# Relations & Their Relatives

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### Jon Awbrey

Aug 2, 2021, 4:40:11 PM8/2/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Relations & Their Relatives • Comment 1
https://inquiryintoinquiry.com/2021/08/02/relations-their-relatives-comment-1/
https://categorytheory.zulipchat.com/#narrow/stream/233104-theory.3A-logic/topic/relation.20theory

All,

I opened a topic on Relation Theory in the Logic stream of Category Theory Zulipchat to discuss the logic of relative
terms and the mathematics of relations as they develop from Peirce's first breakthroughs (1865–1870). As I have
mentioned on a number of occasions, there are radical innovations in this work, probing deeper strata of logic and
mathematics than ever before mined and thus undermining the fundamental nominalism of First Order Logic as we know it.

Resource
========

• Survey of Relation Theory
https://inquiryintoinquiry.com/2020/05/15/survey-of-relation-theory-4/

Regards,

Jon

### Jon Awbrey

Aug 5, 2021, 10:48:23 AM8/5/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Relations & Their Relatives • Comment 2
https://inquiryintoinquiry.com/2021/08/05/relations-their-relatives-comment-2/

All,

Before I forget how I got myself into this particular briar patch — I mean the immediate occasion, not the long ago
straying from the beaten path — it was largely in discussions with Henry Story where he speaks of links between Peirce's
logical graphs and current thinking about string diagrams and bicategories of relations. Now that certainly sounds like
something I ought to get into, if not already witting or wit-not neck deep in it, but there are a few notes of
reservation I know I will eventually have to explain, so I've been working my way up to those.

First I need to set the stage for any properly Peircean discussion of logic and mathematics, and that is the context of
triadic sign relations. I know what you're thinking, “How can we talk about triadic sign relations before we have a
theory of relations in general?” The only way I know to answer that is by putting my programmer hard-hat on and taking
recourse in that practice which starts from the simplest thinkable species of a sort and builds its way back up to the
genus, step by step.

### Jon Awbrey

Aug 6, 2021, 10:40:35 AM8/6/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Relations & Their Relatives • Comment 3
https://inquiryintoinquiry.com/2021/08/06/relations-their-relatives-comment-3/

All,

Recent changes in url-coding on WordPress and the Web
will require me to spend a day or two repairing links
on my blog. Meanwhile, here's a couple of selections
from Peirce's 1870 Logic of Relatives bearing on the
proper use of individuals in mathematics, and thus on
the choice between nominal thinking and real thinking. 😸

• Mathematical Demonstration & the Doctrine of Individuals
https://inquiryintoinquiry.com/2015/02/22/mathematical-demonstration-the-doctrine-of-individuals-1/
https://inquiryintoinquiry.com/2015/02/23/mathematical-demonstration-the-doctrine-of-individuals-2/

Resources
=========
• Peirce’s 1870 “Logic Of Relatives” • Overview
https://inquiryintoinquiry.com/2019/09/24/peirces-1870-logic-of-relatives-overview/

Regards,

Jon

### Jon Awbrey

Aug 6, 2021, 6:04:39 PM8/6/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
All,

I'm reworking my initial blog posts on Relations & Their Relatives.
Looking back over them I think they manage to break ground on the
most-needed concepts in a moderately concrete fashion.

Here is the first one ...

Cf: Relations & Their Relatives • 1
https://inquiryintoinquiry.com/2015/02/17/relations-their-relatives-1/

Sign relations are special cases of triadic relations in much the same way
binary operations in mathematics are special cases of triadic relations.
It amounts to a minor complication that we participate in sign relations
whenever we talk or think about anything else but it still makes sense
to try and tease the separate issues apart as much as we possibly can.

As far as relations in general go, relative terms are often expressed by
slotted frames like “brother of __”, “divisor of __”, and “sum of __ and __”.
Peirce referred to these kinds of incomplete expressions as “rhemes” or
“rhemata” and Frege used the adjective “ungesättigt” or “unsaturated”
to convey more or less the same idea.

Switching the focus to sign relations, it’s fair to ask what kinds
of objects might be denoted by pieces of code like “brother of __”,
“divisor of __”, and “sum of __ and __”. And while we’re at it,
what is this thing called “denotation”, anyway?

