ONT Re: Apposite Purposes Of Logical Languages Objectified (APOLLO)
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There are many diverting matters that I am almost tempted to raise here,
like what it means for a "community of interpretation" (COI) to believe
that they are sensibly conversing about a "genus of objects" (GOO) when
other COI's believe just as firmly that these objects cannot exist, but
I must hew myself to the sticking points of my humbler purposes here,
which are, just to remind myself, somewhat along the following lines:
| What do we want logical syntax to help us do?
| What is the end to which we build a calculus?
| What is the object of propositional calculus?
I've claimed that the object of propositional calculus is a genus
of abstract objects called "logical lattices" or "partial orders",
ignoring for now the differentia of the species among these types,
so let me try to give an initial idea, in a more concrete fashion,
of what sorts of things we are talking about.
We are going to be concerned with two families of lattices,
one for the "functions" on certain domains and one for the
fundamental "geometry" of subdomains of functional domains.
There is of course a natural relationship between the two.
Here is a visual representation of their common structure
for the first two lattices in the series:
| o
| |
| o
| ( )
|
| k = 0, number of elements = 2^2^0 = 2
| o
| / \
| u o o (u)
| \ /
| o
| ( )
|
| k = 1, number of elements = 2^2^1 = 4
The next lattice has sixteen points, so I will have to leave the
visualization of this as an exercise to the reader's imagination.
Another sort of picture is the ordinary venn diagram,
where we can get a little further in Asciiland by
using it as a basis for indicating the lattice
of subspaces in a universe of discourse:
| o-----------------------------------o
| | X |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| o-----------------------------------o
|
| k = 0, number of subspaces = 2^2^0 = 2
| o-----------------------------------o
| | X |
| | o-----------------------o |
| | | U | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | o-----------------------o |
| | |
| o-----------------------------------o
|
| k = 1, number of subspaces = 2^2^1 = 4
| o-----------------------------------o
| | X |
| | o-----------------------o |
| | | U | |
| | | o | |
| | | / \ | |
| | | / \ | |
| | | / V \ | |
| | | / \ | |
| | | / \ | |
| | | / \ | |
| | o-----------------------o |
| | / \ |
| | o o |
| | \ / |
| | \ / |
| | \ / |
| | \ / |
| | \ / |
| | \ / |
| | \ / |
| | \ / |
| | o |
| | |
| o-----------------------------------o
|
| k = 2, number of subspaces = 2^2^2 = 16
| o-----------------------------------o
| | X |
| | o-----------------------o |
| | | U | |
| | | o o | |
| | | / \ / \ | |
| | | / \ / \ | |
| | | / . \ | |
| | | / / \ \ | |
| | | / / \ \ | |
| | | / / \ \ | |
| | o-----------------------o |
| | / / \ \ |
| | o o o o |
| | \ \ / / |
| | \ V \ / W / |
| | \ \ / / |
| | \ \ / / |
| | \ \ / / |
| | \ . / |
| | \ / \ / |
| | \ / \ / |
| | o o |
| | |
| o-----------------------------------o
|
| k = 0, number of subspaces = 2^2^3 = 256
Next time I shall introduce a couple of different languages
for indicating these logical functions and geometric spaces.
Jon Awbrey
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