Triadic Relations

[Jon Awbrey]

Ontolog, Systems Science, Structural Modeling Groups,

Spent the week moving the first load of our furniture to storage —
and boy! are my armchairs tiring! — so I'll have to pick up the
dangling threads later — but here's a bit of material I meant
to post last week ...

The middle ground between relations in general and the sign relations
we need to do logic, inquiry, communication, and so on is occupied by
triadic relations, also called ternary or 3-place relations.

Triadic relations are some of the most pervasive in mathematics,
over and above the importance of sign relations for logic, etc.

Here's a primer with examples from mathematics and semiotics:

• Triadic Relations at InterSciWiki
http://intersci.ss.uci.edu/wiki/index.php/Triadic_relation

• Triadic Relations at Wikiversity
https://en.wikiversity.org/wiki/Triadic_relation

Regards,

Jon

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[joseph simpson]

Jon:

I think we are generally on the same page with respect to the definition of a "mathematical relation."

Another definition of mathematical relation from "Constraint Theory - Multidimensional Mathematical Model Management"  by George Friedman, 2005, is:

".. a relation between a set of variables is defined as that subset within the product set of the variables which satisfies that relation."

A relation between or among a set of variables is not restricted to binary relations (as you point out.)

Further, a relation is a constraint on the variation of the variable values.

Viewing a relation as a constraint on variation directly aligns the definitions of a relation and a system.

A system is defined as a constraint on variation.

Take care and have fun,

Joe

On Fri, Jun 29, 2018 at 8:44 AM, Jon Awbrey <jaw...@att.net> wrote:

Cf: Sign Relations, Triadic Relations, Relations • 6
    https://inquiryintoinquiry.com/2018/06/23/sign-relations-triadic-relations-relations-%e2%80%a2-6/

On 6/19/2018 5:50 PM, joseph simpson wrote:
JS:

Jon:

Thanks for the additional information.

I am using the term "mathematical relation" in a slightly different manner.

For example, a binary relation is a set of ordered pairs of the elements of
some other set.

That is the first definition I learned for binary relations.

Slightly more generally, a binary relation L is a subset of
a cartesian product X × Y of two sets, X and Y.  In symbols,
L ⊆ X × Y.  Of course X and Y could be the same, but that's
not always the case.

I've always used the adjectives, 2-place, binary, and dyadic
pretty much interchangeably in application to relations, but
I developed a bias toward dyadic on account of computational
contexts where binary is reserved for binary numerals.

Once again, partly due to computational exigencies, I would
now regard this first definition as the weak typing version.

The strong typing definition of a k-place relation includes
the cartesian product X_1 × … × X_k as an essential part of
its specification.  This serves to harmonize the definition
of a k-place relation with the use of mathematical category
theory in computer science.

When I get more time, I'll go through the material I linked
on relation theory in a slightly more leisurely manner ...

• Relation Theory
  http://intersci.ss.uci.edu/wiki/index.php/Relation_theory
  https://en.wikiversity.org/wiki/Relation_theory

Regards,

Jon

JS:

The "part-of" natural language relationship is a well known natural
language relationship.

However, is a part allowed to be part-of itself?

In some cases yes, in some cases no.

In the case where a part is not allowed to be part-of itself,
the logical properties for this natural language relationship are:
   - irreflexive
   - asymmetric
   - transitive.

In the case where a part is allowed to be part-of itself,
the logical properties for this natural language relationship are:
  - reflexive
  - symmetric
  - transitive. (equivalence)

    or

  - reflexive
  - asymmetric
  - transitive.

These additional logical (mathematical) relation characteristics
provide a mechanism to more clearly communicate the attributes
of the current part-of natural language relationship of interest.

Natural language and mathematics are different language types.

We have created the Augmented Model-Exchange Isomorphism (AMEI)
to support the creation of a catalog of natural language terms and their
isomorphic structured graphics and mathematical forms. See:

https://www.researchgate.net/publication/272238246_Augmented_Model-Exchange_Isomorphism_Version_11

Take care and have fun,

Joe

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Joe Simpson

“Reasonable people adapt themselves to the world. 

