SUO: Re: IFF LOT Glossary

[Jon Awbrey <jawbrey@att.net>]

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Robert & All,

Now that I think of it -- and I don't know how I could have forgotten this,
since Chomsky_2 or maybe Chomsky_3 makes such a big point of distinguishing
E-languages and I-languages -- not just theories but also languages can also
be given empirically or extensionally, that is, by inspection or observation,
and not just by means of the finite grammars that generate them, which grammars
we have to abduce or induce from the finite data, at any rate.  So anytime that
we are working computably or empirically, no matter whether we are thinking of
languages or theories, we have to keep in mind the distinction between our
finite information sets -- whether they be finite samples of languages,
finite grammars, or finite axiom sets -- and the infinite objects that
our finite data informs us about.

But trying to think about languages and theories in general
is making me light-headed, so let me go back to a concrete
and simple example, say, the one about John Sowa's TLC
that I was working out on the (Examples!)^3 thread.

I will use the diagram that I sketched last time,
and try to flesh it out with more exemplary data.

| Let's say we start with a formal language L c !A!*.
| Written another way, L in Pow(!A!*).  Thus, Pow(!A!*)
| is our first candidate for a "lattice of languages" (LOL)
| over the alphabet !A!.  Let's write LOL(!A!) = Pow(!A!*).
|
| A theory T is a just a subset of L, in symbols, T c L.
| Written another way, T in Pow(L) where L in Pow(!A!*).
| We need to know the relation now between T and !A!.
|
|       *         Pow
| !A! ----- !A!* ----- LOL(!A!)
|            |            |
|            c           elt
|            |            |    Pow
|            L   =====    L   ----- Pow(L)
|            |            |           |
|            c            c          elt
|            |            |           |
|            T   =====    T   =====   T

To formalize John Sowa's "Top Level Categories" (TLC), we take
a (descriptive or ontological) alphabet or lexicon of 25 terms,
!TLC! = {a_1, ..., a_25} = {"Abstract", ..., "Structure"}.

We think of the alphabet as providing us with a "codebook", a filter,
or a template, that we use to code arbitrary elements of experience
that come to us from a source or space that we may call, without
too much loss of generativity, "X".  So !TLC! determines a map,
code : X -> TLC = <|!TLC!|> ~=~ B^25.  For any "predicate" f
about the world X, that is, any f : X -> B, the code map
induces a coded predicate code(f) : TLC -> B given by
the equation code(f)(x) = code(f(x)).

        f
X o----------->o B
   \         ^
    \       /
code \     / code(f)
      \   /
       v /
        o
   TLC ~=~ B^25

code(f)(x) = code(f(x))

So code must be some kind of functor?

code : Cats like (X, X -> B) -> Cats like (B^k, B^k -> B)
code : (objects like X) -> (objects like B^k)
code : (arrows like f)  -> (arrows like code(f))

       code
X o----------->o code(X) ~=~ B^k
  |            |
  |            |
f |            | code(f)
  |            |
  v            v
B o============o B

Need to check whether code preserves composition and identity.
Later.  If not, I'll just call it an "operator" and analyze it
in terms of linear maps anyway.

Please check and see if this is making sense to you so far.
Should be pretty standard notions, though maybe not always
made explicit.

Jon Awbrey

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