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Directory:Jon Awbrey/Papers/Syntactic Transformations (view source)
Revision as of 16:10, 4 February 2009
, 16:10, 4 February 2009→Derived Equivalence Relations: mathematical markup + editing note
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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:
'''Editing Note.''' Need a discussion of converse relations here. Perhaps it would work to introduce the operators that Peirce used for the converse of a dyadic relative <math>\ell,</math> namely, <math>K\ell ~=~ k\!\cdot\!\ell ~=~ \breve\ell.</math>
The dyadic relation <math>L_{IS}\!</math> that is the converse of the connotative relation <math>L_{SI}\!</math> can be defined directly in the following fashion:
{| align="center" cellpadding="8" width="90%"
{| 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>
| <math>\overset{\smile}{\operatorname{Con}(L)} ~=~ L_{IS} ~=~ \{ (i, s) \in I \times S ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \}.</math>
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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.
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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}.
Recall the definition of <math>\operatorname{Den} (L),</math> the denotative component of <math>L,\!</math> in the following form:
{| align="center" cellpadding="8" width="90%"
| <math>\operatorname{Den} (L) ~=~ L_{OS} ~=~ \{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \}.</math>
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Equivalent expressions for this concept are recorded in Definition 10.
Equivalent expressions for this concept are recorded in Definition 10.
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Definition 10
Definition 10