Triadic relation

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In logic, mathematics, and semiotics, a triadic relation is an important special case of a polyadic or finitary relation, one in which the number of places in the relation is three. In other language that is often used, a triadic relation is called a ternary relation. One may also see the adjectives 3-adic, 3-ary, 3-dimensional, 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, L_0\! and L_1,\! that can be described in the following manner.

The first order of business is to define the space in which the relations L_0\! and L_1\! take up residence. This space is constructed as a 3-fold cartesian power in the following way.

The boolean domain is the set \mathbb{B} = \{ 0, 1 \}.\!

The plus sign {}^{\backprime\backprime} + {}^{\prime\prime},\! used in the context of the boolean domain \mathbb{B},\! denotes addition modulo 2. Interpreted for logic, the plus sign can be used to indicate either the boolean operation of exclusive disjunction, \mathrm{XOR} : \mathbb{B} \times \mathbb{B} \to \mathbb{B},\! or the boolean relation of logical inequality, \mathrm{NEQ} \subseteq \mathbb{B} \times \mathbb{B}.\!

The third cartesian power of \mathbb{B}\! is the set \mathbb{B}^3 = \mathbb{B} \times \mathbb{B} \times \mathbb{B} = \{ (x_1, x_2, x_3) : x_j \in \mathbb{B} ~\text{for}~ j = 1, 2, 3 \}.\!

In what follows, the space X \times Y \times Z\! is isomorphic to \mathbb{B} \times \mathbb{B} \times \mathbb{B} ~=~ \mathbb{B}^3.\!

The relation L_0\! is defined as follows:

L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}.~\!

The relation L_0\! is the set of four triples enumerated here:

L_0 ~=~ \{ (0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0) \}.\!

The relation L_1\! is defined as follows:

L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}.~\!

The relation L_1\! is the set of four triples enumerated here:

L_1 ~=~ \{ (0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 1, 1) \}.\!

The triples that make up the relations L_0\! and L_1\! are conveniently arranged in the form of relational data tables, as follows:


L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}\!
X\! Y\! Z\!
0\! 0\! 0\!
0\! 1\! 1\!
1\! 0\! 1\!
1\! 1\! 0\!


L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}\!
X\! Y\! Z\!
0\! 0\! 1\!
0\! 1\! 0\!
1\! 0\! 0\!
1\! 1\! 1\!


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 — let us say \mathrm{Ann}\! and \mathrm{Bob}\! — in using their own proper names, {}^{\backprime\backprime} \mathrm{Ann} {}^{\prime\prime}\! and {}^{\backprime\backprime} \mathrm{Bob} {}^{\prime\prime},\! together with the pronouns, {}^{\backprime\backprime} \mathrm{I} {}^{\prime\prime}\! and {}^{\backprime\backprime} \mathrm{you} {}^{\prime\prime}.\! For brevity, these four signs may be abbreviated to the set \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \}.\! The abstract consideration of how \mathrm{A}\! and \mathrm{B}\! use this set of signs to refer to themselves and each other leads to the contemplation of a pair of 3-adic relations, the sign relations L_\mathrm{A}\! and L_\mathrm{B},\! that reflect the differential use of these signs by \mathrm{A}\! and \mathrm{B},\! respectively.

Each of the sign relations, L_\mathrm{A}\! and L_\mathrm{B},\! consists of eight triples of the form (x, y, z),\! where the object x\! is an element of the object domain O = \{ \mathrm{A}, \mathrm{B} \},\! where the sign y\! is an element of the sign domain S\!, where the interpretant sign z\! is an element of the interpretant domain I,\! and where it happens in this case that S = I = \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \}.\! 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 so far is summarized as follows:

\begin{array}{ccc} L_\mathrm{A}, L_\mathrm{B} & \subseteq & O \times S \times I \\ \\ O & = & \{ \mathrm{A}, \mathrm{B} \} \\ \\ S & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \} \\ \\ I & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \} \\ \\ \end{array}

The relation L_\mathrm{A}\! is the set of eight triples enumerated here:

\begin{array}{cccccc} \{ & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), & \\ & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}) & \}. \end{array}

The triples in L_\mathrm{A}\! represent the way that interpreter \mathrm{A}\! uses signs. For example, the listing of the triple (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime})\! in L_\mathrm{A}\! represents the fact that \mathrm{A}\! uses {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\! to mean the same thing that \mathrm{A}\! uses {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\! to mean, namely, \mathrm{B}.\!

The relation L_\mathrm{B}\! is the set of eight triples enumerated here:

\begin{array}{cccccc} \{ & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), & \\ & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}) & \}. \end{array}

The triples in L_\mathrm{B}\! represent the way that interpreter \mathrm{B}\! uses signs. For example, the listing of the triple (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime})\! in L_\mathrm{B}\! represents the fact that \mathrm{B}\! uses {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\! to mean the same thing that \mathrm{B}\! uses {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\! to mean, namely, \mathrm{B}.\!

The triples that make up the relations L_\mathrm{A}\! and L_\mathrm{B}\! are conveniently arranged in the form of relational data tables, as follows:


L_\mathrm{A} ~=~ \text{Sign Relation of Interpreter A}\!
\text{Object}\! \text{Sign}\! \text{Interpretant}\!
\mathrm{A}\! {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\! {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!
\mathrm{A}\! {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\! {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!
\mathrm{A}\! {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\! {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!
\mathrm{A}\! {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\! {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!
\mathrm{B}\! {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\! {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!
\mathrm{B}\! {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\! {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!
\mathrm{B}\! {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\! {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!
\mathrm{B}\! {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\! {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!


L_\mathrm{B} ~=~ \text{Sign Relation of Interpreter B}\!
\text{Object}\! \text{Sign}\! \text{Interpretant}\!
\mathrm{A}\! {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\! {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!
\mathrm{A}\! {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\! {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!
\mathrm{A}\! {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\! {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!
\mathrm{A}\! {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\! {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!
\mathrm{B}\! {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\! {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!
\mathrm{B}\! {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\! {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!
\mathrm{B}\! {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\! {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!
\mathrm{B}\! {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\! {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!


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