Relations & Their Relatives

[Jon Awbrey]

Relations & Their Relatives • 1
• https://inquiryintoinquiry.com/2024/07/31/relations-their-relatives-1-a/

All,

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 means of 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

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

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

Survey of Relation Theory
• https://inquiryintoinquiry.com/2024/03/23/survey-of-relation-theory-8/

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

Regards,

Jon

cc: https://www.academia.edu/community/Vj80Dj

[Jon Awbrey]

Relations & Their Relatives • 2
• https://inquiryintoinquiry.com/2024/08/01/relations-their-relatives-2-a/

All,

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
for those of us who are fans of what is called “strong typing” in
computer science, such a set is always set in a specific setting,
namely, it’s a subset of a specified cartesian product.

Peirce wrote k‑tuples like (x₁, x₂, …, xₖ₋₁, xₖ) in the form
x₁ : x₂ : … : xₖ₋₁ : xₖ and referred to them as “elementary
k‑adic relatives”. He treated a collection of k‑tuples as
a “logical aggregate” or “logical sum” and regarded them as
being arranged in k‑dimensional arrays.

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

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

Survey of Relation Theory
• https://inquiryintoinquiry.com/2024/03/23/survey-of-relation-theory-8/

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

Regards,

Jon

cc: https://www.academia.edu/community/LmW23m

[Jon Awbrey]

Relations & Their Relatives • 3
• https://inquiryintoinquiry.com/2024/08/02/relations-their-relatives-3-a/

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 mathematical 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 for
ease of reading.)

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,
logical arrays and matrices represent logical transformations.

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

Survey of Relation Theory
• https://inquiryintoinquiry.com/2024/03/23/survey-of-relation-theory-8/

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

Regards,

Jon

cc: https://www.academia.edu/community/VBZ14o

[Jon Awbrey]

Relations & Their Relatives • 4
• https://inquiryintoinquiry.com/2024/08/03/relations-their-relatives-4-a/

From Dyadic to Triadic to Sign Relations —

Peirce's notation for elementary relatives was illustrated
earlier by a dyadic relation from number theory, namely,
the relation written “i|j” for “i divides j”.

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
mathematical 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 for
ease of reading.)

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 of real coefficients in linear algebra represent
linear transformations, matrices of boolean coefficients represent
logical transformations. The capacity of dyadic relations to generate
transformations gives us part of what we need to know about the dynamics
of semiosis inherent in sign relations.

The “divisor of” relation 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 forms an example of a “partial order
relation”, while the “less than or equal to” relation 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 by grasping the generative
sign relation at its core.

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

Survey of Relation Theory
• https://inquiryintoinquiry.com/2024/03/23/survey-of-relation-theory-8/

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

Regards,

Jon

cc: https://www.academia.edu/community/5wb323