Sign Relations • Denotation

One aspect of a sign’s complete meaning concerns the reference a sign has to its objects, which objects are collectively known as the denotation of the sign. In the pragmatic theory of sign relations, denotative references fall within the projection of the sign relation on the plane spanned by its object domain and its sign domain.

The dyadic relation making up the denotative, referent, or semantic aspect of a sign relation $L$ is notated as $\mathrm{Den}(L).$ Information about the denotative aspect of meaning is obtained from $L$ by taking its projection on the object-sign plane. We may visualize this as the “shadow” $L$ casts on the 2-dimensional space whose axes are the object domain $O$ and the sign domain $S.$ The denotative component of a sign relation $L,$ alternatively written in any of forms, $\mathrm{proj}_{OS} L,$ $L_{OS},$ $\mathrm{proj}_{12} L,$ and $L_{12},$ is defined as follows.

$\begin{matrix} \mathrm{Den}(L) & = & \mathrm{proj}_{OS} L & = & \{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \}. \end{matrix}$

Tables 3a and 3b show the denotative components of the sign relations associated with the interpreters $\mathrm{A}$ and $\mathrm{B},$ respectively. The rows of each Table list the ordered pairs $(o, s)$ in the corresponding projections, $\mathrm{Den}(L_\mathrm{A}), \mathrm{Den}(L_\mathrm{B}) \subseteq O \times S.$

Looking to the denotative aspects of $L_\mathrm{A}$ and $L_\mathrm{B},$ various rows of the Tables specify, for example, that $\mathrm{A}$ uses ${}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}$ to denote $\mathrm{A}$ and ${}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}$ to denote $\mathrm{B},$ while $\mathrm{B}$ uses ${}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}$ to denote $\mathrm{B}$ and ${}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}$ to denote $\mathrm{A}.$

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. Archive. Journal. Online.

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cc: Cybernetics • Ontolog • Peirce List (1) (2) • Structural Modeling • Systems Science

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