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Oct 10, 2020, 10:10:19 AM10/10/20

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

Cf: Sign Relations, Triadic Relations, Relation Theory • 1

http://inquiryintoinquiry.com/2020/10/10/sign-relations-triadic-relations-relation-theory-1/

To understand how signs work in Peirce's theory of triadic sign relations,

or “semiotics”, we have to understand, in order of increasing generality,

sign relations, triadic relations, and relations in general, each as

conceived in Peirce's logic of relative terms and the corresponding

mathematics of relations.

Toward that understanding, here are the current versions of articles

I long ago contributed to Wikipedia and continue more lately to develop

at a number of other places.

* Sign Relations ( https://oeis.org/wiki/Sign_relation )

* Triadic Relations ( https://oeis.org/wiki/Triadic_relation )

* Relation Theory ( https://oeis.org/wiki/Relation_theory )

Regards,

Jon

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

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

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

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

http://inquiryintoinquiry.com/2020/10/10/sign-relations-triadic-relations-relation-theory-1/

To understand how signs work in Peirce's theory of triadic sign relations,

or “semiotics”, we have to understand, in order of increasing generality,

sign relations, triadic relations, and relations in general, each as

conceived in Peirce's logic of relative terms and the corresponding

mathematics of relations.

Toward that understanding, here are the current versions of articles

I long ago contributed to Wikipedia and continue more lately to develop

at a number of other places.

* Sign Relations ( https://oeis.org/wiki/Sign_relation )

* Triadic Relations ( https://oeis.org/wiki/Triadic_relation )

* Relation Theory ( https://oeis.org/wiki/Relation_theory )

Regards,

Jon

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

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

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

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

Oct 11, 2020, 1:20:08 PM10/11/20

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

Cf: Sign Relations, Triadic Relations, Relation Theory • Discussion 1

http://inquiryintoinquiry.com/2020/10/11/sign-relations-triadic-relations-relation-theory-discussion-1/

Re: Peirce List ( https://list.iupui.edu/sympa/arc/peirce-l/2020-10/thrd1.html#00004 )

::: Edwina Taborsky ( https://list.iupui.edu/sympa/arc/peirce-l/2020-10/msg00005.html )

ET: I particularly like your comment that “signhood is a role

in a triadic relation, a role that a thing bears or plays

in a given context of relationships — it is not an absolute,

non-relative property of a thing-in-itself, one that it

possesses independently of all relationships to other things”.

ET: I myself emphasize that this context of the role is made up

of relationships (plural) — which gives the triad its capacity

for complexity. Therefore, as we see in Robert Marty's lattice,

a thing is never a thing-in-itself but is an action, a process,

composed of complex relations.

Dear Edwina,

Things grow complex rather quickly once we start to think about

all the roles a sign may play on all the stages where it struts

and frets its parts. There is no unique setting, no one scene,

but concentric and overlapping contexts of relationship all have

their bearing on the sign's significance.

One strategy we have for dealing with these complexities and avoiding

being overwhelmed by them is to build up a stock of well-studied examples,

graded in complexity from the very simplest to the increasingly complex.

The wide world may always present us with situations more complex than

any in our inventory of familiar cases but the better our stock of ready

examples the more aspects of novel situations we can capture and the

greater our odds of coping with them.

Regards,

Jon

http://inquiryintoinquiry.com/2020/10/11/sign-relations-triadic-relations-relation-theory-discussion-1/

Re: Peirce List ( https://list.iupui.edu/sympa/arc/peirce-l/2020-10/thrd1.html#00004 )

::: Edwina Taborsky ( https://list.iupui.edu/sympa/arc/peirce-l/2020-10/msg00005.html )

ET: I particularly like your comment that “signhood is a role

in a triadic relation, a role that a thing bears or plays

in a given context of relationships — it is not an absolute,

non-relative property of a thing-in-itself, one that it

possesses independently of all relationships to other things”.

ET: I myself emphasize that this context of the role is made up

of relationships (plural) — which gives the triad its capacity

for complexity. Therefore, as we see in Robert Marty's lattice,

a thing is never a thing-in-itself but is an action, a process,

composed of complex relations.

Dear Edwina,

Things grow complex rather quickly once we start to think about

all the roles a sign may play on all the stages where it struts

and frets its parts. There is no unique setting, no one scene,

but concentric and overlapping contexts of relationship all have

their bearing on the sign's significance.

