Inquiry Into Inquiry

Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List (1) (2) (3) • Helmut Raulien

The difference between the two definitions of a -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 starting out as follows:

A -place relation is a -tuple …

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 and the codomain share in defining the type of the function

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 a specific property or quality marking 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

• Sign Relations
• Triadic Relations
• Relation Theory
• Peirce’s 1870 Logic Of Relatives : The Wiki Article
• Peirce’s 1870 Logic Of Relatives : The Series Pilot

cc: Peirce List (1) (2) (3)

Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List (1) (2) (3) • 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, and a whole set of ordered tuples that it takes to make up a -place relation.  The language we use to get a handle on the structure of relations goes like this:

Say the variable ranges over the set
and the variable ranges over the set

and the variable ranges over the set

Then the set of all possible -tuples ranges over a set notated as and called the “cartesian product” of the “domains” to

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

1. Some define a relation on the domains to as a subset of the cartesian product in symbols,
2. Others like to make the domains of the relation an explicit part of the definition, saying that a relation 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 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

• Peirce’s 1870 Logic Of Relatives : The Wiki Article
• Peirce’s 1870 Logic Of Relatives : The Series Pilot

cc: Peirce List (1) (2) (3)

Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List (1) (2) (3) • HR • JAS • HR

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 OEIS Wiki site, though in this case it may be better to take them up in order from special to general.

• Sign Relations
• Triadic Relations
• Relation Theory

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

• Relation Reduction

To be continued …

Resources

• Peirce’s 1870 Logic Of Relatives : The Wiki Article
• Peirce’s 1870 Logic Of Relatives : The Series Pilot

cc: Peirce List (1) (2) (3)

Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List (1) (2) (3) • 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 -place relation I’ll always be thinking about a set of -tuples.  Each -tuple has the form:

or, as Peirce often wrote them:

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

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

Using a single letter like to refer to a set of -tuples is already the genesis of an intensional view, since we now think of the elements of as having some property in common, even if it’s only their membership in   When we turn to devising a 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 leads to no end of confusion.

To be continued …

Resources

• Peirce’s 1870 Logic Of Relatives • The Wiki Article
• Peirce’s 1870 Logic Of Relatives • The Series Pilot

cc: Peirce List (1) (2) (3)

Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List (1) (2) (3) • 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 mathematicians and computer scientists 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 encounter in trying to understand Peirce’s logic and semiotics.  We have what might be called different “cognitive styles” or “intellectual inclinations” ranging or swinging between the respective 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

• Peirce’s 1870 Logic Of Relatives • The Wiki Article
• Peirce’s 1870 Logic Of Relatives • The Series Pilot

cc: Peirce List (1) (2) (3)

Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List (1) (2) (3) • Kirsti Määttänen

KM:  Seems valid to me.  But it does not answer the quest for understanding.

The setup described in the previous post is the barest of beginnings.  It is the grounds from which our understanding must grow, if it is to be sustained by the following two provisions:

• MAT.  We take the Methods and Tools Peirce provides us seriously.
• COR.  We take the Context of Research in scientific inquiry seriously.

In practice, of course, we do not take the whole actual universe as our starting point but begin by constructing concrete examples of systems, say, a system defined by its state space and we try to determine what sorts of conditions must satisfy in order for to possess any sort of representation at all of its own structure.

Questions like that have been thoroughly investigated in the case of axiom systems and computational systems, where people speak of a system having a “reflective property”, but there is room to do a lot more work on reflection in full-fledged semiotic systems, those built on triadic sign relations and taking full advantage of their power.

Resources

• Sign Relation
• Triadic Relation
• Semiotic Information
• Peirce’s Logic Of Information

cc: Peirce List (1) (2) (3)