Relations & Their Relatives • Discussion 13

Posted on December 8, 2015 by Jon Awbrey

Re: Peirce List Discussions • (1) • (2)

First off, we need to be clear about the difference between objects and signs:

  • Relations are formal objects of discussion and thought while relative terms are signs employed to denote relations.  (The shorthand term “relative” is short for “relative term”.)
  • The default meaning for “relative term” is general relative term, that is, a term whose denotation extends over many objects.
  • The default meaning for “relation” is general relation, that is, an aggregate, collection, or set of elementary relations.

Next, we need to be clear about the distinction between relatives (= general relatives) and elementary relatives:

  • Note.  There is a distinction in Peirce’s usage between elementary relatives and individual relatives, but if we factor in what he says about the Doctrine of Individuals and recognize that we are dealing with abstract forms then it becomes a “distinction without a difference”.  So I will tend to use the terms interchangeably.

Here is one place where Peirce exhibits his appreciation of the critical difference between relatives in general and elementary or individual relatives.

Chapter 3. The Logic of Relatives

§4. Classification of Relatives

225.   Individual relatives are of one or other of the two forms

\begin{array}{lll} \mathrm{A : A} & \qquad & \mathrm{A : B}, \end{array}

and simple relatives are negatives of one or other of these two forms.

226.   The forms of general relatives are of infinite variety, but the following may be particularly noticed.

Relatives may be divided into those all whose individual aggregants are of the form \mathrm{A : A} and those which contain individuals of the form \mathrm{A : B}.  The former may be called concurrents, the latter opponents.

☞ Peirce’s 1880 “Algebra Of Logic” Chapter 3 • Selection 7

It needs to be appreciated that classifying relations is vastly more complex than classifying elementary or individual relations.

In particular, classifying sign relations is vastly more complex than classifying elementary or individual sign relations, which is just about all the entire literature on sign taxonomy has been able to touch from Peirce’s time to ours.

Resources

Posted in C.S. Peirce, Dyadic Relations, Logic, Logic of Relatives, Mathematics, Peirce, Peirce List, Relation Theory, Semiotics, Sign Relations, Triadic Relations | Tagged , , , , , , , , , , | 11 Comments

Survey of Relation Theory • 2

Posted on November 30, 2015 by Jon Awbrey

In this Survey of blog and wiki posts on Relation Theory, relations are viewed from the perspective of combinatorics, in other words, as a topic in discrete mathematics, with special attention to finite structures and concrete set-theoretic constructions, many of which arise quite naturally in applications.  This approach to relation theory is distinct from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other.

Elements

Relational Concepts

Relation Construction Relation Composition Relation Reduction
Relative Term Sign Relation Triadic Relation
Logic of Relatives Hypostatic Abstraction Continuous Predicate

Illustrations

Blog Series

Peirce’s 1880 “Algebra of Logic” Chapter 3

Resources

Posted in Algebra, Algebra of Logic, C.S. Peirce, Category Theory, Combinatorics, Discrete Mathematics, Duality, Dyadic Relations, Foundations of Mathematics, Graph Theory, Logic, Logic of Relatives, Mathematics, Peirce, Relation Theory, Semiotics, Set Theory, Sign Relational Manifolds, Sign Relations, Surveys, Triadic Relations, Triadicity, Type Theory | Tagged , , , , , , , , , , , , , , , , , , , , , , | 3 Comments

Peirce’s Categories • 3

Posted on November 4, 2015 by Jon Awbrey

Re: Peirce List

Recent travels and other travails (dental work) have scattered my thoughts to the four winds, so let me just document a few bits from my current state of mind in case I can get back to it someday.

Here is the figure I drew to illustrate a recurring theme from Peirce.

Peirce Syllabus

Normative science rests largely on phenomenology and on mathematics;
metaphysics on phenomenology and on normative science.

— Charles Sanders Peirce, Collected Papers, CP 1.186 (1903)
Syllabus : Classification of Sciences (CP 1.180–202, G-1903-2b)

Here are a few resources I find useful by way of establishing a foothold on these shores.

