions. Viewed in

the frame of the sign relation L ⊆ O × X × X, where the

transformations in T are literally “translations” in the

linguistic sense, these “T‑invariant graphs” have the

same denotations in O for both Existential Interpreters

and Entitative Interpreters.

Interpretive Duality as Sign Relation • Orbit Order

• https://inquiryintoinquiry.files.wordpress.com/2021/03/peirce-duality-as-sign-relation-e280a2-orbit-order.png

• https://inquiryintoinquiry.com/2024/05/12/transformations-of-logical-graphs-8/

Semiotic Transformations —

Re: Transformations of Logical Graphs • 2

• https://inquiryintoinquiry.com/2024/05/06/transformations-of-logical-graphs-2/

Turning again to our Table of Orbits let's see

what we can learn about the structure of the

sign relational system in view.

As we saw in Episode 2, the transformation group T = {1, t}

partitions the set X of 16 logical graphs and also the set O

of 16 boolean functions into 10 orbits, all together amounting

to 4 singleton orbits and 6 doubleton orbits.

Points in singleton orbits are called “fixed points” of the

transformation group T : X → X since they are left unchanged,

or changed into themselves, by all group actions. Viewed in

the frame of the sign relation L ⊆ O × X × X, where the

transformations in T are literally “translations” in the

linguistic sense, these “T‑invariant graphs” have the

same denotations in O for both Existential Interpreters

and Entitative Interpreters.

Interpretive Duality as Sign Relation • Orbit Order

• https://inquiryintoinquiry.files.wordpress.com/2021/03/peirce-duality-as-sign-relation-e280a2-orbit-order.png

Resources —

Logic Syllabus

• https://inquiryintoinquiry.com/logic-syllabus/

Survey of Animated Logical Graphs

• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/

Survey of Semiotics, Semiosis, Sign Relations

• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

Regards,

Jon

cc: https://www.academia.edu/community/lPxNmeLogic Syllabus

• https://inquiryintoinquiry.com/logic-syllabus/

Survey of Animated Logical Graphs

• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/

Survey of Semiotics, Semiosis, Sign Relations

• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

Regards,

Jon

May 13, 2024, 1:28:40 PMMay 13

to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG

Transformations of Logical Graphs • 9

• https://inquiryintoinquiry.com/2024/05/13/transformations-of-logical-graphs-9/

Semiotic Transformations —

Last time we took up the four singleton orbits in the action of T

on X and saw each consists of a single logical graph which T fixes,

preserves, or transforms into itself. On that account those four

logical graphs are said to be “self‑dual” or “T‑invariant”.

In general terms, it is useful to think of the entitative and

existential interpretations as two formal languages which happen

to use the same set of signs, each in its own way, to denote the

same set of formal objects. Then T defines the translation between

languages and the self‑dual logical graphs are the points where the

languages coincide, where the same signs denote the same objects in

both. Such constellations of “fixed stars” are indispensable to

navigation between languages, as every argot‑naut discovers in time.

Returning to the case at hand, where T acts on a selection of 16

logical graphs for the 16 boolean functions on two variables, the

following Table shows the values of the denoted boolean function

f : B × B → B for each of the self‑dual logical graphs.

Self-Dual Logical Graphs

• https://inquiryintoinquiry.files.wordpress.com/2021/04/self-dual-logical-graphs.png

The functions indexed here as f₁₂ and f₁₀ are known as the “coordinate

projections” (x, y) ↦ x and (x, y) ↦ y on the 1st and 2nd coordinates,

respectively, and the functions indexed as f₃ and f₅ are the negations

(x, y) ↦ ¬x and (x, y) ↦ ¬y of those projections, respectively.

