Re: Zeroth Order Ontology
Jon Awbrey <[email protected]>
| Newsgroups | gmane.comp.inquiry,gmane.comp.misc.ontology.general |
|---|---|
| Message-ID | <[email protected]> |
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ZOO. Note 14
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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 copied below.
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_2 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 partiall indicated in
the Figure. The lines in the middle indicate that part 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}.
Jon Awbrey
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