Posted in Abstraction, Analogy, C.S. Peirce, Category Theory, Logic, Logic of Relatives, Mathematics, Peirce's Categories, Phenomenology, Philosophy, Pragmatism, Relation Theory, Semiotics, Triadic Relations, Triadicity, Type Theory | Tagged , , , , , , , , , , , , , , , | 11 Comments

Peirce’s Categories • 2

Posted on October 31, 2015 by Jon Awbrey

Re: Peirce List • Jeffrey Brian Downard • Gary Richmond • John Collier

According to Peirce, it is logic that draws on both mathematics and phenomenology.

At any rate, Peirce takes the distinctive position that normative science, which includes logic, “rests largely on” phenomenology and mathematics.  Unless there is a case to be made for a practical difference between drawing on and resting on, as those phrases are intended in the present setting, I would have to say they mean the same thing.

I discussed the relationship among these sciences in a previous post and drew the following figure to illustrate it.


Peirce Syllabus

Normative science rests largely on phenomenology and on mathematics;
metaphysics on phenomenology and on normative science.

— Charles Sanders Peirce, Collected Papers, CP 1.186 (1903)
Syllabus : Classification of Sciences (CP 1.180–202, G-1903-2b)

The following post contains a longer excerpt from Peirce’s Classification of the Sciences.

Posted in Abstraction, C.S. Peirce, Category Theory, Dimensionality, Logic, Logic of Relatives, Mathematics, Peirce, Peirce's Categories, Phenomenology, Pragmatism, Relation Theory, Semiotics, Triadic Relations, Type Theory | Tagged , , , , , , , , , , , , , , | 12 Comments

Peirce’s Categories • 1

Posted on October 30, 2015 by Jon Awbrey

Re: Peirce List • Jeffrey Brian Downard

Just from my experience, the best first approach to questions of firstness, secondness, thirdness, and so on is to regard k-ness as the property that all k-adic relations possess in common.  There is more to say once this first point is appreciated but it is critical to begin from this understanding.

It is best to view k-adic relations as whole sets of k-tuples rather than fixating on single k-tuples at a time since all the most relevant properties of relations are “holistic” properties of whole sets or whole systems that are not reducible to properties of their individual elements.

A k-adic relation and its converses, numbering k! possibilities in all, each bears essentially the same information about the relation of its domains.  This means that fixating on a particular ordering will tend to distract us with inessential features of a particular presentation rather than highlighting the essential properties of the relation in view.

Peirce demonstrates in several places that he appreciates the significance of these facts.

Just my k bits …

cc: Peirce List

Posted in Abstraction, C.S. Peirce, Category Theory, Dimensionality, Logic, Logic of Relatives, Mathematics, Peirce, Peirce's Categories, Phenomenology, Pragmatism, Relation Theory, Semiotics, Triadic Relations, Type Theory | Tagged , , , , , , , , , , , , , , | 10 Comments

What Makes An Object? • 2

Posted on October 26, 2015 by Jon Awbrey

Re: Peirce List Discussison • (1) • (2)

Visual metaphors and perceptual analogies can be instructive — they make for most of my personal favorites — but in logic, mathematics, and science our interest extends through the abductive spectrum, from percept formation to where it shades off to concept formation to where it takes off in theory formation.

Objects in logic and semiotics are any objects of discussion or thought — atoms and atomic clocks, bubbles and bubble chambers, clouds and cohorts, determinants and deuterium, electrons and ellipses, galaxies and ganglia, photons and positrons, quarks and question marks, …, you get the picture.

If we take care that our signs make sense, and by this “sense” mean to say they have objects that are logically consistent, then we pass from the realm of mere semiotics, where literary clutches will dilletate till the twelfth of never on the Madness Of Prince Hamlet (MOPH1) or the Method Of Prince Hamlet (MOPH0), or the taste of Organic Martian Potatoes (OMPs), and leaving all that till the twelfth of never we enter the realm of formal or normative semiotics that we know as logic.

