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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[Ravi Sharma]

Jon thanks good info

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Thanks.
Ravi
(Dr. Ravi Sharma)

 313 204 1740 Mobile 

[Jon Awbrey]

Thanks, Ravi,

I chose those examples of triadic relations to be as simple
as possible without being completely trivial but they already
exemplify many features we need to keep in mind in all the more
complex cases as we use relational models of realistic phenomena
and objective domains.

The mathematical examples are typical of many in linguistic, logical,
and mathematical contexts where we start out with compact, ready-made
axioms, definitions, equations, expressions, formulas, predicates, or
terms that denote the relations of interest. For example, we might be
discussing dyadic relative terms like “parent_of___” or “square_of___”
and triadic relative terms like “giver_of___to___” or “sum_of___and___”.

If we spend the majority of our time in contexts like that we may form
the impression that all the relational concepts we'll ever need can be
requisitioned off-the-shelf from pre-fab stock, no assembly required.

That's a pretty picture of our mental equipment. It may even be true
if we cook the data long enough and fudge the meaning of pre-fab down
to the level of amino acids or quarks or some other bosons on the bus.

As a practical matter, however, research pursued in experimental veins
tends to push the envelope of pre-fab concepts into surprisingly novel
realms of ideas.

I'll discuss the examples of sign relations when I get more time ...

Regards,

Jon

[Ferenc Kovacs]

Ravi, Jon,

For people who do not accept that relations in a natural language is represented by a verb, although the very same relation may be described in other linguistic (grammar) structures and who do not believe that the most generic word and unit of expressing ideas is an object grounded in sensory experience and some mental operations as separation and/or isolation (resulting in a name as well as a concept) it may be difficult to see that relations are operations whether they are indicated expressly or hidden otherwise.

Hence instead of triadic relations you have operands and operators where an operator is also a separator, yet altogether resulting in an operation. The most generic operation follows transitivity, in other words if you have a subject that has to be linked to an object reflecting associations between physical things and their names, objects and their properties, etc. in various sequences produced by the operation transit or change).

An SVO sentence pattern is an example of such transitivity in action. A logical phrase SP and SQ are other morphs with the operator disguised. Operators are normally not indicated as in „ab” (axb) or even in additive relations. In addition for example, what you make note of is the sum itself, and not the number of operations (addition) performed in the calculation (because they are the same in numeric terms). This is as important feature, although totally neglected, because it represents a compact way of denoting symbols and their relations. Nevertheless, when making a computer, you cannot disregard keeping track of the number operations performed.

Consequently, you have various kinds of objects all related by a verb that when fully specified, gives you the details of such a relation that enables you to visualize what takes place/is happening/or was the case. Comparison is the most fundamental mental operation that calls for two objects that are comparable on some grounds despite their differences in other respects. In ancient times comparison was physically performed by overlapping the two items in „space” (on a plane in fact, which was the forerunner of division, both a physical and a mental operation that proved to be very useful in a world where the distribution of goodies was a major component of social life.

The fact that you do (can) not keep track of the number of operations (repetitions of the operand and operators at different times and places) has lead us to two extreme solutions, namely the use of quantors without being able to check if any of those quantors are true or make sense. Fuzzy sets explain how you go round that problem, but interestingly enough the use of quantors is so popular that even syllogisms are built on them as some foolproof ground so logic and reasoning despite the fact that syllogism are nothing but a sort operation base on quantity, which may sound provoking but I need to explain it next time, should anyone be interested.

 Ferenc Kovacs

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[Matthew West]

Dear Colleagues,

 

Ravi, Jon,

 

For people who do not accept that relations in a natural language is represented by a verb, although the very same relation may be described in other linguistic (grammar) structures and who do not believe that the most generic word and unit of expressing ideas is an object grounded in sensory experience and some mental operations as separation and/or isolation (resulting in a name as well as a concept) it may be difficult to see that relations are operations whether they are indicated expressly or hidden otherwise.

