triadic relation

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

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 the

concept

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,

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

𝔹

=

{

0

,

1

}

.

•

The

plus sign

“

+

” denotes

addition

modulo 2.

•

The third cartesian power of

𝔹

is defined as follows:

𝔹

3

=

𝔹

×

𝔹

×

𝔹

=

{

(

x

1

,

x

2

,

x

3

)

:

x

j

∈

𝔹

⁢

for

⁢

j

=

1

,

2

,

3

}

.

In what follows, the space

X

×

Y

×

Z

is isomorphic to

𝔹

×

𝔹

×

𝔹

=

𝔹

3

.

The relation

L

0

is defined as follows:

L

0

=

{

(

x

,

y

,

z

)

∈

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

)

∈

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

)

∈

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

)

∈

𝔹

3

:

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

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. 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

L

A

and

L

B

, that

reflect

the differential use of these signs by

A

and

B

, respectively.

Each of the sign relations,

L

A

and

L

B

, 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:

•

L

A

,

L

B

⊆

O

×

S

×

I

.

•

O

=

{

A

,

B

}

.

•

S

=

{

`

⁢

`

⁢

A

⁢

"

,

`

⁢

`

⁢

B

⁢

"

,

`

⁢

`

⁢

i

⁢

"

,

`

⁢

`

⁢

u

⁢

"

}

.

•

I

=

{

`

⁢

`

⁢

A

⁢

"

,

`

⁢

`

⁢

B

⁢

"

,

`

⁢

`

⁢

i

⁢

"

,

`

⁢

`

⁢

u

⁢

"

}

.

The relation

L

A

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

L

A

represent the way that interpreter

A

uses signs. For example, the listing of the triple

(

B

,

`

⁢

`

⁢

u

⁢

"

,

`

⁢

`

⁢

B

⁢

"

)

in

L

A

represents the fact that

A

uses

`

⁢

`

⁢

B

⁢

"

to mean the same thing that

A

uses

`

⁢

`

⁢

u

⁢

"

to mean, namely,

B

.

The relation

L

B

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

L

B

represent the way that interpreter

B

uses signs. For example, the listing of the triple

(

B

,

`

⁢

`

⁢

i

⁢

"

,

`

⁢

`

⁢

B

⁢

"

)

in

L

B

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

L

A

and

L

B

are conveniently arranged in the form of relational data tables, as follows:

L

A

= 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

⁢

"

L

B

= 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

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