Re: SUO: Re: Examples! Examples! Examples!
Jon,
Didn't the "Ontology" mailing list get created largely to give you a
soapbox for your idiosyncratic retelling of completely standard and
well-understood results in mathematical logic? I thought creation of
that list was a good compromise. Why have you shifted your Basically
Endless Series Of Tutorials (BESOT) back to this list?
Chris Menzel
On Tue, Jun 10, 2003 at 10:40:46PM -0400, Jon Awbrey wrote:
>
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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.
>
> o-----------------------------o-------------------o-----------------------------o
> | Language 1 | Object Domain | Language 2 |
> o-----------------------------o-------------------o-----------------------------o
> | |
> | 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 |
> | |
> o-------------------------------------------------------------------------------o
> 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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