Resources —

Logic Syllabus

• https://inquiryintoinquiry.com/logic-syllabus/

Survey of Animated Logical Graphs

• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/

Survey of Semiotics, Semiosis, Sign Relations

• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

Regards,

Jon

cc: https://www.academia.edu/community/L2g9Gm

• https://inquiryintoinquiry.com/2024/05/13/transformations-of-logical-graphs-9/

Semiotic Transformations —

Last time we took up the four singleton orbits in the action of T

on X and saw each consists of a single logical graph which T fixes,

preserves, or transforms into itself. On that account those four

logical graphs are said to be “self‑dual” or “T‑invariant”.

In general terms, it is useful to think of the entitative and

existential interpretations as two formal languages which happen

to use the same set of signs, each in its own way, to denote the

same set of formal objects. Then T defines the translation between

languages and the self‑dual logical graphs are the points where the

languages coincide, where the same signs denote the same objects in

both. Such constellations of “fixed stars” are indispensable to

navigation between languages, as every argot‑naut discovers in time.

Returning to the case at hand, where T acts on a selection of 16

logical graphs for the 16 boolean functions on two variables, the

following Table shows the values of the denoted boolean function

f : B × B → B for each of the self‑dual logical graphs.

Self-Dual Logical Graphs

• https://inquiryintoinquiry.files.wordpress.com/2021/04/self-dual-logical-graphs.png

The functions indexed here as f₁₂ and f₁₀ are known as the “coordinate

projections” (x, y) ↦ x and (x, y) ↦ y on the 1st and 2nd coordinates,

respectively, and the functions indexed as f₃ and f₅ are the negations

(x, y) ↦ ¬x and (x, y) ↦ ¬y of those projections, respectively.

Resources —

Logic Syllabus

• https://inquiryintoinquiry.com/logic-syllabus/

Survey of Animated Logical Graphs

• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/

Survey of Semiotics, Semiosis, Sign Relations

• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

Regards,

Jon

May 14, 2024, 1:08:47 PMMay 14

to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG

Transformations of Logical Graphs • 10

• https://inquiryintoinquiry.com/2024/05/14/transformations-of-logical-graphs-10/

Semiotic Transformations —

After the four orbits of self‑dual logical graphs we come

to six orbits of dual pairs. In no particular order of

importance, we may start by considering the following two.

• The logical graphs for the “constant functions” f₁₅ and f₀

are dual to each other.

• The logical graphs for the “ampheck functions” f₇ and f₁

are dual to each other.

The values of the constant and ampheck functions for each

(x, y) in B × B and the text expressions of their logical

graphs are given in the following Table.

Constants and Amphecks

• https://inquiryintoinquiry.files.wordpress.com/2021/04/constants-and-amphecks.png

Amphecks • Zeroth Order Logic

• https://oeis.org/wiki/Ampheck

• https://oeis.org/wiki/Zeroth_order_logic

Survey of Animated Logical Graphs

• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/

Survey of Semiotics, Semiosis, Sign Relations

• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

Regards,

Jon

cc: https://www.academia.edu/community/LZ20dN

• https://inquiryintoinquiry.com/2024/05/14/transformations-of-logical-graphs-10/

Semiotic Transformations —

After the four orbits of self‑dual logical graphs we come

to six orbits of dual pairs. In no particular order of

importance, we may start by considering the following two.

• The logical graphs for the “constant functions” f₁₅ and f₀

are dual to each other.

• The logical graphs for the “ampheck functions” f₇ and f₁

are dual to each other.

The values of the constant and ampheck functions for each

(x, y) in B × B and the text expressions of their logical

graphs are given in the following Table.

Constants and Amphecks

• https://inquiryintoinquiry.files.wordpress.com/2021/04/constants-and-amphecks.png

Amphecks • Zeroth Order Logic

• https://oeis.org/wiki/Ampheck

• https://oeis.org/wiki/Zeroth_order_logic

Survey of Animated Logical Graphs

• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/

Survey of Semiotics, Semiosis, Sign Relations

• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

Regards,

Jon

May 15, 2024, 6:05:58 PMMay 15

to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG

Transformations of Logical Graphs • 11

• https://inquiryintoinquiry.com/2024/05/15/transformations-of-logical-graphs-11/

Semiotic Transformations —

Re: Transformations of Logical Graphs • 8

• https://inquiryintoinquiry.com/2024/05/12/transformations-of-logical-graphs-8/

Continuing our scan of the Table in Episode 8, the next two

orbits contain the logical graphs for the boolean functions

f₂, f₁₁, f₄, f₁₃, in that order. A first glance shows the two

orbits have surprisingly intricate structures and relationships

to each other — let's isolate that section for a closer look.

