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Aug 2, 2021, 4:40:11 PMAug 2

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

Invitation Link = https://categorytheory.zulipchat.com/join/zrkfytn4xzt65dn2bcwhkys5/

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

https://inquiryintoinquiry.com/2021/08/02/relations-their-relatives-comment-1/

https://categorytheory.zulipchat.com/#narrow/stream/233104-theory.3A-logic/topic/relation.20theory

Invitation Link = https://categorytheory.zulipchat.com/join/zrkfytn4xzt65dn2bcwhkys5/

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

Aug 5, 2021, 10:48:22 AMAug 5

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.

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.

Aug 6, 2021, 10:40:35 AMAug 6

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

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

Aug 6, 2021, 6:04:39 PMAug 6

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 )

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

• 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

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 )

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

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

• Peirce’s 1870 Logic Of Relatives

Regards,

Jon

Aug 7, 2021, 2:36:25 PMAug 7

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

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

Aug 11, 2021, 6:00:27 PMAug 11

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

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,

these logical arrays and matrices represent logical transformations.

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

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,

these logical arrays and matrices represent logical transformations.

Aug 12, 2021, 12:30:15 PMAug 12

Cf: Relations & Their Relatives • Discussion 1

https://inquiryintoinquiry.com/2015/02/27/relations-their-relatives-discussion-1/

Re: Peirce List

https://web.archive.org/web/20150619134001/http://comments.gmane.org/gmane.science.philosophy.peirce/15704

::: Helmut Raulien

https://web.archive.org/web/20150619133003/http://permalink.gmane.org/gmane.science.philosophy.peirce/15719

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

https://inquiryintoinquiry.com/2015/02/27/relations-their-relatives-discussion-1/

Re: Peirce List

https://web.archive.org/web/20150619134001/http://comments.gmane.org/gmane.science.philosophy.peirce/15704

::: Helmut Raulien

https://web.archive.org/web/20150619133003/http://permalink.gmane.org/gmane.science.philosophy.peirce/15719

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

Aug 15, 2021, 12:24:29 PMAug 15

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

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,

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.

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

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,

these logical arrays and matrices represent logical transformations.

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.

Aug 17, 2021, 9:24:50 AMAug 17

Cf: Relations & Their Relatives • Discussion 2

https://inquiryintoinquiry.com/2015/03/04/relations-their-relatives-discussion-2/

Re: Peirce List

https://web.archive.org/web/20150619134001/http://comments.gmane.org/gmane.science.philosophy.peirce/15704

::: Helmut Raulien

https://web.archive.org/web/20150303194001/http://permalink.gmane.org/gmane.science.philosophy.peirce/15774

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

to open up the additional “elbow room” that triadic relations provide.

https://inquiryintoinquiry.com/2015/03/04/relations-their-relatives-discussion-2/

Re: Peirce List

https://web.archive.org/web/20150619134001/http://comments.gmane.org/gmane.science.philosophy.peirce/15704

::: Helmut Raulien

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

to open up the additional “elbow room” that triadic relations provide.

Aug 17, 2021, 6:34:24 PMAug 17

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

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

Aug 18, 2021, 12:24:08 PMAug 18

Cf: Relations & Their Relatives • Discussion 19

https://inquiryintoinquiry.com/2021/08/18/relations-their-relatives-discussion-19/

Re: Category Theory

https://categorytheory.zulipchat.com/#narrow/stream/233104-theory.3A-logic/topic/relation.20theory

::: Henry Story

https://categorytheory.zulipchat.com/#narrow/stream/233104-theory.3A-logic/topic/relation.20theory/near/249610857

ZulipChat Invitation Link

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

https://inquiryintoinquiry.com/2021/08/18/relations-their-relatives-discussion-19/

Re: Category Theory

https://categorytheory.zulipchat.com/#narrow/stream/233104-theory.3A-logic/topic/relation.20theory

https://categorytheory.zulipchat.com/#narrow/stream/233104-theory.3A-logic/topic/relation.20theory/near/249610857

ZulipChat Invitation Link

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

Aug 19, 2021, 7:45:15 AMAug 19

Cf: Relations & Their Relatives • Discussion 20

https://inquiryintoinquiry.com/2021/08/18/relations-their-relatives-discussion-20/

Regarding Peirce's Formula “Information = Comprehension × Extension”

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/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

https://inquiryintoinquiry.com/2021/08/18/relations-their-relatives-discussion-20/

Regarding Peirce's Formula “Information = Comprehension × Extension”

Re: Category Theory

https://categorytheory.zulipchat.com/#narrow/stream/233104-theory.3A-logic/topic/relation.20theory

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

Regards,

Jon

Sep 13, 2021, 10:48:10 AM (8 days ago) Sep 13

Cf: Relations & Their Relatives • Discussion 21

https://inquiryintoinquiry.com/2021/09/13/relations-their-relatives-discussion-21/

Re: Ontolog Forum

https://groups.google.com/g/ontolog-forum/c/0pDK8IJiFDc

::: Alex Shkotin

https://groups.google.com/g/ontolog-forum/c/0pDK8IJiFDc/m/3YMIAA_UBwAJ

<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.

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

https://inquiryintoinquiry.com/2021/09/13/relations-their-relatives-discussion-21/

Re: Ontolog Forum

https://groups.google.com/g/ontolog-forum/c/0pDK8IJiFDc

::: Alex Shkotin

https://groups.google.com/g/ontolog-forum/c/0pDK8IJiFDc/m/3YMIAA_UBwAJ

<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.

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

Sep 16, 2021, 11:15:25 AM (5 days ago) Sep 16

Cf: Relations & Their Relatives • Discussion 22

https://inquiryintoinquiry.com/2021/09/16/relations-their-relatives-discussion-22/

::: Roberto Rovetto

https://groups.google.com/g/ontolog-forum/c/0pDK8IJiFDc/m/jwg3FbzfBgAJ

https://groups.google.com/g/ontolog-forum/c/0pDK8IJiFDc/m/HrdY9T1QAAAJ

<QUOTE RR:>

What's your view on:

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,

Let me return to your original question and give it better attention.

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

https://inquiryintoinquiry.com/2021/09/16/relations-their-relatives-discussion-22/

::: Roberto Rovetto

https://groups.google.com/g/ontolog-forum/c/0pDK8IJiFDc/m/jwg3FbzfBgAJ

https://groups.google.com/g/ontolog-forum/c/0pDK8IJiFDc/m/HrdY9T1QAAAJ

<QUOTE RR:>

What's your view on:

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,

Let me return to your original question and give it better attention.

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

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