Re: Information = Comprehension x Extension -- Discussion

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ICE.  Discussion Note 24

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To afford the facility but economize the effort of drawing
lots of pictures for the description of logical situations,
it is very useful to have a shorthand notation for calling
to mind the likes of pictures that we have made the effort
of drawing on previous occasions.

To give it a name, I will refer to the universe of discourse
that I sketched last time as [Blue, Color, Red] or [b, c, r].

The universe of discourse [b, c, r] is of two "layers" compact:

   1.  There is a set of "coordinate points"
       that I will notate as X = <|b, c, r|>.

   2.  There is a set of "predicates" or "propositions" p : X -> !B!
       that I will notate as X^ = (X -> !B!) = (<|b, c, r|> -> !B!).

In the language that accompanies venn diagrams, the coordinate points indicate
the "cells" of the universe, that is, the smallest bounded regions of the space,
while the predicates indicate arbitrary areas (also:  domains, regions, subsets)
of the space, insofar as they can be bounded by arcs of the basic set of circles
{Blue, Color, Red}.

When I want to emphasize both the points X and
the propositions X^ = (X -> !B!) of a universe
of discourse, I use a scheme of notation like
X% = [X] = (X, X^) = (X, X -> !B!), and so on.

In the universe of discourse of the present example, we have
X% = [b, c, r] = (<|b, c, r|>, <|b, c, r|> -> !B!), and this
is all just to remind us of the set of cells <|b, c, r|> and
the set of regions that are indicated by the set of possible
2-tone "shadings" p : <|b, c, r|> -> !B!.

Phew!  Hopefully that will stake down enough literal minded
notation in this cerebral hemisphere that we can get back
to the more picturesque hemisphere the next time around.

Jon Awbrey

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