ONT Re: Apposite Purposes Of Logical Languages Objectified (APOLLO)
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We are in dire need of a few concrete examples
of how a propositional calculus can be used to
talk about abstract propositions and thence be
applied to describe aspects of concrete spaces.
Let us go back to a universe marked by two logical features:
| o-----------------------------------o
| | X |
| | o-----------------------o |
| | | U | |
| | | o | |
| | | / \ | |
| | | / \ | |
| | | / V \ | |
| | | / \ | |
| | | / \ | |
| | | / \ | |
| | o-----------------------o |
| | / \ |
| | o o |
| | \ / |
| | \ / |
| | \ / |
| | \ / |
| | \ / |
| | \ / |
| | \ / |
| | \ / |
| | o |
| | |
| o-----------------------------------o
|
| k = 2, number of subspaces = 2^2^2 = 16
Here, our logical lattice has 16 elements,
that may be interpreted as corresponding to
the 16 different boolean functions, that is,
the 16 functions of type B^2 -> B, and that
further correspond to the 16 different ways
of shading a venn diagram on two "circles".
As I said, we may regard this logical lattice as an abstract object,
one that we may take, as a whole and in its parts, as being denoted
by the particular calculus that we may designate as "ZOL(2)", the
"propositional calculus on two basic propositions". As needed,
we may also specify a propositional calculus "ZOL(u, v)" that
is based on the specific propositions u and v.
Being Friday, my brain is running at the speed of molasses,
and so I will need to take a bit of a break before I try
to introduce the specific examples of formal calculi
that I plan to use in the rest of this discussion.
Jon Awbrey
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