Triadic Relations • 2

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 3-place relation among signs, their objects, and their interpreters.

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 \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.  The abstract consideration of how \mathrm{A} and \mathrm{B} use this set of signs leads to the contemplation of a pair of triadic relations, the sign relations L_\mathrm{A} and L_\mathrm{B}, reflecting the differential use of these signs by \mathrm{A} and \mathrm{B}, respectively.

Each of the sign relations L_\mathrm{A} and L_\mathrm{B} consists of eight triples of the form (x, y, z), where the object x belongs to the object domain O = \{ \mathrm{A}, \mathrm{B} \}, the sign y belongs to the sign domain S, the interpretant sign z belongs to the interpretant domain I, and where it happens in this case that S = 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} \}.  The union S \cup I is often referred to as the syntactic domain, but in this case S = I = S \cup I.

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}

The relation L_\mathrm{A} is the following set of eight triples in O \times S \times I.

\begin{array}{cccccc}  \{ &  (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), &  (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), &  (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), &  (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), &  \\  &  (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), &  (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), &  (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), &  (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}) &  \}.  \end{array}

The triples in L_\mathrm{A} represent the way interpreter \mathrm{A} uses signs.  For example, the presence of ( \mathrm{B}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} ) in L_\mathrm{A} says \mathrm{A} uses {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} to mean the same thing \mathrm{A} uses {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} to mean, namely, \mathrm{B}.

The relation L_\mathrm{B} is the following set of eight triples in O \times S \times I.

\begin{array}{cccccc}  \{ &  (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), &  (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), &  (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), &  (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), &  \\  &  (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), &  (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), &  (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), &  (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}) &  \}.  \end{array}

The triples in L_\mathrm{B} represent the way interpreter \mathrm{B} uses signs.  For example, the presence of ( \mathrm{B}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} ) in L_\mathrm{B} says \mathrm{B} uses {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} to mean the same thing \mathrm{B} uses {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} to mean, namely, \mathrm{B}.

The triples in the relations L_\mathrm{A} and L_\mathrm{B} are conveniently arranged in the form of relational data tables, as shown below.

LA = Sign Relation of Interpreter A

LB = Sign Relation of Interpreter B

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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.

cc: CyberneticsOntolog ForumPeirce ListStructural ModelingSystems Science

This entry was posted in C.S. Peirce, Logic, Logic of Relatives, Mathematics, Peirce, Peirce's Categories, Philosophy, Pragmatic Semiotic Information, Pragmatism, Relation Theory, Semiotics, Sign Relations, Thirdness, Triadic Relations, Triadicity and tagged , , , , , , , , , , , , , , . Bookmark the permalink.

1 Response to Triadic Relations • 2

  1. mlibertin says:

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