Changes

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>

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>

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:

1. If E is an arbitrary equivalence relation,

# If <math>E\!</math> is an arbitrary equivalence relation, then the equation <math>x =_E y\!</math> means that <math>(x, y) \in E.</math>

then the equation "x =E y" means that <x, y> C E.

# If <math>L\!</math> is a sign relation such that <math>L_{SI}\!</math> is a SER on <math>S = I,\!</math> then the semiotic equation <math>x =_L y\!</math> means that <math>(x, y) \in L_{SI}.</math>

 

# If <math>L\!</math> is a sign relation such that <math>F\!</math> is its DER on <math>S = I,\!</math> then the denotative equation <math>x ~\overset{L}{=}~ y</math> means that <math>(x, y) \in F,</math> in other words, that <math>\operatorname{Den}(L, x) = \operatorname{Den}(L, y).</math>

2. If R is a sign relation such that RSI is a SER on S = I,

then the semiotic equation "x =R y" means that <x, y> C RSI.

 

3. If R is a sign relation such that F is its DER on S = I,

then the denotative equation "x =R y" means that <x, y> C F,

in other words, that Den(R, x) = Den(R, y).

The uses of square brackets for denoting equivalence classes are recalled and extended in the following ways:

The uses of square brackets for denoting equivalence classes are recalled and extended in the following ways: