SUO: Re: Transformations Of Discourse

[Jon Awbrey <jawbrey@oakland.edu>]

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[ Notation:  Terminological & Interminological --
|
| I will note here only the most pressing bits of notation
| that we need in order to read the present note, and save
| the running accumulation for a tabular form another time.
|
| Given $X$ = {x<1>, ..., x<n>} as an alphabet of logical features:
|
| Define X<j> = <(x<j>), x<j>> = <~x<j>, x<j>> as the "coordinate dimension" j,
| an ordered pair that makes up the oriented space of abstract type B = {0, 1}.
|
| Define X  =  <$X$>
|           =  {<x<1>, ..., x<n>> : x<j> in X<j>}
|           =  X<1> x ... x X<n>
|           =  Prod<j> X<j>,
|
| the set of interpretations, cells, points, vectors
| in the universe of discourse that is based on $X$,
| a space that is of the abstract type B^n.
|
| Define X*  =  {f : linear X -> B}
|
| as the space of "linear propositions" on X,
| also called the "algebraic dual space" of X,
| a space that is also of the abstract type B^n.
|
| Define X^  =  {X -> B}  =  {f : X -> B}
|
| as the space of "boolean functions" on X,
| also called the "truth-valued functions" on X,
| and loosely described as the "propositions" on X,
| a space that enjoys the abstract type of B^n -> B.
|
| Define X°  =  [$X$]
|            =  [x<1>, ..., x<n>]
|            =  <X, X^>
|            =  {X +-> B}
|            =  <X, {X -> B}>
|
| as the "universe of discourse" that is based on the features in $X$,
| a space of the "complex type" <B^n, B^n -> B> = <B^n +-> B> = [B^n].
]

Sorry, but that pained me as much as did you --
I need to take a break and start again later --

Jon Awbrey

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