Resources
=========

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

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

• Peirce’s 1870 Logic Of Relatives
( https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Overview )

Regards,

Jon

### Jon Awbrey

Aug 7, 2021, 2:36:26 PM8/7/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Relations & Their Relatives • 2
https://inquiryintoinquiry.com/2015/02/17/relations-their-relatives-2/

Continuing ...

What is the relationship between “logical relatives”
and “mathematical relations”? The word “relative” used
as a noun in logic is short for “relative term” — as such it
refers to an item of language used to denote a formal object.

What kind of object is that? The way things work in mathematics
we are free to make up a formal object corresponding directly to
the term, so long as we can form a consistent theory of it, but
it’s probably easier and more practical in the long run to relate
the relative term to the kinds of relations ordinarily treated in
mathematics and universally applied in relational databases.

In those contexts a relation is just a set of ordered tuples and
if you are a fan of strong typing like I am, such a set is always
set in a specific setting, namely, it’s a subset of a specified
cartesian product.

Peirce wrote k-tuples (x_1, x_2, ..., x_{k-1}, x_k) in the
form x_1 : x_2 : ... : x_{k-1} : x_k and referred to them as
“elementary k-adic relatives”. He treated a collection of
k-tuples as a “logical aggregate” or “logical sum” and often
regarded them as being arranged in k-dimensional arrays.

Time for some concrete examples, which I will give in the next post.

Regards,

Jon

### Jon Awbrey

Aug 11, 2021, 6:00:26 PM8/11/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Relations & Their Relatives • 3
https://inquiryintoinquiry.com/2015/02/18/relations-their-relatives-3/

All,

Here are two ways of looking at the divisibility relation,
a dyadic relation of fundamental importance in number theory.

Table 1 shows the first few ordered pairs of the relation on
positive integers corresponding to the relative term, “divisor of”.
Thus, the ordered pair i:j appears in the relation if and only if
i divides j, for which the usual notation is i|j.

Table 1. Elementary Relatives for the “Divisor Of” Relation
https://inquiryintoinquiry.files.wordpress.com/2015/02/elementary-relatives-for-the-e2809cdivisor-ofe2809d-relation.png

Table 2 shows the same information in the form of a logical matrix.
This has a coefficient of 1 in row i and column j when i|j, otherwise
it has a coefficient of 0. (The zero entries have been omitted here

Table 2. Logical Matrix for the “Divisor Of” Relation
https://inquiryintoinquiry.files.wordpress.com/2015/02/logical-matrix-for-the-e2809cdivisor-ofe2809d-relation.png

Just as matrices in linear algebra represent linear transformations,
these logical arrays and matrices represent logical transformations.
Elementary Relatives for the “Divisor Of” Relation.png
Logical Matrix for the “Divisor Of” Relation.png

### Jon Awbrey

Aug 12, 2021, 12:30:16 PM8/12/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Relations & Their Relatives • Discussion 1
https://inquiryintoinquiry.com/2015/02/27/relations-their-relatives-discussion-1/

Re: Peirce List
::: Helmut Raulien

The “divisor of” relation signified by x|y is a dyadic relation
on the set of positive integers M and thus may be understood as
a subset of the cartesian product M × M. It is an example of
a “partial order”, while the “less than or equal to” relation
signified by x ≤ y is an example of a “total order” relation.

The mathematics of relations can be applied most felicitously
to semiotics but there we must bump the “adicity” or “arity”
up to three. We take any sign relation L to be subset of a
cartesian product O × S × I, where O is the set of “objects”
under consideration in a given discussion, S is the set of
“signs”, and I is the set of “interpretant signs” involved
in the same discussion.

One thing we need to understand is the sign relation L ⊆ O × S × I
relevant to a given level of discussion may be rather more abstract
than what we would call a “sign process” proper, that is, a structure
extended through a dimension of time. Indeed, many of the most powerful
sign relations generate sign processes through iteration or recursion
or similar operations. In that event, the most penetrating analysis
of the sign process or semiosis in view is achieved through grasping
the generative sign relation at its core.

Regards,

Jon

### Jon Awbrey

Aug 15, 2021, 12:24:30 PM8/15/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Relations & Their Relatives • Review 1
https://inquiryintoinquiry.com/2021/08/15/relations-their-relatives-review-1relations-their-relatives-4/

All,

Peirce’s notation for elementary relatives was illustrated earlier by a
dyadic relation from number theory, namely i|j for i being a divisor of j.

