ONT Re: Zeroth Order Ontology

[Jon Awbrey <jawbrey@att.net>]

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

ZOO.  Note 15

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

I introduced a few items of terminology from the pragmatic theory
of sign relations to discuss the relationship between expressions
in the primary arithmetic L_2 and expressions in a language L_3,
of so-called "epiforms" over L_2.  In my first strike at this
I treated L_2 as an object domain and L_3 as a sign domain.

But if you think about it, you remember that the expressions in L_2
can be evaluated by means of the arithmetic initials I_1 and I_2 to
yield elements of the language <| L_2 | I_1, I_2 |> = { @, | } that
I will now dub "L_0".  I make the patently obvious observation that
L_0, taken as a set, is isomorphic to the 2-point set that we often
denote as B, or !B! in those contexts where it becomes necessary to
use the plain letter alphabet for other purposes.  I record it thus:

   L_0   =   <| L_2 | I_1, I_2 |>   ~=~   { @ , | }   ~=~   B

The operation of evaluating an expression as equal to a value
is very analogous, and perhaps even related as a special case,
to the operation, not necessarily 1-valued, of interpreting a
sign as denoting an object.

In this light, we may treat L_0 = B as an object domain,
L_2 as a sign domain, and L_3 as a kind of higher order
sign domain.  We have a situation of the following form:

         e_3

o    o    o    o    o    o    o    o    o    Higher Sign Domain L_3
         / \
        /   \
       /     \
      /       \
o   e_1   o   e_2   o    o    o    o    o    Sign Domain L_2
      \       /
       \     /
        \   /
         \ /
          o                   o              Object Domain L_0 = B

          @                   |

This is just another way of picturing the fact already noted that:

   [e_1]  =  [e_3 [o/x]]  =  [!e!]  =  [e_3 [|/x]]  =  [e_2].

Jon Awbrey

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