## { Information = Comprehension × Extension } • Discussion 2

Re: Peirce List Discussion • John Sowa

JFS:
A more fundamental term is proposition, which is informally defined as the “meaning” of a sentence.  That meaning is usually analyzed as comprehension (also known as intension) and extension.

The easier-on-the-eyes blog version of my first Discussion post — from which point it is also easier to follow the links to the first six Selections from Peirce — is here:

The word proposition occurs only twice in the first six Selections, once in Selection 2 and once in Selection 4, so maybe it’s worth our pausing to see how Peirce uses the word in this place and time:

The third and last kind of representations are symbols or general representations.  They connote attributes and so connote them as to determine what they denote.  To this class belong all words and all conceptions.  Most combinations of words are also symbols.  A proposition, an argument, even a whole book may be, and should be, a single symbol.  (Peirce 1866, p. 468)

Accordingly, if we are engaged in symbolizing and we come to such a proposition as “Neat, swine, sheep, and deer are herbivorous”, we know firstly that the disjunctive term may be replaced by a true symbol.  (Peirce 1866, p. 469)

For now I’ll just add those two observations to the hopper, and we can take up the issue of propositions in more detail as it arises in the relevant context.

It is good that John Sowa read us the “Freedom Of Interpretation Act” right at the start, as it will serve us in good stead down the road, but again I’ll have to leave its consequences until a few folks have had a chance to delve further into Peirce’s text, at which point I think its significance will become clear.

## { Information = Comprehension × Extension } • Discussion 1

A puzzle in Peirce I have puzzled over for as long as I can remember involves the relationship between his theory of signs, marking the characters of icons, indices, and symbols, and his theory of inquiry, bearing the three inferences of abduction, induction, and deduction.  I have long felt the resolution would lie in his theory of information, epitomized by the formula “Information = Comprehension × Extension”.

Last summer looked ripe for another run at the problem, which I had some years before begun tackling in a series of selections from and comments on Peirce’s “Logic of Science” lectures at Harvard University (1865) and the Lowell Institute (1866).

There’s a working draft of those selections and comments here:

I serialized the selections and comments on my blog as I worked through them.

• First Six Selections from Peirce’s Lectures • (1)(2)(3)(4)(5)(6)

By September I had come to what I imagined was a new understanding of the relationship between the types of signs and the types of inference, at which time I put the whole matter away to cool, it being far harder to judge a new idea when it’s hot.  At any rate, I think a year is long enough to gain a cool eye or two, so I will try sharing the new improved analysis to the wider world.

## The Difference That Makes A Difference That Peirce Makes : 17

Re: Peirce List Discussion • JAGFJFSJLRCJAJFSGF

A rather amusing, if slightly ominous illustration of the point I am trying to make here has just popped up in the daily mayhem.  Let’s call this one:

### Syntax Proposes, Pragmatics Disposes — or — When Does A Question Become A Command?

The big thing that classification maniacs tend to forget about types of signs in a sign relational theory of signs is that they are always interpretive and relative, never essential and absolute.

• An icon is an icon when it’s interpreted as an icon.
• An index is an index when it’s interpreted as an index.
• The same goes for term, sentence, argument by any name.

Category theorists and computer scientists call that “polymorphism” and they study the isomorphisms that relate the various types.

## The Difference That Makes A Difference That Peirce Makes : 16

Re: Peirce List Discussion • JAJFS

JFS:
For those of us who are trying to convince modern students to study Peirce, we need to become bilingual.  We need to show how his terminology and notations map to and from current systems — more importantly, how they point the way to new discoveries and innovations that are obscured by modern methods.

I am also concerned with maintaining avenues of communication and cross-fertilization among various communities of inquiry.  We have to observe the specialized ways that terms are used in particular communities but we cannot capitulate to uses so specialized that they obscure the more general meaning.  In the present case, I am concerned to rescue the beauty of form, as appreciated in classical texts, mathematics, and Peirce’s philosophy, from the anorexia to which it was subjected by a few schools of nominal thought.

A reasonable tactic, then, is simply to say “syntactic form” or “syntactic structure” when that is all one means.