One strategy we have for dealing with these complexities and avoiding

being overwhelmed by them is to build up a stock of well-studied examples,

graded in complexity from the very simplest to the increasingly complex.

The wide world may always present us with situations more complex than

any in our inventory of familiar cases but the better our stock of ready

examples the more aspects of novel situations we can capture and the

greater our odds of coping with them.

Regards,

Jon

Oct 12, 2020, 12:12:09 PM10/12/20

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

Cf: Sign Relations, Triadic Relations, Relation Theory • Discussion 2

http://inquiryintoinquiry.com/2020/10/12/sign-relations-triadic-relations-relation-theory-discussion-2/

::: Edwina Taborsky ( https://list.iupui.edu/sympa/arc/peirce-l/2020-10/msg00007.html

Dear Edwina,

Analytic frameworks, our various theories of categories, sets, sorts, and types,

have their uses but they tend to become à priori, autonomous, top-down, and

top-heavy unless they are supported by a robust population of concrete examples

arising in practical experience, one of the things the maxim of pragmatism

( https://inquiryintoinquiry.com/2008/08/07/pragmatic-maxim/ ) advises us

to remember. That is why Peirce's tackling of information and inquiry is

even-handed with respect to their extensional and intensional sides.

And it's why we need to pay attention when anomalies accumulate and

the population of presenting cases rebels against the dictates of

Procrustean predicates. Times like that tell us we may need to

reconceive our customary conceptual frameworks.

As it happens, I've been thinking a lot lately about a particular class of

sign sequences, namely, proofs in propositional calculus regarded as cases

of sign process, or semiosis. Naturally I've been thinking of delving more

deeply into Robert Marty's work ( https://www.academia.edu/s/4835fb2725 )

on paths through the lattice of sign classes but so far I'm still in the

early stages of that venture.

For what it's worth, here are my blog posts so far on Proof As Semiosis.

• Animated Logical Graphs

https://inquiryintoinquiry.com/2020/08/19/animated-logical-graphs-35/

https://inquiryintoinquiry.com/2020/08/21/animated-logical-graphs-36/

https://inquiryintoinquiry.com/2020/08/22/animated-logical-graphs-37/

https://inquiryintoinquiry.com/2020/08/25/animated-logical-graphs-38/

https://inquiryintoinquiry.com/2020/09/10/animated-logical-graphs-39/

https://inquiryintoinquiry.com/2020/09/26/animated-logical-graphs-40/

https://inquiryintoinquiry.com/2020/09/29/animated-logical-graphs-41/

https://inquiryintoinquiry.com/2020/10/03/animated-logical-graphs-42/

Regards,

Jon

http://inquiryintoinquiry.com/2020/10/12/sign-relations-triadic-relations-relation-theory-discussion-2/

::: Edwina Taborsky ( https://list.iupui.edu/sympa/arc/peirce-l/2020-10/msg00007.html

Dear Edwina,

Analytic frameworks, our various theories of categories, sets, sorts, and types,

have their uses but they tend to become à priori, autonomous, top-down, and

top-heavy unless they are supported by a robust population of concrete examples

arising in practical experience, one of the things the maxim of pragmatism

( https://inquiryintoinquiry.com/2008/08/07/pragmatic-maxim/ ) advises us

to remember. That is why Peirce's tackling of information and inquiry is

even-handed with respect to their extensional and intensional sides.

And it's why we need to pay attention when anomalies accumulate and

the population of presenting cases rebels against the dictates of

Procrustean predicates. Times like that tell us we may need to

reconceive our customary conceptual frameworks.

As it happens, I've been thinking a lot lately about a particular class of

sign sequences, namely, proofs in propositional calculus regarded as cases

of sign process, or semiosis. Naturally I've been thinking of delving more

deeply into Robert Marty's work ( https://www.academia.edu/s/4835fb2725 )

on paths through the lattice of sign classes but so far I'm still in the

early stages of that venture.

For what it's worth, here are my blog posts so far on Proof As Semiosis.

