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| | =====1.3.12.2. Derived Equivalence Relations===== | | =====1.3.12.2. Derived Equivalence Relations===== |
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| − | Signs are "equiferent" if they refer to all and only the same objects, that is, if they have exactly the same denotations. In other language for the same relation, signs are said to be "denotatively equivalent" or "referentially equivalent", but it is probably best to check whether the extension of this concept over the syntactic domain is really a genuine equivalence relation before jumping to the conclusions that are implied by these latter terms.
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| − | To define the "equiference" of signs in terms of their denotations, one says that "x is equiferent to y under R", and writes "x =R y", to mean that Den(R, x) = Den(R, y). Taken in extension, this notion of a relation between signs induces an "equiference relation" on the syntactic domain.
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| − | For each sign relation R, this yields a binary relation Der(R) c SxI that is defined as follows:
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| − | : Der(R) = DerR = {<x, y> C SxI : Den(R, x) = Den(R, y)}.
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| − | These definitions and notations are recorded in the following display.
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| − | <pre>
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| − | Definition 13
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| − | If R c OxSxI,
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| − | then the following are identical subsets of SxI:
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| − | D13a. DerR
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| − | D13b. Der(R)
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| − | D13c. {<x,y> C SxI : DenR|x = DenR|y}
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| − | D13d. {<x,y> C SxI : Den(R, x) = Den(R, y)}
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| − | </pre>
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| | The relation Der(R) is defined and the notation "x =R y" is meaningful in every situation where Den(-,-) makes sense, but it remains to check whether this relation enjoys the properties of an equivalence relation. | | The relation Der(R) is defined and the notation "x =R y" is meaningful in every situation where Den(-,-) makes sense, but it remains to check whether this relation enjoys the properties of an equivalence relation. |