Table 1. Interpretive Duality • Subtractions and Implications

• https://inquiryintoinquiry.files.wordpress.com/2021/05/peirce-duality-e280a2-subtractions-and-implications.png

• The boolean functions f₂ and f₄ are called “subtraction functions”.

• The boolean functions f₁₁ and f₁₃ are called “implication functions”.

• The logical graphs for f₂ and f₁₁ are dual to each other.

• The logical graphs for f₄ and f₁₃ are dual to each other.

The values of the subtraction and implication functions for each

(x, y) in B × B and the text expressions for their logical graphs

• https://inquiryintoinquiry.files.wordpress.com/2021/05/subtractions-and-implications.png

Logical Implication • Zeroth Order Logic

• https://oeis.org/wiki/Logical_implication

• https://inquiryintoinquiry.com/2024/05/15/transformations-of-logical-graphs-11/

Semiotic Transformations —

Re: Transformations of Logical Graphs • 8

• https://inquiryintoinquiry.com/2024/05/12/transformations-of-logical-graphs-8/

Continuing our scan of the Table in Episode 8, the next two

orbits contain the logical graphs for the boolean functions

f₂, f₁₁, f₄, f₁₃, in that order. A first glance shows the two

orbits have surprisingly intricate structures and relationships

to each other — let's isolate that section for a closer look.

Table 1. Interpretive Duality • Subtractions and Implications

• https://inquiryintoinquiry.files.wordpress.com/2021/05/peirce-duality-e280a2-subtractions-and-implications.png

• The boolean functions f₂ and f₄ are called “subtraction functions”.

• The boolean functions f₁₁ and f₁₃ are called “implication functions”.

• The logical graphs for f₂ and f₁₁ are dual to each other.

• The logical graphs for f₄ and f₁₃ are dual to each other.

The values of the subtraction and implication functions for each

(x, y) in B × B and the text expressions for their logical graphs

are given in the following Table.

Table 2. Subtractions and Implications • Truth Table
• https://inquiryintoinquiry.files.wordpress.com/2021/05/subtractions-and-implications.png

Logical Implication • Zeroth Order Logic

• https://oeis.org/wiki/Logical_implication

• https://oeis.org/wiki/Zeroth_order_logic

Survey of Animated Logical Graphs

• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/

Survey of Semiotics, Semiosis, Sign Relations

• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

Regards,

Jon

cc: https://www.academia.edu/community/VDq921Survey of Animated Logical Graphs

• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/

Survey of Semiotics, Semiosis, Sign Relations

• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

Regards,

Jon

May 16, 2024, 4:36:24 PMMay 16

to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG

Transformations of Logical Graphs • 12

• https://inquiryintoinquiry.com/2024/05/16/transformations-of-logical-graphs-12/

Re: Transformations of Logical Graphs • 8 • 9 • 10 • 11

• https://inquiryintoinquiry.com/2024/05/12/transformations-of-logical-graphs-8/

• https://inquiryintoinquiry.com/2024/05/13/transformations-of-logical-graphs-9/

• https://inquiryintoinquiry.com/2024/05/14/transformations-of-logical-graphs-10/

• https://inquiryintoinquiry.com/2024/05/15/transformations-of-logical-graphs-11/

All,

Taking from our wallets an old schedule of orbits, let's

review the classes of logical graphs we've covered so far.

Episode 9. Self-Dual Logical Graphs —

====================================

Episode 9 dealt with four orbits of “self-dual logical graphs”,

whose text expressions are x, y, (x), (y), respectively.

