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Tracing the passage of inquiry through the medium of signs calls for an active, intricate form of cooperation between the converging modes of investigation. Its proper character is best understood by realizing the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject the theory of signs is specialized to treat from comparative and structural points of view.
Tracing the passage of inquiry through the medium of signs calls for an active, intricate form of cooperation between the converging modes of investigation. Its proper character is best understood by realizing the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject the theory of signs is specialized to treat from comparative and structural points of view.
=={{anchor|Examples}}Examples of sign relations==
==Examples of sign relations==
Despite their simplicity, the examples to follow have subtleties of their own and their careful treatment serves to illustrate important issues in the general theory of signs.
Despite their simplicity, the examples to follow have subtleties of their own and their careful treatment serves to illustrate important issues in the general theory of signs.
Already in this elementary context, there are several different meanings that might attach to the project of a <i>formal semiotics</i>, or a formal theory of meaning for signs. In the process of discussing these alternatives, it is useful to introduce a few terms that are occasionally used in the philosophy of language to point out the needed distinctions.
Already in this elementary context, there are several different meanings that might attach to the project of a <i>formal semiotics</i>, or a formal theory of meaning for signs. In the process of discussing these alternatives, it is useful to introduce a few terms that are occasionally used in the philosophy of language to point out the needed distinctions.
=={{anchor|Dyadic Aspects}}Dyadic aspects of sign relations==
==Dyadic aspects of sign relations==
For an arbitrary triadic relation <math>L \subseteq O \times S \times I,</math> whether it happens to be a sign relation or not, there are six dyadic relations obtained by <i>projecting</i> <math>L</math> on one of the planes of the <math>OSI</math>‑space <math>O \times S \times I.</math> The six dyadic projections of a triadic relation <math>L</math> are defined and notated as shown in Table 2.
For an arbitrary triadic relation <math>L \subseteq O \times S \times I,</math> whether it happens to be a sign relation or not, there are six dyadic relations obtained by <i>projecting</i> <math>L</math> on one of the planes of the <math>OSI</math>‑space <math>O \times S \times I.</math> The six dyadic projections of a triadic relation <math>L</math> are defined and notated as shown in Table 2.
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=={{anchor|Semiotic Equivalence Relations 1}}Semiotic equivalence relations==
==Semiotic equivalence relations==
A <i>semiotic equivalence relation</i> (SER) is a special type of equivalence relation arising in the analysis of sign relations. As a general rule, any equivalence relation is closely associated with a family of equivalence classes which partition the underlying set of elements, frequently called the <i>domain</i> or <i>space</i> of the relation. In the case of a SER, the equivalence classes are called <i>semiotic equivalence classes</i> (SECs) and the partition is called a <i>semiotic partition</i> (SEP).
A <i>semiotic equivalence relation</i> (SER) is a special type of equivalence relation arising in the analysis of sign relations. As a general rule, any equivalence relation is closely associated with a family of equivalence classes which partition the underlying set of elements, frequently called the <i>domain</i> or <i>space</i> of the relation. In the case of a SER, the equivalence classes are called <i>semiotic equivalence classes</i> (SECs) and the partition is called a <i>semiotic partition</i> (SEP).
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A few items of notation are useful in discussing equivalence relations in general and semiotic equivalence relations in particular.
A few items of notation are useful in discussing equivalence relations in general and semiotic equivalence relations in particular.