Partly this discussion and partly just the mood I’m in brought to mind a motley assortment of old reminiscences. My first years in college I oscillated (or vacillated) between math and physics, eventually returning to grad school in math, but only after a decade of cycling through majors from communications — of which I recall only a course in Aristotle — to psychology to philosophy to a “radical-liberal arts college” where I got to craft my own Bachelor’s degree in *Mathematical and Philosophical Method*.

But I’m getting ahead of the story. The course in physics took off with a bang right away, moving quickly from classical to relativity to quantum physics. My professors often took a *Read the Masters!* approach, giving us readings in Bohr, Dirac, Feynman, Heisenberg, and others, in addition to our regular textbooks. Among the forces that drove me back to math, I remember Dirac’s algebraic symbolism, Heisenberg’s matrix mechanics, and above all Peirce, especially his use of logical matrices, that made me realize I needed to learn a lot more math before I could comprehend what any of them were talking about.

*To be continued …*

Synchronicity being what it is, here for your contemplation are two pictures from a current discussion on Facebook.

See Tables 8 and 9 in the following article and section:

- Application of Higher Order Propositions to Quantification Theory
- Table 8. Simple Qualifiers of Propositions (Version 1)
- Table 9. Simple Qualifiers of Propositions (Version 2)

My first year at college the university held a cross-campus colloquium taking its theme from C.P. Snow’s *Two Cultures* about the need for and difficulties of cross-disciplinary communication and collaboration in our day. The university had recently created three residential colleges focused on the arts, sciences, and government/history but designed to provide future citizens with an integrated perspective on how these concentrations fit into the bigger picture of the modern world.

Long time passing, I found myself returning to these questions around the turn of the millennium, addressing the “problem of silos” and the “scholarship of integration” from the perspective of Peirce’s and Dewey’s pragmatism and semiotics. Here’s a couple of contributions Susan Awbrey and I made to the area:

- Awbrey, S.M., and Awbrey, J.L. (1999), “Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century”,
*Second International Conference of the Journal ‘Organization’, Re-Organizing Knowledge, Trans-Forming Institutions : Knowing, Knowledge, and the University in the 21st Century*, University of Massachusetts, Amherst, MA. Online.

- Awbrey, S.M., and Awbrey, J.L. (2001), “Conceptual Barriers to Creating Integrative Universities”,
*Organization : The Interdisciplinary Journal of Organization, Theory, and Society*8(2), Sage Publications, London, UK, 269–284. Abstract. Online.

I don’t know if the brands of ontologies being cranked out today are going to be the ultimate answer to these problems, but I do think there are applications of logic, mathematical modeling, and pragmatic semiotics that would certainly help a lot.

Concepts for Peirce are mental symbols, so they fall under the general designation of signs. For triadic sign relations in general, then, we are considering a triadic relation among objects of signs, signs of objects, and what Peirce calls interpretant signs, or interpretants for short. It is critical to regard the designations of objects, signs, and interpretants as relational roles not ontological essences. It is also critical to distinguish (a) extended sign relations, (b) elementary sign relations, (c) the slots of an ordered triple, and (d) the things that fill those slots.

Triangles like the one linked above have long been used to introduce the idea of a triadic sign relation. They have the unintended consequence, however, of leading people to miss all the points I mentioned above. So it’s wise to move quickly on to better pictures and more detailed descriptions.

cc: Cybernetics • Ontolog Forum • Structural Modeling • Systems Science

]]>So here’s just a teaser from Peirce on how concepts evolve from one level of complexity to the next, using incidentally a paradigm from the world of physics.

]]>All sorts of players have given us all sorts of spiel about speech acts over the years, but Peirce stands out from the chorus in giving us generative models of semiosis based on triadic sign relations that maintain a constant relation among signs, their active interpretants in conduct, and pragmata, the objects and objectives of the whole action. Shy of that, the speilerei of Austin and Wittgenstein simply never gets off the ground.

]]>Peirce, of course, took another course …

As fortune has it, I find myself waylaid between bouts of travel, with promises to keep when it comes to Peirce’s information formula, so let me leave this for now with a link to one of the most critical passages in all of Peirce’s explorations:

93. In reference to the doctrine of individuals, two distinctions should be borne in mind. The logical atom, or term not capable of logical division, must be one of which every predicate may be universally affirmed or denied. For, let be such a term. Then, if it is neither true that all is nor that no is it must be true that some is and some is not and therefore may be divided into that is and that is not which is contrary to its nature as a logical atom.

Such a term can be realized neither in thought nor in sense.

Not in sense, because our organs of sense are special — the eye, for example, not immediately informing us of taste, so that an image on the retina is indeterminate in respect to sweetness and non-sweetness. When I see a thing, I do not see that it is not sweet, nor do I see that it is sweet; and therefore what I see is capable of logical division into the sweet and the not sweet. It is customary to assume that visual images are absolutely determinate in respect to color, but even this may be doubted. I know no facts which prove that there is never the least vagueness in the immediate sensation.

In thought, an absolutely determinate term cannot be realized, because, not being given by sense, such a concept would have to be formed by synthesis, and there would be no end to the synthesis because there is no limit to the number of possible predicates.