[MW>] The mistake that people often make is to consider that a relation is something that is represented, rather than it being a way of representing something. That is, it is a mathematical structure that can be used in many different ways to represent many different things. It is not a fundamental ontological category.

 

Regards

Matthew West

[joseph simpson]

It may be useful to talk about "natural language relationships" and "mathematical relations."

A single mathematical relationship may represent many natural language relationships.

A natural language relationship may be used to further define and refine a specific mathematical relation application.

A mathematical relation may be used to further quantify and evaluate a natural language relationship.

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

[Ferenc Kovacs]

On Tuesday, June 19, 2018, 8:32:19 PM GMT+2, Matthew West <dr.matt...@gmail.com> wrote:

Dear Matt,

On Relation:  ... It is not a fundamental ontological category.

The three fairly widely recognized categories of an upper ontology are objects, relations and properties. Objects do have the possibility for verification through sensors of all kinds. Relations and properties are abstractions though, therefore they cannot be falsified that easily. It is also true that while objects tend to lend themselves to common classifications based on properties and relations, no such structures are known for properties and relations.

Unfortunately, relations are deemed to be unrelated to verbs and are used in a simplified manner to fit set theory, relations are not treated as complex patterns with all the specifications carried by verbs. Properties on the other hand allow for inversion with objects again to fit set theory interpretation of the world that is more rationally described by using statements and propositions with fully specified verbs.

 

a relation is something that is represented, rather than it being a way of representing something

A way of representing something (an object for instance) IS REPRESENTING, or performing an operation, not specified in more details, an answer to the how question that calls for a verb, which in turn calls for another verb, or an operation (originally meaning energy) to put in a generic form or relation. We do not know how to break down the web of everything is related to everything else but by comparing objects pairwise. Comparing is relating, and relating is one of the several mental operations that we are not aware of, except for the simplicity of induction, deduction and abduction.

Regards, Ferenc

--

[Jon Awbrey]

Joe,

I'll reply to your remarks first simply because they are the shortest
and they give me the quickest chance of clarifying a few simple points.

I use the word “relation” to mean a special type of mathematical object,
namely, a designated subset included within a cartesian product of sets.

Whatever else this definition of a relation may have going for or against it,
it does single out a class of formal structures that serve in good stead as
intermediary objects between the world of phenomena and our human capacity
for coping with whatever reality may emanate in them. So that is mainly
how I aim to use it here.

When I'm being careful, therefore, I'll try to use words that maintain
a distinction between objects, formal or otherwise, and the symbolic
modifications of media that we use to reference those objects.

For example:

> The mathematical examples are typical of many in linguistic, logical,
> and mathematical contexts where we start out with compact, ready-made
> axioms, definitions, equations, expressions, formulas, predicates, or
> terms that denote the relations of interest. For example, we might be
> discussing dyadic relative terms like “parent_of___” or “square_of___”
> and triadic relative terms like “giver_of___to___” or “sum_of___and___”.

I used “axioms, definitions, equations, expressions, formulas, predicates, terms”
along with “dyadic relative terms” and “triadic relative terms” for various sorts
of symbolic entities that serve to denote or describe formal objects of thought
and discussion, while I tried to reserve “relations” for the objects themselves.

Hope that helps to clear things up ...

Regards,

Jon

On 6/19/2018 2:42 PM, joseph simpson wrote:
> It may be useful to talk about "natural language relationships" and
> "mathematical relations."
>
> A single mathematical relationship may represent many natural language
> relationships.
>
> A natural language relationship may be used to further define and refine a
> specific mathematical relation application.
>
> A mathematical relation may be used to further quantify and evaluate a
> natural language relationship.
>
> Take care and have fun,
>
> Joe
>

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[joseph simpson]

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.

