Triadic relation

In logic, mathematics, and semiotics, a triadic relation or a ternary relation is an important special case of a polyadic or finitary relation, one in which the number of places in the relation is three. One also sees the adjectives 3-adic, 3-ary, 3-dim, or 3-place being used to describe these relations.

Mathematics is positively rife with examples of 3-adic relations, and a sign relation, the arch-idea of the whole field of semiotics, is a special case of a 3-adic relation. Therefore it will be useful to consider a few concrete examples from each of these two realms.

Examples from mathematics
For the sake of topics to be taken up later, it is useful to examine a pair of 3-adic relations in tandem, L0 and L1, that can be described in the following manner.

The first order of business is to define the space in which the relations L0 and L1 take up residence. This space is constructed as a 3-fold cartesian power in the following way.

The boolean domain is the set B = {0, 1}. The plus sign "+", used in the context of the boolean domain B, denotes addition mod 2. Interpreted for logic, this amounts to the same thing as the boolean operation of exclusive-or or not-equal-to.

The third cartesian power of B is B3 = {(x1, x2, x3) : xj in B for j = 1, 2, 3}= B &times; B &times; B.

In what follows, the space X &times; Y &times; Z is isomorphic to B &times; B &times; B $$\cong$$ B3.

The relation L0 is defined as follows:


 * L0 = {(x, y, z) in B3 : x + y + z = 0}.

The relation L0 is the set of four triples enumerated here:


 * L0 = {(0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0)}.

The relation L1 is defined as follows:


 * L1 = {(x, y, z) in B3 : x + y + z = 1}.

The relation L1 is the set of four triples enumerated here:


 * L1 = {(0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 1, 1)}.

The triples that make up the relations L0 and L1 are conveniently arranged in the form of relational data tables, as follows:

Examples from semiotics
The study of signs — the full variety of significant forms of expression — in relation to the things that signs are significant of, and in relation to the beings that signs are significant to, is known as semiotics or the theory of signs. As just described, semiotics treats of a 3-place relation among signs, their objects, and their interpreters.

The term semiosis refers to any activity or process that involves signs. Studies of semiosis that deal with its more abstract form are not concerned with every concrete detail of the entities that act as signs, as objects, or as agents of semiosis, but only with the most salient patterns of relationship among these three roles. In particular, the formal theory of signs does not consider all of the properties of the interpretive agent but only the more striking features of the impressions that signs make on a representative interpreter. In its formal aspects, that impact or influence may be treated as just another sign, called the interpretant sign, or the interpretant for short. Such a 3-adic relation, among objects, signs, and interpretants, is called a sign relation.

For example, consider the aspects of sign use that concern two people, say, Ann and Bob, in using their own proper names, "Ann" and "Bob", and in using the pronouns "I" and "you". For brevity, these four signs may be abbreviated to the set {"A", "B", "i", "u"}. The abstract consideration of how A and B use this set of signs to refer to themselves and to each other leads to the contemplation of a pair of 3-adic relations, the sign relations LA and LB, that reflect the differential use of these signs by A and by B, respectively.

Each of the sign relations, LA and LB, consists of eight triples of the form (x, y, z), where the object x belongs to the object domain O = {A, B}, where the sign y belongs to the sign domain S, where the interpretant sign z belongs to the interpretant domain I, and where it happens in this case that S = I = {"A", "B", "i", "u"}. In general, it is convenient to refer to the union S &cup; I as the "syntactic domain", but in this case S = I = S &cup; I.

The set-up to this point can be summarized as follows:


 * LA, LB &sube; O &times; S &times; I


 * O = {A, B}


 * S = {"A", "B", "i", "u"}


 * I = {"A", "B", "i", "u"}

The relation LA is the set of eight triples enumerated here:


 * {(A, "A", "A"), (A, "A", "i"), (A, "i", "A"), (A, "i", "i"),
 * (B, "B", "B"), (B, "B", "u"), (B, "u", "B"), (B, "u", "u")}.

The triples in LA represent the way that interpreter A uses signs. For example, the listing of the triple (B, "u", "B") in LA represents the fact that A uses "B" to mean the same thing that A uses "u" to mean, namely, B.

The relation LB is the set of eight triples enumerated here:


 * {(A, "A", "A"), (A, "A", "u"), (A, "u", "A"), (A, "u", "u"),
 * (B, "B", "B"), (B, "B", "i"), (B, "i", "B"), (B, "i", "i")}.

The triples in LB represent the way that interpreter B uses signs. For example, the listing of the triple (B, "i", "B") in LB represents the fact that B uses "B" to mean the same thing that B uses "i" to mean, namely, B.

The triples that make up the relations LA and LB are conveniently arranged in the form of relational data tables, as follows: