Cactus Language • Pragmatics 15

Stricture, Strait, Constraint, Information, Complexity

From here it is easy to see how the concatenation of languages is related to the intersection of sets and thus to the conjunction of logical propositions. In the upshot a cartesian product $P \times Q$ is described by a logical proposition $P_{[1]} \land Q_{[2]}$ subject to the following interpretation.

$P_{[1]}$ says there is an element from the set $P$ in the 1st place of the product $P \times Q$ .
$Q_{[2]}$ says there is an element from the set $Q$ in the 2nd place of the product $P \times Q$ .

The integration of those two pieces of information can be taken to specify a yet to be fully determined relation.

In a corresponding fashion at the level of elements, the ordered pair $(p, q)$ is described by a conjunction of propositions, namely $p_{[1]} \land q_{[2]},$ subject to the following interpretation.

$p_{[1]}$ says that $p$ occupies the 1st place of the product element under construction.
$q_{[2]}$ says that $q$ occupies the 2nd place of the product element under construction.

Taking the cartesian product of $P$ and $Q$ or the concatenation of $\mathfrak{L}_1$ and $\mathfrak{L}_2$ in the above manner shifts the level of active construction from the tupling of elements in $P$ and $Q$ or the concatenation of strings in $\mathfrak{L}_1$ and $\mathfrak{L}_2$ to the concatenation of external signs describing those sets or languages.

Thus we pass to a conjunction of indexed propositions $P_{[1]}$ and $Q_{[2]}$ or a conjunction of assertions $(\mathfrak{L}_1)_{[1]}$ and $(\mathfrak{L}_2)_{[2]}$ which mark the indicated sets or languages for insertion in the indicated places of a product set or product language, respectively. On closer examination, we can recognize the subscripting by the indices $``[1]"$ and $``[2]"$ as a type of concatenation, in this case accomplished through the posting of editorial remarks from an external mark‑up language.

Resources

cc: Academia.edu • BlueSky • Laws of Form • Ma^thstodon • Research Gate
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

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