The logical graphs whose text expressions are x, y, (x), (y)

denote the boolean functions f₁₂, f₁₀, f₃, f₅, in that order,

and the value of each function f on each point (x, y) of B × B

is shown in the following Table.

Truth Table for Self-Dual Logical Graphs

• https://inquiryintoinquiry.files.wordpress.com/2021/04/self-dual-logical-graphs.png

Episode 10. Constants and Amphecks —

===================================

Episode 10 dealt with two orbits of logical graphs

called “constants” and “amphecks”, respectively.

The “constant” logical graphs denote the constant functions,

defined as follows.

• f₀ : B × B → 0

• f₁₅ : B × B → 1

Under the Existential Interpretation:

• The function f₀ is denoted by the logical graph

whose text form is “( )”.

• The function f₁₅ is denoted by the logical graph

whose text form is “ ”.

Under the Entitative Interpretation:

• The function f₀ is denoted by the logical graph

whose text form is “ ”.

• The function f₁₅ is denoted by the logical graph

whose text form is “( )”.

The “ampheck” logical graphs denote the ampheck functions,

defined as follows.

• f₁(x, y) = NNOR(x, y).

• f₇(x, y) = NAND(x, y).

Under the Existential Interpretation:

• The function f₁(x, y) = NNOR(x, y) is denoted

by the logical graph with text form (x)(y).

• The function f₇(x, y) = NAND(x, y) is denoted

by the logical graph with text form (xy).

Under the Entitative Interpretation:

• The function f₁(x, y) = NNOR(x, y) is denoted

by the logical graph with text form (xy).

• The function f₇(x, y) = NAND(x, y) is denoted

by the logical graph with text form (x)(y).

The values of the constant and ampheck functions

on the points of B × B are tabulated below.

Truth Table for Constants and Amphecks

• https://inquiryintoinquiry.files.wordpress.com/2021/04/constants-and-amphecks.png

Episode 11. Subtractions and Implications —

==========================================

Episode 11 dealt with two orbits of logical graphs

called “subtractions” and “implications”, respectively.

The “subtraction” logical graphs denote the subtraction functions,

defined as follows.

• f₂(x, y) = y ¬ x.

• f₄(x, y) = x ¬ y.

The “implication” logical graphs denote the implication functions,

defined as follows.

• f₁₁(x, y) = x ⇒ y.

• f₁₃(x, y) = y ⇒ x.

Under the action of the Entitative ↔ Existential duality the

logical graphs for the subtraction f₂ and the implication f₁₁

fall into one orbit while the logical graphs for the subtraction

f₄ and the implication f₁₃ fall into another orbit, making these

2 partitions of the 4 functions “orthogonal” or “transversal”

Truth Table for Subtractions and Implications

• https://inquiryintoinquiry.files.wordpress.com/2021/05/subtractions-and-implications.png

Regards,

Jon

cc: https://www.academia.edu/community/V0yAG4

• https://inquiryintoinquiry.com/2024/05/16/transformations-of-logical-graphs-12/

Re: Transformations of Logical Graphs • 8 • 9 • 10 • 11

• https://inquiryintoinquiry.com/2024/05/12/transformations-of-logical-graphs-8/

• https://inquiryintoinquiry.com/2024/05/13/transformations-of-logical-graphs-9/

• https://inquiryintoinquiry.com/2024/05/14/transformations-of-logical-graphs-10/

• https://inquiryintoinquiry.com/2024/05/15/transformations-of-logical-graphs-11/

All,

Taking from our wallets an old schedule of orbits, let's

review the classes of logical graphs we've covered so far.

Episode 9. Self-Dual Logical Graphs —

====================================

Episode 9 dealt with four orbits of “self-dual logical graphs”,

whose text expressions are x, y, (x), (y), respectively.

The logical graphs whose text expressions are x, y, (x), (y)

denote the boolean functions f₁₂, f₁₀, f₃, f₅, in that order,

and the value of each function f on each point (x, y) of B × B

is shown in the following Table.

