Changes

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 might be able to enjoy, 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

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 might be able to enjoy, 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.

of ''evaluing'' within the broader concept of connotation.

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>

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 us to rewrite the definitions of indicator functions and their fibers as follows:

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

The ''indicator function'' or the ''characteristic function'' of a set <math>Q \in X,</math> written <math>f_Q,\!</math> is the map from <math>X\!</math> to the boolean domain <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \}</math> that is defined in the following ways:

The ''indicator function'' or the ''characteristic function'' of a set <math>Q \subseteq X,</math> written <math>f_Q,\!</math> is the map from <math>X\!</math> to the boolean domain <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \}</math> that is defined in the following ways:

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