Cf: Relations & Their Relatives • 3
https://inquiryintoinquiry.com/2015/02/18/relations-their-relatives-3/

Table 1 shows the first few ordered pairs of the relation
on positive integers corresponding to the relative term,
“divisor of”. Thus, the ordered pair i:j appears in the
relation if and only if i divides j, for which the usual
notation is i|j.

Table 1. Elementary Relatives for the “Divisor Of” Relation
https://inquiryintoinquiry.files.wordpress.com/2015/02/elementary-relatives-for-the-e2809cdivisor-ofe2809d-relation.png

Table 2 shows the same information in the form of a logical matrix.
This has a coefficient of 1 in row i and column j when i|j, otherwise
it has a coefficient of 0. (The zero entries have been omitted here

Table 2. Logical Matrix for the “Divisor Of” Relation
https://inquiryintoinquiry.files.wordpress.com/2015/02/logical-matrix-for-the-e2809cdivisor-ofe2809d-relation.png

Just as matrices in linear algebra represent linear transformations,
these logical arrays and matrices represent logical transformations.

This is a good point — mirabile dictu, the reason I
raised it — to extract the lesson for sign relations.

Cf: Relations & Their Relatives • Discussion 1
https://inquiryintoinquiry.com/2015/02/27/relations-their-relatives-discussion-1/

The “divisor of” relation signified by x|y is a dyadic relation
on the set of positive integers M and thus may be understood as
a subset of the cartesian product M × M. It is an example of
a “partial order”, while the “less than or equal to” relation
signified by x ≤ y is an example of a “total order” relation.

The mathematics of relations can be applied most felicitously
to semiotics but there we must bump the “adicity” or “arity”
up to three. We take any sign relation L to be subset of a
cartesian product O × S × I, where O is the set of “objects”
under consideration in a given discussion, S is the set of
“signs”, and I is the set of “interpretant signs” involved
in the same discussion.

One thing we need to understand is the sign relation L ⊆ O × S × I
relevant to a given level of discussion may be rather more abstract
than what we would call a “sign process” proper, that is, a structure
extended through a dimension of time. Indeed, many of the most powerful
sign relations generate sign processes through iteration or recursion or
similar operations. In that event, the most penetrating analysis of
the sign process or semiosis in view is achieved through grasping
the generative sign relation at its core.

Elementary Relatives for the “Divisor Of” Relation.png
Logical Matrix for the “Divisor Of” Relation.png

### Jon Awbrey

Aug 17, 2021, 9:24:51 AM8/17/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Relations & Their Relatives • Discussion 2
https://inquiryintoinquiry.com/2015/03/04/relations-their-relatives-discussion-2/

In systems theory and engineering there is a well-recognized duality
or complementarity between the dimensions of Control and Information,
frequently cast in terms of “action and perception”, “actuators” and
“detectors”, “effectors” and “sensors”, and a variety of other aliases.
There we find the dual devices of reachability matrices, representing the
operations it takes to put a system in a given state, and observability
matrices, representing the operations it takes to identify a system as
being in a given state.

The appearance of matrices at this point, understood in the sense
of 2-dimensional arrays of coefficients, may clue us to the mainly
dyadic character of the analysis and design that issue from them.
And yet there is every opportunity in systems theory and engineering

### Jon Awbrey

Aug 17, 2021, 6:34:24 PM8/17/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Relations & Their Relatives • Discussion 18
https://inquiryintoinquiry.com/2021/08/17/relations-their-relatives-discussion-18/

Re: Relations & Their Relatives • Review 1
https://inquiryintoinquiry.com/2021/08/15/relations-their-relatives-review-1/
Re: Category Theory
https://categorytheory.zulipchat.com/#narrow/stream/233104-theory.3A-logic/topic/relation.20theory
::: Morgan Rogers
https://categorytheory.zulipchat.com/#narrow/stream/233104-theory.3A-logic/topic/relation.20theory/near/249601853

<QUOTE MR:>
So a “sign process” would be a subset L ⊆ O × S × I × T, where T is a time domain?
</QUOTE>

There are a couple of ways we usually see the concept
of a sign relation L ⊆ O × S × I being applied.

• There is the “translation scenario” where S and I are
two different languages and a large part of L consists
of triples (o, s, i) where s and i are co-referent or
otherwise equivalent signs.