Unreasonable people attempt to adapt the world to themselves. 

All progress, therefore, depends on unreasonable people.”

George Bernard Shaw

[joseph simpson]

Jack stated:

"How do you confirm that your presumed constraint makes the resulting system Fit For Purpose? Friedman did." 

Jack, great question and observation.

It will take some context building to create an environment where the question can be answered.

However, the primary value and operational purpose, I see in Constraint Theory, is the reduction in resources needed to evaluate and analyze any given collection of relations (equations.)

Given a collection of equations that describe systems and/or the needed integrated system behavior, Constraint Theory may be used as a guide to develop an efficient empirical data collection process that supports a specific system design process.

It might be useful to review Chapter 1 in the Constraint Theory book which places Constraint Theory in the realm of a decision support tool.  General information needed to design and produce the product system of interest is encoded into mathematical relations.  I have been thinking about discussing the first chapter, "Motivations," on the Structural Modeling and System Science lists as I believe that George's work is one of the next steps past the ISM software developed by Warfield.

Constraint Theory works with collections of relations.

One of the first steps in the analysis of Chapter 1, Motivations is the categorization of the relation types given in the example of low dimension.

Two relation categories will be used, linear relations and  non-linear relations.

In any case, my plan is to address these issues as the structural modeling material is developed.

Now I will address this area sooner, rather than later.

Bottom line: it is all about reducing cost by applying logical analysis.

Take care, be good to yourself and have fun,

Joe

On Fri, Jun 29, 2018 at 10:20 AM, Jack Ring <jri...@gmail.com> wrote:

How do you confirm that your presumed constraint makes the resulting system Fit For Purpose? Friedman did.

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[joseph simpson]

Jack:

In "Constraint Theory," Chapter 1 Motivations, page 4  an example of low dimension is given.

The context for the example is  "A decision-making manager authorized to initiate the preliminary design of a new system..."

In this example the chief systems engineer defined a "total systems optimization criterion, T."

T is given by the mathematical equation:

  T = PE/C 

   where:

     P = political index of acceptability

     E = system effectiveness

     C = life cycle cost

A few items of interest at this point.

The item of interest (T) is known and encoded in a mathematical relation.

Unlike the initial context in structural modeling (Warfield's work), where the item of interest is unknown and a natural language relationship is used by a group of individuals to determine the structure of the item of interest.

The initial item is described at a high level of abstraction using existing knowledge and a formal mathematical equation.

The item description is further elaborated by providing additional mathematical equations that decrease the level of abstraction as well as add additional existing knowledge to the item of interest description.  Table 1-1, page 5 provides a list of equations and variables.

Table 1-1 lists six (6) equations [three (3) linear equations and three (3) nonlinear equations.].

Table 1-1 lists eight (8) variables, however two (2) variables used in the equations appear to be missing.  These variables are 1) X and 2) Dmax.

On page six (6) the mathematical model is described as having six (6) equations and eight (8) variables.

It appears that the model has six (6) equations and ten (10) variables.  This is a point that is open for discussion. Why were X and Dmax not included in the model analysis.

Page 6 also states, ".. there should be two  "degrees of freedom.""  

Even with the term "degrees of freedom" in quotes it is not clear how the concept of degrees of freedom applies to a mathematical model that is composed of both linear and nonlinear equations.

Clearing up the number of variable in the example model as well as addressing the specific use of "degrees of freedom" in this case will help create a common context to continue the discussion of Constraint Theory.

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Joe Simpson

“Reasonable people adapt themselves to the world. 

Unreasonable people attempt to adapt the world to themselves. 

All progress, therefore, depends on unreasonable people.”

George Bernard Shaw

[joseph simpson]

Jack:

The questions related to the number of variables and the application of the "degrees of freedom" may be put aside as a distraction, for awhile.

