Grammar 3 (concl.)

Returning to the cactus language and fixing the parameter for the moment, we have a language of painted and rooted cactus expressions.  It serves the purpose of efficient accounting to divide the language into two sublanguages.

• The emptily painted and rooted cactus expressions make up the language which consists of a single empty string as its only sentence.

• The significantly painted and rooted cactus expressions make up the language which consists of everything else, namely, all the non‑empty strings in the language

Marking the distinction between empty and significant sentences effectively categorizes each of the three classes of strings as an entity unto itself and conceives the whole of its membership as falling under a distinctive symbol, thereby obtaining the following equation among the three sublanguages.

That makes the disjoint union of and

For brevity in the present case, and to serve as a generic device in similar situations, let be the type of an arbitrary sentence, possibly empty, and let be the type of a non‑empty sentence.

In addition, let be the type of the empty sentence, in effect, the language containing a single empty string, and let a plus sign signify a disjoint union of types.  In the most general type of situation, where the type is permitted to include the empty string, one notes the following relation among types.

With the distinction between empty and significant expressions in mind, we return to the analysis of the cactus language afforded by Grammar 2, and, taking that as a point of departure, explore other avenues of possible improvement in the comprehension of its expressions.

To observe the effects of the alteration as clearly as possible in isolation from other factors it is useful to strip away the higher levels of intermediate organization presented by Grammar 3 and start again with a single intermediate symbol, as used in Grammar 2.  One way to execute that strategy leads to a style of grammar we take up next.

Resources

• Cactus Language • Syntax
• Survey of Animated Logical Graphs
• Survey of Theme One Program

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