Cactus Language • Semantics 5

Last time we reached the threshold of a potential codomain or target space for the kind of semantic function we need at this point, one able to supply logical meanings for the syntactic strings and graphs of a given cactus language. In that pursuit we came to contemplate 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 $j \in J.$

If the set of propositions $\{ q_j : j \in J \}$ is finite then the logical conjunction and logical surjunction can be represented by means of sentential connectives, incorporating the sentences which represent the propositions into finite strings of symbols.

If $J$ is finite, for instance, if $J$ consists of the integers in the interval $j = 1 ~\text{to}~ k,$ and if each proposition $q_j$ is represented by a sentence $s_j,$ then the following forms of expression are possible.

Logical Conjunction: The conjunction $\mathrm{Conj}_j^J q_j$ can be represented by a sentence which is constructed by concatenating the $s_j$ in the following fashion.; $\mathrm{Conj}_j^J q_j ~\leftrightsquigarrow~ s_1 s_2 \ldots s_k.$
Logical Surjunction: The surjunction $\mathrm{Surj}_j^J q_j$ can be represented by a sentence which is constructed by surcatenating the $s_j$ in the following fashion.; $\mathrm{Surj}_j^J q_j ~\leftrightsquigarrow~ \texttt{(} s_1 \texttt{,} s_2 \texttt{,} \ldots \texttt{,} s_k \texttt{)}.$

If one opts for a mode of interpretation which moves more directly from the parse graph of a sentence to the potential logical meaning of both the PARC and the PARCE then the following specifications are in order.

A cactus graph rooted at a particular node is taken to represent what that node represents, namely, its logical denotation.

Denotation of a Node: The logical denotation of a node is the logical conjunction of that node’s arguments, which are defined as the logical denotations of that node’s attachments.; The logical denotation of either a blank symbol or empty node is the boolean value $\underline{1} = \mathrm{true}.$; The logical denotation of the paint $\mathfrak{p}_j$ is the proposition $p_j,$ a proposition regarded as primitive, at least, with respect to the level of analysis represented in the current instance of $\mathfrak{C} (\mathfrak{P}).$
Denotation of a Lobe: The logical denotation of a lobe is the logical surjunction of that lobe’s arguments, which are defined as the logical denotations of that lobe’s appendants.; As a corollary, the logical denotation of the parse graph of $\texttt{()},$ also known as a needle, is the boolean value $\underline{0} = \mathrm{false}.$

Resources

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

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