## Relations & Their Relatives • Discussion 23

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.

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?

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.

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, empirical studies, 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 triadic relations and especially triadic sign relations are paramount.  I will develop Peirce’s pragmatic, semiotic, information-theoretic viewpoint in tandem with the treatment of relation theory.

## Susan Awbrey • I Looked Up

I Looked Up

I looked up and I was old
The loose, wrinkled skin
The lines that leave their traces
I looked up and I was old
The scars of former illness written there
I looked up and I was old
I ask my body why it has forsaken me
It said it has been there through it all
I looked up and I was old
But my spirit is the same
Youth but a breath away
I looked up and I was one
With ancient earth and new born robin
I looked up and my gaze rose
Beyond my mollusk shell to the infinity that is now

❧ Susan Awbrey
September 19, 2021

## Differential Logic • Discussion 15

LA:
Differentials and partial differentials over the real numbers work because one can pick two real numbers that are arbitrarily close to one another.  The difference between any two real numbers can be made as small as desired.  If you have a real function of a real variable then you characterize the change in the function’s real value as the value of the argument changes.  This can be represented as the tangent to the curve representing the function at a given point.

In the Boolean domain there are only two values and they are always one step, unity, apart.  There is nothing to differentiate.  There is no variation in the spacing of the arguments or the function values.  There are no curves in the Boolean domain.  There is nothing to differentiate.

Dear Lyle,

My last post is really just a note-to-self reminding me to get back to work on differential logic, my memory being jogged by a number of posts on the Azimuth Blog.  But if I could nudge a few people to reflect on what the logical analogue of differential calculus ought to look like, that would be a plus.

The short answer to your objection is we don’t need limits in discrete spaces.  We follow the example of the finite difference calculus, using logical analogues of the enlargement operator $\mathrm{E}$ and the difference operator $\mathrm{D}.$

A differential is a locally linear approximation to a function, that is, a linear function which approximates another function at a point.  In boolean spaces, we know what the functions are, we know what the linear functions are, and all we need is a notion of approximation to define differentials.  Yes, there are numerous tricky bits to work out in boolean spaces, but I worked those out in the array of expositions at many different levels of abstraction and detail to which I have linked before, as again below.

## Differential Logic • Comment 7

Re: John BaezCyclic Identity for Partial Derivatives • Maxwell’s Relations (1) (2) (3)

Much fun can be had by trying to do differentials and partial differentials over the boolean domain $\mathbb{B} = \{ 0, 1 \}$ instead of the reals $\mathbb{R}.$

I took a first whack at it in the following project report.

## Relations & Their Relatives • Discussion 22

Re: Ontolog Forum • Roberto Rovetto (1) (2)

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?

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 thread for Relations and Their Relatives.

Regards,

Jon

## Inquiry Into Inquiry • Discussion 3

Re: Peirce List • (1) (2)
Re: Inquiry Into Inquiry • Discussion 2

The fact most neglected about the Neglected Argument is its character as an abductive argument, a “Holy Guess” if you will to believe, and as such the most fallible and mutable of hypotheses a happily fallible creature can create.  Its object is an hypostatic abstraction from human experience and the hypostasis has reality in virtue of whatever properties would be consistently assigned to it.  Does the object of the guess take an active part in human evolution or does human evolution play its part in making and reshaping its best guess?

O time, thou must untangle this, not I.
It is too hard a knot for me t’untie.

Submitted in quality of a case study on the role of abductive inference in inquiry and the role of phenomenology in science.

### Resources

Posted in Anthem, Balance, Inquiry | Tagged , , | 1 Comment

## Relations & Their Relatives • Discussion 21

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.

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.

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

1. The relation itself, which is a mathematical object,
a subset embedded in a cartesian product of several
sets called the “domains” of the relation.
2. The individual $k$-tuple, sometimes called an “elementary relation”,
a single element of the relation and therefore of the cartesian product.
3. The syntactic forms, lexical or graphical or whatever,
used to describe elements and subsets of the relation.
4. The real phenomena and real situations, empirical or quasi-empirical,
which we use mathematical objects such as numbers, sets, functions,
graphs, groups, algebras, manifolds, relations, etc. to model, at least
approximately and well enough to cope with the realities in practice.

## Semiotics, Semiosis, Sign Relations • Discussion 17

Re: Peirce List • Robert Marty (1) (2) (3)

Dear Robert,

I’ve been reviewing the discussions of August on this topic and I think it might be possible to advance our inquiry and even establish new levels of competence in our theory of signs if we examined the main points again and dedicated ourselves to clearing up the subject’s more persistent enigmas.

As I was preparing to recap our earlier discussions, it gradually dawned on me how one issue more than any other is the source of major misunderstanding and a whole lot of “people talking past each other”, as the saying goes.  To put it succinctly if very roughly, it has to do with the difference between people who have tests in search of answers and people who have answers in search of tests.  I say very roughly because it’s clear all of us are all of those people some of the time.  And yet we do see cognitive bifurcations and cultural divides persisting through time and people sorting to one basin or the other for extended periods if not the duration of a lifetime.

I’ll take up a tactic for dealing with that issue next time.

Regards,

Jon

cc: Category Theory • Cybernetics (1) (2)Ontolog Forum • Peirce List (1) (2) (3) (4)
cc: FB | SemeioticsLaws of FormStructural ModelingSystems Science

## Inquiry Into Inquiry • Discussion 2

PC:
Phenomenology is (with math) the underpinning of both scientific inquiry and everyday reasoning.  Improve one’s capability for observation and classification and you improve his/her ability to think and reason.  “Neglected Argument” has interesting things to say about the categories and this process as does “What Pragmatism Is”.

Although the Neglected Argument was one of the first Peirce essays my undergraduate philosophy advisor (who happened to be a Unitarian minister) gave me for contemplation — I remember coming to an unconventional, indirect argument, ontological proof sort of epiphany near the end — I can’t say I’ve paid all that much attention to Peirce’s theodicy since those days, but I can’t recall reading anything he wrote to distinguish his perspective from what is ordinarily called “deism”.  Does he ever declare for the (male personified) anthropomorphic God, so capitalized, of Abraham, Luther, Calvin, or any other, literal, non‑metaphorical theism of that kind?

### Resources

Posted in Anthem, Balance, Inquiry | Tagged , , | 2 Comments

## Inquiry Into Inquiry • Discussion 1

PJ:
I realise, in brief, the extent of your project —

nursing / health care — focus that also happens to have generic utility.

Dear Peter,

I have a few bits of work on my plate at the moment —
I will try to make a fuller reply in a day or two.

My wife, Susan Awbrey, and I have various degrees of acquaintance with nursing research, a smattering for me and much more for her.  Sue’s doctorate is in Educational Systems Design.  She served as an assistant professor and director of learning resources at the Michigan State University College of Nursing through most of the 1980s, collaborating with nurse researchers on issues of instructional design and information technology.  During 1989–91 we both had positions at the University of Texas Medical Branch (UTMB) School of Nursing in Galveston.  I had a faculty associate position consulting on research statistics, computing, and database management, meanwhile doing research on the hot new areas of AI applications to medical knowledge, diagnosis as abductive reasoning, physiological cybernetics, the novice/expert shift, and bridging the gap between qualitative and quantitative research methodologies.

Regards,

Jon

### Resources

Posted in Anthem, Balance, Inquiry | Tagged , , | 1 Comment