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A ''proposition about things in the universe'', more simply, a ''proposition'', is the same thing as an indicator function, that is, a function of the form <math>f : X \to \underline\mathbb{B}.</math>  The convenience of this seemingly redundant usage is that it permits one to refer to an indicator function without having to specify right away, as a part of its only available designation, exactly what set it indicates, even though a proposition is always an indicator function of some subset of the universe, and even though one probably or eventually wants to know which one.

A ''proposition about things in the universe'', for short, a ''proposition'', is the same thing as an indicator function, that is, a function of the form <math>f : X \to \underline\mathbb{B}.</math>  The convenience of this seemingly redundant usage is that it permits one to refer to an indicator function without having to specify right away, as a part of its only available designation, exactly what set it indicates, even though a proposition is always an indicator function of some subset of the universe, and even though one probably or eventually wants to know which one.

According to the stated understandings, a proposition is a function that indicates a set, in the sense that a function associates values with the elements of a domain, some which values can be interpreted to mark out for special consideration a subset of that domain.  The way in which an indicator function is imagined to "indicate" a set can be expressed in terms of the following concepts.

According to the stated understandings, a proposition is a function that indicates a set, in the sense that a function associates values with the elements of a domain, some which values can be interpreted to mark out for special consideration a subset of that domain.  The way in which an indicator function is imagined to "indicate" a set can be expressed in terms of the following concepts.

The ''projective imagination'' of <math>\underline\mathbb{B}^k</math> is the imagination <math>(\pi_1, \ldots, \pi_k).</math>

The ''projective imagination'' of <math>\underline\mathbb{B}^k</math> is the imagination <math>(\pi_1, \ldots, \pi_k).</math>

A ''sentence about things in the universe'', for short, a "sentence", is a sign that denotes a proposition.  In other words, a sentence is any sign that denotes an indicator function, any sign whose object is a function of the form <math>f : X \to \underline\mathbb{B}.</math>

A "sentence about things in the universe", for short, a "sentence", is a sign that denotes a proposition.  In other words, a sentence is any sign that denotes an indicator function, any sign whose object is a function of the form f : U �> B.

 

To emphasize the empirical contingency of this definition, one can say that a sentence is any sign that is interpreted as naming a proposition, any sign that is taken to denote an indicator function, or any sign whose object happens to be a function of the form <math>f : X \to \underline\mathbb{B}.</math>

To emphasize the empirical contingency of this definition, one can say that a sentence is any sign that is interpreted as naming a proposition, any sign that is taken to denote an indicator function, or any sign whose object happens to be a function of the form f : U �> B.

An ''expression'' is a type of sign, for instance, a term or a sentence, that has a value.  In this conception of an expression, I am deliberately leaving a number of options open, like whether it amounts to a term or to a sentence and whether it ought to be accounted as denoting a value or as connoting a value.  Perhaps the expression has different values under different lights, and perhaps it relates to them differently in different respects.  In the end, what one calls an expression matters less than where its value lies.  Of course, no matter whether one calls an expression a ''term'' or a ''sentence'', if the value is an element in <math>\underline\mathbb{B},</math> then the expression affords the option of being treated as a sentence, meaning that it is subject to assertion and composition in the same way that any sentence is, having its value figure into the values of larger expressions through the linkages of sentential connectives, and allowing the consideration of what things in what universe the corresponding proposition indicates.

Expressions with this degree of flexibility in the types under which they can be interpreted are difficult to translate from their formal settings into more natural contexts.  Indeed, the whole issue can be difficult to talk about, or even to think about, since the grammatical categories of sentences and noun phrases are not so fluid in natural language settings are they can made in artificial arenas.

Expressions with this degree of flexibility in the types under which they can be interpreted are difficult to translate from their formal settings into more natural contexts.  Indeed, the whole issue can be difficult to talk about, or even to think about, since the grammatical categories of sentences and noun phrases are not so fluid in natural language settings are they can made in artificial arenas.

To finesse the issue of whether an expression denotes or connotes its value, or else to create general term that covers what both possibilities have in common, one can say that an expression "evalues" its value.

To finesse the issue of whether an expression denotes or connotes its value, or else to create general term that covers what both possibilities have in common, one can say that an expression ''evalues'' its value.

An "assertion" is just a sentence that is being used in a certain way, namely, to indicate the indication of the indicator function that the sentence is usually used to denote.  In other words, an assertion is a sentence that is being converted to a certain use or being interpreted in a certain role, and one whose immediate denotation is being pursued to its substantive indication, specifically, the fiber of truth of the proposition that the sentence potentially denotes.  Thus, an assertion is a sentence that is held to denote the set of things in the universe of which the sentence is true.

An ''assertion'' is just a sentence that is being used in a certain way, namely, to indicate the indication of the indicator function that the sentence is usually used to denote.  In other words, an assertion is a sentence that is being converted to a certain use or being interpreted in a certain role, and one whose immediate denotation is being pursued to its substantive indication, specifically, the fiber of truth of the proposition that the sentence potentially denotes.  Thus, an assertion is a sentence that is held to denote the set of things in the universe of which the sentence is true.

Taken in a context of communication, an assertion is basically a request that the interpreter consider the things for which the sentence is true, in other words, to find the fiber of truth in the associated proposition, or to invert the indicator function that is denoted by the sentence with respect to its possible value of truth.

Taken in a context of communication, an assertion is basically a request that the interpreter consider the things for which the sentence is true, in other words, to find the fiber of truth in the associated proposition, or to invert the indicator function that is denoted by the sentence with respect to its possible value of truth.

A "denial" of a sentence S is an assertion of its negation (S).  It acts as a request to think about the things for which the sentence is false, in other words, to find the fiber of falsity in the indicted proposition, or to invert the indicator function that is denoted by the sentence with respect to its possible value of falsity.

A "denial" of a sentence S is an assertion of its negation (S).  It acts as a request to think about the things for which the sentence is false, in other words, to find the fiber of falsity in the indicted proposition, or to invert the indicator function that is denoted by the sentence with respect to its possible value of falsity.