SUO: Re: Examples! Examples! Examples!
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EEE. Note 10
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Logical Equivalence Classes
To flesh out the sketches that I drew last time,
intended to suggest how quotient structures on
spaces of signs can mirror the forms of spaces
of objects, Figure 7 fills in a sample of the
concrete details for two particular languages
for zeroth order logic, Language 1 being one
of the more familiar syntaxes and Language 2
being the Cactus Language earlier described.
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| Language 1 | Object Domain | Language 2 |
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| |
| o----------o o----------o |
| /| "T" |\ 1 /| " " |\ |
| / | "x => x" |~\~~~~~~~~~~~~~~~~o~~~~~~~~~~~~~~~~/~| "(x(x))" | \ |
| / | ... | \ / \ / | ... | \ |
| / o----------o \ / \ / o----------o \ |
| / \ / \ / \ |
| / o----------o / \ / o----------o |
| / | "x" | / \ x / | "x" | |
| / | "T => x" |~~~~~~/~~~~~~~~~~~o~~~~/~~~~~~~~~~~~~| "((x))" | |
| / | | / / / | ... | |
| o----------o o----------o / / o----------o o----------o |
| | "~x" | / / / | "(x)" | / |
| | "x => F" |~~~~~~~~~~~~~/~~~~o~~~~~~~~~~~/~~~~~~| "(x(()))"| / |
| | ... | / (x) \ / | | / |
| o----------o / \ / o----------o / |
| \ / \ / \ / |
| \ o----------o / \ / \ o----------o / |
| \ | "F" | / \ / \ | "()" | / |
| \ | "x & ~x" |~/~~~~~~~~~~~~~~~~o~~~~~~~~~~~~~~~~\~| "x(x)" | / |
| \| ... |/ 0 \| ... |/ |
| o----------o o----------o |
| |
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Figure 7. Lattice of Objects Inducing a Diversity of Sign Partitions
In this illustration, the object structure is the lattice of four propositions
(X^, =>) that inheres in the 1-dimensional universe of discourse X% = [x], and
the quotient structures on the syntactic spaces are induced by the equivalence
relation of logical equivalence (<=>). Whether these amount to REC's or SEC's
is a little bit of a chicken or the egg question -- for the time being it will
do to call them "logical equivalence classes" (LEC's). One of the things that
we ought to observe right off is that the correspondence from signs to objects
is infinity-to-one in its cardinality.
Jon Awbrey
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