Sign Relations, Triadic Relations, Relation Theory

[Jon Awbrey]

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

[Jon Awbrey]

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

[Jon Awbrey]

Cf: Sign Relations, Triadic Relations, Relation Theory • Discussion 2
http://inquiryintoinquiry.com/2020/10/12/sign-relations-triadic-relations-relation-theory-discussion-2/

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/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

[Jon Awbrey]

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
=========

[Jon Awbrey]

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>

[Jon Awbrey]

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>

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>

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,