12,080
edits
Changes
MyWikiBiz, Author Your Legacy — Wednesday April 09, 2025
Jump to navigationJump to searchDirectory: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
Line 1,585:
Line 1,585:
<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.
−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>
Line 1,619:
Line 1,619:
<br>
<br>
−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
Line 1,645:
Line 1,647:
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}.