Another way of looking at this specific knowledge encoding problem is to identify and apply a structuring natural language relationship that may provide some insight into the structure of the group of equations.

The current problem space is populated with two basic object types, 1) variables and 2) equations.

The variables are a basic type not composed of any other types.

The equations are a composite type composed of variables, constants, numeric values and mathematical operators.

We will focus on the basic type, variables.

What is the structuring relationship of interest between and among the given variables?

There are a few candidate structuring natural language relationships that could be used, but "dependent-on" was selected for the initial analysis.

The natural language relationship dependent-on has the following logical properties:

  - irreflexive

  - asymmetric

  - transitive

Starting with "T" at the top of a sheet of paper, a directed graph of the variable dependencies may be constructed.

Once the variable dependencies are encoded in this manner, recursive cycles of dependencies appear in some variable combinations.

If you trace the dependency thread from the T at the top of the page down a single thread to a terminal variable, without encountering a "dependency cycle", then you arrive at a variable terminal set of P, M, A, Dmax and X.

The three variables, P, M, and A are included in the book as part of the allowable computation group.  T = f(E,P), T = f(M,P) and T = f(A,P).

Due to the fact that X and Dmax were not included in the variable set in the book, they are not addressed.

It should be easy to check the and see if X and Dmax are, in fact allowed in the computations.

In any case, this is a direct connection between the work of John and George.

There are some simple evaluation and parsing rules that might be applied to this general "dependency tree" algorithm to engage a wide range of computational system types.

By selecting the proper natural language relationship, the structure of a group of equations may be evaluated and analyzed.

In this case, the components of the total system structural graph that are directed acyclic graphs appear to hold the key to "computational allowability".

The components of the total system structure that create dependency cycles appear to hold the key to computational constraint generation.

Mathematics provides a wide range of valuable knowledge encoding techniques.

Using structural modeling to more closely align natural language relationships  with formal mathematical relations has the capability to provide insight and increase communication in a number of areas.

[Jack Ring]

Joseph,

Rather than variables and equations the fundamentals may be Operands and Operators. Each can be simple, compound or complex. What you describe strikes me as a special case, not a system perspective.

Is this possible?

Jack

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[joseph simpson]

Jack:

The concepts and terms were selected to support the primary context of "Triadic Relations."

The selected terms are the terms used by George in Constraint Theory.

A system is a group of objects organized by a given relationship.

In this case, the selected objects are "variables."

In this case, the natural language organizing relationship is "dependent on."

Take care and have fun,

Joe

Jack

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[Jack Ring]

Joe,

Be clear that George was addressing a model as in a simulator so the terms he selected make sense in that context. 

When addressing a general model of system structure I suggest using more generic terms.

Your choice, of course.

Jack

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[joseph simpson]

Jack:

A brief discussion of some aspects of the example of low dimension from Chapter 1 of "Constraint Theory," is included in Technical Report 23.

Technical Report 23 is scheduled to be posted to Research Gate in a day or two.

A key, fundamental aspect of the example evaluation is the language used in the evaluation and communication of the evaluation results.

The referenced example has many specific issues when evaluated in formal mathematical terms.

For example, are the six presented equations in the collection of equations.

Two of the equations appear to address the same variable.

Does this make a duplicate entry in the collection of equations and therefore not a set?

The collection of equations contains both linear and nonlinear equations.

Does this mix of equation types matter when formal mathematics is used to evaluate the collection of equations?

If we use mathematics as the object language of general systems and natural language as the metalanguage to discuss the system object language, then these questions may need to be answered using a language other than mathematics.

Structural modeling uses natural language to evaluate system structure.

In this case, all of the listed issues associated with formal mathematical evaluation of the collection of equations become irrelevant when the "depends on" natural language relationship is used to analyze the collection of equations.

The missing variables are highlighted in this natural language analysis. 

The duplicate variable equations are combined into one factor.