## The Difference That Makes A Difference That Peirce Makes : 15

Re: Peirce List Discussion • JAGF

One could hardly dispute the importance of logical implication relations like $A \Rightarrow B.$  Their set-theoretic analogues are subset relations like $A \subseteq B,$ which are almost the canonical way of expressing constraint, determination, information, and so on.  There is moreover a deep analogy or isomorphism between propositions like $A \Rightarrow B$ and functional types like $A \to B$ of considerable importance in the theory of computation.  That is probably enough to earn implications a primary and fundamental status but there are several reasons we might stop short of claiming these order relations are exclusively primary and fundamental.

For one thing, implication in existential graphs is expressed in a compound form, as $\texttt{(} A \texttt{(} B \texttt{))},$ “not A without B”.  For another, there is Peirce’s own discovery of the amphecks, the logical connectives expressed by “not both” and “both not”, respectively, which appear to have a primary and fundamental status all their own.  Lastly, implicational inferences are in general information-losing while the fundamental operations in Peirce’s logical graphs, either entitative or existential, give us the option of equational rules of inference, that is, information-preserving steps.

Just a few things to think about …

## The Difference That Makes A Difference That Peirce Makes : 14

Re: Peirce List Discussion : John Sowa

We find ourselves at the thresh-old of yet another recurring discussion, this time concerning Peirce’s use of the adjectives formal and quasi-necessary with normative connotations, all of which I think is clear from the following sample of texts:

As it happens, we had pretty much this same discussion regarding the meanings of formal, normal and peculiar, about this time five years ago, as the following instance, among others, shows:

The most general meaning of “formal” is “concerned with form”,
but the Latin “forma” can mean “beauty” in addition to “form”,
so perhaps a normative “goodness of form” enters at this root.

The Latin word “norma” literally means a “carpenter’s square”.
The Greek “gnomon” is a sundial pointer taking a similar form.
The most general meaning of “normative” is “having to do with
what a person ought to do”, but a pragmatic interpretation of
ethical imperatives tends to treat that as “having to do with
what a person ought to do in order to achieve a given object”,
so another formula might be “relating to the good that befits
a being of our kind, what must be done in order to bring that
good into being, and how to tell the signs that show the way”.

Defining logic as formal or normative semiotic differentiates
logic from other species of semiotic under the general theory
of signs, leaving a niche open for descriptive semiotic, just
to mention the obvious branch. This brings us to the question:

How does a concern with form, or goodness of form, along with
the question of what is required to achieve an object, modify
our perspective on sign relations in a way that duly marks it
as a logical point of view?

If I had to add any finer point now, I would take pains to point out that formal in the sense of concerned with form can mean either syntactic form or objective form and that it’s good form to keep that distinction in mind.

## The Difference That Makes A Difference That Peirce Makes : 13

I would like to return to a point where the paths of discussion began to diverge and then bifurcated so chaotically that I could not track them further, namely, here:

Re: Peirce List Discussion • JAGFJA

I imagine different readers derive different morals from the passage Gary Fuhrman quoted.  It resonates for me with a host of themes going back to my Vita Nuova in many dimensions of life during my first years of college.  But memories from fifty years ago are hard to put in order and so what comes more freshly to mind are later harvests of those seeds.

One of those outgrowths was the work I did applying Peirce’s paradigm to fundamental problems in AI, or Intelligent Systems Engineering as my advisor in Systems Engineering preferred to call it.  I posted a link to a section from one of my project reports:

Many distractions kept me from following up at the time, so I’ll copy here the introduction of that section with the aim of moving forward from there:

### Functional Logic : Inquiry and Analogy

#### Functional Conception of Quantification Theory

Up till now quantification theory has been based on the assumption of individual variables ranging over universal collections of perfectly determinate elements.  Merely to write down quantified formulas like $\forall_{x \in X} f(x)$ and $\exists_{x \in X} f(x)$ involves a subscription to such notions, as shown by the membership relations invoked in their indices.  Reflected on pragmatic and constructive principles, however, these ideas begin to appear as problematic hypotheses whose warrants are not beyond question, projects of exhaustive determination that overreach the powers of finite information and control to manage.  Therefore, it is worth considering how we might shift the scene of quantification theory closer to familiar ground, toward the predicates themselves that represent our continuing acquaintance with phenomena.