Re: John Corcoran • Semiotic Triangle • My Comment
The following passage is very instructive on several points, illuminating especially the relationship between interpreters (sign‑using agents) and interpretant signs.
We are all, then, sufficiently familiar with the fact that many words have much implication; but I think we need to reflect upon the circumstance that every word implies some proposition or, what is the same thing, every word, concept, symbol has an equivalent term — or one which has become identified with it, — in short, has an interpretant.
Consider, what a word or symbol is; it is a sort of representation. Now a representation is something which stands for something. I will not undertake to analyze, this evening, this conception of standing for something — but, it is sufficiently plain that it involves the standing to something for something. A thing cannot stand for something without standing to something for that something. Now, what is this that a word stands to? Is it a person?
We usually say that the word homme stands to a Frenchman for man. It would be a little more precise to say that it stands to the Frenchman’s mind — to his memory. It is still more accurate to say that it addresses a particular remembrance or image in that memory. And what image, what remembrance? Plainly, the one which is the mental equivalent of the word homme — in short, its interpretant. Whatever a word addresses then or stands to, is its interpretant or identified symbol. Conversely, every interpretant is addressed by the word; for were it not so, did it not as it were overhear what the word says, how could it interpret what it says.
There are doubtless some who cannot understand this metaphorical argument. I wish to show that the relation of a word to that which it addresses is the same as its relation to its equivalent or identified terms. For that purpose, I first show that whatever a word addresses is an equivalent term, — its mental equivalent. I next show that, since the intelligent reception of a term is the being addressed by that term, and since the explication of a term’s implication is the intelligent reception of that term, that the interpretant or equivalent of a term which as we have already seen explicates the implication of a term is addressed by the term.
The interpretant of a term, then, and that which it stands to are identical. Hence, since it is of the very essence of a symbol that it should stand to something, every symbol — every word and every conception — must have an interpretant — or what is the same thing, must have information or implication.
(Peirce 1866, Lowell Lecture 7, CE 1, 466–467).
My study of Peirce’s information formula, “Information = Comprehension × Extension”, provides a measure of context for the above passage.
There’s additional discussion in the following article and section.
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.
Resources
- C.S. Peirce • Upon Logical Comprehension and Extension
- My Notes • Information = Comprehension × Extension
is a formal structure that satisfies a very general definition, on the same order of generality as a mathematical group or geometry. So any consideration of what a particular sign relation contains will be very context-dependent.
one sign in the sign domain 
could be the name “Socrates” and one interpretant in the interpretant domain 
could be the name “Socrates”.
where 
is the object, 
is the sign, and 
is the interpretant sign (usually shortened to interpretant). In other words, a specific sign relation 
is a subset of the cartesian product 
where 
is the object domain, 
the partial differential operator 
sends a function 
with fiber 
to a function 
whose fiber 
consists of all the places where a change in the value of 
makes a change in the value of 
This provides the basis for a prospective object domain 
to be constructed out of its elements as our description of the universe develops.
and interpretant domain 
are needed for the ongoing discussion and inquiry.
and 
respectively. The rectangular area represents a set or space 
usually called the universe of discourse, though viewed from the angle of Peircean semiotics it is really just the ground level of a more complex object domain 
that has the property 
Figure 2 shows this area shaded blue. We may think of the shading in the diagram as indicating the corresponding subset of the universe, in other words, associating a distinctive value with it.
represents the subset of 
Figure 3 shows this area shaded blue. We may think of the shading in the diagram as indicating the corresponding subset of the universe, in other words, associating a distinctive value with it.
but not the property 
is true. Further, we may interpret either the propositional form 
or the corresponding logical graph as indicating the same subset of the universe as the shading in the Venn diagram.
is true. Finally, we may interpret either the propositional form 
or the corresponding logical graph as indicating the same subset of the universe as the shading in the Venn diagram.
have or have not existence in it. This marks an interpretive transition from the functional interpretation to the relational interpretation of all these styles of signs.
and various regions of soap film tinted with the indicative color. I see the transformation from Figure 5 to Figure 6 as occurring when a pin pops the untinted space of the first and the region collapses to give the arrangement of extant regions in the final diagram. This is the sort of diagram we usually draw to indicate a subset relation, in this case showing the set 
where 
where 
Inquiry Into Inquiry 