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A ''partial equivalence relation'' (PER) on a set <math>X\!</math> is a relation <math>L \subseteq X \times X\!</math> that is an equivalence relation on its domain of definition <math>\operatorname{Quo} (L) \subseteq X.\!</math> In this situation, <math>[x]_L\!</math> is empty for each <math>x\!</math> in <math>X\!</math> that is not in <math>\operatorname{Quo} (L).\!</math> Another way of reaching the same concept is to call a PER a dyadic relation that is symmetric and transitive, but not necessarily reflexive. Like the “self-identical elements” of old that epitomized the very definition of self-consistent existence in classical logic, the property of being a self-related or self-equivalent element in the purview of a PER on <math>X\!</math> singles out the members of <math>\operatorname{Quo} (L)\!</math> as those for which a properly meaningful existence can be contemplated.
A ''partial equivalence relation'' (PER) on a set <math>X\!</math> is a relation <math>L \subseteq X \times X\!</math> that is an equivalence relation on its domain of definition <math>\operatorname{Quo} (L) \subseteq X.\!</math> In this situation, <math>[x]_L\!</math> is empty for each <math>x\!</math> in <math>X\!</math> that is not in <math>\operatorname{Quo} (L).\!</math> Another way of reaching the same concept is to call a PER a dyadic relation that is symmetric and transitive, but not necessarily reflexive. Like the “self-identical elements” of old that epitomized the very definition of self-consistent existence in classical logic, the property of being a self-related or self-equivalent element in the purview of a PER on <math>X\!</math> singles out the members of <math>\operatorname{Quo} (L)\!</math> as those for which a properly meaningful existence can be contemplated.
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.
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In applying the equivalence class notation to a sign relation R, the definitions and examples considered so far only cover the case where the connotative component RSI is a total equivalence relation on the whole syntactic domain S. The next job is to adapt this usage to PERs.
In applying the equivalence class notation to a sign relation R, the definitions and examples considered so far only cover the case where the connotative component RSI is a total equivalence relation on the whole syntactic domain S. The next job is to adapt this usage to PERs.
