Cf: Animated Logical Graphs • 16
https://inquiryintoinquiry.com/2019/07/07/animated-logical-graphs-16/
All,
In lieu of a field study requirement for my bachelor’s degree I spent
a couple years in a host of state and university libraries reading
everything I could find by and about Peirce, poring most memorably
through the reels of microfilmed Peirce manuscripts Michigan State
had at the time, all in trying to track down some hint of a clue to
a puzzling passage in Peirce’s “Simplest Mathematics”, most acutely
coming to a head with that bizarre line of type at CP 4.306, which
the editors of the Collected Papers, no doubt compromised by the
typographer’s resistance to cutting new symbols, transmogrified
into a script more cryptic than even the manuscript’s original
hieroglyphic.
I found one key to the mystery in Peirce’s use of operator variables,
which he and his students Christine Ladd-Franklin and O.H. Mitchell
explored in 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 ∨ ¬B
¬(A ∨ B) = ¬A ∧ ¬B
We could capture the common form of these two rules in a single formula
by taking “O₁” and “O₂” as variable names ranging over a set of logical
operators, then asking what substitutions for O₁ and O₂ would satisfy
the following equation:
¬(A O₁ B) = ¬A O₂ ¬B
We already know two solutions to this operator equation, namely,
(O₁, O₂) = (∧, ∨) and (O₁, O₂) = (∨, ∧). Wouldn’t it be just
like Peirce to ask if there are others?
Having broached the subject of logical operator variables,
I will leave it for now in the same way Peirce himself did:
<QUOTE CSP:>
I shall not further enlarge upon this matter at this point,
although the conception mentioned opens a wide field; because
it cannot be set in its proper light without overstepping the
limits of dichotomic mathematics. (Collected Papers, CP 4.306).
</QUOTE>
Further exploration of operator variables and operator invariants
treads on grounds traditionally known as “second intentional logic”
and “opens a wide field”, as Peirce says. For now, however, I will
tend to that corner of the field where our garden variety logical
graphs grow, observing the ways operative variations and operative
themes naturally develop on those grounds.
Regards,
Jon