If one takes the view that PARCEs and PARCs amount to a pair of intertranslatable representations for the same domain of objects then denotation brackets of the form can be used to indicate the logical denotation of a sentence or the logical denotation of a cactus

The relations connecting sentences, graphs, and propositions are shown in the next two Tables.

Between the sentences and the propositions run the graph‑theoretic data structures which arise in the process of parsing sentences and catalyze their potential for expressing logical propositions or indicator functions.  The graph‑theoretic medium supplies an intermediate form of representation between the linguistic sentences and the indicator functions, not only rendering the possibilities of connection between them more readily conceivable in fact but facilitating the necessary translations on a practical basis.

In each Table the passage from the first to the middle column articulates the mechanics of parsing cactus language sentences into graph‑theoretic data structures while the passage from the middle to the last column articulates the semantics of interpreting cactus graphs as logical propositions or indicator functions.

Aside from their common topic, the two Tables present slightly different ways of drawing the maps which go to make up the full semantic transformation.

Semantic Translation • Functional Form

The first Table shows the functional associations connecting each domain with the next, taking the triple of a sentence a cactus and a proposition as basic data, and fixing the rest by recursion on those ingredients.

Semantic Translation • Equational Form

The second Table records the transitions in the form of equations, treating sentences and graphs as alternative types of signs and generalizing the denotation bracket to indicate the proposition denoted by either type.

It should be clear at this point that either scheme of translation puts the triples of sentences, graphs, and propositions roughly in the roles of signs, interpretants, and objects, respectively, of a triadic sign relation.  Indeed, the roughly can be rendered exactly as soon as the domains of a suitable sign relation are specified precisely.

Resources

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

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