• There is the “transition scenario” where S = I and we have triples of
the form (o, s, s′) where s′ is the next state of s in a sign process.
As it happens, a concept of process is more basic than a concept of time,
since the latter involves reference to a standard process commonly known
as a “clock”.

Regards,

Jon

### Jon Awbrey

Aug 18, 2021, 12:24:09 PM8/18/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Relations & Their Relatives • Discussion 19
https://inquiryintoinquiry.com/2021/08/18/relations-their-relatives-discussion-19/
::: Henry Story
https://categorytheory.zulipchat.com/#narrow/stream/233104-theory.3A-logic/topic/relation.20theory/near/249610857

https://categorytheory.zulipchat.com/join/p5wlq72e6nb5i6mdgkwaje7b/

<QUOTE HS:>
Could one not say that Frege also had a three part relation? I guess:
for singular terms their Sense and Reference. […] His argument could be
explained very simply. Imagine you start with a theory of language where
words only have referents. Then since in point of fact Hesperus = Phosphorus,
The Morning Star = The Evening Star, the simple theory of meaning would not
allow one to explain how the discovery that they both were the planet Venus,
came to be such a big event. So sense cannot be reduced to reference.
Equalities can have informational content.
</QUOTE>

Yes, Peirce's take on semiotics is often compared with Frege's parsing
of Sinn und Bedeutung. There's a long tradition concerned with the
extension and intension of concepts and terms, also denotation and
connotation, though the latter tends to be somewhat fuzzier from
one commentator to the next. The following paper by Peirce gives
one of his characteristically thoroughgoing historical and technical
surveys of the question.

• C.S. Peirce (1867) • Upon Logical Comprehension and Extension
( https://peirce.sitehost.iu.edu/writings/v2/w2/w2_06/v2_06.htm )

The duality, inverse proportion, or reciprocal relation between
extension and intension is the generic form of the more specialized
galois correspondences we find in mathematics. Peirce preferred the
more exact term “comprehension” for a compound of many intensions.
In his Lectures on the Logic of Science (Harvard 1865, Lowell Institute
1866) he proposed his newfangled concept of “information” to integrate
the dual aspects of comprehension and extension, saying the measures
of comprehension and extension are inversely proportional only when the
measure of information is constant. The fundamental principle governing
his “laws of information” could thus be expressed in the following formula.

• Information = Comprehension × Extension

The development of Peirce's information formula is
discussed in my ongoing study notes, consisting of
selections from Peirce's 1865–1866 Lectures on the
Logic of Science and my commentary on them.

• Information = Comprehension × Extension
( https://oeis.org/wiki/Information_%3D_Comprehension_%C3%97_Extension )

Regards,

Jon

### Jon Awbrey

Aug 19, 2021, 7:45:15 AM8/19/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Relations & Their Relatives • Discussion 20
https://inquiryintoinquiry.com/2021/08/18/relations-their-relatives-discussion-20/

Regarding Peirce's Formula “Information = Comprehension × Extension”
::: Morgan Rogers
https://categorytheory.zulipchat.com/#narrow/stream/233104-theory.3A-logic/topic/relation.20theory/near/249716869

<QUOTE MR:>
Care to make any of this more precise?
[The above] formula, for example?
</QUOTE>

Yes, it will take some care to make it all more precise,
and I’ve cared enough to work on it when I get a chance.
I initially came to Peirce’s 1865–1866 lectures in grad
school from the direction of graph-&-group theory in
connection with a 19th century device called a “table
of marks”, out which a lot of work on group characters
and group representations developed.

A table of marks for a transformation group (G, X) is an
incidence matrix with 1 in the (g, x) cell if g fixes x
and 0 otherwise. I could see Peirce’s formula was based
on a logical analogue of those incidence matrices so that
gave me at least a little stable ground to inch forward on.

The development of Peirce’s information formula is discussed
in my ongoing study notes, consisting of selections from
Peirce’s 1865–1866 Lectures on the Logic of Science and
my commentary on them.

• Information = Comprehension × Extension
https://oeis.org/wiki/Information_%3D_Comprehension_%C3%97_Extension

Regards,

Jon

### Jon Awbrey

Sep 13, 2021, 10:48:09 AM9/13/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Relations & Their Relatives • Discussion 21
https://inquiryintoinquiry.com/2021/09/13/relations-their-relatives-discussion-21/

Re: Ontolog Forum
::: Alex Shkotin

<QUOTE AS:>
Let me underline an important point: first of all, we have found in
nature and society one or another relation and ask how many members
each example of this relation can have? i.e. arity is a feature of
relation itself. So […] we come here to the logic of relations and
its discovery. For me, examples of relations of different arity from
one or another domain would be great.
</QUOTE>

Here’s a first introduction to k-adic or k-ary relations
from a mathematical perspective.