Truth Table for Self-Dual Logical Graphs

• https://inquiryintoinquiry.files.wordpress.com/2021/04/self-dual-logical-graphs.png

Episode 10. Constants and Amphecks —

===================================

Episode 10 dealt with two orbits of logical graphs

called “constants” and “amphecks”, respectively.

The “constant” logical graphs denote the constant functions,

defined as follows.

• f₀ : B × B → 0

• f₁₅ : B × B → 1

Under the Existential Interpretation:

• The function f₀ is denoted by the logical graph

whose text form is “( )”.

• The function f₁₅ is denoted by the logical graph

whose text form is “ ”.

Under the Entitative Interpretation:

• The function f₀ is denoted by the logical graph

whose text form is “ ”.

• The function f₁₅ is denoted by the logical graph

whose text form is “( )”.

The “ampheck” logical graphs denote the ampheck functions,

defined as follows.

• f₁(x, y) = NNOR(x, y).

• f₇(x, y) = NAND(x, y).

Under the Existential Interpretation:

• The function f₁(x, y) = NNOR(x, y) is denoted

by the logical graph with text form (x)(y).

• The function f₇(x, y) = NAND(x, y) is denoted

by the logical graph with text form (xy).

Under the Entitative Interpretation:

• The function f₁(x, y) = NNOR(x, y) is denoted

by the logical graph with text form (xy).

• The function f₇(x, y) = NAND(x, y) is denoted

by the logical graph with text form (x)(y).

The values of the constant and ampheck functions

Truth Table for Constants and Amphecks

• https://inquiryintoinquiry.files.wordpress.com/2021/04/constants-and-amphecks.png

Episode 11. Subtractions and Implications —

==========================================

Episode 11 dealt with two orbits of logical graphs

called “subtractions” and “implications”, respectively.

The “subtraction” logical graphs denote the subtraction functions,

defined as follows.

• f₂(x, y) = y ¬ x.

• f₄(x, y) = x ¬ y.

The “implication” logical graphs denote the implication functions,

defined as follows.

• f₁₁(x, y) = x ⇒ y.

• f₁₃(x, y) = y ⇒ x.

Under the action of the Entitative ↔ Existential duality the

logical graphs for the subtraction f₂ and the implication f₁₁

fall into one orbit while the logical graphs for the subtraction

f₄ and the implication f₁₃ fall into another orbit, making these

2 partitions of the 4 functions “orthogonal” or “transversal”

to each other.

The values of the subtraction and implication functions

on the points of B × B are tabulated below.
The values of the subtraction and implication functions

Truth Table for Subtractions and Implications

• https://inquiryintoinquiry.files.wordpress.com/2021/05/subtractions-and-implications.png

Regards,

Jon

cc: https://www.academia.edu/community/V0yAG4

May 19, 2024, 8:48:30 AMMay 19

to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG

Transformations of Logical Graphs • 13

• https://inquiryintoinquiry.com/2024/05/18/transformations-of-logical-graphs-13/

Semiotic Transformations —

Re: Transformations of Logical Graphs • 8

• https://inquiryintoinquiry.com/2024/05/12/transformations-of-logical-graphs-8/

Continuing our scan of the Table in Episode 8, the next orbit

contains the logical graphs for the boolean functions f₈ and f₁₄.

Interpretive Duality • Conjunction and Disjunction

• https://inquiryintoinquiry.files.wordpress.com/2021/05/peirce-duality-e280a2-conjunction-and-disjunction.png

The boolean functions f₈ and f₁₄ are called “logical conjunction” and

“logical disjunction”, respectively. The values taken by f₈ and f₁₄

for each pair of arguments (x, y) in B × B and the text expressions

for their logical graphs are given in the following Table.

