## Signspiel • 1

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.

## Individuality, Identity, Teridentity • 1

Some problems cannot be solved in the paradigms where they first appear, which is why we keep recurring to them without quite freeing ourselves from the loops in which they ensnare us.  Questions about the supposed uniqueness of supposed individuals and the dyadic relation of identity are as old as the ship of Theseus and the morning and evening star(s) we steer by.

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:

### Selection from C.S. Peirce, “Logic Of Relatives” (1870), CP 3.45–149

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 $\mathrm{A}$ be such a term.  Then, if it is neither true that all $\mathrm{A}$ is $\mathrm{X}$ nor that no $\mathrm{A}$ is $\mathrm{X},$ it must be true that some $\mathrm{A}$ is $\mathrm{X}$ and some $\mathrm{A}$ is not $\mathrm{X};$  and therefore $\mathrm{A}$ may be divided into $\mathrm{A}$ that is $\mathrm{X}$ and $\mathrm{A}$ that is not $\mathrm{X},$ 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 $I$ be a term which in consequence of such neglect becomes indivisible in that discourse, we have in that discourse,

$[I] = 1.$

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.

### Note

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, $[t].$

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

### References

• 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–.

## { Information = Comprehension × Extension } • Discussion 17

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.

(Peirce 1866, p. 467)

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.

### Reference

• 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.

## { Information = Comprehension × Extension } • Discussion 16

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.

### Reference

• 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.

## { Information = Comprehension × Extension } • Discussion 15

I am roughly at the halfway point of my comments on Peirce’s information formula, having just finished up the link between abductive inference and iconic reference.  The discussion of induction and indexicals will follow pretty much the same pattern, though there are a few wrinkles having to do with a number of interesting differences between Peirce’s early and later accounts of indices.

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.

### Reference

• 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.

## { Information = Comprehension × Extension } • Discussion 14

Information and optimization go hand in hand — discovering the laws or constraints naturally governing the systems in which we live is a big part of moving toward our hearts’ desires within them.  I’m engaged in trying to clear up a few old puzzles about information at present but the dual relationship of information and control in cybernetic systems is never far from my mind.  At any rate, here’s a sampling of thoughts along those lines I thought I might add to the mix.

## { Information = Comprehension × Extension } • Discussion 13

As much as I incline toward Fisher’s views over those of Neyman and Pearson, I always find these controversies driving me back to Peirce.  It’s my personal sense there’s no chance (or hope) of resolving the issues until we get clear about the distinct roles of abductive, deductive, and inductive inference and quit confounding abduction and induction the way mainstream statistics has always done.