ONT Re: Inquiry Driven Systems
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The sign relations P and Q are shown in the next two Tables:
Table 1. Sign Relation P
o---------------o---------------o---------------o
| Object | Sign | Interpretant |
o---------------o---------------o---------------o
| p | s^1 | s^1 |
| p | s^1 | s^2 |
| p | s^2 | s^1 |
| p | s^2 | s^2 |
o---------------o---------------o---------------o
| q | s^4 | s^4 |
| q | s^4 | s^3 |
| q | s^3 | s^4 |
| q | s^3 | s^3 |
o---------------o---------------o---------------o
Table 2. Sign Relation Q
o---------------o---------------o---------------o
| Object | Sign | Interpretant |
o---------------o---------------o---------------o
| p | s^1 | s^1 |
| p | s^1 | s^3 |
| p | s^3 | s^1 |
| p | s^3 | s^3 |
o---------------o---------------o---------------o
| q | s^4 | s^4 |
| q | s^4 | s^2 |
| q | s^2 | s^4 |
| q | s^2 | s^2 |
o---------------o---------------o---------------o
One way to read these 3-adic sign relations is to recognize that
the 2-adic relations formed by the sign and interpretant columns
constitute equivalence relations, and this property allows us to
graph the pertinent facts as shown in the next couple of Figures:
o-----------------------------o-----------------------------o
| Objective Framework | Interpretive Framework |
o-----------------------------o-----------------------------o
| P |
| s^1 |
| · ^ |
| · | |
| · | |
| · v |
| p · · · · s^2 |
| |
| |
| |
| |
| q · · · · s^3 |
| · ^ |
| · | |
| · | |
| · v |
| s^4 |
| |
o-----------------------------------------------------------o
Figure 3. Sign Relation P
o-----------------------------o-----------------------------o
| Objective Framework | Interpretive Framework |
o-----------------------------o-----------------------------o
| Q |
| s^1 |
| · ^ |
| · | |
| · | |
| · v |
| p · · · · s^3 |
| |
| |
| |
| |
| q · · · · s^2 |
| · ^ |
| · | |
| · | |
| · v |
| s^4 |
| |
o-----------------------------------------------------------o
Figure 4. Sign Relation Q
These are the kinds of pictures that we like to see when it comes to
the relationships between objects and signs. A double arrow between
a pair of signs indicates that they are equivalent to each other in
the associated equivalence relation. Since an equivalence relation
on a set partitions it into disjoint exhaustive parts, another way
of describing the situation in each Figure is to say that P and Q,
or their projections on the subspace S x I, each partitions the
set of signs into subsets that correspond to the objects in O.
Except for the annoying circumstance that each sign relation
does this in a different way, leaving us with two partitions
instead of just one, this would be a perfect picture of how
we naturally tend to think that signs and objects ought to
behave in regard to each other.
Jon Awbrey
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