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

Here's a few additional resources and assorted discussions
with folks around the web.

• Survey of Relation Theory
( https://inquiryintoinquiry.com/2020/05/15/survey-of-relation-theory-4/ )

More than anything else it is critical to understand
the differences among the following things.

• The relation itself, which is a mathematical object,
a subset embedded in a cartesian product of several
sets called the “domains” of the relation.

• The individual k-tuple, sometimes called an “elementary relation”,
a single element of the relation and hence the cartesian product.

• The syntactic forms, lexical or graphical or whatever,
used to describe elements and subsets of the relation.

• The real phenomena and real situations, empirical or quasi-empirical,
we use mathematical objects such as numbers, sets, functions, groups,
algebras, manifolds, relations, etc. to model, at least approximately
and well enough to cope with their realities in practice.

Regards,

Jon

### Jon Awbrey

Sep 16, 2021, 11:15:26 AM9/16/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Relations & Their Relatives • Discussion 22
https://inquiryintoinquiry.com/2021/09/16/relations-their-relatives-discussion-22/
::: Roberto Rovetto

<QUOTE RR:>

When to create a greater-than-binary relation rather than a binary relation?

Consider: You want to represent some information, statement, or knowledge,
without necessarily being forced to limit to binary relations. A common
example is when wanting to reference time. And “between” is greater than
binary. What are other pieces of knowledge that you'd want assert a ternary,
or greater than binary relation to capture it accurately?

Do you have any rules of thumb for knowing when
to assert n-ary relations greater than binary?
</QUOTE>

Dear Roberto,

You have probably noticed you got a wide variety of answers coming
from a diversity of conceptual frameworks and philosophical paradigms.
It gradually dawned on me some years ago these differences are most likely
matters of taste about which all dispute is futile, however much we go ahead
and do it anyway. So I'll just say what I've found works best in my particular
applications of interest, namely, applying relational logic to mathematics and
research sciences.

To avoid the kinds of culture clashes I remember from the Standard Upper Ontology
Lists and other ancestors of this Forum at the turn of the millennium, I'll develop
the rest of this line of inquiry on the Relations & Their Relatives thread
( https://groups.google.com/g/ontolog-forum/c/cL22KqWr8PI ) and, as I usually do,
post better-formatted copy on my blog ( https://inquiryintoinquiry.com/ ).

Regards,

Jon

### Jon Awbrey

Sep 21, 2021, 3:14:23 PM9/21/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Relations & Their Relatives • Discussion 23
https://inquiryintoinquiry.com/2021/09/21/relations-their-relatives-discussion-23/
::: Alex Shkotin

All,

Having lost my concentration to another round of
home reconstruction disruption, let me loop back
to the texts from Roberto and Alex which drew me
into this discussion last week.

<QUOTE RR:>

When to create a greater-than-binary relation rather than a binary relation?

Consider: You want to represent some information, statement, or knowledge, without necessarily being forced to limit to
binary relations. A common example is when wanting to reference time. And “between” is greater than binary. What are
other pieces of knowledge that you'd want assert a ternary, or greater than binary relation to capture it accurately?

Do you have any rules of thumb for knowing when to assert n-ary relations greater than binary?
</QUOTE>

<QUOTE AS:>
Let me underline an important point: first of all, we have found in nature and society one or another relation and ask
how many members each example of this relation can have? i.e. arity is a feature of relation itself. So […] we come
here to the logic of relations and its discovery. For me, examples of relations of different arity from one or another
domain would be great.
</QUOTE>

I will take up k-adic or k-ary relations from a mathematical perspective
and I will treat them from the standpoint of one whose “customers” over
his actually getting paid years were academic, education, health, and
research science units or investigators engaged in gathering data by
means of experiments or survey instruments and analyzing those data
according to the protocols of qualitative observation methods or
quantitative statistical hypothesis testing, all toward the purpose
of discovering reproducible facts about their research domains and
subject populations.

A sidelong but critically necessary reflection on the research scene
comes from the Peircean perspective on scientific inquiry, in which