Cactus Language • Semantics 4

Words spoken are symbols or signs (symbola) of affections or impressions (pathemata) of the soul (psyche); written words are the signs of words spoken. As writing, so also is speech not the same for all races of men. But the mental affections themselves, of which these words are primarily signs (semeia), are the same for the whole of mankind, as are also the objects (pragmata) of which those affections are representations or likenesses, images, copies (homoiomata). (Aristotle, De Interp. i. 16^a4).

At this point we have two distinct dialects, scripts, or modes of presentation for the typical cactus language $\mathfrak{C} (\mathfrak{P}),$ each of which needs to be interpreted, that is to say, equipped with a semantic function defined on its domain.

$\textsc{parce} (\mathfrak{P})$: There is the language of strings in $\textsc{parce} (\mathfrak{P}),$ the painted and rooted cactus expressions collectively forming the language $\mathfrak{L} = \mathfrak{C} (\mathfrak{P}) \subseteq \mathfrak{A}^* = (\mathfrak{M} \cup \mathfrak{P})^*.$
$\textsc{parc} (\mathfrak{P})$: There is the language of graphs in $\textsc{parc} (\mathfrak{P}),$ the painted and rooted cacti themselves, a family of graphs or species of data structures formed by parsing the language of strings.

Those two modalities of formal language, like written and spoken natural languages, are meant to have compatible interpretations, which means it is generally sufficient to give the meanings of just one or the other.

All that remains is to provide a codomain or target space for the intended semantic function, that is, to supply a suitable range of logical meanings for the memberships of those languages to map into. One way to do that proceeds by making the following definitions.

Logical Conjunction: The conjunction $\mathrm{Conj}_j^J q_j$ of a set of propositions $\{ q_j : j \in J \}$ is a proposition which is true if and only if every one of the $q_j$ is true.; $\mathrm{Conj}_j^J q_j$ is true $\Leftrightarrow$ $q_j$ is true for every $j \in J.$
Logical Surjunction: The surjunction $\mathrm{Surj}_j^J q_j$ of a set of propositions $\{ q_j : j \in J \}$ is a proposition which is true if and only if exactly one of the $q_j$ is untrue.; $\mathrm{Surj}_j^J q_j$ is true $\Leftrightarrow$ $q_j$ is untrue for unique $j \in J.$

Resources

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

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