• Animated Logical Graphs

https://inquiryintoinquiry.com/2020/08/19/animated-logical-graphs-35/

https://inquiryintoinquiry.com/2020/08/21/animated-logical-graphs-36/

https://inquiryintoinquiry.com/2020/08/22/animated-logical-graphs-37/

https://inquiryintoinquiry.com/2020/08/25/animated-logical-graphs-38/

https://inquiryintoinquiry.com/2020/09/10/animated-logical-graphs-39/

https://inquiryintoinquiry.com/2020/09/26/animated-logical-graphs-40/

https://inquiryintoinquiry.com/2020/09/29/animated-logical-graphs-41/

https://inquiryintoinquiry.com/2020/10/03/animated-logical-graphs-42/

Regards,

Jon

Dec 29, 2020, 3:40:17 PM12/29/20

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

Cf: Sign Relations, Triadic Relations, Relation Theory • Discussion 3

http://inquiryintoinquiry.com/2020/12/29/sign-relations-triadic-relations-relation-theory-discussion-3/

All,

I decided the issue I discussed under “Adic Versus Tomic” comes up

often enough to deserve a revised text and upgraded figure so I put

a better copy on my blog, adding it to a series on Relation Theory

where I might have an easier time finding it again. Transcript below,

but see the link above for a more readable copy

Re: Peirce List

https://list.iupui.edu/sympa/arc/peirce-l/2020-11/thrd1.html#00022

::: Helmut Raulien

https://list.iupui.edu/sympa/arc/peirce-l/2020-11/msg00022.html

HR: As Peircean semiotics is a three-valued logic, I think it

bears relevance for the discussion about multiple-valued logic.

Dear Helmut,

The distinction between “k-adic” (involving a span of k dimensions) and “k-tomic” (involving a range of k values) is one

of the earliest questions I remember discussing on the Peirce List and the panoply of other lists we ranged across in

those heady surfer days. It is critical not to confuse the two aspects of multiplicity. In some cases it is possible

to observe what mathematicians call a projective relationship between the two aspects but that does not make them identical.

I'm adding a lightly edited excerpt from one of those earlier discussions as I think it introduces the issues about as

well as I could manage today.

Arisbe List

===========

https://web.archive.org/web/20150211184002/http://stderr.org/pipermail/arisbe/

Re: Inquiry Into Isms • k-adic versus k-tomic

https://web.archive.org/web/20141219190201/http://stderr.org/pipermail/arisbe/2001-August/thread.html#878

Jon Awbrey • 21 Aug 2001

https://web.archive.org/web/20141005035422/http://stderr.org/pipermail/arisbe/2001-August/000878.html

Here is an old note I've been looking for since we started on this bit about isms, as I feel I managed to express in it

my point of view that the key to integrating variant perspectives is to treat their contrasting values as axes or

dimensions rather than so many points on a line to be selected among, each in exclusion of all the others. To express

it briefly, it is the difference between k-tomic decisions among terminal values and k-adic dimensions of extended

variation.

Standard Upper Ontology List • Dyads

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

https://web.archive.org/web/20010312172719/http://suo.ieee.org/email/threads.html#02488

Jon Awbrey • 06 Dec 2000

https://web.archive.org/web/20010512091429/http://suo.ieee.org/email/msg02488.html

Jon Awbrey • 08 Dec 2000

https://web.archive.org/web/20010416032310/http://suo.ieee.org/email/msg02518.html

Jon Awbrey:

I think we need to distinguish “dichotomous thinking” from “dyadic thinking”. One has to do with the number of values,

{0, 1}, {F, T}, {evil, good}, and so on, one imposes on the cosmos, the other with the number of dimensions a person

puts on the face of the deep, that is to say, the number of independent axes in the frame of reference one projects on

the scene or otherwise puts up to put the cosmos on.

Tom Gollier:

Your transmission kind of faded out after the “number of values”, but do you mean a difference between, say, two values

of truth and falsity on the one hand, and all things being divided into subjects and predicates, functions and

arguments, and such as that on the other? If so, I'd like to second the notion, as not only are the two values much

less odious, if no less rigorous, in their applications, but they're often maligned as naive or simplistic by arguments

which actually should be applied to the idea, naive and simplistic in the extreme, that there are only two kinds of things.

Jon Awbrey:

There may be a connection — I will have to think about it — but trichotomic, dichotomic, monocotyledonic, whatever,

refer to the number of values, 3, 2, 1, whatever, in the range of a function. In contrast, triadic, dyadic, monadic, as

a series, refer to the number of independent dimensions involved in a relation, which could be represented as the axes

of a coordinate frame or the columns of a data table. As the appearance of the word “independent” should clue you in,

this will be one of those parti-colored woods in which the interpretive paths of mathematicians and normal folks are

likely to diverge.

