[Inquiry] Re: Futures Of Logical Graphs

[Jon Awbrey]

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FOLG.  Note 38

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Cybernetics List, Peirce List,

In lieu of a field study requirement for my bachelor's degree program
I spent a couple of years in several state and university libraries
reading everything I could find by and about Peirce, poring most
memorably through the 38 reels of microfilmed manuscripts that
Michigan State had at the time, all in trying to track down
some hint of a clue to some puzzling passages that I read
in Peirce's "Simplest Mathematics", most acutely coming
to a head with that bizzare line of type in CP 4.306,
that the editors of CP, no doubt compromised by the
typographer's resistance to cutting new symbols,
transmogrified into an averse verse that's even
more cryptic than the manuscript hieroglyphic.

The first key to the mystery is discovered in Peirce's
use of "operator variables", that he and his students
Christine Ladd-Franklin and O.H. Mitchell explored in
some depth.  I will shortly discuss this theme as it
affects logical graphs, but it may be useful to give
a shorter and sweeter explanation of how the basic
idea typically arises in common logical practice.

Think of De Morgan's rules:

   ~[A & B]  =  ~A v ~B

   ~[A v B]  =  ~A & ~B

We could capture the common form of these two rules in a single formula
by letting "X" and "Y" be variable names that range over a pre-selected
set of logical operators, and then by asking what X and Y would satsify:

   ~[A X B]  =  ~A Y ~B

We already know two solutions to this "operator equation", specifically,
<X, Y> = <&, v> and <X, Y> = <v, &>.  Wouldn't it be just like Peirce to
ask if there are others?  I will leave that as an exercise for the reader.

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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