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| | =====1.3.12.2. Derived Equivalence Relations===== | | =====1.3.12.2. Derived Equivalence Relations===== |
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| − | One seeks a method of general application for approaching the individual sign relation, a way to select an aspect of its form, to analyze it with regard to its intrinsic structure, and to classify it in comparison with other sign relations. With respect to a particular sign relation, one approach that presents itself is to examine the relation between signs and interpretants that is given directly by its connotative component and to compare it with the various forms of derived, indirect, mediate, or peripheral relationships that can be found to exist among signs and interpretants by way of secondary considerations or subsequent studies. Of especial interest are the relationships among signs and interpretants that can be obtained by working through the collections of objects that they commonly or severally denote.
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| − | A classic way of showing that two sets are equal is to show that every element of the first belongs to the second and that every element of the second belongs to the first. The problem with this strategy is that one can exhaust a considerable amount of time trying to prove that two sets are equal before it occurs to one to look for a counterexample, that is, an element of the first that does not belong to the second or an element of the second that does not belong to the first, in cases where that is precisely what one ought to be seeking. It would be nice if there were a more balanced, impartial, neutral, or nonchalant way to go about this task, one that did not require such an undue commitment to either side, a technique that helps to pinpoint the counterexamples when they exist, and a method that keeps in mind the original relation of "proving that" and "showing that" to probing, testing, and seeing "whether".
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| − | A different way of seeing that two sets are equal, or of seeing whether two sets are equal, is based on the following observation:
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| − | <pre>
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| − | Two sets are equal as sets
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| − | <=> the indicator functions of these sets are equal as functions
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| − | <=> the values of these functions are equal on all domain elements.
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| − | </pre>
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| − | It is important to notice the hidden quantifier, of a universal kind, that lurks in all three equivalent statements but is only revealed in the last.
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| − | In making the next set of definitions and in using the corresponding terminology it is taken for granted that all of the references of signs are relative to a particular sign relation R c OxSxI that either remains to be specified or is already understood. Further, I continue to assume that S = I, in which case this set is called the "syntactic domain" of R.
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| − | In the following definitions let R c OxSxI, let S = I, and let x, y C S.
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| − | Recall the definition of Con(R), the connotative component of R, in the following form:
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| − | : Con(R) = RSI = {< s, i> C SxI : <o, s, i> C R for some o C O}.
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| − | Equivalent expressions for this concept are recorded in Definition 8.
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| − | <pre>
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| − | Definition 8
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| − | If R c OxSxI,
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| − | then the following are identical subsets of SxI:
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| − | D8a. RSI
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| − | D8b. ConR
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| − | D8c. Con(R)
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| − | D8d. PrSI(R)
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| − | D8e. {< s, i> C SxI : <o, s, i> C R for some o C O}
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| − | </pre>
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| | The dyadic relation RIS that constitutes the converse of the connotative relation RSI can be defined directly in the following fashion: | | The dyadic relation RIS that constitutes the converse of the connotative relation RSI can be defined directly in the following fashion: |
