As a general rule Peirce avoided in advance many of the problems bedeviling later philosophies of science in the 20th Century. Doing so came rather naturally to him as he rarely succumbed to the cycloptic species of reductionism afflicting so many modern isms.

In particular, it’s not so much that Peirce sought a way to jam together the extensions and intensions of concepts and other symbols, the stuffs empiricism and rationalism are made on, as he grasped the whole body of information, the “synthetic unity of apperception” of which extensions and intensions are but the facets or lower dimensional projections.

That integral core of information borne by signs is the prize we’ll keep in view, stereoscopically, as we make our way into the texts of the 1860s.

]]>Here’s a gloss on “comprehension” I copied from Runes some years ago, plus a bit of commentary I added at the time.

**Comprehension.** The sum of characteristics which connote a class notion symbolized by a general term. Also, the features common to a number of instances or objects. Thus, the *connotation* (*qv*) or *intension* (*qv*) of a concept. (Otto F. Kraushaar, in D.D. Runes (ed.) *Dictionary of Philosophy*, 1962).

Re: Systems Science • Kent Palmer

It may be useful to clear up a couple of technical points about Peirce’s terminology before getting to the main business.

Here I’m using *extension* and *intension* in the conventional sense that concerns the extension and intension of a concept or term. We define a concept or term in extension by giving all the things falling under it. We define a concept or term in intension by giving all the properties it implies or falls under. The parallel morphology of the words “extension” and “intension” is pleasing enough to ear and eye that we usually just let it go at that, but punctilious pedants long ago noticed the fly in the ointment that an extension is a many, a set of things, while an intension is a one, a single property or quality, and so they insist on the technical term *comprehension* to signify a collection of intensions. In practice, of course, context of application will determine how picky we need to be.

Re: Ontolog Forum • JS

I’ve been looking at the record of past discussions and thinking about what I’d like to accomplish this year. Just off hand, I see a lot of discussion points I didn’t get a chance to give their due at the time and it looks like many of those are worth revisiting. Some of the more troublesome points may seem tangential at first but they have a tendency to recur if not dealt with, so it’s critical to address them as best one can.

More to the main arc of this thread, I need to keep developing the implications of Peirce’s ideas about information, in particular, the way information integrates the extensional and intensional aspects of logic and thus helps to solve many old puzzles about the nature of scientific inference and inquiry in general.

]]>What is information and how does it impact the spectrum of activities that answer to the name of inquiry?

Setting out on his lifelong quest to explore and explain the “Logic of Science”, C.S. Peirce pierced the veil of historical confusions surrounding the issue and fixed on what he called the “laws of information” as the key to answering how science works. This was in 1865 and 1866, detailed in his lectures at Harvard University and the Lowell Institute.

Fast forward to the present and I see the Big Question as follows. Comparing Peirce’s theory of information, however much it remains in a rough-hewn state, with Shannon’s paradigm so pervasively informing the ongoing revolution in our understanding and use of information today, I have reason to believe Peirce’s idea is more general in principle and has the potential, with due development, to resolve many mysteries still bedeviling our grasp of inference, information, and inquiry.

- InterSciWiki • Information = Comprehension × Extension

- Blog Series • { Information = Comprehension × Extension }

- Blog Series • Pragmatic Semiotic Information (Ψ)

- Semiotic Theory Of Information • (1) • (2) • (3) • (4) • (5) • (6)
- Icon Index Symbol • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) (16) • (17)
- Sign Relations • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12)
- Sign Relations, Triadic Relations, Relation Theory • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11)

- Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”. Online.
- Awbrey, J.L., and Awbrey, S.M. (1992), “Interpretation as Action : The Risk of Inquiry”,
*The Eleventh International Human Science Research Conference*, Oakland University, Rochester, Michigan. - Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”,
*Inquiry : Critical Thinking Across the Disciplines*15(1), pp. 40–52. Archive. Journal. Online.

It doesn’t help with the question of semiogenesis, which is no doubt lost to the mists of history, but Peirce being Peirce naturally discerned three shades of signs in this respect: Tone, Token, Type. I collected a few excerpts here:

The distinction between Original and Copy figures frequently in Plato, with echoes of still more ancient voices. Aristotle on Categories gives an example where a word meaning both a live animal and its true-to-life image must be shorn of ambiguity prior to appearing in a court of logic. Aristotle on Interpretation distinguishes objects from their copies, images, likenesses in the mind:

Words spoken are symbols or signs (*symbola*) of affections or impressions (*pathemata*) of the soul (*psyche*); written words are the signs of words spoken. As writing, so also is speech not the same for all races of men. But the mental affections themselves, of which these words are primarily signs (*semeia*), are the same for the whole of mankind, as are also the objects (*pragmata*) of which those affections are representations or likenesses, images, copies (*homoiomata*). (Aristotle, *De Interp.* i. 16^{a}4).

