## Icon Index Symbol • 11

### Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List Discussion • Helmut Raulien

HR:
A simple triadic or n-adic relation, I think, belongs to secondness, and has only two modes:  the quality, e.g. function or caprice (intension), and the resulting set of tuples (extension).

There is a kind of Peircean “secondness” involved in the use of set theory, indeed, there are several kinds of dyadic relations in the mix, all intimately related.  Letting $X$ be the universe of discourse, there is the dyadic elementhood or membership relation $x \in X,$ there is the dyadic subset relation $A \subseteq X,$ and every subset $A \subseteq X$ has a characteristic or indicator function $f_{A} : X \to \{ 0, 1 \}$ with $f_{A}(x) = 1$ if $x \in A$ and $f_{A}(x) = 0$ if $x \notin A.$  So one could say there is secondness afoot in the extensions of whatever symbols one uses to demarcate or distinguish portions of the universe.  As it usually turns out, though, if you know enough to invoke secondness, you usually know enough to say something more specific about the dyadic relations you have in mind.

This is a very old theme.  The very word existence, whether by folk etymology or not, is said to mean standing out, the way a subset stands out against its ground.  It’s a nice image if nothing else.  In another connection, some take the prevalence of these set-theoretic dyadic relations, along with the assumption of set theory’s foundational status, as proving all structure is ultimately dyadic.

Well, I have my reasons to doubt that …

## Icon Index Symbol • 10

### Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List Discussion • Helmut Raulien

HR:
I am not so sure, if thirdness is about any triadic relation.

It may be more a matter of exegetic strategy than anything else but it’s convenient to attribute thirdness to all triadic relations, differentiating their genus in specific and individual cases according to how generic or genuine their triadicity may be.

HR:
The categories in Peirce’s “new list” of them are quality, relation, representation.

Peirce’s paper “On a New List of Categories” is from 1867, before he had worked out his Logic of Relatives to its full strength, and he is still thinking of relation as limited to dyadic relations, as many in some quarters of logic still do today.  In his 1870 Logic of Relatives he refers to the “three grand classes” of logical terms as absolute terms, simple relative terms, and conjugative terms.

Now logical terms are of three grand classes.

The first embraces those whose logical form involves only the conception of quality, and which therefore represent a thing simply as “a ──”.  These discriminate objects in the most rudimentary way, which does not involve any consciousness of discrimination.  They regard an object as it is in itself as such (quale);  for example, as horse, tree, or man.  These are absolute terms.

The second class embraces terms whose logical form involves the conception of relation, and which require the addition of another term to complete the denotation.  These discriminate objects with a distinct consciousness of discrimination.  They regard an object as over against another, that is as relative;  as father of, lover of, or servant of.  These are simple relative terms.

The third class embraces terms whose logical form involves the conception of bringing things into relation, and which require the addition of more than one term to complete the denotation.  They discriminate not only with consciousness of discrimination, but with consciousness of its origin.  They regard an object as medium or third between two others, that is as conjugative;  as giver of ── to ──, or buyer of ── for ── from ──.  These may be termed conjugative terms.

The conjugative term involves the conception of third, the relative that of second or other, the absolute term simply considers an object.  No fourth class of terms exists involving the conception of fourth, because when that of third is introduced, since it involves the conception of bringing objects into relation, all higher numbers are given at once, inasmuch as the conception of bringing objects into relation is independent of the number of members of the relationship.  Whether this reason for the fact that there is no fourth class of terms fundamentally different from the third is satisfactory or not, the fact itself is made perfectly evident by the study of the logic of relatives.  (CP 3.63).

HR:
Maybe representation is a very special kind of triadic relation.

If representation refers to the class of sign relations then those are marked out from the general class of triadic relations by a definition that specifies the roles that signs, their interpretant signs, and their objects play within the bounds of a sign relation.  Not too incidentally, Peirce gives one of his more consequential definitions of a sign relation in the process of defining logic:

Logic will here be defined as formal semiotic.  A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time.  Namely, a sign is something, A, which brings something, B, its interpretant sign determined or created by it, into the same sort of correspondence with something, C, its object, as that in which itself stands to C.  It is from this definition, together with a definition of “formal”, that I deduce mathematically the principles of logic.  I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has virtually been quite generally held, though not generally recognized.  (NEM 4, 20–21).

### 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. (1902), “Parts of Carnegie Application” (L 75), published in Carolyn Eisele (ed., 1976), The New Elements of Mathematics by Charles S. Peirce, vol. 4, 13–73.  Online.
• 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–.

## Icon Index Symbol • 9

### Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List Discussion • Helmut Raulien

I confess I have never found going on about Firstness Secondness Thirdness all that useful in any practical situation.

