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Directory:Jon Awbrey/Papers/Syntactic Transformations (view source)
Revision as of 15:02, 4 February 2009
, 15:02, 4 February 2009→Derived Equivalence Relations: mathematical markup
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.
In making the next set of definitions and in using the corresponding terminology it is taken for granted that all of the references of signs are relative to a particular sign relation <math>L \subseteq O \times S \times I</math> that either remains to be specified or is already understood. Further, I continue to assume that <math>S = I,</math> in which case this set is called the ''syntactic domain'' of <math>L.\!</math>
In making the next set of definitions and in using the corresponding terminology it is taken for granted that all of the references of signs are relative to a particular sign relation <math>L \subseteq O \times S \times I</math> that either remains to be specified or is already understood. Further, I continue to assume that <math>S = I,\!</math> in which case this set is called the ''syntactic domain'' of <math>L.\!</math>
In the following definitions, let <math>L \subseteq O \times S \times I,</math> let <math>S = I,\!</math> and let <math>x, y \in S.\!</math>
In the following definitions, let <math>L \subseteq O \times S \times I,</math> let <math>S = I,\!</math> and let <math>x, y \in S.\!</math>
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The dyadic relation <math>L_{IS}\!</math> that makes up the converse of the connotative relation <math>L_{SI}\!</math> 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}.
{| align="center" cellpadding="8" width="90%"
| <math>\widehat{\operatorname{Con} (L)} ~=~ L_{IS} ~=~ \{ (i, s) \in I \times S ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \}.</math>
|}
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>
<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}.