A particular type of misunderstanding may arise when people imbued in the different ways of thinking try to communicate

with each other. The following figure illustrates the situation for the case where k = 2.

Figure. Dyadic Versus Dichotomic

https://inquiryintoinquiry.files.wordpress.com/2020/12/dyadic-versus-dichotomic.png

This shows how the “number of values” thinker projects the indications of the “number of axes” thinker onto the linear

spectrum of admitted directions, oppositions, or values, tending to reduce the mutually complementing dimensions to a

tug-of-war of strife-torn exclusions and polarizations.

Even when the tomic thinker tries to achieve a balance, a form of equilibrium, or a compromising harmony, the distortion

due to this style of projection will always render the resulting system untenable.

Probably my bias is evident.

But I think it is safe to say, for whatever else it might be good, tomic thinking is of limited use in trying to

understand Peirce's thought.

Just to mention one of the settings where this theme has arisen in my studies recently, you may enjoy the exercise of

reading, in the light of this projective template, Susan Haack's Evidence and Inquiry, where she strives to achieve a

balance or a compromise between foundationalism and coherentism, that is, more or less, objectivism and relativism, and

with some attempt to incorporate the insights of Peirce's POV. But a tomic thinker, per se, will not be able to

comprehend what the heck Peirce was talking about.

Resources

=========

http://inquiryintoinquiry.com/2020/12/29/sign-relations-triadic-relations-relation-theory-discussion-3/

All,

I decided the issue I discussed under “Adic Versus Tomic” comes up

often enough to deserve a revised text and upgraded figure so I put

a better copy on my blog, adding it to a series on Relation Theory

where I might have an easier time finding it again. Transcript below,

but see the link above for a more readable copy

Re: Peirce List

https://list.iupui.edu/sympa/arc/peirce-l/2020-11/thrd1.html#00022

::: Helmut Raulien

https://list.iupui.edu/sympa/arc/peirce-l/2020-11/msg00022.html

HR: As Peircean semiotics is a three-valued logic, I think it

bears relevance for the discussion about multiple-valued logic.

Dear Helmut,

The distinction between “k-adic” (involving a span of k dimensions) and “k-tomic” (involving a range of k values) is one

of the earliest questions I remember discussing on the Peirce List and the panoply of other lists we ranged across in

those heady surfer days. It is critical not to confuse the two aspects of multiplicity. In some cases it is possible

to observe what mathematicians call a projective relationship between the two aspects but that does not make them identical.

I'm adding a lightly edited excerpt from one of those earlier discussions as I think it introduces the issues about as

well as I could manage today.

Arisbe List

===========

https://web.archive.org/web/20150211184002/http://stderr.org/pipermail/arisbe/

Re: Inquiry Into Isms • k-adic versus k-tomic

https://web.archive.org/web/20141219190201/http://stderr.org/pipermail/arisbe/2001-August/thread.html#878

Jon Awbrey • 21 Aug 2001

https://web.archive.org/web/20141005035422/http://stderr.org/pipermail/arisbe/2001-August/000878.html

Here is an old note I've been looking for since we started on this bit about isms, as I feel I managed to express in it

my point of view that the key to integrating variant perspectives is to treat their contrasting values as axes or

dimensions rather than so many points on a line to be selected among, each in exclusion of all the others. To express

it briefly, it is the difference between k-tomic decisions among terminal values and k-adic dimensions of extended

variation.

Standard Upper Ontology List • Dyads

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

https://web.archive.org/web/20010312172719/http://suo.ieee.org/email/threads.html#02488

Jon Awbrey • 06 Dec 2000

https://web.archive.org/web/20010512091429/http://suo.ieee.org/email/msg02488.html

Jon Awbrey • 08 Dec 2000

https://web.archive.org/web/20010416032310/http://suo.ieee.org/email/msg02518.html

Jon Awbrey:

I think we need to distinguish “dichotomous thinking” from “dyadic thinking”. One has to do with the number of values,

{0, 1}, {F, T}, {evil, good}, and so on, one imposes on the cosmos, the other with the number of dimensions a person

puts on the face of the deep, that is to say, the number of independent axes in the frame of reference one projects on

the scene or otherwise puts up to put the cosmos on.

