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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, and that can be described in the following manner.

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

The boolean domain is the set

The plus sign Failed to parse (syntax error): ^{\backprime\backprime} + ^{\prime\prime},

used in the context of the boolean domain  denotes addition modulo 2.  Interpreted for logic, the plus sign can be used to indicate either the boolean operation of exclusive disjunction,  or the boolean relation of logical inequality, 

The third cartesian power of is the set Failed to parse (unknown function\text): \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 is isomorphic to

The relation is defined as follows:

The relation is the set of four triples enumerated here:

The relation is defined as follows:

The relation is the set of four triples enumerated here:

The triples that make up the relations and 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 — let us say and — in using their own proper names, Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{Ann} ^{\prime\prime}

and Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{Bob} ^{\prime\prime},
together with the pronouns, Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{I} ^{\prime\prime}
and Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{you} ^{\prime\prime}.
 For brevity, these four signs may be abbreviated to the set Failed to parse (unknown function\backprime): \{ \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime} \, \}.
 The abstract consideration of how  and  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  and  that reflect the differential use of these signs by  and  respectively.

Each of the sign relations, and consists of eight triples of the form where the object is an element of the object domain where the sign is an element of the sign domain where the interpretant sign is an element of the interpretant domain and where it happens in this case that Failed to parse (unknown function\backprime): S = I = \{ \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime} \, \}.

 In general, it is convenient to refer to the union  as the syntactic domain, but in this case 

The set-up so far is summarized as follows:

The relation is the set of eight triples enumerated here:

The triples in represent the way that interpreter uses signs. For example, the listing of the triple Failed to parse (unknown function\backprime): (\operatorname{B}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime})

in  represents the fact that  uses Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}
to mean the same thing that  uses Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}
to mean, namely, 

The relation is the set of eight triples enumerated here:

The triples in represent the way that interpreter uses signs. For example, the listing of the triple Failed to parse (unknown function\backprime): (\operatorname{B}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime})

in  represents the fact that  uses Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}
to mean the same thing that  uses Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}
to mean, namely, 

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

	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}
	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}
	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}
	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}
	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}
	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}
	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}
	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}

	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}
	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}
	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}
	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}
	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}
	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}
	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}
	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}	Failed to parse (syntax error): ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}

Syllabus

Focal nodes

Peer nodes

Logical operators

Related topics

Relational concepts

Information, Inquiry

Related articles

• Jon Awbrey, “Semiotic Information”

• Jon Awbrey, “Introduction To Inquiry Driven Systems”

• Jon Awbrey, “Prospects For Inquiry Driven Systems”

• Jon Awbrey, “Inquiry Driven Systems : Inquiry Into Inquiry”

• Jon Awbrey, “Propositional Equation Reasoning Systems”

• Jon Awbrey, “Differential Logic : Introduction”

• Jon Awbrey, “Differential Propositional Calculus”

• Jon Awbrey, “Differential Logic and Dynamic Systems”

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.