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If <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> is a boolean function on <math>k\!</math> variables, then it is possible to define a mapping <math>F^\$ : (X \to \underline\mathbb{B})^k \to (X \to \underline\mathbb{B}),</math> in effect, an operation that takes <math>k\!</math> propositions into a single proposition, where <math>F^\$</math> satisfies the following conditions:

If <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> is a boolean function on <math>k\!</math> variables, then it is possible to define a mapping <math>F^\$ : (X \to \underline\mathbb{B})^k \to (X \to \underline\mathbb{B}),</math> in effect, an operation that takes <math>k\!</math> propositions into a single proposition, where <math>F^\$</math> satisfies the following conditions:

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<math>\begin{array}{lcl}

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Thus, <math>F^\$</math> is what a propositional connective denotes, a particular way of connecting the propositions that are denoted by a number of sentences into a proposition that is denoted by a single sentence.

Thus, <math>F^\$</math> is what a propositional connective denotes, a particular way of connecting the propositions that are denoted by a number of sentences into a proposition that is denoted by a single sentence.

<pre>

Now <math>^{\backprime\backprime} f_Q \, ^{\prime\prime}</math> is sign that denotes the proposition <math>f_Q,\!</math> and it certainly seems like a sufficient sign for it.  Why is there is a need to recognize any other signs of it?

Now "fX" is sign that denotes the proposition fX, and it certainly seems like a sufficient sign for it.  Why is there is a need to recognize any other signs of it?

 

If one takes a sentence as a type of sign that denotes a proposition and a proposition as a type of function whose values serve to indicate a set, then one needs a way to grasp the overall relation between the sentence and the set as taking place within a "higher order" (HO) sign relation.

If one takes a sentence as a type of sign that denotes a proposition and a proposition as a type of function whose values serve to indicate a set, then one needs a way to grasp the overall relation between the sentence and the set as taking place within a higher order sign relation.

Roughly sketched, the relations of denotation and indication that exist among sets, propositions, sentences, and values can be diagrammed as in Table 10.

Roughly sketched, the relations of denotation and indication that exist among sets, propositions, sentences, and values can be diagrammed as in Table&nbsp;11.

Table 10.  Levels of Indication

Table 11.  Levels of Indication

Object

Object

Sign

Sign

~X 0 "0"

~X 0 "0"

Strictly speaking, a proposition is too abstract to be a sign, and so the contents of Table 10 have to be taken with the indicated grains of salt.  Propositions, as indicator functions, are abstract mathematical objects, not any kinds of syntactic elements, and so propositions cannot literally constitute the orders of concrete signs that remain of ultimate interest in the pragmatic theory of signs, or in any theory of effective meaning.  Therefore, it needs to be understood that a proposition f can be said to "indicate" a set X only insofar as the values of 1 and 0 that it assigns to the elements of the universe U are positive and negative indications, respectively, of the elements in X, and thus indications of the set X and of its complement ~X = U - X, respectively.  It is actually these values, when rendered by a concrete implementation of the indicator function f, that are the actual signs of the objects that are inside the set X and the objects that are outside the set X, respectively.

Strictly speaking, a proposition is too abstract to be a sign, and so the contents of Table 11 have to be taken with the indicated grains of salt.  Propositions, as indicator functions, are abstract mathematical objects, not any kinds of syntactic elements, and so propositions cannot literally constitute the orders of concrete signs that remain of ultimate interest in the pragmatic theory of signs, or in any theory of effective meaning.  Therefore, it needs to be understood that a proposition f can be said to "indicate" a set X only insofar as the values of 1 and 0 that it assigns to the elements of the universe U are positive and negative indications, respectively, of the elements in X, and thus indications of the set X and of its complement ~X = U - X, respectively.  It is actually these values, when rendered by a concrete implementation of the indicator function f, that are the actual signs of the objects that are inside the set X and the objects that are outside the set X, respectively.

In order to deal with the HO sign relations that are involved in this situation, I introduce a couple of new notations:

In order to deal with the HO sign relations that are involved in this situation, I introduce a couple of new notations:

1. To mark the relation of denotation between a sentence S and the proposition that it denotes, let the "spiny bracket" notation "[S]" be used for "the indicator function denoted by the sentence S".

1. To mark the relation of denotation between a sentence S and the proposition that it denotes, let the "spiny bracket" notation "[S]" be used for "the indicator function denoted by the sentence S".