Tom Gollier:

Your transmission kind of faded out after the “number of values”, but do you mean a difference between, say, two values

of truth and falsity on the one hand, and all things being divided into subjects and predicates, functions and

arguments, and such as that on the other? If so, I'd like to second the notion, as not only are the two values much

less odious, if no less rigorous, in their applications, but they're often maligned as naive or simplistic by arguments

which actually should be applied to the idea, naive and simplistic in the extreme, that there are only two kinds of things.

Jon Awbrey:

There may be a connection — I will have to think about it — but trichotomic, dichotomic, monocotyledonic, whatever,

refer to the number of values, 3, 2, 1, whatever, in the range of a function. In contrast, triadic, dyadic, monadic, as

a series, refer to the number of independent dimensions involved in a relation, which could be represented as the axes

of a coordinate frame or the columns of a data table. As the appearance of the word “independent” should clue you in,

this will be one of those parti-colored woods in which the interpretive paths of mathematicians and normal folks are

likely to diverge.

A particular type of misunderstanding may arise when people imbued in the different ways of thinking try to communicate

with each other. The following figure illustrates the situation for the case where k = 2.

Figure. Dyadic Versus Dichotomic

https://inquiryintoinquiry.files.wordpress.com/2020/12/dyadic-versus-dichotomic.png

This shows how the “number of values” thinker projects the indications of the “number of axes” thinker onto the linear

spectrum of admitted directions, oppositions, or values, tending to reduce the mutually complementing dimensions to a

tug-of-war of strife-torn exclusions and polarizations.

Even when the tomic thinker tries to achieve a balance, a form of equilibrium, or a compromising harmony, the distortion

due to this style of projection will always render the resulting system untenable.

Probably my bias is evident.

But I think it is safe to say, for whatever else it might be good, tomic thinking is of limited use in trying to

understand Peirce's thought.

Just to mention one of the settings where this theme has arisen in my studies recently, you may enjoy the exercise of

reading, in the light of this projective template, Susan Haack's Evidence and Inquiry, where she strives to achieve a

balance or a compromise between foundationalism and coherentism, that is, more or less, objectivism and relativism, and

with some attempt to incorporate the insights of Peirce's POV. But a tomic thinker, per se, will not be able to

comprehend what the heck Peirce was talking about.

Resources

=========

Dec 31, 2020, 1:15:24 PM12/31/20

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

Cf: Sign Relations, Triadic Relations, Relation Theory • Discussion 4

http://inquiryintoinquiry.com/2020/12/31/sign-relations-triadic-relations-relation-theory-discussion-4/

Re: Previous Post

https://inquiryintoinquiry.com/2020/12/29/sign-relations-triadic-relations-relation-theory-discussion-3/

Re: Cybernetics

https://groups.google.com/g/cybcom/c/Wyz1oRmturc

::: Cliff Joslyn

https://groups.google.com/g/cybcom/c/Wyz1oRmturc/m/nRfUd8SLBwAJ

Dear Cliff,

Many thanks for your thoughtful reply. I copied a transcript

to my blog (see link above) to take up first thing next year.

Here's hoping we all have a better one!

Regards,

Jon

Cliff Joslyn wrote:

<QUOTE>

I think what you have is sound, and can be described in a number of ways.

In years past in seeking ways to both qualify and quantify variety in

systems I characterized this distinction as between “dimensional variety”

and “cardinal variety”. Thankfully, this seems straightforward from

a mathematical perspective, namely in a standard relational system

S = ×_{i=1}^k X_i, where the X_i are dimensions (something that can

vary), typically cast as sets, so that × here is Cartesian product.

Here k is the dimensional variety (number of dimensions, k-adicity),

while n_i = |X_i| is the cardinal variety (cardinality of dimension i,

n_i-tomicity (n_i-tonicity, actually?)). One might think of the two

most classic examples:

• Multiadic diatom/nic: Maximal (finite) dimensionality, minimal

non-trivial cardinality: The bit string (b_1, b_2, …, b_k) where

there are k Boolean dimensions X_i = {0, 1}. One can imagine k → ∞,

an infinite bit string, even moreso.

• Diadic infini-omic: Minimal non-trivial dimensionality,

maximal cardinality: The Cartesian plane ℝ², where there

are 2 real dimensions.

There's another quantity you didn't mention, which is the

overall “variety” or size of the system, so ∏_{n=1}^k n_i,

which is itself a well-formed expression (only) if there

are a finite number of finite dimensions.

