## 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.

## The Difference That Makes A Difference That Peirce Makes : 12

Re: Peirce List Discussion • Kirsti Määttänen

I have a sense of what Peirce meant by the “Logic of Science” and what Dewey meant by calling Logic the “Theory of Inquiry”.  If that’s logic in the narrow sense and not Logic in the Grandest Metaphysical Sense then it’s been enough for me, ever since I said farewell to the foundational crises of my youth and set to work on tools to help us reason.

That is what logic means to me.

## The Difference That Makes A Difference That Peirce Makes : 11

Re: Peirce List Discussion • KMJAJFS

The most important difference between linguistics and logic is that linguistics is descriptive while logic is normative.

Yes, some grammarians try to treat grammar as prescriptive, but most in modern times have given up on that and realize that usage will have its day and win out in the long run.  And even when grammar appears to dictate form it does so only on the plane of signs, sans objects, and so remains a flat affair.

It is only logic that inhabits all three dimensions $O \times S \times I$ of sign relations, inquiring into how we ought to conduct our transactions with signs in order to realize their objectives.  A normative science has different aims even when it looks on the same materials as a descriptive science.  So logic may have abstractions from language among its materials but it is more than abstract linguistics — it is an augmentation of language.

## The Difference That Makes A Difference That Peirce Makes : 10

Re: Peirce List Discussion • Jon Alan Schmidt

JA:
As I am realizing more and more in recent years, analyzing and classifying signs as a substitute for analyzing and classifying objects is the first slip of a slide into nominalism, namely, the idea that the essence or reality of objects is contained in the signs we use to describe them.

The operative phrase in what I wrote is “as a substitute for”.  We always have the task of classifying signs and classifying objects but the problems arise when your favorite ism thinks that half the work will do double duty.  It hardly ever does.

Dyadic forms of correspondence between syntactic structures and objective functions are always nice when you can get them and it’s always worth taking advantage of them when they occur.  It would make things a whole lot simpler if the forms of signs always mirrored the forms of their objects.  That is one of the attractions of Fregean compositionality and Russell’s isomorphism theory and it’s one of the reasons programming language designers keep to the realm of context-free languages for as long as they can.  Taking the Chomsky–Schützenberger Hierarchy as our first rough guide to the complexity of formal languages and the competencies demanded of their processors, we run into a critical point at the threshold between context-free and context-sensitive languages where the mirror of language breaks and the triadic nature of genuine symbols can no longer be avoided.