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The sign <math>^{\backprime\backprime} f(x) ^{\prime\prime}</math> manifestly names the value <math>f(x).\!</math>  This is a value that can be seen in many lights.  It is, at turns:  (1) the value that the proposition <math>f\!</math> has at the point <math>x,\!</math> in other words, that it bears at the point where it is evaluated, and that it takes on with respect to the argument or the object that the whole proposition is taken to be about, (2) the value that the proposition <math>f\!</math> not only takes up the point <math>x,\!</math> but that it carries, conveys, transfers, or transports into the setting <math>^{\backprime\backprime} \{ x \in X : \underline{~~~} \} ^{\prime\prime}</math> or into any other context of discourse where <math>f\!</math> is meant to be evaluated, (3) the value that the sign <math>^{\backprime\backprime} f(x) ^{\prime\prime}</math> has in the context where it is placed, that it stands for in the context where it stands, and that it continues to stand for in this context just so long as the same proposition <math>f\!</math> and the same object <math>x\!</math> are borne in mind, and last but not least, (4) the value that the sign <math>^{\backprime\backprime} f(x) ^{\prime\prime}</math> represents to its full interpretive context as being its own logical interpretant, namely, the value that it signifies as its canonical connotation to any interpreter of the sign that is cognizant of the context in which it appears.

The sign <math>^{\backprime\backprime} f(x) ^{\prime\prime}</math> manifestly names the value <math>f(x).\!</math>  This is a value that can be seen in many lights.  It is, at turns:  (1) the value that the proposition <math>f\!</math> has at the point <math>x,\!</math> in other words, that it bears at the point where it is evaluated, and that it takes on with respect to the argument or the object that the whole proposition is taken to be about, (2) the value that the proposition <math>f\!</math> not only takes up the point <math>x,\!</math> but that it carries, conveys, transfers, or transports into the setting <math>^{\backprime\backprime} \{ x \in X : \underline{~~~} \} ^{\prime\prime}</math> or into any other context of discourse where <math>f\!</math> is meant to be evaluated, (3) the value that the sign <math>^{\backprime\backprime} f(x) ^{\prime\prime}</math> has in the context where it is placed, that it stands for in the context where it stands, and that it continues to stand for in this context just so long as the same proposition <math>f\!</math> and the same object <math>x\!</math> are borne in mind, and last but not least, (4) the value that the sign <math>^{\backprime\backprime} f(x) ^{\prime\prime}</math> represents to its full interpretive context as being its own logical interpretant, namely, the value that it signifies as its canonical connotation to any interpreter of the sign that is cognizant of the context in which it appears.

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The sentence <math>^{\backprime\backprime} f(x) = \underline{1} ^{\prime\prime}</math> indirectly names what the sign <math>^{\backprime\backprime} f(x) ^{\prime\prime}</math> more directly names, that is, the value <math>f(x).\!</math> In other words, the sentence <math>^{\backprime\backprime} f(x) = \underline{1} ^{\prime\prime}</math> has the same value to its interpretive context that the sign <math>^{\backprime\backprime} f(x) ^{\prime\prime}</math> imparts to any comparable context, each by way of its respective evaluation for the same <math>x \in X.</math>

The sentence "f(u) = 1" indirectly names what the sign "f(u)" more directly names, that is, the value f(u).  In other words, the sentence "f(u) = 1" has the same value to its interpretive context that the sign "f(u)" imparts to any comparable context, each by way of its respective evaluation for the same u ? U.

 

What is the relation among connoting, denoting, and ''evaluing'', where the last term is coined to describe all the ways of bearing, conveying, developing, or evolving a value in, to, or into an interpretive context? In other words, when a sign is evaluated to a particular value, one can say that the sign ''evalues'' that value, using the verb in a way that is categorically analogous or grammatically conjugate to the times when one says that a sign ''connotes'' an idea or that a sign ''denotes'' an object.  This does little more than provide the discussion with a ''weasel word'', a term that is designed to avoid the main issue, to put off deciding the exact relation between formal signs and formal values, and ultimately to finesse the question about the nature of formal values, the question whether they are more akin to conceptual signs and figurative ideas or the kinds of literal objects and platonic ideas that are independent of the mind.

