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).

Reference

Posted in C.S. Peirce, Constraint, Determination, Discovery, Dyadic Relations, Fixation of Belief, Inference, Information, Inquiry, Intentional Objects, Intentionality, Law of Nature, Logic, Logic of Science, Objects Objectives Objectivity, Peirce, Philosophy, Pragmata, Pragmatism, Scientific Inquiry, Semeiosis, Semiotics, Sign Relations, Triadic Relations, Triadicity | Tagged , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

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.
Posted in C.S. Peirce, Determination, Discovery, Dyadic Relations, Fixation of Belief, Inference, Information, Inquiry, Intentional Objects, Intentionality, Logic, Logic of Science, Objects Objectives Objectivity, Peirce, Philosophy, Pragmata, Pragmatism, Scientific Inquiry, Semeiosis, Semiotics, Sign Relations, Triadic Relations, Triadicity | Tagged , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

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 …

Resources

Posted in Abduction, Algorithms, Artificial Intelligence, Automated Research Tools, C.S. Peirce, Cognitive Science, Computer Science, Data Structures, Deduction, Functional Logic, Icon Index Symbol, Induction, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Intelligent Systems, Interpretive Frameworks, Knowledge Representation, Logic, Logic of Relatives, Logic of Science, Logical Graphs, Mathematics, Objective Frameworks, Peirce, Relation Theory, Relative Membership, Scientific Method, Semiotics, Set Theory, Sign Relations, Systems Theory, Triadic Relations | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

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 …

Resources

Posted in Abduction, Algorithms, Artificial Intelligence, Automated Research Tools, C.S. Peirce, Cognitive Science, Computer Science, Data Structures, Deduction, Functional Logic, Icon Index Symbol, Induction, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Intelligent Systems, Interpretive Frameworks, Knowledge Representation, Logic, Logic of Relatives, Logic of Science, Logical Graphs, Mathematics, Objective Frameworks, Peirce, Relation Theory, Relative Membership, Scientific Method, Semiotics, Set Theory, Sign Relations, Systems Theory, Triadic Relations | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | 1 Comment

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 …

Resources

Posted in Abduction, Algorithms, Artificial Intelligence, Automated Research Tools, C.S. Peirce, Cognitive Science, Computer Science, Data Structures, Deduction, Functional Logic, Icon Index Symbol, Induction, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Intelligent Systems, Interpretive Frameworks, Knowledge Representation, Logic, Logic of Relatives, Logic of Science, Logical Graphs, Mathematics, Objective Frameworks, Peirce, Relation Theory, Relative Membership, Scientific Method, Semiotics, Set Theory, Sign Relations, Systems Theory, Triadic Relations | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

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 …

Resources

Posted in Abduction, Algorithms, Artificial Intelligence, Automated Research Tools, C.S. Peirce, Cognitive Science, Computer Science, Data Structures, Deduction, Functional Logic, Icon Index Symbol, Induction, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Intelligent Systems, Interpretive Frameworks, Knowledge Representation, Logic, Logic of Relatives, Logic of Science, Logical Graphs, Mathematics, Objective Frameworks, Peirce, Relation Theory, Relative Membership, Scientific Method, Semiotics, Set Theory, Sign Relations, Systems Theory, Triadic Relations | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Icon Index Symbol • 4

Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List Discussion • Helmut Raulien

One thing I always say at these junctures is that people really ought to take Peirce’s advice and study his logic of relative terms and its relation to what most mathematics and computer science folks these days would call the mathematical theory of relations.  Personally I find his 1870 Logic of Relatives very instructive, partly because he gives such concrete and simple examples of every abstract abstrusity and partly because he maintains a healthy balance between the extensional and intensional views of things, drawing on both our empiricist and rationalist ways of thinking.

Thereby hangs another hang up, a problem people often have with understanding Peirce’s logic and semiotics.  We have what might be called diverse “cognitive styles” or “intellectual inclinations” that range or swing between the above two poles.  I doubt if there’s anything like pure types in the human arena, but thinkers do tend to lean in one direction or the other, at least, at any given moment.  As a rule, though, we are almost always operating at two different levels of abstraction, whether we are aware of it or not, and our task is to get better at doing that, through increased awareness of how thought works.  There is the level of intension, or rational concepts, and there is the level of extension, or empirical cases.

To be continued …

Resources

Posted in Abduction, Algorithms, Artificial Intelligence, Automated Research Tools, C.S. Peirce, Cognitive Science, Computer Science, Data Structures, Deduction, Functional Logic, Icon Index Symbol, Induction, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Intelligent Systems, Interpretive Frameworks, Knowledge Representation, Logic, Logic of Relatives, Logic of Science, Logical Graphs, Mathematics, Objective Frameworks, Peirce, Relation Theory, Relative Membership, Scientific Method, Semiotics, Set Theory, Sign Relations, Systems Theory, Triadic Relations | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment