Changes

# To mark the relation of denotation between a sentence <math>s\!</math> and the proposition that it denotes, let the ''drop'' notation <math>\downharpoonleft s \downharpoonright</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 ''drop'' notation <math>\downharpoonleft s \downharpoonright</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 ''lift'' notation <math>\upharpoonleft X \upharpoonright</math> be used for ''the indicator function of the set <math>X.\!</math>''

# To mark the relation of denotation between a proposition <math>q\!</math> and the set that it indicates, let the ''lift'' notation <math>\upharpoonleft Q \upharpoonright</math> be used for ''the indicator function of the set <math>Q.\!</math>''

Notice that the drop operator <math>\downharpoonleft \cdots \downharpoonright</math> takes one "downstream", in accord with the direction of denotation, from a sign to its object, while the lift operator <math>\upharpoonleft \cdots \upharpoonright</math> takes one "upstream", against the direction of denotation, and thus from an object to its sign.

Notice that the drop operator <math>\downharpoonleft \cdots \downharpoonright</math> takes one "downstream", in accord with the direction of denotation, from a sign to its object, while the lift operator <math>\upharpoonleft \cdots \upharpoonright</math> takes one "upstream", against the 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:

# The ''up hooks'' <math>\upharpoonleft \cdots \upharpoonright</math> are placed around the name of a sentence <math>s,\!</math> as in the expression <math>\upharpoonleft s \upharpoonright,</math>, or else around a token appearance of the sentence itself, to serve as a name for the proposition that <math>s\!</math> denotes.

# Let the ''down hooks'' <math>\downharpoonleft \cdots \downharpoonright</math> be placed around the name of a sentence <math>s,\!</math> as in the expression <math>^{\backprime\backprime} \downharpoonleft s \downharpoonright \, ^{\prime\prime},</math> or else around a token appearance of the sentence itself, to serve as a name for the proposition that <math>s\!</math> denotes.

# The ''down hooks'' <math>\downharpoonleft \cdots \downharpoonright</math> are placed around a name of a set <math>X,\!</math> as in the expression <math>\downharpoonleft X \downharpoonright,</math> to serve as a name for the indicator function <math>f_X.\!</math>

# Let the ''up hooks'' <math>\upharpoonleft \cdots \upharpoonright</math> be placed around a name of a set <math>Q,\!</math> as in the expression <math>^{\backprime\backprime} \upharpoonleft Q \upharpoonright \, ^{\prime\prime},</math> to serve as a name for the indicator function <math>f_Q.\!</math>

Table 12 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 12.  Illustrations of Notation

Table 12.  Illustrations of Notation

Object Sign Higher Order Sign

Object Sign Higher Order Sign