For an arbitrary triadic relation $L \subseteq O \times S \times I,$ whether it is a sign relation or not, there are six dyadic relations obtained by projecting $L$ on one of the planes of the $OSI$-space $O \times S \times I.$  The six dyadic projections of a triadic relation $L$ are defined and notated as shown in Table 2.

$\text{Table 2.} ~~ \text{Dyadic Aspects of Triadic Relations}$

By way of unpacking the set-theoretic notation, here is what the first definition says in ordinary language.

The dyadic relation resulting from the projection of $L$ on the $OS$-plane $O \times S$ is written briefly as $L_{OS}$ or written more fully as $\mathrm{proj}_{OS}(L)$ and is defined as the set of all ordered pairs $(o, s)$ in the cartesian product $O \times S$ for which there exists an ordered triple $(o, s, i)$ in $L$ for some interpretant $i$ in the interpretant domain $I.$

In the case where $L$ is a sign relation, which it becomes by satisfying one of the definitions of a sign relation, some of the dyadic aspects of $L$ can be recognized as formalizing aspects of sign meaning which have received their share of attention from students of signs over the centuries, and thus they can be associated with traditional concepts and terminology.  Of course, traditions may vary as to the precise formation and usage of such concepts and terms.  Other aspects of meaning have not received their fair share of attention, and thus remain anonymous on the contemporary scene of sign studies.

References

• Peirce, C.S. (1902), “Parts of Carnegie Application” (L 75), in Carolyn Eisele (ed., 1976), The New Elements of Mathematics by Charles S. Peirce, vol. 4, 13–73.  Online.
• Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), pp. 40–52.  ArchiveJournalOnline.

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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 • Peirce List (1) (2)Structural ModelingSystems Science

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