ONT Re: Zeroth Order Ontology
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ZOO 17
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In: ZOO. http://suo.ieee.org/ontology/thrd1.html#05155
Hx: ZOO 14. http://suo.ieee.org/ontology/msg05195.html
Hx: Errata. http://suo.ieee.org/ontology/msg05196.html
Hx: ZOO 15. http://suo.ieee.org/ontology/msg05197.html
A month to the day having passed since I last left off on this line,
let me go back to the last couple of notes and refresh my memory as
to where I was in this babbling brook of intermittent consciousness.
Here is the corrected synopsis of the last two notes in this series:
Last time we formed the notion of an "epiformula", or an "epiform" for short,
that is nothing but a more or less iconic or schematic sign that "denotes" or
"indicates" all the members in a set of purely arithmetic logical expressions.
Loosely speaking, it "names" a set of formulas in the primary arithmetic L_2.
For example, we contemplated the epiform e_3 that is shown as follows:
o-----------------------------------------------------------o
| Epiform e_3 |
o-----------------------------------------------------------o
| |
| x o o x x o x |
| | | | |
| x o o x x o x |
| \ / | |
| o---------o |
| | |
| | |
| @ |
| |
o-----------------------------------------------------------o
| ( ( x ( x )) (( x ) x ) ( ( x ( x x ) x ) )) |
o-----------------------------------------------------------o
Taken as a sign, e_3 denotes exactly the two formulas e_1 and e_2.
o-----------------------------------------------------------o
| Expression e_1 |
o-----------------------------------------------------------o
| |
| o o o |
| | | | |
| o o o |
| \ / | |
| o---------o |
| | |
| | |
| @ |
| |
o-----------------------------------------------------------o
| ( (()) (()) ( (()) )) |
o-----------------------------------------------------------o
o-----------------------------------------------------------o
| Expression e_2 |
o-----------------------------------------------------------o
| |
| o o o o |
| \ / \ / |
| o o o o o o o |
| \| |/ \|/ |
| o o o |
| \ / | |
| o---------o |
| | |
| | |
| @ |
| |
o-----------------------------------------------------------o
| ( (()(()))((())()) ( (()(()())()) )) |
o-----------------------------------------------------------o
Consider the following way of drawing what we
are doing when we make use of such an epiform:
o-----------------------------o-----------------------------o
| Object Domain L_2 | Sign Domain L_3 |
o-----------------------------o-----------------------------o
| |
| o o |
| |
| o o |
| |
| e_1 o o |
| \ |
| o \ o |
| \ |
| o \ o |
| \ |
| o \ o |
| \ |
| o o e_3 |
| / |
| o / o |
| / |
| o / o |
| / |
| o / o |
| / |
| e_2 o o |
| |
| o o |
| |
| o o |
| |
o-----------------------------------------------------------o
Here, the expressions e_1 and e_2, as points in the space L_2,
and the epiform e_3, as a point in the space L_3 of comparable
epiforms, are represented as nodes in the appropriate parts of
a "bigraph", or bipartite graph, that is partially sketched in
the Figure. The lines in the middle indicate the piece of the
2-adic denotation relation that affects the points in question.
Thus, we say that e_3 denotes or indicates all of the elements
in the set {e_1, e_2}, and only those, or else we can say that
the extension of the sign e_3 is the set {e_1, e_2}. Strictly
speaking, we must not say that e_3 denotes the set {e_1, e_2},
as that is a different type of situation entirely, but we may
often say in loose speech that e_3 "names" the set {e_1, e_3}.
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,
the 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
refer to as B, or as !B! in those contexts where it is 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:
o-----------------------------------------------------------o
| |
| e_3 |
| |
| o o o o o o o o o Epiforms L_3 |
| / \ |
| / \ |
| / \ |
| / \ |
| o e_1 o e_2 o o o o o Formulae L_2 |
| \ / |
| \ / |
| \ / |
| \ / |
| o o Objects L_0 |
| |
o-----------------------------------------------------------o
| |
| o |
| | |
| @ @ B |
| |
o-----------------------------------------------------------o
This is just another way of picturing the fact already noted that:
[e_1] = [e_3 [o/x]] = [!e!] = [e_3 [|/x]] = [e_2].
So let us now try to push on from there.
Jon Awbrey
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http://www.cs.bsu.edu/homepages/mighty/history.html
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