2. To mark the relation of denotation between a proposition P and the set that it indicates, let the "spiny brace" notation "{X}" be used for "the indicator function of the set X".

2. To mark the relation of denotation between a proposition P and the set that it indicates, let the "spiny brace" notation "{X}" be used for "the indicator function of the set X".

Notice that the spiny bracket operator "[ ]" takes one "downstream", in accord with the usual direction of denotation, from a sign to its object, while the spiny brace operator "{ }" takes one "upstream", against the usual direction of denotation, and thus from an object to its sign.

Notice that the spiny bracket operator "[ ]" takes one "downstream", in accord with the usual direction of denotation, from a sign to its object, while the spiny brace operator "{ }" takes one "upstream", against the usual direction of denotation, and thus from an object to its sign.

In order to make these notations useful in practice, it is necessary to note of a couple of their finer points, points that might otherwise seem too fine to take much trouble over.  For this reason, I express their usage a bit more carefully as follows:

In order to make these notations useful in practice, it is necessary to note of a couple of their finer points, points that might otherwise seem too fine to take much trouble over.  For this reason, I express their usage a bit more carefully as follows:

1. Let "spiny brackets", like "[ ]", be placed around a name of a sentence S, as in the expression "[S]", or else around a token appearance of the sentence itself, to serve as a name for the proposition that S denotes.

1. Let "spiny brackets", like "[ ]", be placed around a name of a sentence S, as in the expression "[S]", or else around a token appearance of the sentence itself, to serve as a name for the proposition that S denotes.

2. Let "spiny braces", like "{ }", be placed around a name of a set X, as in the expression "{X}", to serve as a name for the indicator function fX.

2. Let "spiny braces", like "{ }", be placed around a name of a set X, as in the expression "{X}", to serve as a name for the indicator function fX.

Table 11 illustrates the use of this notation, listing in each column several different but equivalent ways of referring to the same entity.

Table 12 illustrates the use of this notation, listing in each column several different but equivalent ways of referring to the same entity.

Table 11.  Illustrations of Notation

Table 12.  Illustrations of Notation

Object Sign Higher Order Sign

Object Sign Higher Order Sign

Set Proposition Sentence

Set Proposition Sentence

In particular, one can observe the following relations and formulas, all of a purely notational character:

In particular, one can observe the following relations and formulas, all of a purely notational character:

1. If the sentence S denotes the proposition P : U -> B, then [S] = P.

1. If the sentence S denotes the proposition P : U -> B, then [S] = P.

2. If the sentence S denotes the proposition P : U -> B

2. If the sentence S denotes the proposition P : U -> B

such that |P| = P-1(1) = X c U, then [S] = P = fX = {X}.

such that |P| = P-1(1) = X c U, then [S] = P = fX = {X}.

3. X = {u C U : u C X}

3. X = {u C U : u C X}

= |{X}| = {X}-1(1)

= |{X}| = {X}-1(1)

= |fX| = fX-1(1).

= |fX| = fX-1(1).

4. {X} = { {u C U : u C X} }

4. {X} = { {u C U : u C X} }

= [u C X]

= [u C X]

= fX.

= fX.

Now if a sentence S really denotes a proposition P, and if the notation "[S]" is merely meant to supply another name for the proposition that S already denotes, then why is there any need for the additional notation?  It is because the interpretive mind habitually races from the sentence S, through the proposition P that it denotes, and on to the set X = P-1(1) that the proposition P indicates, often jumping to the conclusion that the set X is the only thing that the sentence S is intended to denote.  This HO sign situation and the mind's inclination when placed within its setting calls for a linguistic mechanism or a notational device that is capable of analyzing the compound action and controlling its articulate performance, and this requires a way to interrupt the flow of assertion that typically takes place from S to P to X.

Now if a sentence S really denotes a proposition P, and if the notation "[S]" is merely meant to supply another name for the proposition that S already denotes, then why is there any need for the additional notation?  It is because the interpretive mind habitually races from the sentence S, through the proposition P that it denotes, and on to the set X = P-1(1) that the proposition P indicates, often jumping to the conclusion that the set X is the only thing that the sentence S is intended to denote.  This HO sign situation and the mind's inclination when placed within its setting calls for a linguistic mechanism or a notational device that is capable of analyzing the compound action and controlling its articulate performance, and this requires a way to interrupt the flow of assertion that typically takes place from S to P to X.