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

[John F Sowa]

Dear Matthew, Jon, Ravi, John Y, and Joe,

In March, I was the respondent for a talk on "The necessity of genuine
triadic relations" by Bill McCurdy at a philosophy convention in
San Diego. Since Bill and I are both Peirce fans, we were in strong
agreement. See http://jfsowa.com/talks/triads.pdf

MW (in a response to Jon and Ravi)

> The mistake that people often make is to consider that a relation
> is something that is represented, rather than it being a way of
> representing something. That is, it is a mathematical structure
> that can be used in many different ways to represent many different
> things. It is not a fundamental ontological category.

I agree. To illustrate the issues, note the two different ways
of representing the sentence "Sue gives a child a book" in slide 3
of triads.pdf.

On the left is a conceptual graph (CG), which represents the
verb 'gives' by a triadic relation named Gives. On the right is
a CG with a concept of type Give, which is linked by three dyadic
relations (Agent, Recipient, and Theme) to concepts for Sue, a
child, and a book.

This example illustrates three points:

1. For linguists, you can choose to represent verbs by a relation
or to a node that could represent the verb 'give' or the gerund
Giviing. The option depends on whatever ontology you choose.

2. For logicians, it illustrates the point that any triadic relatioin
can be replaced by three dyadic relations.

3. And for graph theorists, it shows that there is an unavoidable
triadic connection that will always be there, no matter which
choice of ontology you happen to choose.

JA

> 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.

I agree. But if you say that to a logician, you'll get the response,
"Any triadic relation can be replaced by three dyadic relations."

See slide 2 for the kind of response that Peirce, Bill McCurdy, and I
would make to that point: Peirce used the term 'triad', not triadic
relation. If you replace all triadic relations with dyadic relations,
the graph will still contain irreducible triadic nodes. In existential
graphs, you will get a "teridentity", which consists of three lines
linked together. In CGs, you get a concept node with three links.

No matter how you transform the graph, you'll always have something
that represents that triadic notion. See the rest of triads.pdf.

MW
> Existence here is not about whether you can kick something, but
> about whether you can talk about it.

I agree. And that raises many important issues about logic and
ontology that would take too long to explain in this note.

Short summary: languages and logics consist of signs, and some
of those signs express propositions, some of which are laws.
For more, see http://jfsowa.com/pubs/signs.pdf

JY
> I find that having separate classes for Information Entity and
> Information Realization helps to know what I am representing.
> The fact that it is dependent continuant helps but may not be
> the dominant characteristic. So there is nothing preventing
> addition of these classes as subclasses in BFO?

Barry classified information artifacts as dependent continuants.
That makes the information contained in the book "depend" on
the printed or otherwise represented copies.

But as I said in a previous note, that is a half-vast theory of
signs. It lets you talk about sign tokens (copies of information),
but not about sign types (the kinds of information contained in
those sign tokens). Philosophers from antiquity to the present
have gone into huge amounts of detail about signs and systems of
signs such as languages. See http://jfsowa.com/pubs/signs.pdf

In that article, I quote the philosopher David Armstrong, who
mentioned a “distinction that practically all contemporary
philosophers accept... It is the distinction between token and
type” by Charles Sanders Peirce.

Joe S

> It may be useful to talk about "natural language relationships"
> and "mathematical relations." A single mathematical relationship
> may represent many natural language relationships. A natural
> language relationship may be used to further define and refine
> a specific mathematical relation application. A mathematical
> relation may be used to further quantify and evaluate a natural
> language relationship.

Yes. There are many ways to translate an NL text to some version
of logic for different purposes in different contexts. Slide 2
of triads.pdf is just one example.

John

[Jon Awbrey]

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:

⁂

[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

[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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[John F Sowa]

On 6/29/2018 11:44 AM, Jon Awbrey wrote:
> a binary relation L is a subset of a cartesian product X × Y
> of two sets, X and Y. In symbols, L ⊆ X × Y.

That is a purely extensional definition. It doesn't allow you
to distinguish relations that happen to have different intensions,
but the same extension.

