The important thing now is to extend our perspective beyond one sign at a time and one object, sign, interpretant at a time to comprehending a sign relation as a specified set of object, sign, interpretant triples embedded in the set of all possible triples in a specified context.
In my mind’s eye, no doubt influenced by my early interest in Gestalt Psychology, I always picture a sign relation as a gestalt composed of figure and ground. The triples in the sign relation form a figure set in relief against the background of all possible triples and the triples left over form the ground of the gestalt.
From a mathematical point of view, the set of possible triples is a cartesian product of the following form.
Here, is the object domain, the set of objects under discussion, is the sign domain, the specified set of signs, and is the interpretant domain, the specified set of interpretants.
On this canvass, in this frame, any number of sign relations might be set as figures and each of them would be delimited as a salient subset of the cartesian product in view. Letting be any such sign relation, mathematical convention provides the following description of its relation to the set of possible triples.
It’s important to note at this point that the specified cartesian product and the specified subset of it are equally critical parts of the sign relation’s definition.
Well, it took a lot longer to set the scene than I thought it would when I got up this morning, so I’ll break here and get back to your specific comments when I next get a chance.
- 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.