Truth Table for Conjunction and Disjunction

• https://inquiryintoinquiry.com/wp-content/uploads/2024/05/conjunction-and-disjunction.png

Logical Conjunction • Logical Disjunction

• https://oeis.org/wiki/Logical_conjunction

• https://oeis.org/wiki/Logical_disjunction

Survey of Animated Logical Graphs

• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/

Survey of Semiotics, Semiosis, Sign Relations

• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

Regards,

Jon

cc: https://www.academia.edu/community/Vj8Qb8

• https://inquiryintoinquiry.com/2024/05/18/transformations-of-logical-graphs-13/

Semiotic Transformations —

Re: Transformations of Logical Graphs • 8

Continuing our scan of the Table in Episode 8, the next orbit

contains the logical graphs for the boolean functions f₈ and f₁₄.

Interpretive Duality • Conjunction and Disjunction

• https://inquiryintoinquiry.files.wordpress.com/2021/05/peirce-duality-e280a2-conjunction-and-disjunction.png

The boolean functions f₈ and f₁₄ are called “logical conjunction” and

“logical disjunction”, respectively. The values taken by f₈ and f₁₄

for each pair of arguments (x, y) in B × B and the text expressions

for their logical graphs are given in the following Table.

Truth Table for Conjunction and Disjunction

• https://inquiryintoinquiry.com/wp-content/uploads/2024/05/conjunction-and-disjunction.png

Logical Conjunction • Logical Disjunction

• https://oeis.org/wiki/Logical_conjunction

• https://oeis.org/wiki/Logical_disjunction

Survey of Animated Logical Graphs

• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/

Survey of Semiotics, Semiosis, Sign Relations

• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

Regards,

Jon

May 21, 2024, 2:56:25 PMMay 21

to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG

Transformations of Logical Graphs • 14

• https://inquiryintoinquiry.com/2024/05/21/transformations-of-logical-graphs-14/

Semiotic Transformations —

Re: Transformations of Logical Graphs • 8

• https://inquiryintoinquiry.com/2024/05/12/transformations-of-logical-graphs-8/

Completing our scan of the Table in Episode 8, the last

orbit up for consideration contains the logical graphs

• https://inquiryintoinquiry.files.wordpress.com/2021/05/peirce-duality-e280a2-difference-and-equality.png

The boolean functions f₆ and f₆ are known as “logical difference”

and “logical equality”, respectively. The values taken by f₆ and f₆

• https://inquiryintoinquiry.com/wp-content/uploads/2024/05/truth-table-e280a2-difference-and-equality.png

Logical Difference • Logical Equality

• https://oeis.org/wiki/Exclusive_disjunction

• https://oeis.org/wiki/Logical_equality

Survey of Animated Logical Graphs

• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/

Survey of Semiotics, Semiosis, Sign Relations

• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

Regards,

Jon

cc: https://www.academia.edu/community/laAAnW

• https://inquiryintoinquiry.com/2024/05/21/transformations-of-logical-graphs-14/

Semiotic Transformations —

Re: Transformations of Logical Graphs • 8

• https://inquiryintoinquiry.com/2024/05/12/transformations-of-logical-graphs-8/

orbit up for consideration contains the logical graphs

for the boolean functions f₆ and f₆.

Interpretive Duality • Difference and Equality
• https://inquiryintoinquiry.files.wordpress.com/2021/05/peirce-duality-e280a2-difference-and-equality.png

The boolean functions f₆ and f₆ are known as “logical difference”

and “logical equality”, respectively. The values taken by f₆ and f₆

for each pair of arguments (x, y) in B × B and the text expressions

for their logical graphs are given in the following Table.

Truth Table for Difference and Equality
for their logical graphs are given in the following Table.

• https://inquiryintoinquiry.com/wp-content/uploads/2024/05/truth-table-e280a2-difference-and-equality.png

Logical Difference • Logical Equality

• https://oeis.org/wiki/Exclusive_disjunction

• https://oeis.org/wiki/Logical_equality

Survey of Animated Logical Graphs

• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/

Survey of Semiotics, Semiosis, Sign Relations

• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

Regards,

Jon

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