## Animated Logical Graphs • 22

The step of controlled reflection we just took can be repeated at will, as suggested by the following series of forms:

Written inline, we have the series ${}^{\backprime\backprime} \texttt{(} a \texttt{)} {}^{\prime\prime},$ ${}^{\backprime\backprime} \texttt{(} a \texttt{,} b \texttt{)} {}^{\prime\prime},$ ${}^{\backprime\backprime} \texttt{(} a \texttt{,} b \texttt{,} c \texttt{)} {}^{\prime\prime},$ ${}^{\backprime\backprime} \texttt{(} a \texttt{,} b \texttt{,} c \texttt{,} d \texttt{)} {}^{\prime\prime},$ and so on, whose general form is ${}^{\backprime\backprime} \texttt{(} x_1 \texttt{,} x_2 \texttt{,} \ldots \texttt{,} x_k \texttt{)} {}^{\prime\prime}.$  With this move we have passed beyond the graph-theoretical form of rooted trees to what graph theorists know as rooted cacti.

I will discuss this cactus language and its logical interpretations next.

## Animated Logical Graphs • 21

A funny thing just happened.  Let’s see if we can tell where.  We started with the algebraic expression ${}^{\backprime\backprime} \texttt{(} a \texttt{)} {}^{\prime\prime},$ in which the operand ${}^{\backprime\backprime} a {}^{\prime\prime}$ suggests the contemplated absence or presence of any arithmetic expression or its value, then we contemplated the absence or presence of the operator ${}^{\backprime\backprime} \texttt{(} ~ \texttt{)} {}^{\prime\prime}$ in ${}^{\backprime\backprime} \texttt{(} a \texttt{)} {}^{\prime\prime}$ to be indicated by a cross or a space, respectively, for the value of a newly introduced variable, ${}^{\backprime\backprime} b {}^{\prime\prime},$ placed in a new slot of a newly extended operator form, as suggested by this picture:

What happened here is this.  Our contemplation of a constant operator as being potentially variable gave rise to the contemplation of a newly introduced but otherwise quite ordinary operand variable, albeit in a newly-fashioned formula.  In its interpretation for logic the newly formed operation may be viewed as an extension of ordinary negation, one in which the negation of the first variable is controlled by the value of the second variable.  Thus, we may regard this development as marking a form of controlled reflection, or a form of reflective control.  From here on out we will use the inline syntax ${}^{\backprime\backprime} \texttt{(} a \texttt{,} b \texttt{)} {}^{\prime\prime}$ for the corresponding operation on two variables, whose operation table is given below:

$\begin{array}{|c|c|c|} \hline a & b & \texttt{(} a \texttt{,} b \texttt{)} \\ \hline\hline \texttt{Space} & \texttt{Space} & \texttt{Cross} \\ \texttt{Space} & \texttt{Cross} & \texttt{Space} \\ \texttt{Cross} & \texttt{Space} & \texttt{Space} \\ \texttt{Cross} & \texttt{Cross} & \texttt{Cross} \\ \hline \end{array}$

• The Entitative Interpretation $(\mathrm{En}),$ for which Space = False and Cross = True, calls this operation equivalence.
• The Existential Interpretation $(\mathrm{Ex}),$ for which Space = True and Cross = False, calls this operation distinction.

## Animated Logical Graphs • 20

Another tactic I tried by way of porting operator variables into logical graphs and laws of form was to hollow out a leg of Spencer-Brown’s crosses, gnomons, markers, whatever you wish to call them, as shown below:

The initial idea I had in mind was the same as before, that the operator over $q$ would be counted as absent when $p$ evaluates to a space and present when $p$ evaluates to a cross.

However, much in the same way that operators with a shade of negativity tend to be more generative than the purely positive brand, it turned out more useful to reverse this initial polarity of operation, letting the operator over $q$ be counted as absent when $p$ evaluates to a cross and present when $p$ evaluates to a space.

So that is the convention I’ll adopt from here on.

## Animated Logical Graphs • 19

We have encountered the question of how to extend our formal calculus to take account of operator variables.

