Triadic relation

Of triadic Being the multitude of forms is so terrific that I have usually shrunk from the task of enumerating them;  and for the present purpose such an enumeration would be worse than superfluous:  it would be a great inconvenience.

— C.S. Peirce, Collected Papers, CP 6.347

A triadic relation (or ternary relation) is a special case of a polyadic or finitary relation, one in which the number of places in the relation is three.  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 triadic relations and the field of semiotics is rich in its harvest of sign relations, which are special cases of triadic relations.  In either subject, as Peirce observes, the multitude of forms is truly terrific, so it's best to begin with concrete examples just complex enough to illustrate the distinctive features of each type.  The discussion to follow takes up a pair of simple but instructive examples from each of the realms of mathematics and semiotics.

Examples from mathematics

For the sake of topics to be taken up later, it is useful to examine a pair of triadic relations in tandem.  In what follows we construct two triadic relations, \(L_0\) and \(L_1,\) each of which is a subset of the same cartesian product \(X \times Y \times Z.\)  The structures of \(L_0\) and \(L_1\) can be described in the following way.

Each space \(X, Y, Z\) is isomorphic to the boolean domain \(\mathbb{B} = \{ 0, 1 \}\) so \(L_0\) and \(L_1\) are subsets of the cartesian power \(\mathbb{B} \times \mathbb{B} \times \mathbb{B}\) or the boolean cube \(\mathbb{B}^3.\)

The operation of boolean addition, \(+ : \mathbb{B} \times \mathbb{B} \to \mathbb{B},\) is equivalent to addition modulo 2, where \(0\) acts in the usual manner but \(1 + 1 = 0.\)  In its logical interpretation, the plus sign can be used to indicate either the boolean operation of exclusive disjunction or the boolean relation of logical inequality.

The relation \(L_0\) is defined by the following formula.

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

The relation \(L_0\) is the following set of four triples in \(\mathbb{B}^3.\)

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

The relation \(L_1\) is defined by the following formula.

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

The relation \(L_1\) is the following set of four triples in \(\mathbb{B}^3.\)

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

The triples in the relations \(L_0\) and \(L_1\) are conveniently arranged in the form of relational data tables, as shown below.

\(L_0 ~=~ \{ (x, y, z) : x + y + z = 0 \}\)

\(x\)	\(y\)	\(z\)
\(0\)	\(0\)	\(0\)
\(0\)	\(1\)	\(1\)
\(1\)	\(0\)	\(1\)
\(1\)	\(1\)	\(0\)

\(L_1 ~=~ \{ (x, y, z) : x + y + z = 1 \}\)

\(x\)	\(y\)	\(z\)
\(0\)	\(0\)	\(1\)
\(0\)	\(1\)	\(0\)
\(1\)	\(0\)	\(0\)
\(1\)	\(1\)	\(1\)

Examples from semiotics

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

The term semiosis refers to any activity or process involving signs.  Studies of semiosis focusing on its abstract form are not concerned with every concrete detail of the entities acting as signs, as objects, or as agents of semiosis, but only with the most salient patterns of relationship among those three roles.  In particular, the formal theory of signs does not consider all the properties of the interpretive agent but only the more striking features of the impressions signs make on a representative interpreter.  From a formal point of view this impact or influence may be treated as just another sign, called the interpretant sign, or the interpretant for short.  A triadic relation of this type, among objects, signs, and interpretants, is called a sign relation.

For example, consider the aspects of sign use involved when two people, say \(\mathrm{Ann}\) and \(\mathrm{Bob},\) use their own proper names, \({}^{\backprime\backprime} \mathrm{Ann} {}^{\prime\prime}\) and \({}^{\backprime\backprime} \mathrm{Bob} {}^{\prime\prime},\) along with the pronouns, \({}^{\backprime\backprime} \mathrm{I} {}^{\prime\prime}\) and \({}^{\backprime\backprime} \mathrm{you} {}^{\prime\prime},\) to refer to themselves and each other.  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 leads to the contemplation of a pair of triadic relations, the sign relations \(L_\mathrm{A}\) and \(L_\mathrm{B},\) reflecting 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 \(O = \{ \mathrm{A}, \mathrm{B} \},\) the sign \(y\) is an element of the sign domain \(S,\) the interpretant sign \(z\) is an element of the interpretant domain \(I,\) and where it happens in this case that \(S = I = \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.\)  The union \(S \cup I\) is often referred to as the syntactic domain, but in this case \(S = I = S \cup I.\)

The set-up so far is summarized as follows:

\(\begin{array}{ccc} L_\mathrm{A}, L_\mathrm{B} & \subseteq & O \times S \times I \\[5pt] O & = & \{ \mathrm{A}, \mathrm{B} \} \\[5pt] S & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \} \\[5pt] I & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \} \end{array}\)

The relation \(L_\mathrm{A}\) is the following set of eight triples in \(O \times S \times I.\)

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

The triples in \(L_\mathrm{A}\) represent the way interpreter \(\mathrm{A}\) uses signs.  For example, the presence of \(( \mathrm{B}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} )\) in \(L_\mathrm{A}\) tells us \(\mathrm{A}\) uses \({}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\) to mean the same thing \(\mathrm{A}\) uses \({}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\) to mean, namely, \(\mathrm{B}.\)

The relation \(L_\mathrm{B}\) is the following set of eight triples in \(O \times S \times I.\)

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

The triples in \(L_\mathrm{B}\) represent the way interpreter \(\mathrm{B}\) uses signs.  For example, the presence of \(( \mathrm{B}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} )\) in \(L_\mathrm{B}\) tells us \(\mathrm{B}\) uses \({}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\) to mean the same thing \(\mathrm{B}\) uses \({}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\) to mean, namely, \(\mathrm{B}.\)

The triples in the relations \(L_\mathrm{A}\) and \(L_\mathrm{B}\) are conveniently arranged in the form of relational data tables, as shown below.

\(L_\mathrm{A} ~=~ \text{Sign Relation of Interpreter A}\)

\(\text{Object}\)	\(\text{Sign}\)	\(\text{Interpretant}\)
\(\mathrm{A}\)	\({}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\)	\({}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\)
\(\mathrm{A}\)	\({}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\)	\({}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\)
\(\mathrm{A}\)	\({}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\)	\({}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\)
\(\mathrm{A}\)	\({}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\)	\({}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\)
\(\mathrm{B}\)	\({}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\)	\({}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\)
\(\mathrm{B}\)	\({}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\)	\({}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\)
\(\mathrm{B}\)	\({}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\)	\({}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\)
\(\mathrm{B}\)	\({}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\)	\({}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\)

\(L_\mathrm{B} ~=~ \text{Sign Relation of Interpreter B}\)

\(\text{Object}\)	\(\text{Sign}\)	\(\text{Interpretant}\)
\(\mathrm{A}\)	\({}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\)	\({}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\)
\(\mathrm{A}\)	\({}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\)	\({}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\)
\(\mathrm{A}\)	\({}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\)	\({}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\)
\(\mathrm{A}\)	\({}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\)	\({}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\)
\(\mathrm{B}\)	\({}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\)	\({}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\)
\(\mathrm{B}\)	\({}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\)	\({}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\)
\(\mathrm{B}\)	\({}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\)	\({}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\)
\(\mathrm{B}\)	\({}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\)	\({}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\)

Resources

• Logic Syllabus

Document history

Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.

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• Triadic Relation, OEIS Wiki
• Triadic Relation, InterSciWiki
• Triadic Relation, MyWikiBiz

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• Triadic Relation, PlanetMath
• Triadic Relation, Wikiversity
• Triadic Relation, Wikipedia

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