Changes

<p>'''Symmetric property.'''</p>

<p>'''Symmetric property.'''</p>

<p>Does <math>x ~\overset{L}{=}~ y ~\Rightarrow~ y ~\overset{L}{=}~ x</math> for all <math>x, y \in S</math>?</p>

<p>Does <math>x ~\overset{L}{=}~ y</math> imply <math>y ~\overset{L}{=}~ x</math> for all <math>x, y \in S</math>?</p>

<p>In effect, does <math>\operatorname{Den}(L, x) = \operatorname{Den}(L, y)</math> imply <math>\operatorname{Den}(L, y) = \operatorname{Den}(L, x)</math> for all signs <math>x\!</math> and <math>y\!</math> in the syntactic domain <math>S\!</math>?</p>

<p>In effect, does <math>\operatorname{Den}(L, x) = \operatorname{Den}(L, y)</math> imply <math>\operatorname{Den}(L, y) = \operatorname{Den}(L, x)</math> for all signs <math>x\!</math> and <math>y\!</math> in the syntactic domain <math>S\!</math>?</p>

It should be clear at this point that any question about the equiference of signs reduces to a question about the equality of sets, specifically, the sets that are indexed by these signs.  As a result, so long as these sets are well-defined, the issue of whether equiference relations induce equivalence relations on their syntactic domains is almost as trivial as it initially appears.

It should be clear at this point that any question about the equiference of signs reduces to a question about the equality of sets, specifically, the sets that are indexed by these signs.  As a result, so long as these sets are well-defined, the issue of whether equiference relations induce equivalence relations on their syntactic domains is almost as trivial as it initially appears.

Taken in its set-theoretic extension, a relation of equiference induces a ''denotative equivalence relation'' (DER) on its syntactic domain <math>S = I.\!</math> This leads to the formation of ''denotative equivalence classes'' (DECs), ''denotative partitions'' (DEPs), and ''denotative equations'' (DEQs) on the syntactic domain.  But what does it mean for signs to be equiferent?

Taken in its set-theoretic extension, a relation of equiference induces a "denotative equivalence relation" (DER) on its syntactic domain S = I.  This leads to the formation of "denotative equivalence classes" (DEC's), "denotative partitions" (DEP's), and "denotative equations" (DEQ's) on the syntactic domain.  But what does it mean for signs to be equiferent?

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.

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".

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".