In the days when I scribbled these things on the backs of computer punchcards, the first thing I tried was drawing big loopy script characters, placing some inside the loops of others.  Lower case alphas, betas, gammas, deltas, and so on worked best.  Graphics like these conveyed the idea that a character-shaped boundary drawn around another space can be viewed as absent or present depending on whether the formal value of the character is unmarked or marked.  The same idea can be conveyed by attaching characters directly to the edges of graphs.

Here is how we might suggest an algebraic expression of the form ${}^{\backprime\backprime} \texttt{(} q \texttt{)} {}^{\prime\prime}$ where the absence or presence of the operator ${}^{\backprime\backprime} \texttt{(} ~ \texttt{)} {}^{\prime\prime}$ depends on the value of the algebraic expression ${}^{\backprime\backprime} p {}^{\prime\prime},$ the operator ${}^{\backprime\backprime} \texttt{(} ~ \texttt{)} {}^{\prime\prime}$ being absent whenever $p$ is unmarked and present whenever $p$ is marked.

It was obvious to me from the outset that this sort of tactic would need a lot of work to become a usable calculus, especially when it came time to feed those punchcards back into the computer.

## Animated Logical Graphs • 18

We had been contemplating the penultimately simple algebraic expression ${}^{\backprime\backprime} \texttt{(} a \texttt{)} {}^{\prime\prime}$ as a name for a set of arithmetic expressions, namely, $\texttt{(} a \texttt{)} = \{ \,\texttt{()}\, , \,\texttt{(())}\, \},$ taking the equality sign in the appropriate sense.

Then we asked the corresponding question about the operator ${}^{\backprime\backprime} \texttt{(} ~ \texttt{)} {}^{\prime\prime}.$  The above selection of arithmetic expressions is what it means to contemplate the absence or presence of the operand ${}^{\backprime\backprime} a {}^{\prime\prime}$ in the algebraic expression ${}^{\backprime\backprime} \texttt{(} a \texttt{)} {}^{\prime\prime}.$  But what would it mean to contemplate the absence or presence of the operator ${}^{\backprime\backprime} \texttt{(} ~ \texttt{)} {}^{\prime\prime}$ in the algebraic expression ${}^{\backprime\backprime} \texttt{(} a \texttt{)} {}^{\prime\prime}?$

Clearly, a variation between the absence and the presence of the operator ${}^{\backprime\backprime} \texttt{(} ~ \texttt{)} {}^{\prime\prime}$ in the algebraic expression ${}^{\backprime\backprime} \texttt{(} a \texttt{)} {}^{\prime\prime}$ refers to a variation between the algebraic expressions ${}^{\backprime\backprime} a {}^{\prime\prime}$ and ${}^{\backprime\backprime} \texttt{(} a \texttt{)} {}^{\prime\prime},$ respectively, somewhat as pictured below:

But how shall we signify such variations in a coherent calculus?

## Animated Logical Graphs • 17

To get a clearer view of the relation between primary arithmetic and primary algebra consider the following extremely simple algebraic expression:

In this expression the variable name ${}^{\backprime\backprime} a {}^{\prime\prime}$ appears as an operand name.  In functional terms, ${}^{\backprime\backprime} a {}^{\prime\prime}$ is called an argument name, but it’s best to avoid the potentially confusing connotations of the word argument here, since it also refers in logical discussions to a more or less specific pattern of reasoning.

As we discussed, the algebraic variable name indicates the contemplated absence or presence of any arithmetic expression taking its place in the surrounding template, which expression is proxied well enough by its formal value, and of which values we know but two.  Thus, the given algebraic expression varies between these two choices:

The above selection of arithmetic expressions is what it means to contemplate the absence or presence of the operand ${}^{\backprime\backprime} a {}^{\prime\prime}$ in the algebraic expression ${}^{\backprime\backprime} \texttt{(} a \texttt{)} {}^{\prime\prime}.$  But what would it mean to contemplate the absence or presence of the operator ${}^{\backprime\backprime} \texttt{(} ~ \texttt{)} {}^{\prime\prime}$ in the algebraic expression ${}^{\backprime\backprime} \texttt{(} a \texttt{)} {}^{\prime\prime}?$

That is the question I’ll take up next.

## 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.