Changes

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:

# 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>''

# 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{\lbrack} s \underline{\rbrack}</math> be used for ''the indicator function denoted by the sentence <math>s.\!</math>''

# 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{\lbrace} X \underline{\rbrace}</math> be used for ''the indicator function of the set <math>X.\!</math>''

# 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 underlined bracket operator <math>\underline{\lbrack} ~ \underline{\rbrack}</math> takes one "downstream", in accord with the usual direction of denotation, from a sign to its object, while the underlined brace operator <math>\underline{\lbrace} ~ \underline{\rbrace}</math> 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.