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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.

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, ... .

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 <math>k = 0, 1, 2, 3, \ldots.</math>

<pre>

<ol style="list-style-type:decimal">

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

 

    concatenation of the k sentences that fill those places.

<p>The ''blank connective'' of <math>k\!</math> places is signified by the concatenation of the <math>k\!</math> sentences that fill those places.</p>

 

<p>For the special case of <math>k = 0,\!</math> the blank connective is taken to be an empty string or a blank symbol &mdash; it does not matter which, since both are assigned the same denotation among propositions.</p>

 

<p>For the generic case of <math>k > 0,\!</math> the blank connective takes the form <math>s_1 \cdot \ldots \cdot s_k.</math> In the type of data that is called a ''text'', the use of the center dot <math>(\cdot)</math> is generally supplanted by whatever number of spaces and line breaks serve to improve the readability of the resulting text.</p></li>

 

<p>The ''bound connective'' of <math>k\!</math> places is signified by the surcatenation of the <math>k\!</math> sentences that fill those places.</p>

    For the special case of k = 0, the "blank connective" is taken to

<p>For the special case of <math>k = 0,\!</math> the bound connective is taken to be an empty closure &mdash; an expression enjoying one of the forms <math>\underline{(} \underline{)}, \, \underline{(} ~ \underline{)}, \, \underline{(} ~~ \underline{)}, \, \ldots</math> with any number of blank symbols between the parentheses &mdash; all of which are assigned the same logical denotation among propositions.</p>

    be an empty string or a blank symbol -- it does not matter which,

    since both are assigned the same denotation among propositions.

    For the generic case of k > 0, the "blank connective" takes

2.  The "bound connective" of k places is signified by the

<p>For the generic case of <math>k > 0,\!</math> the bound connective takes the form <math>\underline{(} s_1, \ldots, s_k \underline{)}.</math></p></li>

    surcatenation of the k sentences that fill those places.

</ol>

    For the generic case of k > 0, the "bound connective" takes the

At this point, there are actually two different "dialects", "scripts",

At this point, there are actually two different "dialects", "scripts",

or "modes" of presentation for the cactus language that need to be

or "modes" of presentation for the cactus language that need to be