Examples of Sign Relations
Because the examples to follow have been artificially constructed to be as simple as possible, their detailed elaboration can run the risk of trivializing the whole theory of sign relations. Despite their simplicity, however, these examples have subtleties of their own, and their careful treatment will serve to illustrate many important issues in the general theory of signs.
Imagine a discussion between two people, Ann and Bob, and attend only to that aspect of their interpretive practice that involves the use of the following nouns and pronouns: "Ann", "Bob", "I", "you".
The object domain of this discussion fragment is the set of two people {Ann, Bob}. The syntactic domain or the sign system that is involved in their discussion is limited to the set of four signs {"Ann", "Bob", "I", "You"}.
In their discussion, Ann and Bob are not only the passive objects of nominative and accusative references but also the active interpreters of the language that they use. The system of interpretation (SOI) associated with each language user can be represented in the form of an individual three-place relation called the sign relation of that interpreter.
Understood in terms of its set-theoretic extension, a sign relation L is a subset of a cartesian product O × S × I. Here, O, S, I are three sets that are known as the object domain, the sign domain, and the interpretant domain, respectively, of the sign relation L ⊆ O × S × I.
Broadly speaking, the three domains of a sign relation can be any sets whatsoever, but the kinds of sign relations that are typically contemplated in a computational setting are usually constrained to having I ⊆ S. In this case, interpretants are just a special variety of signs, and this makes it convenient to lump signs and interpretants together into a single class called the syntactic domain. In the forthcoming examples, S and I are identical as sets, so the same elements manifest themselves in two different roles of the sign relations in question. When it is necessary to refer to the whole set of objects and signs in the union of the domains O, S, I for a given sign relation L, one may refer to this set as the world of L and write W = WL = O ∪ S ∪ I.
To facilitate an interest in the abstract structures of sign relations, and to keep the notations as brief as possible as the examples become more complicated, it serves to introduce the following general notations:
-
O = Object Domain S = Sign Domain I = Interpretant Domain
Introducing a few abbreviations for use in considering the present Example, we have the following data:
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O = {Ann, Bob} = {A, B} S = {"Ann", "Bob", "I", "You"} = {"A", "B", "i", "u"} I = {"Ann", "Bob", "I", "You"} = {"A", "B", "i", "u"}
In the present Example, S = I = Syntactic Domain.
The next two Tables give the sign relations associated with the interpreters A and B, respectively, putting them in the form of relational databases. Thus, the rows of each Table list the ordered triples of the form (o, s, i) that make up the corresponding sign relations, LA and LB ⊆ O × S × I. It is often tempting to use the same names for objects and for relations involving these objects, but it is best to avoid this in a first approach, taking up the issues that this practice raises after the less problematic features of these relations have been treated.
| Object | Sign | Interpretant |
|---|---|---|
| A | "A" | "A" |
| A | "A" | "i" |
| A | "i" | "A" |
| A | "i" | "i" |
| B | "B" | "B" |
| B | "B" | "u" |
| B | "u" | "B" |
| B | "u" | "u" |
| Object | Sign | Interpretant |
|---|---|---|
| A | "A" | "A" |
| A | "A" | "u" |
| A | "u" | "A" |
| A | "u" | "u" |
| B | "B" | "B" |
| B | "B" | "i" |
| B | "i" | "B" |
| B | "i" | "i" |
These Tables codify a rudimentary level of interpretive practice for the agents A and B, and provide a basis for formalizing the initial semantics that is appropriate to their common syntactic domain. Each row of a Table names an object and two co-referent signs, making up an ordered triple of the form (o, s, i) that is called an elementary relation, that is, one element of the relation's set-theoretic extension.
Already in this elementary context, there are several different meanings that might attach to the project of a formal semiotics, or a formal theory of meaning for signs. In the process of discussing these alternatives, it is useful to introduce a few terms that are occasionally used in the philosophy of language to point out the needed distinctions.
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