All Process, No Paradox : 2

These are the forms of time, which imitates eternity and revolves according to a law of number.

Re: Lou Kauffman • Iterants, Imaginaries, and Matrices

As serendipity would have it, Lou Kauffman, who knows a lot about the lines of work that Charles Sanders Peirce and George Spencer Brown carried out on graphical syntaxes for logic, just last month opened a new blog and his very first post touched on perennial questions of logic and time — Logos and Chronos — that have puzzled the wits of everyone who has thought about them for as long as anyone can remember. Just locally and recently these questions have arisen in the following contexts:

Alpha Now, Omega Later • (1) • (2) • (3) • (4) • (5) • (6) • (7)
Objects, Models, Theories • (1) • (2) • (3) • (4)
Peirce List • (1) • (2) • (3)

Kauffman’s treatment of logic, paradox, time, and imaginary truth values led me to make the following comments that I think are very close to what I’d been struggling to say before.

Let me get some notational matters out of the way before continuing.

I use $\mathbb{B}$ for a generic 2-point set, usually $\{ 0, 1 \}$ and usually but not always interpreted for logic so that $0 = \text{false}$ and $1 = \text{true}.$ I use “teletype” parentheses $\texttt{(} \ldots \texttt{)}$ for negation, so that $\texttt{(} x \texttt{)} = \lnot x$ for $x ~\text{in}~ \mathbb{B}.$ Later on I’ll be using teletype format lists $\texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)}$ for minimal negation operators.

As long as we’re reading as a boolean variable $(x \in \mathbb{B})$ the equation $x = \texttt{(} x \texttt{)}$ is not paradoxical but simply false. As an algebraic structure $\mathbb{B}$ can be extended in many ways but it remains a separate question what sort of application, if any, such extensions might have to the normative science of logic.

On the other hand, the assignment statement $x := \texttt{(} x \texttt{)}$ makes perfect sense in computational contexts. The effect of the assignment operation on the value of the variable is commonly expressed in time series notation as $x' = \texttt{(} x \texttt{)}$ and the same change is expressed even more succinctly by defining $\mathrm{d}x = x' - x$ and writing $\mathrm{d}x = 1.$

Now suppose we are observing the time evolution of a system with a boolean state variable $x : X \to \mathbb{B}$ and what we observe is the following time series:

$\begin{array}{c|c} t & x \\ \hline 0 & 0 \\ 1 & 1 \\ 2 & 0 \\ 3 & 1 \\ 4 & 0 \\ 5 & 1 \\ 6 & 0 \\ 7 & 1 \\ 8 & 0 \\ 9 & 1 \\ \ldots & \ldots \end{array}$

Computing the first differences we get:

$\begin{array}{c|cc} t & x & \mathrm{d}x \\ \hline 0 & 0 & 1 \\ 1 & 1 & 1 \\ 2 & 0 & 1 \\ 3 & 1 & 1 \\ 4 & 0 & 1 \\ 5 & 1 & 1 \\ 6 & 0 & 1 \\ 7 & 1 & 1 \\ 8 & 0 & 1 \\ 9 & 1 & 1 \\ \ldots & \ldots & \ldots \end{array}$

Computing the second differences we get:

$\begin{array}{c|cccc} t & x & \mathrm{d}x & \mathrm{d}^2 x & \ldots \\ \hline 0 & 0 & 1 & 0 & \ldots \\ 1 & 1 & 1 & 0 & \ldots \\ 2 & 0 & 1 & 0 & \ldots \\ 3 & 1 & 1 & 0 & \ldots \\ 4 & 0 & 1 & 0 & \ldots \\ 5 & 1 & 1 & 0 & \ldots \\ 6 & 0 & 1 & 0 & \ldots \\ 7 & 1 & 1 & 0 & \ldots \\ 8 & 0 & 1 & 0 & \ldots \\ 9 & 1 & 1 & 0 & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots \end{array}$

This leads to thinking of the system as having an extended state $(x, \mathrm{d}x, \mathrm{d}^2 x, \ldots, \mathrm{d}^k x),$ and this additional language gives us the facility of describing state transitions in terms of the various orders of differences. For example, the rule $x' = \texttt{(} x \texttt{)}$ can now be expressed by the rule $\mathrm{d}x = 1.$