The task before us is to specify a semantic function for the cactus language in other words, to define a mapping from the space of syntactic expressions to a space of logical statements which “interprets” each expression of as an expression which says something, an expression which bears a meaning, in short, an expression which denotes a proposition, and is in the end a sign of an indicator function.

When the syntactic expressions of a formal language are given a referent significance in logical terms, for example, as denoting propositions or indicator functions, then each form of syntactic combination takes on a corresponding form of logical significance.

A handy way of providing a logical interpretation for the expressions of any given cactus language is to introduce a family of operators on indicator functions called propositional connectives, to be distinguished from the associated family of syntactic combinations called sentential connectives, where the relationship between the two realms of connection is exactly that between objects on the one hand and their signs on the other.

A propositional connective, as an entity of a well‑defined functional and operational type, can be treated in every way as a logical or mathematical object and thus as the type of object which can be denoted by the corresponding form of syntactic entity, namely, the sentential connective appropriate to the case at hand.

There are two basic types of connectives, called the blank connectives and the bound connectives, respectively, with one connective of each type for each natural number

Blank Connective

The blank connective of places is signified by the concatenation of the sentences filling those places.

For the initial case the blank connective is an empty string or a blank symbol, both of which have the same denotation among propositions.

For the generic case the blank connective takes the form   In the type of data called a text, the use of the center dot “⋅” is generally supplanted by whatever number of spaces and line breaks serve to improve the readability of the resulting text.

Bound Connective

The bound connective of places is signified by the surcatenation of the sentences filling those places.

For the initial case the bound connective is an empty closure, an expression taking one of the forms with any number of spaces between the parentheses, all of which have the same denotation among propositions.

For the generic case the bound connective takes the form

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