Changes

Strictly speaking, a proposition is too abstract to be a sign, and so the contents of Table&nbsp;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 <math>f\!</math> can be said to "indicate" a set <math>Q\!</math> only insofar as the values of <math>\underline{1}</math> and <math>\underline{0}</math> that it assigns to the elements of the universe <math>X\!</math> are positive and negative indications, respectively, of the elements in <math>Q,\!</math> and thus indications of the set <math>Q\!</math> and of its complement <math>{}^{_\sim} Q = X\!-\!Q,</math> respectively.  It is actually these values, when rendered by a concrete implementation of the indicator function <math>f,\!</math> that are the actual signs of the objects that are inside the set <math>Q\!</math> and the objects that are outside the set <math>Q,\!</math> respectively.

Strictly speaking, a proposition is too abstract to be a sign, and so the contents of Table&nbsp;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 <math>f\!</math> can be said to "indicate" a set <math>Q\!</math> only insofar as the values of <math>\underline{1}</math> and <math>\underline{0}</math> that it assigns to the elements of the universe <math>X\!</math> are positive and negative indications, respectively, of the elements in <math>Q,\!</math> and thus indications of the set <math>Q\!</math> and of its complement <math>{}^{_\sim} Q = X\!-\!Q,</math> respectively.  It is actually these values, when rendered by a concrete implementation of the indicator function <math>f,\!</math> that are the actual signs of the objects that are inside the set <math>Q\!</math> and the objects that are outside the set <math>Q,\!</math> respectively.

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

In order to deal with the higher order 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".

# To mark the relation of denotation between a sentence <math>s\!</math> and the proposition that it denotes, let the ''underlined bracket'' notation <math>\underline{[} s \underline{]}</math> be used for ''the indicator function denoted by the sentence <math>s.\!</math>''

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".

# To mark the relation of denotation between a proposition <math>p\!</math> and the set that it indicates, let the ''underlined brace'' notation <math>\underline{ \{ } X \underline{ \} }</math> be used for ''the indicator function of the set <math>X.\!</math>''

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: