Cactus Language • Pragmatics 14

Stricture, Strait, Constraint, Information, Complexity

To give a concrete example of strictures and straits in action, let us institute a frame of discussion where the number of places in a relation is bounded at two and the variety of sets under active consideration is limited to the subsets $P$ and $Q$ of a universe $X.$ Under those conditions one may use the following sorts of expression as schematic strictures.

$\begin{matrix} ``X" & ``P" & ``Q" \\[4pt] ``X \times X" & ``X \times P" & ``X \times Q" \\[4pt] ``P \times X" & ``P \times P" & ``P \times Q" \\[4pt] ``Q \times X" & ``Q \times P" & ``Q \times Q" \end{matrix}$

The above strictures and their corresponding straits are stratified according to the amounts of information they contain, or the levels of constraint they impose, as shown in the following table.

$\begin{array}{lcccc} \text{High:} & ``P \times P" & ``P \times Q" & ``Q \times P" & ``Q \times Q" \\[4pt] \text{Med:} & ``P" & ``X \times P" & ``P \times X" \\[4pt] \text{Med:} & ``Q" & ``X \times Q" & ``Q \times X" \\[4pt] \text{Low:} & ``X" & ``X \times X" \end{array}$

In that framework, the complex strait $P \times Q$ can be defined in terms of the simpler straits $P \times X$ and $X \times Q$ as the following set‑theoretic intersection.

$\begin{array}{lllll} P \times Q & = & P \times X & \cap & X \times Q \end{array}$

Resources

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