</QUOTE>

http://inquiryintoinquiry.com/2020/12/31/sign-relations-triadic-relations-relation-theory-discussion-4/

Re: Previous Post

https://inquiryintoinquiry.com/2020/12/29/sign-relations-triadic-relations-relation-theory-discussion-3/

Re: Cybernetics

https://groups.google.com/g/cybcom/c/Wyz1oRmturc

::: Cliff Joslyn

https://groups.google.com/g/cybcom/c/Wyz1oRmturc/m/nRfUd8SLBwAJ

Dear Cliff,

Many thanks for your thoughtful reply. I copied a transcript

to my blog (see link above) to take up first thing next year.

Here's hoping we all have a better one!

Regards,

Jon

Cliff Joslyn wrote:

<QUOTE>

I think what you have is sound, and can be described in a number of ways.

In years past in seeking ways to both qualify and quantify variety in

systems I characterized this distinction as between “dimensional variety”

and “cardinal variety”. Thankfully, this seems straightforward from

a mathematical perspective, namely in a standard relational system

S = ×_{i=1}^k X_i, where the X_i are dimensions (something that can

vary), typically cast as sets, so that × here is Cartesian product.

Here k is the dimensional variety (number of dimensions, k-adicity),

while n_i = |X_i| is the cardinal variety (cardinality of dimension i,

n_i-tomicity (n_i-tonicity, actually?)). One might think of the two

most classic examples:

• Multiadic diatom/nic: Maximal (finite) dimensionality, minimal

non-trivial cardinality: The bit string (b_1, b_2, …, b_k) where

there are k Boolean dimensions X_i = {0, 1}. One can imagine k → ∞,

an infinite bit string, even moreso.

• Diadic infini-omic: Minimal non-trivial dimensionality,

maximal cardinality: The Cartesian plane ℝ², where there

are 2 real dimensions.

There's another quantity you didn't mention, which is the

overall “variety” or size of the system, so ∏_{n=1}^k n_i,

which is itself a well-formed expression (only) if there

are a finite number of finite dimensions.

</QUOTE>

Jan 16, 2021, 11:20:29 AMJan 16

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

Cf: Sign Relations, Triadic Relations, Relation Theory • Discussion 5

http://inquiryintoinquiry.com/2021/01/15/sign-relations-triadic-relations-relation-theory-discussion-5/

Re: Cybernetics

( https://groups.google.com/g/cybcom/c/Wyz1oRmturc )

::: Cliff Joslyn

( https://groups.google.com/g/cybcom/c/Wyz1oRmturc/m/nRfUd8SLBwAJ )

Dear Cliff,

I’m still collecting my wits from the mind-numbing events

of the past two weeks so I’ll copy your last remarks here

and work through them step by step.

CJ: <QUOTE>

Relational systems are just the context we need. It is usual to

begin at a moderate level of generality by considering a space X

of the following form.

X = ×_{i=1}^k X_i = X_1 × X_2 × ... × X_{k-1} × X_k.

(I’ll use X instead of S here because I want to save the letter “S” for

sign domains when we come to the special case of sign relational systems.)

We can now define a “relation” L as a subset of a cartesian product.

L ⊆ X_1 × X_2 × ... × X_{k-1} × X_k.

There are two common ways of understanding the subset symbol “⊆”

in this context. Using language from computer science I’ll call

them the “weak typing” and “strong typing” interpretations.

• Under “weak typing” conventions L is just a set which happens to be

a subset of the cartesian product X_1 × X_2 × ... × X_{k-1} × X_k but

which could just as easily be cast as a subset of any other qualified

superset. The mention of a particular cartesian product is accessory

but not necessary to the definition of the relation itself.

• Under “strong typing” conventions the cartesian product

X_1 × X_2 × ... × X_{k-1} × X_k in the type-casting

L ⊆ X_1 × X_2 × ... × X_{k-1} × X_k is an essential part of the

definition of L. Employing a conventional mathematical idiom, a

k-adic relation over the nonempty sets X_1, X_2, ..., X_{k-1}, X_k

is defined as a (k+1)-tuple (L, X_1, X_2, ..., X_{k-1}, X_k) where

L is a subset of X_1 × X_2 × ... × X_{k-1} × X_k.

We have at this point opened two fronts of interest in cybernetics,

namely, the generation of variety and the recognition of constraint.

