triadic relation


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 relationMathworldPlanetmath is three. In other languagePlanetmathPlanetmath 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 the conceptMathworldPlanetmath of a sign relation, a special case of a 3-adic relation, is fundamental to the field of semiotics, or the theory of signs. Therefore it will be useful to consider a few concrete examples from each of these two realms.

1 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 “+” denotes additionPlanetmathPlanetmath modulo 2.

  • •

    The third cartesian power of 𝔹 is defined as follows:

    𝔹3=𝔹×𝔹×𝔹={(x1,x2,x3):xj∈𝔹⁢for⁢j=1,2,3}.

In what follows, the space X×Y×Z is isomorphic to 𝔹×𝔹×𝔹=𝔹3.

The relation L0 is defined as follows:

L0={(x,y,z)∈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)∈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:

L0={(x,y,z)∈𝔹3:x+y+z=0}
 X Y Z
0 0 0
0 1 1
1 0 1
1 1 0
L1={(x,y,z)∈𝔹3:x+y+z=1}
X Y Z
0 0 1
0 1 0
1 0 0
1 1 1

2 Examples from semiotics

The study of signs — the full varietyMathworldPlanetmath 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. 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 simplicity, these four signs may be abbreviated to form 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 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∪I as the syntactic domain, but in this case S=I=S∪I.

The set-up so far can be summarized as follows:

  • •

    LA,LB⊆O×S×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:

LA = Sign Relation of Interpreter A
 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⁢"
LB = Sign Relation of Interpreter B
 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⁢"
Title triadic relation
Canonical name TriadicRelation
Date of creation 2013-03-22 17:47:53
Last modified on 2013-03-22 17:47:53
Owner Jon Awbrey (15246)
Last modified by Jon Awbrey (15246)
Numerical id 30
Author Jon Awbrey (15246)
Entry type Topic
Classification msc 94A15
Classification msc 94A05
Classification msc 03G15
Classification msc 03E20
Classification msc 03B42
Classification msc 03B10
Synonym ternary relation
Related topic LogicalMatrix
Related topic RelationTheory
Related topic SignRelation
Related topic SignRelationalComplex
Related topic SemioticEquivalenceRelation