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EEE. Note 8
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Example 1. John Sowa's "Top Level Categories" (cont.)
Just for orientation, Figure 4 sketches the hi and lo points of
the "unreduced extensional lattices" for the TLC example, namely,
the lattice of subsets Pow(A) and the lattice of propositions A^.
I have written the coordinate 25-tuples of A as commafree strings.
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| | |
| A | (( )) |
| | |
| <|a_1, ..., a_25|> | 1 : A -> B |
| | |
| {<0000000000000000000000000>, | {<0000000000000000000000000> ~> 1, |
| <0000000000000000000000001>, | <0000000000000000000000001> ~> 1, |
| ... | ... |
| <1111111111111111111111110>, | <1111111111111111111111110> ~> 1, |
| <1111111111111111111111111>} | <1111111111111111111111111> ~> 1} |
| | |
| @ | @ |
| / \ | / \ |
| / \ | / \ |
| / \ | / \ |
| / \ | / \ |
| / 2^A \ | / A^ \ |
| / \ | / \ |
| / Pow (A) \ | / (A --> B) \ |
| ... ... | ... ... |
| \ 2^(2^25) / | \ 2^(2^25) / |
| \ subsets / | \ functions / |
| \ / | \ / |
| \ / | \ / |
| \ / | \ / |
| \ / | \ / |
| \ / | \ / |
| @ | @ |
| | |
| { } | ( ) |
| | |
| | 0 : A -> B |
| | |
| | {<0000000000000000000000000> ~> 0, |
| | <0000000000000000000000001> ~> 0, |
| | ... |
| | <1111111111111111111111110> ~> 0, |
| | <1111111111111111111111111> ~> 0} |
| | |
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Figure 4. Lattice of Subsets Pow(A) and Lattice of Propositions A^ = (A -> B)
That should be enough, lattice hope, about the spaces of "interpretations",
truth value assignments, or propositional models for a while. Next we need
to establish a base camp for the assault on the summit of TLC's theory space.
Jon Awbrey
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