Cactus Language • Pragmatics 16

Stricture, Strait, Constraint, Information, Complexity

The ways in which strictures and straits at different levels of complexity relate to one another can be given systematic treatment by introducing the following pair of definitions.

Excerpt of a Stricture

The $j^\text{th}$ excerpt of a stricture $``S_1 \times \ldots \times S_k",$ regarded in a frame of discussion where the number of places is bounded by $k,$ is a stricture of the form $``X \times \ldots \times S_j \times \ldots \times X".$

The $j^\text{th}$ excerpt can be written more briefly in context as the stricture $``(S_j)_{[j]}",$ an assertion which places the $j^\text{th}$ set in the $j^\text{th}$ place of the product.

Extract of a Strait

The $j^\text{th}$ extract of a strait $S_1 \times \ldots \times S_k,$ regarded in a frame of discussion where the number of places is bounded by $k,$ is a strait of the form $X \times \ldots \times S_j \times \ldots \times X.$

The $j^\text{th}$ extract can be denoted more briefly in context by the stricture $``(S_j)_{[j]}",$ an assertion which places the $j^\text{th}$ set in the $j^\text{th}$ place of the product.

Using the above definitions, a stricture of the form $``S_1 \times \ldots \times S_k"$ can be expressed in terms of simpler strictures, namely, as the following conjunction of its individual excerpts.

$\begin{array}{lll} ``S_1 \times \ldots \times S_k" & = & ``(S_1)_{[1]}" \land \ldots \land ``(S_k)_{[k]}" \end{array}$

In a similar vein, a strait of the form $S_1 \times \ldots \times S_k$ can be expressed in terms of simpler straits, namely, as the following intersection of its individual extracts.

$\begin{array}{lll} S_1 \times \ldots \times S_k & = & (S_1)_{[1]} \cap \ldots \cap (S_k)_{[k]} \end{array}$

Resources

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