These questions are confounded by the presence of certain peculiarities in formal discussions, especially by the fact that an equivalence class of signs is tantamount to a formal object.  This has the effect of allowing an abstract connotation to work as a formal denotation.  In other words, if the purpose of a sign is merely to lead its interpreter up to a sign in an equivalence class of signs, then it follows that this equivalence class is the object of the sign, that connotation can achieve denotation, at least, to some degree, and that the interpretant domain collapses with the object domain, at least, in some respect, all things being relative to the sign relation that embeds the discussion.

These questions are confounded by the presence of certain peculiarities in formal discussions, especially by the fact that an equivalence class of signs is tantamount to a formal object.  This has the effect of allowing an abstract connotation to work as a formal denotation.  In other words, if the purpose of a sign is merely to lead its interpreter up to a sign in an equivalence class of signs, then it follows that this equivalence class is the object of the sign, that connotation can achieve denotation, at least, to some degree, and that the interpretant domain collapses with the object domain, at least, in some respect, all things being relative to the sign relation that embeds the discussion.

Introducing the realm of "values" is a stopgap measure that temporarily permits the discussion to avoid certain singularities in the embedding sign relation, and allowing the process of "evaluation" as a compromise mode of signification between connotation and denotation only manages to steer around a topic that eventually has to be mapped in full, but these strategies do allow the discussion to proceed a little further without having to answer questions that are too difficult to be settled fully or even tackled directly at this point.  As far as the relations among connoting, denoting, and evaluing are concerned, it is possible that all of these constitute independent dimensions of significance that a sign can have, but since the notion of connotation is already generic enough to contain multitudes of subspecies, I am going to subsume, on a tentative basis, all of the conceivable modes of "evaluing" within the broader concept of connotation.

Introducing the realm of ''values'' is a stopgap measure that temporarily permits the discussion to avoid certain singularities in the embedding sign relation, and allowing the process of ''evaluation'' as a compromise mode of signification between connotation and denotation only manages to steer around a topic that eventually has to be mapped in full, but these strategies do allow the discussion to proceed a little further without having to answer questions that are too difficult to be settled fully or even tackled directly at this point.  As far as the relations among connoting, denoting, and evaluing are concerned, it is possible that all of these constitute independent dimensions of significance that a sign can have, but since the notion of connotation is already generic enough to contain multitudes of subspecies, I am going to subsume, on a tentative basis, all of the conceivable modes of ''evaluing'' within the broader concept of connotation.

With this degree of flexibility in mind, one can say that the sentence "f(u) = 1" latently connotes what the sign "f(u)" patently connotes.  Taken in abstraction, both syntactic entities fall into an equivalence class of signs that constitutes an abstract object, a thing of value that is "identified by" the sign "f(u)", and thus an object that might as well be "identified with" the value f(u).

With this degree of flexibility in mind, one can say that the sentence <math>^{\backprime\backprime} f(x) = \underline{1} ^{\prime\prime}</math> latently connotes what the sign <math>^{\backprime\backprime} f(x) ^{\prime\prime}</math> patently connotes.  Taken in abstraction, both syntactic entities fall into an equivalence class of signs that constitutes an abstract object, a thing of value that is ''identified by'' the sign <math>^{\backprime\backprime} f(x) ^{\prime\prime},</math> and thus an object that might as well be ''identified with'' the value <math>f(x).\!</math>

The upshot of this whole discussion of evaluation is that it allows one to rewrite the definitions of indicator functions and their fibers as follows:

The upshot of this whole discussion of evaluation is that it allows one to rewrite the definitions of indicator functions and their fibers as follows:

The "indicator function" or the "characteristic function" of a set X ? U, written "fX", is the map from U to the boolean domain B = {0, 1} that is defined in the following ways:

The "indicator function" or the "characteristic function" of a set X ? U, written "fX", is the map from U to the boolean domain B = {0, 1} that is defined in the following ways: