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A ''moderate equivalence relation'' (MER) on the ''modus'' <math>M \subseteq X\!</math> is a relation on <math>X\!</math> whose restriction to <math>M\!</math> is an equivalence relation on <math>M.\!</math> In symbols, <math>L \subseteq X \times X\!</math> such that <math>L|M \subseteq M \times M\!</math> is an equivalence relation. Notice that the subset of restriction, or modus <math>M,\!</math> is a part of the definition, so the same relation <math>L\!</math> on <math>X\!</math> could be a MER or not depending on the choice of <math>M.\!</math> In spite of how it sounds, a moderate equivalence relation can have more ordered pairs in it than the ordinary sort of equivalence relation on the same set.
A ''moderate equivalence relation'' (MER) on the ''modus'' <math>M \subseteq X\!</math> is a relation on <math>X\!</math> whose restriction to <math>M\!</math> is an equivalence relation on <math>M.\!</math> In symbols, <math>L \subseteq X \times X\!</math> such that <math>L|M \subseteq M \times M\!</math> is an equivalence relation. Notice that the subset of restriction, or modus <math>M,\!</math> is a part of the definition, so the same relation <math>L\!</math> on <math>X\!</math> could be a MER or not depending on the choice of <math>M.\!</math> In spite of how it sounds, a moderate equivalence relation can have more ordered pairs in it than the ordinary sort of equivalence relation on the same set.
In applying the equivalence class notation to a sign relation <math>L,\!</math> the definitions and examples considered so far cover only the case where the connotative component <math>L_{SI}\!</math> is a total equivalence relation on the whole syntactic domain <math>S.\!</math> The next job is to adapt this usage to PERs.
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If R is a sign relation whose syntactic projection RSI is a PER on S, then I still write "[s]R" for the "equivalence class of s under RSI". But now, [s]R can be empty if s has no interpretant, that is, if s lies outside the "adequately meaningful" subset of the syntactic domain, where synonymy and equivalence of meaning are defined. Otherwise, if s has an i then it also has an o, by the definition of RSI. In this case, there is a triple <o, s, i> C R, and it is permissible to let [o]R = [s]R.
If R is a sign relation whose syntactic projection RSI is a PER on S, then I still write "[s]R" for the "equivalence class of s under RSI". But now, [s]R can be empty if s has no interpretant, that is, if s lies outside the "adequately meaningful" subset of the syntactic domain, where synonymy and equivalence of meaning are defined. Otherwise, if s has an i then it also has an o, by the definition of RSI. In this case, there is a triple <o, s, i> C R, and it is permissible to let [o]R = [s]R.
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