From a Peircean semiotic perspective we can distinguish an object domain and a semiotic plane, so we can have three types of type/token relations: (1) within the object domain, (2) between objects and signs, (3) within the semiotic plane. We could subtilize further but this much is enough for a start.

Type/token relations of type (1) are very common in mathematics and go back to the origins of mathematical thought. These days computer science is rife with them. I’ve seen a lot of confusion about this in Peircean circles as it’s not always grasped that type/token relations are not always all about signs. It can help to speak of types versus instances or instantiations instead.

Aristotle covers type/token relations of types (2) and (3) in *De Interp.*, the latter since he recognizes signs of signs in the clause, “written words are the signs of words spoken”.

- Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”,
*Inquiry : Critical Thinking Across the Disciplines*15(1), pp. 40–52. Archive. Journal. Online. - Limited Mark Universes • Peirce’s Note “On a Limited Universe of Marks”

The sense of *reduction* operative in complexity theory has its roots in Aristotle’s απαγωγη, variously translated as *abduction* or *reduction* and sometimes glossed as *retroduction* by C.S. Peirce. See my project report on Inquiry and Analogy for a discussion, especially the following section:

It’s been a while since I started this thread, with many sidetrips and tangents, so let me go back to the top and expand on the motivations I expressed there, addressing a few issues that have arisen in the meantime.

People interested in category theory as applied to systems may wish to check out the following article, reporting work I carried out while engaged in a systems engineering program at Oakland University.

- Differential Logic and Dynamic Systems

This article develops a differential extension of propositional calculus and applies it to a context of problems arising in dynamic systems. The work pursued here is coordinated with a parallel application that focuses on neural network systems, but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.

Category theory, as working mathematicians understand it, is one of the chief conceptual frameworks for the development of mathematics today, the other being set theory. Every graduate math course I ever took began with a two- or three-week review of set theory and category theory before launching into the main subject matter. From a logical point of view, however, category theory has a history stretching back to Aristotle.

I once started writing a sketch on the “Precursors of Category Theory”, collecting a sample of historical landmarks through the centuries, from Aristotle to category theory’s modern mathematical avatars. Here’s a link to a survey page on my blog:

*To be continued …*

- Differential Logic • Primer
- Differential Logic • Introduction
- Differential Logic and Dynamic Systems

cc: Structural Modeling • Ontolog Forum • Laws of Form • Cybernetics

]]>Let’s stand back from the picture and see how the dimensions of syntax, semantics, and pragmatics look from a pragmatic semiotic or sign relational perspective.

is an object domain, a set of elements under view in a given discussion. Depending on the application we might be calling it a universe of discourse, a population, a sample space, a state space, or any number of other things.

and are sets of signs related to by means of a triadic relation, If the triadic relation satisfies a set of conditions set down in a definition of a sign relation then we say is a sign relation.

Peirce’s best definitions of a sign relation are pretty minimal in what they demand and cover a wide range of cases from barely formed to highly structured.

Let’s move on to the more structured types of sign relations forming our ultimate practical interest.

In a typical case like that, is a formal language defined by a formal grammar.

Generally speaking, we might think of as being more loosely defined in its own right but when it comes to formal investigations the so-called interpretant sign domain will also be a formal language. Here the cases divide into two broad sorts.

- We use this case to discuss transitions in time from one sign to the next.
- We use this case to discuss translations from one language to another.

*To be continued …*

- Differential Logic • Primer
- Differential Logic • Introduction
- Differential Logic and Dynamic Systems

cc: Structural Modeling • Ontolog Forum • Laws of Form • Cybernetics

]]>A few of my readers are racing well ahead of me, exploring a range of different roads, but I’ll be making a dogged effort to stick to my math-bio-graphical narrative this time around, and try to tell how I came to climb down from logical trees and learned to love logical cacti.

As far as the logical ballpark goes, this is all just classical propositional logic, what my old circle used to call “zeroth order logic”, alluding to its basemental status for every storey built on it. (But I have since found that others use that term for other things, so usage varies as it usually does.)

When it comes to semantics, the class of formal or mathematical objects residing among the referents of our propositional signs, I’m content for most purposes to say they’re all the same, namely, Boolean functions of abstract type where and is a non-negative integer. Although we’re likely to have other sorts of meanings in mind, this class of models suffices for a ready check on logical consistency and serves us well, especially in practical applications.

The upshot is — I’m aiming for innovation solely in the syntactic sphere, the end being only to discover/invent a better syntax for the same realm of logical objects.

*To be continued …*

- Differential Logic • Primer
- Differential Logic • Introduction
- Differential Logic and Dynamic Systems

cc: Structural Modeling • Ontolog Forum • Laws of Form • Cybernetics

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