There’s more detail on this brand of relation theory in the resource

article linked below. I’ll be taking the strong typing approach to

relations from this point on, largely because it comports more

naturally with category theory & by which virtue it enjoys more

immediate applications to systems and their transformations.

But my eye-brain system is going fuzzy on me now,

so I’ll break here and continue later ...

Regards,

Jon

Resources

=========

• Relation Theory ( https://oeis.org/wiki/Relation_theory )

• Triadic Relations ( https://oeis.org/wiki/Triadic_relation )

• Sign Relations ( https://oeis.org/wiki/Sign_relation )

> overall “variety” or size of the system, so ∏_{i=1}^k n_i,

http://inquiryintoinquiry.com/2021/01/15/sign-relations-triadic-relations-relation-theory-discussion-5/

Re: Cybernetics

( https://groups.google.com/g/cybcom/c/Wyz1oRmturc )

::: Cliff Joslyn

( https://groups.google.com/g/cybcom/c/Wyz1oRmturc/m/nRfUd8SLBwAJ )

Dear Cliff,

I’m still collecting my wits from the mind-numbing events

of the past two weeks so I’ll copy your last remarks here

and work through them step by step.

CJ: <QUOTE>

I think what you have is sound, and can be described in a number of ways.

In years past in seeking ways to both qualify and quantify variety in

systems I characterized this distinction as between “dimensional variety”

and “cardinal variety”. Thankfully, this seems straightforward from

a mathematical perspective, namely in a standard relational system

S = ×_{i=1}^k X_i, where the X_i are dimensions (something that can

vary), typically cast as sets, so that × here is Cartesian product.

</QUOTE>
In years past in seeking ways to both qualify and quantify variety in

systems I characterized this distinction as between “dimensional variety”

and “cardinal variety”. Thankfully, this seems straightforward from

a mathematical perspective, namely in a standard relational system

S = ×_{i=1}^k X_i, where the X_i are dimensions (something that can

vary), typically cast as sets, so that × here is Cartesian product.

Relational systems are just the context we need. It is usual to

begin at a moderate level of generality by considering a space X

of the following form.

X = ×_{i=1}^k X_i = X_1 × X_2 × ... × X_{k-1} × X_k.

(I’ll use X instead of S here because I want to save the letter “S” for

sign domains when we come to the special case of sign relational systems.)

We can now define a “relation” L as a subset of a cartesian product.

L ⊆ X_1 × X_2 × ... × X_{k-1} × X_k.

There are two common ways of understanding the subset symbol “⊆”

in this context. Using language from computer science I’ll call

them the “weak typing” and “strong typing” interpretations.

• Under “weak typing” conventions L is just a set which happens to be

a subset of the cartesian product X_1 × X_2 × ... × X_{k-1} × X_k but

which could just as easily be cast as a subset of any other qualified

superset. The mention of a particular cartesian product is accessory

but not necessary to the definition of the relation itself.

• Under “strong typing” conventions the cartesian product

X_1 × X_2 × ... × X_{k-1} × X_k in the type-casting

L ⊆ X_1 × X_2 × ... × X_{k-1} × X_k is an essential part of the

definition of L. Employing a conventional mathematical idiom, a

k-adic relation over the nonempty sets X_1, X_2, ..., X_{k-1}, X_k

is defined as a (k+1)-tuple (L, X_1, X_2, ..., X_{k-1}, X_k) where

L is a subset of X_1 × X_2 × ... × X_{k-1} × X_k.

We have at this point opened two fronts of interest in cybernetics,

namely, the generation of variety and the recognition of constraint.

There’s more detail on this brand of relation theory in the resource

article linked below. I’ll be taking the strong typing approach to

relations from this point on, largely because it comports more

naturally with category theory & by which virtue it enjoys more

immediate applications to systems and their transformations.

But my eye-brain system is going fuzzy on me now,

so I’ll break here and continue later ...

Regards,

Jon

Resources

=========

• Relation Theory ( https://oeis.org/wiki/Relation_theory )

• Triadic Relations ( https://oeis.org/wiki/Triadic_relation )

• Sign Relations ( https://oeis.org/wiki/Sign_relation )

Jan 17, 2021, 10:30:19 AMJan 17

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

Azamat, Helmut, John, All ...

We (some of us) have doggedly chased our assorted masters down this road

so many times before I could hardly e-numerate them all especially since

the lion's share of their spoors long ago disapparated from the live web.

