Changes

On entering a context where a logical interpretation is intended for the sentences of a formal language there are a few conventions that make it easier to make the translation from abstract syntactic forms to their intended semantic senses.  Although these conventions are expressed in unnecessarily colorful terms, from a purely abstract point of view, they do provide a useful array of connotations that help to negotiate what is otherwise a difficult transition.  This terminology is introduced as the need for it arises in the process of interpreting the cactus language.

On entering a context where a logical interpretation is intended for the sentences of a formal language there are a few conventions that make it easier to make the translation from abstract syntactic forms to their intended semantic senses.  Although these conventions are expressed in unnecessarily colorful terms, from a purely abstract point of view, they do provide a useful array of connotations that help to negotiate what is otherwise a difficult transition.  This terminology is introduced as the need for it arises in the process of interpreting the cactus language.

The task of this Subsection is to specify a ''semantic function'' for the sentences of the cactus language !L! = !C!(!P!), in other words, to define a mapping that "interprets" each sentence of !C!(!P!) as a sentence that says something, as a sentence that bears a meaning, in short, as a sentence that denotes a proposition, and thus as a sign of an indicator function.  When the syntactic sentences 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.

The task of this Subsection is to specify a ''semantic function'' for the sentences of the cactus language <math>\mathfrak{L} = \mathfrak{C}(\mathfrak{P}),</math> in other words, to define a mapping that "interprets" each sentence of <math>\mathfrak{C}(\mathfrak{P})</math> as a sentence that says something, as a sentence that bears a meaning, in short, as a sentence that denotes a proposition, and thus as a sign of an indicator function.  When the syntactic sentences 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.

By way of providing a logical interpretation for the cactus language, I introduce a family of operators on indicator functions that are called ''propositional connectives'', and I distinguish these from the associated family of syntactic combinations that are called ''sentential connectives'', where the relationship between these two realms of connection is exactly that between objects and their signs.  A propositional connective, as an entity of a well-defined functional and operational type, can be treated in every way as a logical or a mathematical object, and thus as the type of object that can be denoted by the corresponding form of syntactic entity, namely, the sentential connective that is appropriate to the case in question.

By way of providing a logical interpretation for the cactus language, I introduce a family of operators on indicator functions that are called ''propositional connectives'', and I distinguish these from the associated family of syntactic combinations that are called ''sentential connectives'', where the relationship between these two realms of connection is exactly that between objects and their signs.  A propositional connective, as an entity of a well-defined functional and operational type, can be treated in every way as a logical or a mathematical object, and thus as the type of object that can be denoted by the corresponding form of syntactic entity, namely, the sentential connective that is appropriate to the case in question.