Animated Logical Graphs • 16

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:

$\begin{array}{lll} \lnot (A \land B) & = & \lnot A \lor \lnot B \\[4px] \lnot (A \lor B) & = & \lnot A \land \lnot B \end{array}$

We could capture the common form of these two rules in a single formula by taking ${}^{\backprime\backprime} O_1 {}^{\prime\prime}$ and ${}^{\backprime\backprime} O_2 {}^{\prime\prime}$ as variable names ranging over a set of logical operators, and then by asking what substitutions for $O_1$ and $O_2$ would satisfy the following equation:

$\begin{array}{lll} \lnot (A ~O_1~ B) & = & \lnot A ~O_2~ \lnot B \end{array}$

We already know two solutions to this operator equation, namely, $(O_1, O_2) = (\land, \lor)$ and $(O_1, O_2) = (\lor, \land).$  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:

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).

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.

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