But here's just a trace I found of an early relic ...

ONT Inquiry Into Symbolization (29 Aug 2001)

### http://web.archive.org/web/20070320034430/http://suo.ieee.org/ontology/thrd46.html#03201

(1) http://web.archive.org/web/20081204194050/http://suo.ieee.org/ontology/msg03201.html

(2) http://web.archive.org/web/20081204194631/http://suo.ieee.org/ontology/msg03202.html

(3) http://web.archive.org/web/20080906121059/http://suo.ieee.org/ontology/msg03204.html

(4) http://web.archive.org/web/20081011051100/http://suo.ieee.org/ontology/msg03234.html

In the middle of a different discussion now,

will get back to this later when I'm able ...

Regards,

Jon

On 1/17/2021 3:13 AM, Azamat Abdoullaev wrote:> John wrote:

> "Diogenes the Cynic plucked a chicken/cock and threw it into Plato's

> Academy while shouting "Here is Plato's man."

>

> Thanks, John. It is one of the funniest and smartest history memes.

> To extend it a bit.

> When asked about the origin of his epithet, *cynic/*dog, Diogenes replied

> that it was given to him because he “fawns upon those who give him anything

> and barks at those who give him nothing.”

>

> It implies that operational meanings or definitions could be more

> significant than an intension/extension or representation/reference or

> connotation/denotation dichotomy.

>

> On Sun, Jan 17, 2021 at 7:43 AM John F. Sowa <so...@bestweb.net> wrote:

>

>> Helmut,

>>

>> The distinction between intension and extension is important for every

>> version of logic since antiquity. The oldest example is "rational animal"

>> vs. "featherless biped" -- those are two terms with different intensions,

>> but the same extension. Diogenes the Cynic plucked a chicken and threw it

>> into Plato's Academy while shouting "Here is Plato's man."

>>

>> Alonzo Church, who wrote that excerpt I cited, had been the editor of the

>> Journal of Symbolic Logic for many years.

>>

>> It's just as important for the latest work in computer science for both

>> theory and applications.

>>

>> John

>>

We (some of us) have doggedly chased our assorted masters down this road

so many times before I could hardly e-numerate them all especially since

the lion's share of their spoors long ago disapparated from the live web.

But here's just a trace I found of an early relic ...

ONT Inquiry Into Symbolization (29 Aug 2001)

### http://web.archive.org/web/20070320034430/http://suo.ieee.org/ontology/thrd46.html#03201

(1) http://web.archive.org/web/20081204194050/http://suo.ieee.org/ontology/msg03201.html

(2) http://web.archive.org/web/20081204194631/http://suo.ieee.org/ontology/msg03202.html

(3) http://web.archive.org/web/20080906121059/http://suo.ieee.org/ontology/msg03204.html

(4) http://web.archive.org/web/20081011051100/http://suo.ieee.org/ontology/msg03234.html

In the middle of a different discussion now,

will get back to this later when I'm able ...

Regards,

Jon

On 1/17/2021 3:13 AM, Azamat Abdoullaev wrote:> John wrote:

> "Diogenes the Cynic plucked a chicken/cock and threw it into Plato's

> Academy while shouting "Here is Plato's man."

>

> Thanks, John. It is one of the funniest and smartest history memes.

> To extend it a bit.

> When asked about the origin of his epithet, *cynic/*dog, Diogenes replied

> that it was given to him because he “fawns upon those who give him anything

> and barks at those who give him nothing.”

>

> It implies that operational meanings or definitions could be more

> significant than an intension/extension or representation/reference or

> connotation/denotation dichotomy.

>

> On Sun, Jan 17, 2021 at 7:43 AM John F. Sowa <so...@bestweb.net> wrote:

>

>> Helmut,

>>

>> The distinction between intension and extension is important for every

>> version of logic since antiquity. The oldest example is "rational animal"

>> vs. "featherless biped" -- those are two terms with different intensions,

>> but the same extension. Diogenes the Cynic plucked a chicken and threw it

>> into Plato's Academy while shouting "Here is Plato's man."

>>

>> Alonzo Church, who wrote that excerpt I cited, had been the editor of the

>> Journal of Symbolic Logic for many years.

>>

>> It's just as important for the latest work in computer science for both

>> theory and applications.

>>

>> John

>>

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