A logical atom, then, like a point in space, would involve for its precise determination an endless process. We can only say, in a general way, that a term, however determinate, may be made more determinate still, but not that it can be made absolutely determinate. Such a term as “the second Philip of Macedon” is still capable of logical division — into Philip drunk and Philip sober, for example; but we call it individual because that which is denoted by it is in only one place at one time. It is a term not *absolutely* indivisible, but indivisible as long as we neglect differences of time and the differences which accompany them. Such differences we habitually disregard in the logical division of substances. In the division of relations, etc., we do not, of course, disregard these differences, but we disregard some others. There is nothing to prevent almost any sort of difference from being conventionally neglected in some discourse, and if be a term which in consequence of such neglect becomes indivisible in that discourse, we have in that discourse,

This distinction between the absolutely indivisible and that which is one in number from a particular point of view is shadowed forth in the two words *individual* (τὸ ἄτομον) and *singular* (τὸ καθ᾿ ἕκαστον); but as those who have used the word *individual* have not been aware that absolute individuality is merely ideal, it has come to be used in a more general sense.

Peirce explains his use of the square bracket notation at CP 3.65.

I propose to denote the number of a logical term by enclosing the term in square brackets, thus,

The *number* of an absolute term, as in the case of is defined as the number of individuals it denotes.

- Peirce, C.S. (1870), “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic”,
*Memoirs of the American Academy of Arts and Sciences*9, 317–378, 26 January 1870. Reprinted,*Collected Papers*3.45–149,*Chronological Edition*2, 359–429. Online (1) (2) (3). - Peirce, C.S.,
*Collected Papers of Charles Sanders Peirce*, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958. - Peirce, C.S.,
*Writings of Charles S. Peirce : A Chronological Edition*, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.

- Peirce’s 1870 Logic Of Relatives
- C.S. Peirce • Doctrine Of Individuals
- Mathematical Demonstration and the Doctrine of Individuals • (1) • (2)

We are in the middle of trying to work out what Peirce had in mind with his concept of information. He appears to have developed it from purely logical considerations, if logic can remain pure in applying itself to experience, and he thinks it solves “the puzzle of the validity of scientific inference”.

I am going, next, to show that inference is symbolization and that the puzzle of the validity of scientific inference lies merely in this superfluous comprehension and is therefore entirely removed by a consideration of the laws of *information*.

We will eventually come to the task of seeing how a theory of information born in that environment relates to concepts of information in common use today, sprouted as they were from the needs of telegraph operators to detect and correct errors of transmission through noisy channels of communication. As I see it, Peirce’s concept of information is potentially deeper and more general than concepts of information based on quantitative measures of probability and quantifiable statistics of messages. This is possible because the qualitative properties of spaces studied in topology are deeper and more general than the quantitative properties of spaces bearing real-valued measures.

All in good time, though. We have a ways to go understanding Peirce’s idea before we can say how the two paradigms compare.

- Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in
*Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866*, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

- This Blog • Survey of Pragmatic Semiotic Information
- My Notes • Information = Comprehension × Extension
- C.S. Peirce • Upon Logical Comprehension and Extension

To understand the purpose of Peirce’s lecture hall illustrations I think we need to consider how these sorts of expository examples come into being. Having crafted a few myself the technique is much like the Art of the Story Problem I remember from my days teaching math. We have a universe of discourse circumscribed by a particular subject matter, say linear algebra, plane geometry, the quadratic formula, or the like, and we have a set of methods that work well enough in that context to recommend their use to others. The methods themselves have been abstracted and formalized over the years, if not millennia, to the point of being detached from everyday life and potential practice, so we flesh them out with names and local habitations and narrative figures designed to tutor nature — or at least the students thereof.

The main thing we want from our stock examples and story problems is to show how it’s possible to bring a body of abstract ideas to bear on ordinary practical affairs. We are thus reversing to a degree the process by which a formalized subject matter is abstracted from a host of concrete situations, but only to a degree, as dredging up the mass of adventitious and conflicting details would be too distracting. Instead we stipulate a hypothetical state of affairs whose concrete structure falls under the class of ideal structures studied in our formal subject matter.

- Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in
*Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866*, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

- This Blog • Survey of Pragmatic Semiotic Information
- My Notes • Information = Comprehension × Extension
- C.S. Peirce • Upon Logical Comprehension and Extension

The rest of this post is slightly tangent to the topic at hand, but I couldn’t resist saying a few more words about the duality of information and control once other discussions brought the issue to mind.

⁂

Viewing systems topics like change, control, dynamics, goals, objectives, optimization, process, purpose and so on in the light of the information dimension opens up a wide field of investigation. It’s been my custom to cultivate that field layer by layer, working up from the most basic layer with a modicum of utility, namely, propositional calculus. This is the layer of qualitative description underlying every layer of quantitative description.

Propositional calculus is the level of logic we’ve been using in our present discussion to describe various classes of entities populating a given universe of discourse. Whether we call the corresponding descriptors *predicates*, *propositions*, or *terms* is of no importance for present purposes so long as we are using them solely as symbols in a symbolic calculus following a specific set of rules.

Extending the layer of propositional calculus from its coverage of static situations to the description of time-evolving states can be done fairly easily. One follows the model of physics, where dealing with change made little progress until the development of differential calculus. The analogous medium at the logical level is the differential extension of propositional calculus, or “differential propositional calculus”, for short. See the following resource for a gentle introduction.

- Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in
*Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866*, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

- This Blog • Survey of Pragmatic Semiotic Information
- My Notes • Information = Comprehension × Extension
- C.S. Peirce • Upon Logical Comprehension and Extension