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Notice that this is not the same thing as being ''semiotically equivalent'', in the sense of belonging to a single ''semiotic equivalence class'' (SEC), falling into the same part of a ''semiotic partition'' (SEP), or having a ''semiotic equation'' (SEQ) between them.  It is only when very felicitous conditions obtain, establishing a concord between the denotative and the connotative components of a sign relation, that these two ideas coalesce.
 
Notice that this is not the same thing as being ''semiotically equivalent'', in the sense of belonging to a single ''semiotic equivalence class'' (SEC), falling into the same part of a ''semiotic partition'' (SEP), or having a ''semiotic equation'' (SEQ) between them.  It is only when very felicitous conditions obtain, establishing a concord between the denotative and the connotative components of a sign relation, that these two ideas coalesce.
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In general, there is no necessity that the equiference of signs, that is, their denotational equivalence or their referential equivalence, induces the same equivalence relation on the syntactic domain as that defined by their semiotic equivalence, even though this state of accord seems like an especially desirable situation.  This makes it necessary to find a distinctive nomenclature for these structures, for which I adopt the term ''denotative equivalence relations'' (DERs).  In their train they bring the allied structures of ''denotative equivalence classes'' (DECs) and ''denotative partitions'' (DEPs), while the corresponding statements of ''denotative equations'' (DEQs) are expressible in the form <math>x ~\overset{L}{=}~ y.</math>
    
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In general, there is no necessity that the equiference of signs, that is, their denotational equivalence or their referential equivalence, induces the same equivalence relation on the syntactic domain as that defined by their semiotic equivalence, even though this state of accord seems like an especially desirable situation.  This makes it necessary to find a distinctive nomenclature for these structures, for which I adopt the term "denotative equivalence relations" (DER's).  In their train they bring the allied structures of "denotative equivalence classes" (DEC's) and "denotative partitions" (DEP's), while the corresponding statements of "denotative equations" (DEQ's) are expressible in the form "x =R y".
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The uses of the equal sign for denoting equations or equivalences are recalled and extended in the following ways:
 
The uses of the equal sign for denoting equations or equivalences are recalled and extended in the following ways:
  
Jon Awbrey
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