For example, people used to think that the relation swan(x) and the
relation white-swan(x) were identical because they never saw a swan
of any other color. But when they discovered black swans, they had
to recognize that the two definitions represented distinct sets.

More generally, a definition of a relation R(x1,...,xN) by intension
is a rule or criterion for determining whether R is true of some
N-tuple. But a definition by extension is just a set of N-tuples.

You can have multiple relations with distinct intensions, but the
same extension. For more about intensions and extensions, the
logician Alonzo Church wrote a very nice three-page discussion:
http://jfsowa.com/logic/alonzo.htm

In those three pages, Church talks about definitions of functions,
but the same principle holds for definitions of relations.

John

[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.

[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

On Sun, Jul 1, 2018 at 10:31 AM, Jack Ring <jri...@gmail.com> wrote:

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:

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

On Mon, Jul 2, 2018 at 10:13 AM, Jack Ring <jri...@gmail.com> wrote:

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

[Jon Awbrey]

Ferenc, Ontologists ...

Our thoughts live in natural and artificial languages the
way fish swim in natural and artificial bodies of water.

One of the lessons most strikingly impressed on me by my first year physics course
or the mass of collateral reading I did at the time was to guard against the errors
that arise from “projecting the properties and structures of any language or symbol
system on the external world”. This was mentioned especially often in discussions
of quantum mechanics — it was a common observation that our difficulties grasping
wave-particle duality might be due to our previous conditioning to see the world
through the lenses of our subject-predicate languages and logics. Soon after, I
learned about the Sapir-Whorf hypothesis, and today I lump all those cautionary
tales under the heading of GRAM (“Grammar Recycled As Metaphysics”).

Back to packing ...
More later ...

Jon

On 6/19/2018 1:31 PM, 'Ferenc Kovacs' via ontolog-forum wrote:
> Ravi, Jon, For people who do not accept that relations in a natural language is represented by a verb, although the

> very same relation may be described in other linguistic(grammar) structures and who do not believe that the most

[Ferenc Kovacs]

Jon

As I have written before the fact that we can move about in space and (time), both physically (in fact) and mentally enables us to switch points of views that are described in geometric terms resulting in points, two of which connect as a line and three of which connect as a plane. Lines have a bipolar setup, but moving on up and down is the same as a circle, or a cycle. Words are invented to represent the end points and experience shows that such a setup is contrast required for recognition of points in language too. Movement is thus closed, may be bidirectional and is the most likely structure of associations, the elements or repetition, substitution and recursion. Moving on a line connecting three points means moving on a plane that cannot be represented by words arranged in a planar fashion in a natural language manifested linearly.

The reason or explanation may be that the number of linguistic units may not be arbitrarily (more than two or three), they may not be connected in any kind of chunks at will either, because in thinking you need to follow a particular direction and  you must stay in the confinement of a single plane (three points). It is true whether you manipulate words or numbers, it is easy to repeat addition and multiplication, etc. of twos, but not threes which requires changing directions or leaving the plane of communicative operations. As I once already quoted this passage:

" \Every morning in the early part of the above-cited month, on my coming down to breakfast, your (then) little brother William Edwin, and yourself, used to ask me: `Well, Papa, can you multiplytriplets?' Whereto I was always obliged to reply, with a sad shake of the head: `No, I can only addand subtract them.' " The problem, of course, was that there exists no 3-dimensional normed division algebra. He really needed a 4-dimensional algebra.

i2 = j2 = k2 = ijk = 􀀀1:

 He showed that they were a normed division algebra and used this to express the product of two sums of eight perfect squares as another sum of eight perfect squares: the `eight squares theorem' [51 http://www.ams.org/journals/bull/2002-39-02/S0273-0979-01-00934-X/S0273-0979-01-00934-X.pdf

 Ferenc

Ferenc Kovacs

Cell phone: 36-70-6784000 

Landline:36-87-782831

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[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?

[Ronald Stamper]

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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