Posted in C.S. Peirce, Interpretation, Interpretive Frameworks, Intuition, Logic, Logic of Relatives, Manifolds, Mathematics, Objective Frameworks, Peirce, Peirce List, Physics, Pragmata, Pragmatism, Process, Process Thinking, Relation Theory, Semiosis, Semiotics, Sign Relational Manifolds, Sign Relations, Triadic Relations | Tagged , , , , , , , , , , , , , , , , , , , , , | 1 Comment

What Makes An Object? • 1

Posted on October 25, 2015 by Jon Awbrey

Re: Gary Fuhrman • Seeing Things

What makes an object is a perennial question.

I can remember my physics professors bringing it up in a really big way when I was still just a freshman in college.  They always cautioned us then about extrapolating our everyday intuitions about everyday objects beyond their native realms.

Anyone who has been graced or grazed by a modicum of process thinking, say Whitehead or Bucky Fuller, is aware of the trade-off between process thinking and product thinking that rules our descriptions of every domain of phenomena, but in a retrograde time like the one we are currently experiencing it takes a mighty effort to recollect the way that hidebound objects are precipitated from more primal processes.

Here’s an old post I happened on that may apply here:

Posted in C.S. Peirce, Interpretation, Interpretive Frameworks, Intuition, Logic, Logic of Relatives, Manifolds, Mathematics, Objective Frameworks, Peirce, Peirce List, Physics, Pragmata, Pragmatism, Process, Process Thinking, Relation Theory, Semiosis, Semiotics, Sign Relational Manifolds, Sign Relations, Triadic Relations | Tagged , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Forgetfulness Of Purpose • 8

Posted on October 12, 2015 by Jon Awbrey

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

Just to review, we were looking at Ashby’s example of a regulation game as given below.

\begin{array}{cc|ccc} \multicolumn{5}{c}{\text{Table 11/3/1}} \\[4pt] & & & R & \\ & & \alpha & \beta & \gamma \\ \hline & 1 & b & a & c \\ D & 2 & a & c & b \\ & 3 & c & b & a \end{array}

I observed that this gives us a triadic relation G_1 \subseteq D \times R \times O whose triples are listed next.

\begin{matrix} D & R & O \\ \hline 1 & \alpha & b \\ 1 & \beta & a \\ 1 & \gamma & c \\ 2 & \alpha & a \\ 2 & \beta & c \\ 2 & \gamma & b \\ 3 & \alpha & c \\ 3 & \beta & b \\ 3 & \gamma & a \end{matrix}

Sungchul Ji raised a question about the irreducibility of G_1 as a triadic relation, suggesting that it would imply a degree of “communication” or “transfer of information” between D and R as constrained by the relation G_1.

For the moment I remain puzzled how we ought to measure various orders and quantities of information that affect the play of regulation games.  That question evidently depends on the kinds of structures that are key to the regulation task.  The quickest way to gain information on those scores is probably to read a little further in Ashby’s text, so I will set a goal of doing that.

On the other hand, the irreducibility of G_1 as a triadic relation can be investigated more directly, without considering the information question, so I will try that tack first.

Reference

  • Ashby, W.R. (1956), An Introduction to Cybernetics, Chapman and Hall, London, UK.  Republished by Methuen and Company, London, UK, 1964.  Online.
Posted in Anamnesis, Ashby, C.S. Peirce, Cybernetics, Memory, Peirce, Pragmata, Purpose, Systems Theory | Tagged , , , , , , , , | 1 Comment

Forgetfulness Of Purpose • 7

Posted on October 7, 2015 by Jon Awbrey

Re: Peirce List Discussion • Sungchul Ji

I invited readers to consider Ashby’s example of a regulation game as a triadic relation G_1 \subseteq D \times R \times O whose triples (d, r, o) are given by either one of the following tables.

