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=====1.3.12.2. Derived Equivalence Relations=====
=====1.3.12.2. Derived Equivalence Relations=====
One seeks a method of general application for approaching the individual sign relation, a way to select an aspect of its form, to analyze it with regard to its intrinsic structure, and to classify it in comparison with other sign relations. With respect to a particular sign relation, one approach that presents itself is to examine the relation between signs and interpretants that is given directly by its connotative component and to compare it with the various forms of derived, indirect, mediate, or peripheral relationships that can be found to exist among signs and interpretants by way of secondary considerations or subsequent studies. Of especial interest are the relationships among signs and interpretants that can be obtained by working through the collections of objects that they commonly or severally denote.
One seeks a method of general application for approaching the individual sign relation, a way to select an aspect of its form, to analyze it with regard to its intrinsic structure, and to classify it in comparison with other sign relations. With respect to a particular sign relation, one approach that presents itself is to examine the relation between signs and interpretants that is given directly by its connotative component and to compare it with the various forms of derived, indirect, mediate, or peripheral relationships that can be found to exist among signs and interpretants by way of secondary considerations or subsequent studies. Of especial interest are the relationships among signs and interpretants that can be obtained by working through the collections of objects that they commonly or severally denote.
A different way of seeing that two sets are equal, or of seeing whether two sets are equal, is based on the following observation:
A different way of seeing that two sets are equal, or of seeing whether two sets are equal, is based on the following observation:
<pre>
Two sets are equal as sets
Two sets are equal as sets
<=> the values of these functions are equal on all domain elements.
<=> the values of these functions are equal on all domain elements.
</pre>
It is important to notice the hidden quantifier, of a universal kind, that lurks in all three equivalent statements but is only revealed in the last.
It is important to notice the hidden quantifier, of a universal kind, that lurks in all three equivalent statements but is only revealed in the last.
Recall the definition of Con(R), the connotative component of R, in the following form:
Recall the definition of Con(R), the connotative component of R, in the following form:
Con(R) = RSI = {< s, i> C SxI : <o, s, i> C R for some o C O}.
: Con(R) = RSI = {< s, i> C SxI : <o, s, i> C R for some o C O}.
Equivalent expressions for this concept are recorded in Definition 8.
Equivalent expressions for this concept are recorded in Definition 8.
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Definition 8
Definition 8
D8e. {< s, i> C SxI : <o, s, i> C R for some o C O}
D8e. {< s, i> C SxI : <o, s, i> C R for some o C O}
</pre>
The dyadic relation RIS that constitutes the converse of the connotative relation RSI can be defined directly in the following fashion:
The dyadic relation RIS that constitutes the converse of the connotative relation RSI can be defined directly in the following fashion:
Con(R)^ = RIS = {< i, s> C IxS : <o, s, i> C R for some o C O}.
: Con(R)^ = RIS = {< i, s> C IxS : <o, s, i> C R for some o C O}.
A few of the many different expressions for this concept are recorded in Definition 9.
A few of the many different expressions for this concept are recorded in Definition 9.
<pre>
Definition 9
Definition 9
D9g. {< i, s> C IxS : <o, s, i> C R for some o C O}
D9g. {< i, s> C IxS : <o, s, i> C R for some o C O}
</pre>
Recall the definition of Den(R), the denotative component of R, in the following form:
Recall the definition of Den(R), the denotative component of R, in the following form:
Den(R) = ROS = {<o, s> C OxS : <o, s, i> C R for some i C I}.
: Den(R) = ROS = {<o, s> C OxS : <o, s, i> C R for some i C I}.
Equivalent expressions for this concept are recorded in Definition 10.
Equivalent expressions for this concept are recorded in Definition 10.
<pre>
Definition 10
Definition 10
D10e. {<o, s> C OxS : <o, s, i> C R for some i C I}
D10e. {<o, s> C OxS : <o, s, i> C R for some i C I}
</pre>
The dyadic relation RSO that constitutes the converse of the denotative relation ROS can be defined directly in the following fashion:
The dyadic relation RSO that constitutes the converse of the denotative relation ROS can be defined directly in the following fashion:
Den(R)^ = RSO = {< s, o> C SxO : <o, s, i> C R for some i C I}.
: Den(R)^ = RSO = {< s, o> C SxO : <o, s, i> C R for some i C I}.
A few of the many different expressions for this concept are recorded in Definition 11.
