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The ''boolean domain'' is the set <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \}</math> of two logical values, whose elements can be read as "false" and "true", or as "falsity" and "truth", respectively.

The ''boolean domain'' is the set <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \}</math> of two logical values, whose elements can be read as "false" and "true", or as "falsity" and "truth", respectively.

At this point, I cannot tell whether the distinction between these two domains is slight or significant, and so this question must evolve its own answer, while I pursue a larger inquiry by means of its hypothesis.  The weight of the matter appears to increase as the investigation moves from abstract, algebraic, and formal settings to contexts where logical semantics, natural language syntax, and concrete categories of grammar are compelling considerations.  Speaking roughly or abstractly enough, it is often acceptable to identify these two domains, and up until this point there has rarely appeared to be a sufficient reason to keep their concepts separately in mind.  The boolean domain <math>\underline\mathbb{B}</math> comes with at least two operations, though often under different names and always included in a number of others, that are analogous to the field operations of the binary domain <math>\mathbb{B},</math> and operations that are isomorphic to the rest of the boolean operations in <math>\underline\mathbb{B}</math> can always be built on the binary basis of <math>\mathbb{B}.</math> Of course, as sets of the same cardinality, the domains <math>\mathbb{B}</math> and <math>\underline\mathbb{B}</math> and all of the structures that can be built on them become isomorphic at a high enough level of abstraction.  Consequently, the main reason for making this distinction in the present context appears to be a matter more of grammar than an issue of logical or mathematical substance, namely, so that the signs <math>^{\backprime\backprime} \underline{0} ^{\prime\prime}</math> and <math>^{\backprime\backprime} \underline{1} ^{\prime\prime}</math> can appear with some semblance of syntactic legitimacy in linguistic contexts that call for a grammatical sentence or a sentence surrogate to represent the classes of sentences that are ''always false'' and ''always true'', respectively.  The signs <math>^{\backprime\backprime} 0 ^{\prime\prime}</math> and <math>^{\backprime\backprime} 1 ^{\prime\prime},</math> customarily read as nouns but not as sentences, fail to be suitable for this purpose.  Whether these scruples, that are needed to conform to a particular choice of natural language context, are ultimately important, is another thing that remains to be determined.

At this point, I cannot tell whether the distinction between these two domains is slight or significant, and so this question must evolve its own answer, while I pursue a larger inquiry by means of its hypothesis.  The weight of the matter appears to increase as the investigation moves from abstract, algebraic, and formal settings to contexts where logical semantics, natural language syntax, and concrete categories of grammar are compelling considerations.  Speaking roughly or abstractly enough, it is often acceptable to identify these two domains, and up until this point there has rarely appeared to be a sufficient reason to keep their concepts separately in mind.  The boolean domain <math>\underline\mathbb{B}</math> comes with at least two operations, though often under different names and always included in a number of others, that are analogous to the field operations of the binary domain <math>\mathbb{B},</math> and operations that are isomorphic to the rest of the boolean operations in <math>\underline\mathbb{B}</math> can always be built on the binary basis of <math>\mathbb{B}.</math>

 

Of course, as sets of the same cardinality, the domains <math>\mathbb{B}</math> and <math>\underline\mathbb{B}</math> and all of the structures that can be built on them become isomorphic at a high enough level of abstraction.  Consequently, the main reason for making this distinction in the present context appears to be a matter more of grammar than an issue of logical or mathematical substance, namely, so that the signs <math>^{\backprime\backprime} \underline{0} ^{\prime\prime}</math> and <math>^{\backprime\backprime} \underline{1} ^{\prime\prime}</math> can appear with some semblance of syntactic legitimacy in linguistic contexts that call for a grammatical sentence or a sentence surrogate to represent the classes of sentences that are ''always false'' and ''always true'', respectively.  The signs <math>^{\backprime\backprime} 0 ^{\prime\prime}</math> and <math>^{\backprime\backprime} 1 ^{\prime\prime},</math> customarily read as nouns but not as sentences, fail to be suitable for this purpose.  Whether these scruples, that are needed to conform to a particular choice of natural language context, are ultimately important, is another thing that remains to be determined.

The ''negation'' of a value <math>x\!</math> in <math>\underline\mathbb{B},</math> written <math>^{\backprime\backprime} \underline{(} x \underline{)} ^{\prime\prime}</math> or <math>^{\backprime\backprime} \lnot x ^{\prime\prime}</math> and read as <math>^{\backprime\backprime} \operatorname{not}\ x ^{\prime\prime},</math> is the boolean value <math>\underline{(} x \underline{)} \in \underline\mathbb{B}</math> that is <math>\underline{1}</math> when <math>x\!</math> is <math>\underline{0}</math> and <math>\underline{0}</math> when <math>x\!</math> is <math>\underline{1}.</math>  Negation is a monadic operation on boolean values, that is, a function of the form <math>f : \underline\mathbb{B} \to \underline\mathbb{B},</math> as shown in Table&nbsp;8.

The ''negation'' of a value <math>x\!</math> in <math>\underline\mathbb{B},</math> written <math>^{\backprime\backprime} \underline{(} x \underline{)} ^{\prime\prime}</math> or <math>^{\backprime\backprime} \lnot x ^{\prime\prime}</math> and read as <math>^{\backprime\backprime} \operatorname{not}\ x ^{\prime\prime},</math> is the boolean value <math>\underline{(} x \underline{)} \in \underline\mathbb{B}</math> that is <math>\underline{1}</math> when <math>x\!</math> is <math>\underline{0}</math> and <math>\underline{0}</math> when <math>x\!</math> is <math>\underline{1}.</math>  Negation is a monadic operation on boolean values, that is, a function of the form <math>f : \underline\mathbb{B} \to \underline\mathbb{B},</math> as shown in Table&nbsp;8.

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 ''fiber'' of a codomain element <math>y \in Y\!</math> under a function <math>f : X \to Y</math> is the subset of the domain <math>X\!</math> that is mapped onto <math>y,\!</math> in short, it is <math>f^{-1} (y) \subseteq X.</math>

The ''fiber'' of a codomain element <math>y \in Y\!</math> under a function <math>f : X \to Y</math> is the subset of the domain <math>X\!</math> that is mapped onto <math>y,\!</math> in short, it is <math>f^{-1} (y) \subseteq X.</math> In other language that is often used, the fiber of <math>y\!</math> under <math>f\!</math> is called the ''antecedent set'', the ''inverse image'', the ''level set'', or the ''pre-image'' of <math>y\!</math> under <math>f.\!</math>  All of these equivalent concepts are defined as follows:

 

In other language that is often used, the fiber of <math>y\!</math> under <math>f\!</math> is called the ''antecedent set'', the ''inverse image'', the ''level set'', or the ''pre-image'' of <math>y\!</math> under <math>f.\!</math>  All of these equivalent concepts are defined as follows:

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