\begin{array}{cc|ccc} \multicolumn{5}{c}{\text{Table 11/3/1}} \\[4pt] & & & R & \\ & & \alpha & \beta & \gamma \\ \hline & 1 & b & a & c \\ D & 2 & a & c & b \\ & 3 & c & b & a \end{array}

\begin{matrix} D & R & O \\ \hline 1 & \alpha & b \\ 1 & \beta & a \\ 1 & \gamma & c \\ 2 & \alpha & a \\ 2 & \beta & c \\ 2 & \gamma & b \\ 3 & \alpha & c \\ 3 & \beta & b \\ 3 & \gamma & a \end{matrix}

Sungchul Ji asked a rather good question about the degree of “communication” or “transfer of information” between D and R within G_1 and its bearing on the irreducibility of G_1 as a triadic relation.  Given the light that concrete examples can throw on abstract questions, I thought it worth the effort to work through a detailed answer and I began as follows:

Ashby cast this example at a high level of abstraction in order to isolate the “bare bones” structure of the situation but we know the intended interpretation makes D the source of disturbances arising from the environment and R the regulator who has to counteract the disturbances in order to maintain the goal.

With regard to the question of communication or transfer of information between D and R, it is clear that R has to make different choices depending on what D does, so there is a degree of information transmission between D and R to that extent.  In effect, R is trying to cancel the variety that D creates.

Careful discussions of irreducibility will require us to work from explicit definitions and here it turns out there are several different notions of irreducibility that are often confounded in common use.

Looking to the case at hand, G_1 \subseteq D \times R \times O is a triadic relation consisting of 9 triples in the larger set D \times R \times O that consists of 3 \times 3 \times 3 = 27 triples.

We could reasonably use the ratio 3/9 = 1/3 as one measure of constraint, determination, information, selection, or “law” involved in carving G_1 from the uncarved block D \times R \times O.

If we prefer an additive measure, we could extract the exponents from the ratio 3^1/3^3 and use their difference as a measure of the information it takes to select G_1 from D \times R \times O.  Peirce pulled this very trick in some manuscripts I saw one time, getting a simple version of our modern day logarithmic measure of information.

To be continued …

Reference

  • Ashby, W.R. (1956), An Introduction to Cybernetics, Chapman and Hall, London, UK.  Republished by Methuen and Company, London, UK, 1964.  Online.
Posted in Anamnesis, Ashby, C.S. Peirce, Cybernetics, Memory, Peirce, Pragmata, Purpose, Systems Theory | Tagged , , , , , , , , | 3 Comments

Forgetfulness Of Purpose • 6

Posted on October 5, 2015 by Jon Awbrey

Just enough time for an incidental observation.

Consider the table Ashby uses to describe his first example of a regulation game.

\begin{array}{cc|ccc} \multicolumn{5}{c}{\text{Table 11/3/1}} \\[4pt] & & & R & \\ & & \alpha & \beta & \gamma \\ \hline & 1 & b & a & c \\ D & 2 & a & c & b \\ & 3 & c & b & a \end{array}

A table like that is a compact way of describing a triadic relation, in this case a relation L \subseteq D \times R \times O whose triples are displayed in the following relational data table.

\begin{matrix} D & R & O \\ \hline 1 & \alpha & b \\ 1 & \beta & a \\ 1 & \gamma & c \\ 2 & \alpha & a \\ 2 & \beta & c \\ 2 & \gamma & b \\ 3 & \alpha & c \\ 3 & \beta & b \\ 3 & \gamma & a \end{matrix}

May you find food for thought, a late night snack on those tables.

Reference

  • Ashby, W.R. (1956), An Introduction to Cybernetics, Chapman and Hall, London, UK.  Republished by Methuen and Company, London, UK, 1964.  Online.
Posted in Anamnesis, Ashby, C.S. Peirce, Cybernetics, Memory, Peirce, Pragmata, Purpose, Systems Theory | Tagged , , , , , , , , | 1 Comment