Cactus Language • Preliminaries 5

The easiest way to define the language $\mathfrak{C}(\mathfrak{P})$ is to indicate the general run of operations required to construct the greater share of its sentences from the designated few which require a special election.

To do that we introduce a family of operations called syntactic connectives on the strings of $\mathfrak{A}^*.$ If the strings on which they operate are already sentences of $\mathfrak{C}(\mathfrak{P})$ then the operations amount to sentential connectives. If the syntactic sentences, viewed as abstract strings of uninterpreted signs, are provided with a semantics where they denote propositions, in other words, indicator functions on a universe of discourse, then the operations amount to propositional connectives.

Rather than presenting the most concise description of cactus languages right from the beginning, it aids comprehension to develop a picture of their forms in gradual stages, starting with the most natural ways of viewing their elements, if somewhat at a distance, and working through the most easily grasped impressions of their structures, if not always the sharpest acquaintances with their details.

We begin by defining two sets of basic operations on strings of $\mathfrak{A}^*.$

Concatenation

The concatenation of one string $s_1$ is the string $s_1.$

The concatenation of two strings $s_1, s_2$ is the string ${s_1 \cdot s_2}.$

The concatenation of $k$ strings $(s_j)_{j = 1}^k$ is the string ${s_1 \cdot \ldots \cdot s_k}.$

Surcatenation

The surcatenation of one string $s_1$ is the string $``\text{(}" \cdot s_1 \cdot ``\text{)}".$

The surcatenation of two strings $s_1, s_2$ is the string $``\text{(}" \cdot s_1 \cdot ``\text{,}" \cdot s_2 \cdot ``\text{)}".$

The surcatenation of $k$ strings $(s_j)_{j = 1}^k$ is the string $``\text{(}" \cdot s_1 \cdot ``\text{,}" \cdot \ldots \cdot ``\text{,}" \cdot s_k \cdot ``\text{)}".$

Resources

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

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