[Inquiry] Re: Logic Of Relatives

[Jon Awbrey]

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LOR.  Note 10

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Peirce in 1870 is five years down the road from the Peirce of 1865-1866
who lectured extensively on the role of sign relations in the logic of
scientific inquiry, articulating their involvement in the three types
of inference, and inventing the concept of "information" to explain
what it is that signs convey in the process.  By this time, then,
the semiotic or sign relational approach to logic is so implicit
in his way of working that he does not always take the trouble
to point out its distinctive features at each and every turn.
So let's take a moment to draw out a few of these characters.

Sign relations, like any non-trivial brand of 3-adic relations,
can become overwhelming to think about once the cardinality of
the object, sign, and interpretant domains or the complexity
of the relation itself ascends beyond the simplest examples.
Furthermore, most of the strategies that we would normally
use to control the complexity, like neglecting one of the
domains, in effect, projecting the 3-adic sign relation
onto one of its 2-adic faces, or focusing on a single
ordered triple of the form <o, s, i> at a time, can
result in our receiving a distorted impression of
the sign relation's true nature and structure.

I find that it helps me to draw, or at least to imagine drawing,
diagrams of the following form, where I can keep tabs on what's
an object, what's a sign, and what's an interpretant sign, for
a selected set of sign-relational triples.

Here is how I would picture Peirce's example of equivalent terms:
v = p, where "v" denotes the Vice-President of the United States,
and "p" denotes the President of the Senate of the United States.

o-----------------------------o-----------------------------o
|  Objective Framework (OF)   | Interpretive Framework (IF) |
o-----------------------------o-----------------------------o
|           Objects           |            Signs            |
o-----------------------------o-----------------------------o
|                                                           |
|                                 o "v"                     |
|                                /                          |
|                               /                           |
|                              /                            |
|           o ... o-----------@                             |
|                              \                            |
|                               \                           |
|                                \                          |
|                                 o "p"                     |
|                                                           |
o-----------------------------o-----------------------------o

Depending on whether we interpret the terms "v" and "p" as applying to
persons who hold these offices at one particular time or as applying to
all those persons who have held these offices over an extended period of
history, their denotations may be either singular of plural, respectively.

As a shortcut technique for indicating general denotations or plural referents,
I will use the "elliptic convention" that represents these by means of figures
like "o o o" or "o ... o", placed at the object ends of sign relational triads.

For a more complex example, here is how I would picture Peirce's example
of an equivalence between terms that comes about by applying one of the
distributive laws, for relative multiplication over absolute summation.

o-----------------------------o-----------------------------o
|  Objective Framework (OF)   | Interpretive Framework (IF) |
o-----------------------------o-----------------------------o
|           Objects           |            Signs            |
o-----------------------------o-----------------------------o
|                                                           |
|                                 o "'s'(m +, w)"           |
|                                /                          |
|                               /                           |
|                              /                            |
|           o ... o-----------@                             |
|                              \                            |
|                               \                           |
|                                \                          |
|                                 o "'s'm +, 's'w"          |
|                                                           |
o-----------------------------o-----------------------------o

Jon Awbrey

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