Transformations of Logical Graphs
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Jon Awbrey
May 5, 2024, 4:16:43 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
Jon Awbrey
May 6, 2024, 2:15:29 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
Jon Awbrey
May 7, 2024, 3:40:41 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
Jon Awbrey
May 8, 2024, 12:00:30 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
Jon Awbrey
May 9, 2024, 4:30:35 PM (14 days ago) May 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
Jon Awbrey
May 11, 2024, 7:04:28 AM (12 days ago) May 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
Jon Awbrey
May 12, 2024, 12:16:47 AM (12 days ago) May 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
Jon Awbrey
May 12, 2024, 8:45:28 PM (11 days ago) May 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
Jon Awbrey
May 13, 2024, 1:28:41 PM (10 days ago) May 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
Jon Awbrey
May 14, 2024, 1:08:47 PM (9 days ago) May 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
Jon Awbrey
May 15, 2024, 6:05:58 PM (8 days ago) May 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
Jon Awbrey
May 16, 2024, 4:36:24 PM (7 days ago) May 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
Jon Awbrey
May 19, 2024, 8:48:29 AM (4 days ago) May 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
Jon Awbrey
May 21, 2024, 2:56:25 PM (2 days ago) May 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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