## { 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.

## { Information = Comprehension × Extension } Revisited • Comment 5

Let’s stay with Peirce’s example of abductive inference a little longer and try to clear up the more troublesome confusions that tend to arise.

Figure 1 shows the implication ordering of logical terms in the form of a lattice diagram.

Figure 1. Conjunctive Term z, Taken as Predicate

One thing needs to be stressed at this point.  It is important to recognize the conjunctive term itself — namely, the syntactic string “spherical bright fragrant juicy tropical fruit” — is not an icon but a symbol.  It has its place in a formal system of symbols, for example, a propositional calculus, where it would normally be interpreted as a logical conjunction of six elementary propositions, denoting anything in the universe of discourse with all six of the corresponding properties.  The symbol denotes objects which may be taken as icons of oranges by virtue of their bearing those six properties in common with oranges.  But there are no objects denoted by the symbol which aren’t already oranges themselves.  Thus we observe a natural reduction in the denotation of the symbol, consisting in the absence of cases outside of oranges which have all the properties indicated.

The above analysis provides another way to understand the abductive inference from the Fact $x \Rightarrow z$ and the Rule $y \Rightarrow z$ to the Case $x \Rightarrow y.$  The lack of any cases which are $z$ and not $y$ is expressed by the implication $z \Rightarrow y.$  Taking this together with the Rule $y \Rightarrow z$ gives the logical equivalence $y = z.$  But this reduces the Case $x \Rightarrow y$ to the Fact $x \Rightarrow z$ and so the Case is justified.

Viewed in the light of the above analysis, Peirce’s example of abductive reasoning exhibits an especially strong form of inference, almost deductive in character.  Do all abductive arguments take this form, or may there be weaker styles of abductive reasoning which enjoy their own levels of plausibility?  That must remain an open question at this point.

### 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 } Revisited • Comment 4

Many things still puzzle me about Peirce’s account at this point.  I indicated a few of them by means of question marks at several places in the last two Figures.  There is nothing for it but returning to the text and trying once more to follow the reasoning.

Let’s go back to Peirce’s example of abductive inference and try to get a clearer picture of why he connects it with conjunctive terms and iconic signs.

Figure 1 shows the implication ordering of logical terms in the form of a lattice diagram.

Figure 1. Conjunctive Term z, Taken as Predicate

The relationship between conjunctive terms and iconic signs may be understood as follows.  If there is anything with all the properties described by the conjunctive term — spherical bright fragrant juicy tropical fruit — then sign users may use that thing as an icon of an orange, precisely by virtue of the fact it shares those properties with an orange.  But the only natural examples of things with all those properties are oranges themselves, so the only thing qualified to serve as a natural icon of an orange by virtue of those very properties is that orange itself or another orange.

### 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.