## Triadic Forms of Constraint, Determination, Interaction • 2

Re: Peirce List Discussion • GR

Here’s one way of stating what I call a constraint:

• The set $L$ is constrained to a subset of the set $M.$

Here’s one way of stating a triadic constraint:

• The set $L$ is a subset of the cartesian product $X \times Y \times Z.$

So any way we define a triadic relation we are stating or imposing a triadic constraint.

In particular, any way we define a sign relation we are stating or imposing a triadic constraint of the form:

• $L \subseteq O \times S \times I.$

where:

• $O$ is the set of all objects under discussion,
• $S$ is the set of all signs under discussion, and
• $I$ is the set of all interpretant signs under discussion.

The concepts of constraint, definition, determination, lawfulness, ruliness, and so on all have their basis in the idea that one set is contained as a subset of another set.

Among the next questions that may occur to us, we might ask:

• What bearings do these types of global constraints have on various local settings we might select?

And conversely:

• To what extent do various types of local constraints combine to constrain or determine various types of global constraint?

There are by the way such things as mutual constraints, indeed, they are very common, and not just in matters of human bondage.  So, for instance, the fact that objects constrain or determine signs in a given sign relation does not exclude the possibility that signs constrain or determine objects within the same sign relation.

## Triadic Forms of Constraint, Determination, Interaction • 1

Re: Peirce List Discussion • JAGRJAJBD

There are many places where Peirce uses the word object in the full pragmatic sense, so much so that it demands a very selective attention not to remark them.  I cited a couple at the top of this discussion but perhaps the most critical locution for the sake of pragmatism is stated here:

Consider what effects that might conceivably have practical bearings you conceive the objects of your conception to have.  Then, your conception of those effects is the whole of your conception of the object.  (CP 5.438).

## The object of reasoning is to find out …

No longer wondered what I would do in life but defined my object.
— C.S. Peirce (1861), “My Life, written for the Class-Book”, (CE 1, 3)

The object of reasoning is to find out, from the consideration of what we already know, something else which we do not know.
— C.S. Peirce (1877), “The Fixation of Belief”, (CP 5.365)

If the object of an investigation is to find out something we do not know then the clues we discover along the way are the signs that determine that object.

People will continue to be confused about determination so long as they can think of no other forms of it but the analytic-behaviorist-causal-dyadic-temporal, object-as-stimulus and sign-as-response variety.  It is true that ordinary language biases us toward billiard-ball styles of dyadic determination, but there are triadic forms of constraint, determination, and interaction that are not captured by S-R chains of that order.  A pragmatic-semiotic object is anything we talk or think about, and semiosis does not conduct its transactions within the bounds of object as cue, sign as cue ball, and interpretants as solids, stripes, or pockets.

### References

• Peirce, C.S. (1859–1861), “My Life, written for the Class-Book”, pp. 1–3 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
• Peirce, C.S. (1877), “The Fixation Of Belief”, Popular Science Monthly 12 (Nov 1877), pp. 1–15.  Reprinted in Collected Papers, CP 5.358–387.  Online.

## Icon Index Symbol • 8

### Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List Discussion • Helmut Raulien

The difference between the two definitions of a $k$-place relation in the previous post is sometimes described as decontextualized versus contextualized or, in computer science lingo, weak typing versus strong typing.  The second definition is typically expressed in a peculiar mathematical idiom that starts out as follows:

A $k$-place relation is a $(k\!+\!1)$-tuple $(X_1, \ldots, X_k, L)$

That way of defining relations is a natural generalization of the way functions are defined in the mathematical subject of category theory, where the domain $X$ and the codomain $Y$ share in defining the type $X \to Y$ of the function $f : X \to Y.$

The threshold between arbitrary, artificial, or random kinds of relations and those selected for due consideration as reasonable, proper, or natural kinds tends to shift from context to context.  We usually have in mind some property or quality that marks the latter class as proper objects of contemplation relative to the end in view, and so this relates to both the intensional and the intentional views of subject matters.

To be continued …

## Icon Index Symbol • 7

### Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List Discussion • Helmut Raulien

Looking back over many previous discussions, I think one of the main things keeping people from being on the same page, or even being able to understand what others write on their individual pages, is the question of what makes a relation.

There’s a big difference between a single ordered tuple, say, $(a_1, a_2, \ldots, a_k),$ and a whole set of ordered tuples that it takes to make up a $k$-place relation.  The language we use to get a handle on the structure of relations goes like this:

Say the variable $x_1$ ranges over the set $X_1,$
and the variable $x_2$ ranges over the set $X_2,$
$\cdots$
and the variable $x_k$ ranges over the set $X_k.$

Then the set of all possible $k$-tuples $(x_1, x_2, \ldots, x_k)$ ranges over a set that is notated as $X_1 \times X_2 \times \ldots \times X_k$ and called the “cartesian product” of the “domains” $X_1$ to $X_k.$

There are two different ways in common use of defining a $k$-place relation.

1. Some define a relation $L$ on the domains $X_1$ to $X_k$ as a subset of the cartesian product $X_1 \times \ldots \times X_k,$ in symbols, $L \subseteq X_1 \times \ldots \times X_k.$
2. Others like to make the domains of the relation an explicit part of the definition, saying that a relation $L$ is a list of domains plus a subset of their cartesian product.

Sounds like a mess but it’s usually pretty easy to translate between the two conventions, so long as one watches out for the difference.

By way of a geometric image, the cartesian product $X_1 \times \ldots \times X_k$ may be viewed as a space in which many different relations reside, each one cutting a different figure in that space.

To be continued …

## Icon Index Symbol • 6

### Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List Discussion • HRJASHR

I think it would be a good idea to continue reviewing basic concepts and get better acquainted with the relational context needed to ground the higher level functions, properties, and structures we might wish to think about.  Once we understand what relations are, then we can drill down to triadic relations, and then sign relations will fall more readily within our grasp.

I’ve written up introductions to these topics on a number of occasions and the latest editions can be found on the InterSciWiki site, though in this case it may be better to take them up in order from special to general:

The related question of Relational Reducibility, in its Compositional and Projective aspects, is treated in the following article:

To be continued …

## Icon Index Symbol • 5

### Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List Discussion • Helmut Raulien

Given that sign relations are special cases of triadic relations, we can get significant insight into the structures of both cases by examining a few simple examples of triadic relations, without getting distracted by all the extra features that come into play with sign relations.

When I’m talking about a $k$-place relation $L$ I’ll always be thinking about a set of $k$-tuples.  Each $k$-tuple has the form:

$(x_1, x_2, \ldots, x_{k-1}, x_k),$

or, as Peirce often wrote them:

$x_1 : x_2 : \ldots : x_{k-1} : x_k.$

Of course, $L$ could be a set of one $k$-tuple but that would be counted a trivial case.

That sums up the extensional view of $k$-place relations, so far as we need it for now.

Using a single letter like $L"$ to refer to a set of $k$-tuples is already the genesis of an intensional view, since we now think of the elements of $L$ as having some property in common, even if it’s only their membership in $L.$  When we turn to devising some sort of formalism for working with relations in general, whether it’s an algebra, logical calculus, or graph-theoretic notation, it’s in the nature of the task to “unify the manifold”, to represent a many as a one, to express a set of many tuples by means of a single sign.  That can be a great convenience, producing formalisms of significant power, but failing to discern the many in the one can lead to no end of confusion.

To be continued …