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If a sentence denotes a proposition <math>f : X \to \underline\mathbb{B},</math> then the ''value'' of the sentence with regard to <math>x \in X</math> is the value <math>f(x)\!</math> of the proposition at <math>x,\!</math> where <math>^{\backprime\backprime} \underline{0} ^{\prime\prime}</math> is interpreted as ''false'' and <math>^{\backprime\backprime} \underline{1} ^{\prime\prime}</math> is interpreted as ''true''.

If a sentence denotes a proposition <math>f : X \to \underline\mathbb{B},</math> then the ''value'' of the sentence with regard to <math>x \in X</math> is the value <math>f(x)\!</math> of the proposition at <math>x,\!</math> where <math>^{\backprime\backprime} \underline{0} ^{\prime\prime}</math> is interpreted as ''false'' and <math>^{\backprime\backprime} \underline{1} ^{\prime\prime}</math> is interpreted as ''true''.

Since the value of a sentence or a proposition depends on the universe of discourse to which it is referred, and since it also depends on the element of the universe with regard to which it is evaluated, it is usual to say that a sentence or a proposition ''refers'' to a universe and to its elements, though perhaps in a variety of different senses.  Furthermore, a proposition, acting in the role of as an indicator function, ''refers'' to the elements that it ''indicates'', namely, the elements on which it takes a positive value.  In order to sort out the possible confusions that are capable of arising here, I need to examine how these various notions of reference are related to the notion of denotation that is used in the pragmatic theory of sign relations.

Since the value of a sentence or a proposition depends on the universe of discourse to which it is "referred", and since it also depends on the element of the universe with regard to which it is evaluated, it is usual to say that a sentence or a proposition "refers" to a universe and to its elements, though perhaps in a variety of different senses.  Furthermore, a proposition, acting in the role of as an indicator function, "refers" to the elements that it "indicates", namely, the elements on which it takes a positive value.  In order to sort out the possible confusions that are capable of arising here, I need to examine how these various notions of reference are related to the notion of denotation that is used in the pragmatic theory of sign relations.

One way to resolve the various senses of "reference" that arise in this setting is to make the following sorts of distinctions among them.  Let the reference of a sentence or a proposition to a universe of discourse, the one that it acquires by way of taking on any interpretation at all, be taken as its "general reference", the kind of reference that one can safely ignore as irrelevant, at least, so long as one stays immersed in only one context of discourse or only one moment of discussion.  Let the references that an indicator function f has to the elements on which it evaluates to 0 be called its "negative references".  Let the references that an indicator function f has to the elements on which it evaluates to 1 be called its "positive references" or its "indications".  Finally, unspecified references to the "references" of a sentence, a proposition, or an indicator function can be taken by default as references to their specific, positive references.

One way to resolve the various senses of ''reference'' that arise in this setting is to make the following sorts of distinctions among them.  Let the reference of a sentence or a proposition to a universe of discourse, the one that it acquires by way of taking on any interpretation at all, be taken as its ''general reference'', the kind of reference that one can safely ignore as irrelevant, at least, so long as one stays immersed in only one context of discourse or only one moment of discussion.  Let the references that an indicator function <math>f\!</math> has to the elements on which it evaluates to <math>\underline{0}</math> be called its ''negative references''.  Let the references that an indicator function <math>f\!</math> has to the elements on which it evaluates to <math>\underline{1}</math> be called its ''positive references'' or its ''indications''.  Finally, unspecified references to the "references" of a sentence, a proposition, or an indicator function can be taken by default as references to their specific, positive references.

The universe of discourse for a sentence, the set whose elements the sentence is interpreted to be about, is not a property of the sentence by itself, but of the sentence in the presence of its interpretation.  Independently of how many explicit variables a sentence contains, its value can always be interpreted as depending on any number of implicit variables.  For instance, even a sentence with no explicit variable, a constant expression like "0" or "1", can be taken to denote a constant proposition of the form c : U �> B. Whether or not it has an explicit variable, I always take a sentence as referring to a proposition, one whose values refer to elements of a universe U.

The universe of discourse for a sentence, the set whose elements the sentence is interpreted to be about, is not a property of the sentence by itself, but of the sentence in the presence of its interpretation.  Independently of how many explicit variables a sentence contains, its value can always be interpreted as depending on any number of implicit variables.  For instance, even a sentence with no explicit variable, a constant expression like <math>^{\backprime\backprime} \underline{0} ^{\prime\prime}</math> or <math>^{\backprime\backprime} \underline{1} ^{\prime\prime},</math> can be taken to denote a constant proposition of the form <math>c : X \to \underline\mathbb{B}.</math>  Whether or not it has an explicit variable, I always take a sentence as referring to a proposition, one whose values refer to elements of a universe <math>X.\!</math>

Notice that the letters "P" and "Q", interpreted as signs that denote indicator functions P, Q : U �> B, have the character of sentences in relation to propositions, at least, they have the same status in this abstract discussion as genuine sentences have in concrete discussions.  This illustrates the relation between sentences and propositions as a special case of the relation between signs and objects.

Notice that the letters "P" and "Q", interpreted as signs that denote indicator functions P, Q : U �> B, have the character of sentences in relation to propositions, at least, they have the same status in this abstract discussion as genuine sentences have in concrete discussions.  This illustrates the relation between sentences and propositions as a special case of the relation between signs and objects.