Changes

|}

|}

In this Subsection, I describe a particular semantics for the painted cactus language, telling what meanings I aim to attach to its bare syntactic forms.  This supplies an ''interpretation'' for this parametric family of formal languages, but it is good to remember that it forms just one of many such interpretations that are conceivable and even viable.  In deed, the distinction between the object domain and the sign domain can be observed in the fact that many languages can be deployed to depict the same set of objects and that any language worth its salt is bound to to give rise to many different forms of interpretive saliency.

In this Subsection, I describe a particular semantics for the

painted cactus language, telling what meanings I aim to attach

to its bare syntactic forms.  This supplies an "interpretation"

for this parametric family of formal languages, but it is good

to remember that it forms just one of many such interpretations

that are conceivable and even viable.  In deed, the distinction

between the object domain and the sign domain can be observed in

the fact that many languages can be deployed to depict the same

set of objects and that any language worth its salt is bound to

to give rise to many different forms of interpretive saliency.

In formal settings, it is common to speak of "interpretation" as if it

In formal settings, it is common to speak of interpretation as if it created a direct connection between the signs of a formal language and the objects of the intended domain, in other words, as if it determined the denotative component of a sign relation.  But a closer attention to what goes on reveals that the process of interpretation is more indirect, that what it does is to provide each sign of a prospectively meaningful source language with a translation into an already established target language, where ''already established'' means that its relationship to pragmatic objects is taken for granted at the moment in question.

created a direct connection between the signs of a formal language and

the objects of the intended domain, in other words, as if it determined

the denotative component of a sign relation.  But a closer attention to

what goes on reveals that the process of interpretation is more indirect,

that what it does is to provide each sign of a prospectively meaningful

source language with a translation into an already established target

language, where "already established" means that its relationship to

pragmatic objects is taken for granted at the moment in question.

With this in mind, it is clear that interpretation is an affair of signs

With this in mind, it is clear that interpretation is an affair of signs that at best respects the objects of all of the signs that enter into it, and so it is the connotative aspect of semiotics that is at stake here. There is nothing wrong with my saying that I interpret a sentence of a formal language as a sign that refers to a function or to a proposition, so long as you understand that this reference is likely to be achieved by way of more familiar and perhaps less formal signs that you already take to denote those objects.

that at best respects the objects of all of the signs that enter into it,

and so it is the connotative aspect of semiotics that is at stake here.

There is nothing wrong with my saying that I interpret a sentence of a

formal language as a sign that refers to a function or to a proposition,

so long as you understand that this reference is likely to be achieved

by way of more familiar and perhaps less formal signs that you already

take to denote those objects.

On entering a context where a logical interpretation is intended for the

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.

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 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 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.

By way of providing a logical interpretation for the cactus language,

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.

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.

There are two basic types of connectives, called the "blank connectives"

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 k = 0, 1, 2, 3, ... .

and the "bound connectives", respectively, with one connective of each

type for each natural number k = 0, 1, 2, 3, ... .

1.  The "blank connective" of k places is signified by the

1.  The "blank connective" of k places is signified by the

     concatenation of the k sentences that fill those places.

     concatenation of the k sentences that fill those places.