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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]]''.

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 &mdash; let us say <math>\operatorname{Ann}</math> and <math>\operatorname{Bob}\!</math> &mdash; in the use of their own proper names, <math>^{\backprime\backprime} \operatorname{Ann} ^{\prime\prime}</math> and <math>^{\backprime\backprime} \operatorname{Bob} ^{\prime\prime},</math> together with the pronouns, <math>^{\backprime\backprime} \operatorname{I} ^{\prime\prime}</math> and <math>^{\backprime\backprime} \operatorname{you} ^{\prime\prime}.</math>  For brevity, these four signs may be abbreviated to the set <math>\{ \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime} \, \}.</math>  The abstract consideration of how <math>\operatorname{A}</math> and <math>\operatorname{B}</math> 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 <math>L_\operatorname{A}</math> and <math>L_\operatorname{B},</math> that reflect the differential use of these signs by <math>\operatorname{A}</math> and <math>\operatorname{B},</math> respectively.

For example, consider the aspects of sign use that concern two people &mdash; let us say <math>\operatorname{Ann}</math> and <math>\operatorname{Bob}\!</math> &mdash; in using their own proper names, <math>^{\backprime\backprime} \operatorname{Ann} ^{\prime\prime}</math> and <math>^{\backprime\backprime} \operatorname{Bob} ^{\prime\prime},</math> together with the pronouns, <math>^{\backprime\backprime} \operatorname{I} ^{\prime\prime}</math> and <math>^{\backprime\backprime} \operatorname{you} ^{\prime\prime}.</math>  For brevity, these four signs may be abbreviated to the set <math>\{ \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime} \, \}.</math>  The abstract consideration of how <math>\operatorname{A}</math> and <math>\operatorname{B}</math> 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 <math>L_\operatorname{A}</math> and <math>L_\operatorname{B},</math> that reflect the differential use of these signs by <math>\operatorname{A}</math> and <math>\operatorname{B},</math> respectively.

Each of the sign relations, <math>L_\operatorname{A}</math> and <math>L_\operatorname{B},</math> consists of eight triples of the form <math>(x, y, z),\!</math> where the ''object'' <math>x\!</math> is an element of the ''object domain'' <math>O = \{ \operatorname{A}, \operatorname{B} \},</math> where the ''sign'' <math>y\!</math> is an element of the ''sign domain'' <math>S\!,</math> where the ''interpretant sign'' <math>z\!</math> is an element of the interpretant domain <math>I,\!</math> and where it happens in this case that <math>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} \, \}.</math>  In general, it is convenient to refer to the union <math>S \cup I</math> as the ''syntactic domain'', but in this case <math>S ~=~ I ~=~ S \cup I.</math>

Each of the sign relations, <math>L_\operatorname{A}</math> and <math>L_\operatorname{B},</math> consists of eight triples of the form <math>(x, y, z),\!</math> where the ''object'' <math>x\!</math> is an element of the ''object domain'' <math>O = \{ \operatorname{A}, \operatorname{B} \},</math> where the ''sign'' <math>y\!</math> is an element of the ''sign domain'' <math>S\!,</math> where the ''interpretant sign'' <math>z\!</math> is an element of the interpretant domain <math>I,\!</math> and where it happens in this case that <math>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} \, \}.</math>  In general, it is convenient to refer to the union <math>S \cup I</math> as the ''syntactic domain'', but in this case <math>S ~=~ I ~=~ S \cup I.</math>

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The relation '''L'''_A is the set of eight triples enumerated here:

The relation <math>L_\operatorname{A}</math> is the set of eight triples enumerated here:

: {(A, "A", "A"), (A, "A", "i"), (A, "i", "A"), (A, "i", "i"),

{| align="center" cellpadding="8" width="90%"

: &nbsp;(B, "B", "B"), (B, "B", "u"), (B, "u", "B"), (B, "u", "u")}.

(\operatorname{A}, \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}), &

(\operatorname{A}, \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}), &

(\operatorname{A}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}), &

(\operatorname{A}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}), &

&

(\operatorname{B}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}), &

(\operatorname{B}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}), &

(\operatorname{B}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}), &

(\operatorname{B}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}) &

\}.

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 triples in <math>L_\operatorname{A}</math> represent the way that interpreter <math>\operatorname{A}</math> uses signs.  For example, the listing of the triple <math>(\operatorname{B}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime})</math> in <math>L_\operatorname{A}</math> represents the fact that <math>\operatorname{A}</math> uses <math>^{\backprime\backprime} \operatorname{B} ^{\prime\prime}</math> to mean the same thing that <math>\operatorname{A}</math> uses <math>^{\backprime\backprime} \operatorname{u} ^{\prime\prime}</math> to mean, namely, <math>\operatorname{B}.</math>

The relation '''L'''_B is the set of eight triples enumerated here:

The relation '''L'''_B is the set of eight triples enumerated here: