## All Process, No Paradox • 2

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

As serendipity would have it, Lou Kauffman, who knows a lot about the lines of inquiry Charles Sanders Peirce and George Spencer Brown pursued into graphical syntaxes for logic, just last month opened a blog and his very first post touched on perennial questions of logic and time — Logos and Chronos — puzzling 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:

Kauffman’s treatment of logic, paradox, time, and imaginary truth values led me to make the following comments 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 typically but not always interpreted for logic so that $0 = \mathrm{false}$ and $1 = \mathrm{true}.$  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 $x$ 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 $x$ 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 $X$ 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 $X$ 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 $\mathrm{d}x = 1.$

The following article has a few more examples along these lines.

### 5 Responses to All Process, No Paradox • 2

1. gary lasseter says:

i will get back to you when i have time.

• Jon Awbrey says:

I don’t have time …
Time has me …

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