semiotic equivalence relation

semiotic equivalence relation

Note.

The intended forms of the Tables are visible only in the Page Image view.

A

semiotic equivalence relation

(SER) is a type of

equivalence relation

that arises in the

analysis

of

sign relations

. Any equivalence relation is closely associated with a corresponding set of

equivalence classes

that

partition

the underlying set of

elements

, commonly called the

domain

or the

space

of the

relation

. In the case of a SER, the equivalence classes are called

semiotic equivalence classes

(SEC’s) and the partition is called a

semiotic partition

(SEP).

Contents:

1

Examples

2

Notation

1

Examples

Tables 1 and 2 list the triples of two finite relations,

L

A

and

L

B

, that are introduced in the entries on

triadic relations

and sign relations as examples of both those

concepts

.

Table 1. Sign Relation

L

A

Table 2. Sign Relation

L

B

Object

Sign

Interpretant

Object

Sign

Interpretant

A

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The sign relations

L

A

and

L

B

have many interesting

properties

that are not possessed by sign relations in general. Some of these properties have to do with the relation between signs and their interpretant signs, as reflected in the projections of the sign relations

L

A

and

L

B

on the

S

⁢

I

-plane, notated as

proj

S

⁢

I

⁡

L

A

and

proj

S

⁢

I

⁡

L

B

, respectively. The dyadic relations on

S

×

I

that are

induced

by these projections are also referred to as

connotative

components

(

http://planetmath.org/Connotation

)

of the corresponding sign relations, notated as

Con

⁡

(

L

A

)

and

Con

⁡

(

L

B

)

, respectively. The discussion of these properties is made a little easier by using the following abbreviations.

•

A

S

⁢

I

:=

Con

⁡

(

L

A

)

:=

proj

S

⁢

I

⁡

L

A

.

•

B

S

⁢

I

:=

Con

⁡

(

L

B

)

:=

proj

S

⁢

I

⁡

L

B

.

Tables 3 and 4 exhibit the connotative components of

L

A

and

L

B

, respectively.

Table 3.

A

S

⁢

I

Table 4.

B

S

⁢

I

Sign

Interpretant

Sign

Interpretant

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A

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One nice property possessed by the sign relations

L

A

and

L

B

is that their connotative components

A

S

⁢

I

and

B

S

⁢

I

constitute a pair of equivalence relations on their common syntactic domain

S

=

I

. It is convenient to refer to such a

structure

as a

semiotic equivalence relation

(SER) since it equates signs that mean the same thing to some interpreter. Each of the SER’s,

A

S

⁢

I

,

B

S

⁢

I

⊆

S

×

I

≅

S

×

S

partitions the whole

collection

of signs into

semiotic equivalence classes

(SEC’s). This makes for a strong form of representation in that the structure of the participants’ common object domain

{

A

,

B

}

is reflected or reconstructed, part for part, in the structure of each of their

semiotic partitions

(SEP’s) of the syntactic domain

{

`

`

A

"

,

`

`

B

"

,

`

`

i

"

,

`

`

u

"

,

}

. But it needs to be noted that the semiotic partitions for

A

and

B

are not the same, indeed, they are

orthogonal

to each other. This makes it difficult to interpret either one of the partitions or equivalence relations on the syntactic domain as corresponding to any sort of objective structure or

invariant

reality,

independent

of the

individual

interpreter’s point of view (POV).

Information

about the different patterns of semiotic equivalence induced by the interpreters

A

and

B

is summarized in Tables 5 and 6. The form of these Tables should suffice to explain what is meant by saying that the SEP’s for

A

and

B

are orthogonal to each other.

Table 5. SEP for

A

Table 6. SEP for

B

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2

Notation

A few items of notation are useful in discussing equivalence relations in general and semiotic equivalence relations in particular.

As a general consideration, if

E

is an equivalence relation on a set

X

, then every element

x

of

X

belongs to a unique equivalence class under

E

called “the equivalence class of

x

under

E

”. Convention provides the

square bracket notation

for denoting this equivalence class, either in the subscripted form

[

x

]

E

or in the simpler form

[

x

]

when the

subscript

E

is understood. A statement that the elements

x

and

y

are

equivalent

under

E

is called an

equation

and can be expressed in numerous ways, for example, any of the following equivalent statements:

•

(

x

,

y

)

∈

E

•

x

∈

[

y

]

E

•

y

∈

[

x

]

E

•

[

x

]

E

=

[

y

]

E

•

x

=

E

y

Thus we have the following

definitions

:

•

[

x

]

E

:=

{

y

∈

X

:

(

x

,

y

)

∈

E

}

.

•

x

=

E

y

⇔

(

x

,

y

)

∈

E

.

In the application to sign relations it is useful to extend the square bracket notation in the following ways. If

L

is a sign relation whose connotative component or syntactic projection

L

S

⁢

I

is an equivalence relation on

S

, let

[

s

]

L

be the equivalence class of

s

under

L

S

⁢

I

. That is to say,

[

s

]

L

:=

[

s

]

L

S

⁢

I

. A statement that the signs

x

and

y

are equivalent under a semiotic equivalence relation

L

S

⁢

I

is called a

semiotic equation

(SEQ) and can be written in either of the forms,

[

x

]

L

=

[

y

]

L

or

x

=

L

y

.

In many situations there is one further adaptation of the square bracket notation that can be useful. Namely, when there is known to exist a particular triple

(

o

,

s

,

i

)

∈

L

, it is permissible to let

[

o

]

L

:=

[

s

]

L

. These

modifications

are designed to make the notation for semiotic equivalence classes harmonize as well as possible with the frequent use of similar devices for the denotations of signs and

expressions

.

The SER for interpreter

A

yields the following semiotic equations:

[

`

⁢

`

⁢

A

⁢

"

]

A

=

[

`

⁢

`

⁢

i

⁢

"

]

A

[

`

⁢

`

⁢

B

⁢

"

]

A

=

[

`

⁢

`

⁢

u

⁢

"

]

A

or

`

⁢

`

⁢

A

⁢

"

=

A

`

⁢

`

⁢

i

⁢

"

`

⁢

`

⁢

B

⁢

"

=

A

`

⁢

`

⁢

u

⁢

"

and the semiotic partition:

{

{

`

⁢

`

⁢

A

⁢

"

,

`

⁢

`

⁢

i

⁢

"

}

,

{

`

⁢

`

⁢

B

⁢

"

,

`

⁢

`

⁢

u

⁢

"

}

}

.

The SER for interpreter

B

yields the following semiotic equations:

[

`

⁢

`

⁢

A

⁢

"

]

B

=

[

`

⁢

`

⁢

u

⁢

"

]

B

[

`

⁢

`

⁢

B

⁢

"

]

B

=

[

`

⁢

`

⁢

i

⁢

"

]

B

or

`

⁢

`

⁢

A

⁢

"

=

B

`

⁢

`

⁢

u

⁢

"

`

⁢

`

⁢

B

⁢

"

=

B

`

⁢

`

⁢

i

⁢

"

and the semiotic partition:

{

{

`

⁢

`

⁢

A

⁢

"

,

`

⁢

`

⁢

u

⁢

"

}

,

{

`

⁢

`

⁢

B

⁢

"

,

`

⁢

`

⁢

i

⁢

"

}

}

.

Title

semiotic equivalence relation

Canonical

name

SemioticEquivalenceRelation

Date of creation

2013-10-28 23:54:03

Last modified on

2013-10-28 23:54:03

Owner

Jon Awbrey (15246)

Last modified by

Jon Awbrey (15246)

Numerical id

27

Author

Jon Awbrey (15246)

Entry type

Definition

Classification

msc 94A05

Classification

msc 03G15

Classification

msc 03E20

Classification

msc 03B42

Classification

msc 03B10

Classification

msc 94A15

Synonym

semiotic congruence

Related topic

RelationTheory

Related topic

SignRelation

Related topic

TriadicRelation

Defines

semiotic equation

Defines

semiotic equivalence

Defines

semiotic equivalence class

Defines

semiotic partition

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