In logic, mathematics, and semiotics, a triadic relation or a ternary relation is an important special case of a polyadic or finitary relation, one in which the number of places in the relation is three. One also sees the adjectives 3-adic, 3-ary, 3-dim, or 3-place being used to describe these relations. Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of criteria for the evaluation of arguments, although the exact definition of logic is a matter of controversy among philosophers. ... Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... Semiotics, or semiology, is the study of signs, both individually and grouped in sign systems. ... In mathematics, an n-ary relation (or n-place relation or often simply relation) is a generalization of binary relations such as = and < which occur in statements such as 5 < 6 or 2 + 2 = 4. It is the fundamental notion in the relational model for databases. ...

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. A sign relation is the basic construct in the theory of signs, or semiotic theory, as developed by Charles Sanders Peirce (1839-1914). ...

Examples from mathematics

For the sake of topics to be taken up later, it is useful to examine a pair of 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 "+", used in the context of the boolean domain B, denotes addition mod 2. Interpreted for logic, this amounts to the same thing as the boolean operation of exclusive-or or not-equal-to.

The third cartesian power of B is B³ = {(x₁, x₂, x₃) : x_j in B for j = 1, 2, 3}= B × B × B.

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) in 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) in 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: It has been suggested that this article or section be merged into Relational model. ...

X	Y	Z
0	0	0
0	1	1
1	0	1
1	1	0

X	Y	Z
0	0	1
0	1	0
1	0	0
1	1	1

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. Semiotics, or semiology, is the study of signs, both individually and grouped in sign systems. ...

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. Semiosis is a term introduced by Charles Peirce. ... A sign relation is the basic construct in the theory of signs, or semiotic theory, as developed by Charles Sanders Peirce (1839-1914). ...

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 brevity, these four signs may be abbreviated to 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 by 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 to this point 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: It has been suggested that this article or section be merged into Relational model. ...

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"

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"

Operations on triadic relations

Properties of triadic relations

References

Bibliography

See also

In mathematics, an n-ary relation (or n-place relation or often simply relation) is a generalization of binary relations such as = and < which occur in statements such as 5 < 6 or 2 + 2 = 4. It is the fundamental notion in the relational model for databases. ... The theory of relations treats the subject matter of relations in its combinatorial aspect, as distinguished from, though related to, its more properly logical study on one side and its more generally mathematical study on another. ... In mathematics, the concept of binary relation, sometimes called dyadic relation, is exemplified by such ideas as is greater than and is equal to in arithmetic, or is congruent to in geometry, or is an element of or is a subset of in set theory. ... In mathematics, the Cartesian product (or direct product) of two sets X and Y, denoted X × Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y: The Cartesian product is named after René Descartes... Logic of relatives, short for logic of relative terms, is a term used to cover the study of relations in their logical, philosphical, or semiotic aspects, as distinguised from, though closely coordinated with, their more properly formal, mathematical, or objective aspects. ... In mathematics, a projection is any one of several different types of functions, mappings, operations, or transformations, for example, the following: A set-theoretic operation typified by the jth projection map, written , that takes an element of the cartesian product to the value . ... A sign relation is the basic construct in the theory of signs, or semiotic theory, as developed by Charles Sanders Peirce (1839-1914). ...

External links