The composite variable E is highlighted and shown to be comprised of other variables.

In my mind it more about language type than the level of abstraction associated with the terms.

In this specific case we have associated logical properties with our natural language relationship.

Correct logic has always been a feature of proper, effective rhetoric. 

Logic appears to be an area that spans mathematics and natural language.

In any case, the brief evaluation of Constraint Theory in Technical Report 23, is just a start.

Take care, be good to yourself and have fun,

Joe

[joseph simpson]

Jack:

During the presentation of Technical Report 23, "System Structure and Behavior," it was noted that the current presentation of Constraint Theory could be improved.

The report -- available at:

  https://www.researchgate.net/publication/326146126_System_Structure_and_Behavior

-- began to analyze the Constraint Theory examples.

If there is interest, we could continue the evaluation of this material in more detail.

Would that be a valuable exercise?

[Jon Awbrey]

Ronald, All …

Carpet installers have taken over my office — just barely have WiFi and phone, so being briefest for now …

Concrete examples are very instructive in this arena, and when it comes to refutations a single counter-example serves to puncture many a fallacious general proposition. 

Here’s a couple of articles, illustrated with concrete examples of a fundamental character, that provide a bare minimum introduction to triadic relations along with their reducibility and irreducibility properties.

• Triadic Relations 

http://intersci.ss.uci.edu/wiki/index.php/Triadic_relation

• Relation Reduction 

http://intersci.ss.uci.edu/wiki/index.php/Relation_reduction

Regards,

Jon

http://inquiryintoinquiry.com

On Jul 12, 2018, at 7:20 AM, Ronald Stamper <stamper...@gmail.com> wrote:

Dear colleagues,

May I call upon your interest in triadic relations to assist me?

A key result of our research programme is an approach to semantics that is proving very successful in practice.

Nevertheless, as Refutationists, following Popper, we always look for ways to falsify our hypotheses and theory.

One key hypothesis states that, in our semantics, triadic and higher order relations are not needed; they are always

composed of binary relationship, not of an artificial character, but ‘real’ physical, social or semiological relationships.

I would greatly appreciate any suggestions of triadic relationships that threaten to refute that hypothesis.

So far, we have not found one.

Regards,

Ronald Stamper

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[Jon Awbrey]

Ravi, Ronald, All ...

Apologies for the scattered reply but it's been a chaotic summer.

In pursuing applications of pragmatic semiotics to scientific research
the following distinctions are critical.

We have the relational roles known as Object, Sign, and Interpretant Sign.
These are places or roles a thing may occupy in a given moment in a given
context. They are not absolute essences or fixed ontological characters.

We can formalize the “moment” mentioned above as an ordered triple (o, s, i),
where o is the object, s is the sign, and i is the interpretant sign in view.

We can formalize the “context” mentioned above as a set of ordered triples,
each one having the form (o, s, i). This set is called a “sign relation”.

We can formalize a given sign relation L as a subset of a cartesian product,
L ⊆ O × S × I, where O is the set of objects under consideration in a given
context, S is the set of signs, and I is the set of interpretant signs being
considered in the same context.

It is critically important to distinguish the triples (o, s, i), which may be
called “elementary sign relations”, from the sign relation proper, L ⊆ O×S×I.
Among other things, this is important because sets have properties that their
elements do not and it amounts to a category mistake to confuse the 2 levels.
In particular, the properties of reducibility and irreducibility are defined
at the level of whole sign relations, not their individual elements.

Another very important distinction we have to keep in mind is the difference
between the formal objects we are discussing and the formal signs and syntax
we use to discuss them. I'll talk to that point more next time.

Regards,

Jon


On 7/14/2018 3:04 PM, Ravi Sharma wrote:
> What is the difference between Predicates, triples (as special case!) and
> triadic relations?
> Sanskrit grammar has 8 Cases (these are modifications of Noun or Object)
> and German 4 and I think Russian might have 6? Are these related to Triadic?
> Regards

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