In the development of Cactus Language to date the following two species of graphs have been instrumental.

• Painted And Rooted Cacti (PARCAI).
• Painted And Rooted Conifers (PARCOI).

It suffices to begin with the first class of data structures, developing their properties and uses in full, leaving discussion of the latter class to a part of the project where their distinctive features are key to developments at that stage.  Partly because the two species are so closely related and partly for the sake of brevity, we’ll always use the genus name “PARC” to denote the corresponding cacti.

To provide a computational middle ground between sentences seen as syntactic strings and propositions seen as indicator functions the language designer must not only supply a medium for the expression of propositions but also link the assertion of sentences to a means for inverting the indicator functions, that is, for computing the fibers or inverse images of the propositions.

Given a body of conceivable propositions we need a way to follow the threads of their indications from their object domain to their values for the mind and a way to follow those same threads back again.  Moreover, we need to implement both ways of proceeding in computational form.  Thus we need programs for tracing the clues sentences provide from the universe of their objects to the signs of their values and, in turn, from signs to objects.  Ultimately, we need to render propositions so functional as indicators of sets and so essential for examining the equality of sets as to give a rule for the practical conceivability of sets.  Tackling that task requires us to introduce a number of new definitions and a collection of additional notational devices, to which we now turn.

Resources

• Cactus Language • Overview
• 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