Triadic Relations • Examples from Semiotics
The study of signs — the full variety of significant forms of expression — in relation to all the affairs signs are significant of, and in relation to all the beings signs are significant to, is known as semiotics or the theory of signs. As described, semiotics treats of a 
The term semiosis refers to any activity or process involving signs. Studies of semiosis focusing on its abstract form are not concerned with every concrete detail of the entities acting as signs, as objects, or as agents of semiosis, but only with the most salient patterns of relationship among those three roles. In particular, the formal theory of signs does not consider all the properties of the interpretive agent but only the more striking features of the impressions signs make on a representative interpreter. From a formal point of view this impact or influence may be treated as just another sign, called the interpretant sign, or the interpretant for short. A triadic relation of this type, among objects, signs, and interpretants, is called a sign relation.
For example, consider the aspects of sign use involved when two people, say Ann and Bob, use their own proper names, “Ann” and “Bob”, along with the pronouns, “I” and “you”, to refer to themselves and each other. For brevity, these four signs may be abbreviated to the set 
Each of the sign relations 
The set-up so far is summarized as follows:
![\begin{array}{ccc} L_\mathrm{A}, L_\mathrm{B} & \subseteq & O \times S \times I \\[8pt] O & = & \{ \mathrm{A}, \mathrm{B} \} \\[8pt] S & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \} \\[8pt] I & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Bccc%7D++L_%5Cmathrm%7BA%7D%2C+L_%5Cmathrm%7BB%7D+%26+%5Csubseteq+%26+O+%5Ctimes+S+%5Ctimes+I++%5C%5C%5B8pt%5D++O+%26+%3D+%26+%5C%7B+%5Cmathrm%7BA%7D%2C+%5Cmathrm%7BB%7D+%5C%7D++%5C%5C%5B8pt%5D++S+%26+%3D+%26+%5C%7B+%5C%2C+%7B%7D%5E%7B%5Cbackprime%5Cbackprime%7D+%5Cmathrm%7BA%7D+%7B%7D%5E%7B%5Cprime%5Cprime%7D%2C+%5C%2C+%7B%7D%5E%7B%5Cbackprime%5Cbackprime%7D+%5Cmathrm%7BB%7D+%7B%7D%5E%7B%5Cprime%5Cprime%7D%2C+%5C%2C+%7B%7D%5E%7B%5Cbackprime%5Cbackprime%7D+%5Cmathrm%7Bi%7D+%7B%7D%5E%7B%5Cprime%5Cprime%7D%2C+%5C%2C+%7B%7D%5E%7B%5Cbackprime%5Cbackprime%7D+%5Cmathrm%7Bu%7D+%7B%7D%5E%7B%5Cprime%5Cprime%7D+%5C%2C+%5C%7D++%5C%5C%5B8pt%5D++I+%26+%3D+%26+%5C%7B+%5C%2C+%7B%7D%5E%7B%5Cbackprime%5Cbackprime%7D+%5Cmathrm%7BA%7D+%7B%7D%5E%7B%5Cprime%5Cprime%7D%2C+%5C%2C+%7B%7D%5E%7B%5Cbackprime%5Cbackprime%7D+%5Cmathrm%7BB%7D+%7B%7D%5E%7B%5Cprime%5Cprime%7D%2C+%5C%2C+%7B%7D%5E%7B%5Cbackprime%5Cbackprime%7D+%5Cmathrm%7Bi%7D+%7B%7D%5E%7B%5Cprime%5Cprime%7D%2C+%5C%2C+%7B%7D%5E%7B%5Cbackprime%5Cbackprime%7D+%5Cmathrm%7Bu%7D+%7B%7D%5E%7B%5Cprime%5Cprime%7D+%5C%2C+%5C%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=0&c=20201002)
The relation
The triples in 
The relation
The triples in 
The triples in the relations
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.
- Triadic Relation • OEIS Wiki
- Triadic Relation • InterSciWiki
- Triadic Relation • MyWikiBiz
- Triadic Relation • Wikiversity
- Triadic Relation • Wikipedia
cc: Category Theory • Cybernetics (1) (2) • Ontolog (1) (2)
cc: Structural Modeling (1) (2) • Systems Science (1) (2)
cc: FB | Relation Theory • Laws of Form • Peirce List (1) (2)
and 
each of which is a subset of the same cartesian product 
The structures of 
can be described in the following way.
is isomorphic to the 
so 
or the boolean cube 
is equivalent to addition modulo 2, where 
acts in the usual manner but 
In its logical interpretation, the plus sign can be used to indicate either the boolean operation of 
be an arbitrary 2‑adic relation. The following properties of 
can be defined.
then 
This is sometimes indicated by giving 
and writing 
Thus we have the following definition.
which happens to be total at 
indicated by writing 
To say a relation 
-regular at 
Thus, we may formalize the following definition.
we have the following additional definitions.
-relation in a 
-relation’s tuples reminds me Etter’s paper “
denote the same object, represented in his logical graphs as a node of degree three, and at first I thought you might be talking about that.
read as 
is more like the expressions I use to discuss “equivalence relations from a particular point of view”, following one of Peirce’s more radical innovations from his 1870 “Logic of Relatives”.![[a]_e](https://s0.wp.com/latex.php?latex=%5Ba%5D_e&bg=ffffff&fg=000000&s=0&c=20201002)
for the equivalence class of an element 
in an equivalence relation 
we can express the above idea in one of the following forms.![\begin{matrix} [y = z]_x & \text{or} & [y]_x = [z]_x & \text{or} & y =_x z \end{matrix}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Bmatrix%7D++%5By+%3D+z%5D_x+%26+%5Ctext%7Bor%7D+%26+%5By%5D_x+%3D+%5Bz%5D_x+%26+%5Ctext%7Bor%7D+%26+y+%3D_x+z++%5Cend%7Bmatrix%7D&bg=ffffff&fg=000000&s=0&c=20201002)
and a subset 
is a subset of 
at 
or the 
-flag of 
a mathematical object with the following definition.
-regular at 
if and only if the cardinality of the local flag 
is 
— to write it in symbols, if and only if 
for all 
-regular at 
-regular at 
-flag of 
of the local flag 
is said to be 
where 
is true for all 
is 
where 
with 
domain.
then 
Inquiry Into Inquiry 