SUO: Re: IFF LOT Glossary

[Jon Awbrey <jawbrey@att.net>]

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

SUO Working Group,

Let me clean up the current workspace a bit before trying to move on.

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 a 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))(code(x)) = code(f(x)) = f(x),
given that the code map acts as the identity on B.

Given the open nature of X, this is the sort of equation
that can hold approximately at best, but I do not know
how to formalize that, or even whether it can be done.

Here are three ways of looking at how the code map operates:

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

           f
  X  o---------->o B
     |           =
     |           =
code |           = code
     |           =
     v           =
 TLC o===========o B
        code(f)

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

Defining Equation:  (code(f))(code(x)) = code(f(x)) = f(x).

To get a language with the power of ZOL over !TLC!, we adjoin the extra marks:

   !M!  =  {m_1, m_2, m_3, m_4}

   m_1  =  " "  =  blank

   m_2  =  "("  =  links

   m_3  =  ","  =  comma

   m_4  =  ")"  =  right

Take the augmented alphabet !A! = !A!(!TLC!) = !M! |_| !TLC!.

There is a formal language L = !C!(!TLC!) c !A!*, a grammar for which is known,
and a parser for which has been written.  Each expression of L can be read to
denote one of the functions f : TLC -> B, conveniently called a "proposition".
For any f : TLC -> B, we also write:  f in TLC^ = (TLC -> B) = {f : TLC -> B}.
And, of course, by virtue of the fact that f : TLC -> B "indicates" a subset
of TLC, we can also read each expression of L as denoting such a subset.

Here are the mugshots of our usual suspects:

      !TLC!
X o---------->o !TLC!(X) = <|!TLC!|> = TLC ~=~ B^25
  |           |
  |           |
f |           | !TLC!(f)
  |           |
  v           v
B o===========o !TLC!(B) = B
      !TLC!

!A! = !M! |_| !TLC!

      *         Pow
!A! ----- !A!* ----- LOL(!A!) = Pow(!A!*)
           |            |
           c           elt
           |            |    Pow
           L   =====    L   ----- Pow(L)
           |            |           |
           c            c          elt
           |            |           |
           T   =====    T   =====   T

I am here experimentally assigning double duty to alphabets,
for example, using an indication of the alphabet !TLC! as an
indication of the associated code map !TLC! : X -> <|!TLC!|>,
and will continue to do so as long as it causes no confusion.
Otherwise, notations like code_!TLC! : (X, X^) -> (TLC, TLC^)
are always available and might become a practical necessity.

As examples of theories about TLC, we spent some time looking at a couple, both
of which can be expressed as singleton theories, that is, as sets that consist
of single expressions from L.  We examined T_!a! = {!a!} and T_!c! = {!c!},
where !a! is shown in Table 20 and !c! is shown in Table 21, opere sitato:

EEE 32.  http://suo.ieee.org/email/msg10094.html

The next task is to get a clear idea of just what it means
to state an axiom !t! or a theory T from a language L that
is given by experience or grammar as a subset of some !A!*.

After that, I would like to explore various notions of closure,
gestalt or logical, that come into the spiel of logical sprach.

Jon Awbrey

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o