A few of the many different expressions for this concept are recorded in Definition 11.
<pre>
Definition 11
Definition 11
D11g. {< s, o> C SxO : <o, s, i> C R for some i C I}
D11g. {< s, o> C SxO : <o, s, i> C R for some i C I}
</pre>
The "denotation of x in R", written "Den(R, x)", is defined as follows:
The "denotation of x in R", written "Den(R, x)", is defined as follows:
Den(R, x) = {o C O : <o, x> C Den(R)}.
: Den(R, x) = {o C O : <o, x> C Den(R)}.
In other words:
In other words:
Den(R, x) = {o C O : <o, x, i> C R for some i C I}.
: Den(R, x) = {o C O : <o, x, i> C R for some i C I}.
Equivalent expressions for this concept are recorded in Definition 12.
Equivalent expressions for this concept are recorded in Definition 12.
<pre>
Definition 12
Definition 12
D12h. {o C O : <o, x, i> C R for some i C I}
D12h. {o C O : <o, x, i> C R for some i C I}
</pre>
Signs are "equiferent" if they refer to all and only the same objects, that is, if they have exactly the same denotations. In other language for the same relation, signs are said to be "denotatively equivalent" or "referentially equivalent", but it is probably best to check whether the extension of this concept over the syntactic domain is really a genuine equivalence relation before jumpimg to the conclusions that are implied by these latter terms.
Signs are "equiferent" if they refer to all and only the same objects, that is, if they have exactly the same denotations. In other language for the same relation, signs are said to be "denotatively equivalent" or "referentially equivalent", but it is probably best to check whether the extension of this concept over the syntactic domain is really a genuine equivalence relation before jumpimg to the conclusions that are implied by these latter terms.
For each sign relation R, this yields a binary relation Der(R) c SxI that is defined as follows:
For each sign relation R, this yields a binary relation Der(R) c SxI that is defined as follows:
Der(R) = DerR = {<x, y> C SxI : Den(R, x) = Den(R, y)}.
: Der(R) = DerR = {<x, y> C SxI : Den(R, x) = Den(R, y)}.
These definitions and notations are recorded in the following display.
These definitions and notations are recorded in the following display.
<pre>
Definition 13
Definition 13
D13d. {<x,y> C SxI : Den(R, x) = Den(R, y)}
D13d. {<x,y> C SxI : Den(R, x) = Den(R, y)}
</pre>
The relation Der(R) is defined and the notation "x =R y" is meaningful in every situation where Den(-,-) makes sense, but it remains to check whether this relation enjoys the properties of an equivalence relation.
The relation Der(R) is defined and the notation "x =R y" is meaningful in every situation where Den(-,-) makes sense, but it remains to check whether this relation enjoys the properties of an equivalence relation.
# Reflexive property. Is it true that x =R x for every x C S = I? By definition, x =R x if and only if Den(R, x) = Den(R, x). Thus, the reflexive property holds in any setting where the denotations Den(R, x) are defined for all signs x in the syntactic domain of R.
# Symmetric property. Does x =R y => y =R x for all x, y C S? In effect, does Den(R, x) = Den(R, y) imply Den(R, y) = Den(R, x) for all signs x and y in the syntactic domain S? Yes, so long as the sets Den(R, x) and Den(R, y) are well-defined, a fact which is already being assumed.
# Transitive property. Does x =R y & y =R z => x =R z for all x, y, z C S? To belabor the point, does Den(R, x) = Den(R, y) and Den(R, y) = Den(R, z) imply Den(R, x) = Den(R, z) for all x, y, z in S? Yes, again, under the stated conditions.
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.
By applying the form of Fact 1 to the special case where X = Den(R, x) and Y = Den(R, y), one obtains the following facts.
By applying the form of Fact 1 to the special case where X = Den(R, x) and Y = Den(R, y), one obtains the following facts.
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Fact 2.1
Fact 2.1
} :D12a
} :D12a
</pre>
<pre>
Fact 2.2
Fact 2.2
} :Log
} :Log
F2.2e. {DerR} = {<x, y, v> C SxIxB :
F2.2e. {DerR} = {<x, y, v> C SxIxB :
} :Log
} :Log
F2.2f. {DerR} = {<x, y, v> C SxIxB :
F2.2f. {DerR} = {<x, y, v> C SxIxB :
} :$
} :$
</pre>
<pre>
Fact 2.3
Fact 2.3