• Firstness means you have a monadic predicate in mind as relevant to a phenomenon, problem, or other subject matter.
• Secondness means you have a dyadic relation in mind as bearing on the situation at hand.
• Thirdness means you have a triadic relation in mind relative to the same end.

After that one may consider the fine points of generic versus degenerate cases, and that is all well and good, but until you venture to say exactly which monadic, dyadic, or triadic predicate you have in mind, you haven’t really said that much at all.

What I do think is interesting in all this is the fact that Peirce, from 1865 on, maintains in the background of his thought the idea that information is the solid substance borne by concepts and symbols, while comprehension and extension are its complementary aspects, its shadows.

I have been studying Peirce’s way of integrating comprehension and extension in the form of information for quite a while, and there is my set of excerpts and comments on this page:

But I just ran across a shorter sketch of the main ideas that I must have begun some time ago but not yet finished:

It has the advantage of presenting a nicely self-explanatory figure right up front.  At any rate, try taking a look at that.

## Triadic Forms of Constraint, Determination, Interaction • 3

In the beginning was the three-pointed star,
One smile of light across the empty face;
One bough of bone across the rooting air,
The substance forked that marrowed the first sun;
And, burning ciphers on the round of space,
Heaven and hell mixed as they spun.

Re: Peirce List Discussion • GRJC

I have every reason to suppose triadic relations are the very fabric of the universe, and for all I know every triadic relation has the potential to serve as a sign relation in one measure or another.

Triadic relations do not evolve from lower species but are present from the beginning.  Symbols do not emerge from icons and indices but icons and indices devolve from their generic precursors in the triadic matrix.

## Triadic Forms of Constraint, Determination, Interaction • 2

Re: Peirce List Discussion • Gary Richmond

Here’s one way of stating what I call a constraint:

• The set $L$ is constrained to a subset of the set $M.$

Here’s one way of stating a triadic constraint:

• The set $L$ is a subset of the cartesian product $X \times Y \times Z.$

So any way we define a triadic relation we are stating or imposing a triadic constraint.

In particular, any way we define a sign relation we are stating or imposing a triadic constraint of the form:

• $L \subseteq O \times S \times I.$

where:

• $O$ is the set of all objects under discussion,
• $S$ is the set of all signs under discussion, and
• $I$ is the set of all interpretant signs under discussion.

The concepts of constraint, definition, determination, lawfulness, ruliness, and so on all have their basis in the idea that one set is contained as a subset of another set.

Among the next questions that may occur to us, we might ask:

• What bearings do these types of global constraints have on various local settings we might select?

And conversely:

• To what extent do various types of local constraints combine to constrain or determine various types of global constraint?

There are by the way such things as mutual constraints, indeed, they are very common, and not just in matters of human bondage.  So, for instance, the fact that objects constrain or determine signs in a given sign relation does not exclude the possibility that signs constrain or determine objects within the same sign relation.

## Triadic Forms of Constraint, Determination, Interaction • 1

Re: Peirce List Discussion • JAGRJAJBD

There are many places where Peirce uses the word object in the full pragmatic sense, so much so that it demands a very selective attention not to remark them.  I cited a couple at the top of this discussion but perhaps the most critical locution for the sake of pragmatism is stated here:

Consider what effects that might conceivably have practical bearings you conceive the objects of your conception to have.  Then, your conception of those effects is the whole of your conception of the object.  (CP 5.438).

## The object of reasoning is to find out …

No longer wondered what I would do in life but defined my object.
— C.S. Peirce (1861), “My Life, written for the Class-Book”, (CE 1, 3)

The object of reasoning is to find out, from the consideration of what we already know, something else which we do not know.
— C.S. Peirce (1877), “The Fixation of Belief”, (CP 5.365)

If the object of an investigation is to find out something we do not know then the clues we discover along the way are the signs that determine that object.

People will continue to be confused about determination so long as they can think of no other forms of it but the analytic-behaviorist-causal-dyadic-temporal, object-as-stimulus and sign-as-response variety.  It is true that ordinary language biases us toward billiard-ball styles of dyadic determination, but there are triadic forms of constraint, determination, and interaction that are not captured by S-R chains of that order.  A pragmatic-semiotic object is anything we talk or think about, and semiosis does not conduct its transactions within the bounds of object as cue, sign as cue ball, and interpretants as solids, stripes, or pockets.

### References

• Peirce, C.S. (1859–1861), “My Life, written for the Class-Book”, pp. 1–3 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
• Peirce, C.S. (1877), “The Fixation Of Belief”, Popular Science Monthly 12 (Nov 1877), pp. 1–15.  Reprinted in Collected Papers, CP 5.358–387.  Online.