Changes

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 <math>x,\!</math> for <math>x \in \underline\mathbb{B},</math> written <math>^{\backprime\backprime} \underline{(} x \underline{)} ^{\prime\prime}</math> and read <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>  In other words, negation is a monadic operation on boolean values, or a function of the form <math>\underline{(} \cdot \underline{)} : \underline\mathbb{B} \to \underline\mathbb{B}.</math>

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.

It is convenient to transport the product and the sum operations of <math>\mathbb{B}</math> into the logical setting of <math>\underline\mathbb{B},</math> where they can be symbolized by signs of the same character, underlined as necessary to avoid confusion.  This yields the following definitions of a ''product'' and a ''sum'' in <math>\underline\mathbb{B}</math> and leads to the following forms of multiplication and addition tables.

 

 

 

It is convenient to transport the product and the sum operations of <math>\mathbb{B}</math> into the logical setting of <math>\underline\mathbb{B},</math> where they can be symbolized by signs of the same character.  This yields the following definitions of a ''product'' and a ''sum'' in <math>\underline\mathbb{B}</math> and leads to the following forms of multiplication and addition tables.

The ''product'' of <math>x\!</math> and <math>y,\!</math> for values <math>x, y \in \underline\mathbb{B},</math> is given by Table&nbsp;8.  Viewed as a function of logical values, <math>\cdot : \underline\mathbb{B} \times \underline\mathbb{B} \to \underline\mathbb{B},</math> this corresponds to the logical operation that is commonly called ''conjunction'' and otherwise expressed as <math>^{\backprime\backprime} x ~\operatorname{and}~ y ^{\prime\prime}.</math> In accord with common practice, the multiplication sign <math>^{\backprime\backprime} \cdot ^{\prime\prime}</math> is often omitted from written expressions.

The ''product'' <math>x \cdot y</math> of two values <math>x\!</math> and <math>y\!</math> in <math>\underline\mathbb{B}</math> is given by Table&nbsp;9.  As a dyadic operation on boolean values, that is, a function of the form <math>f : \underline\mathbb{B} \times \underline\mathbb{B} \to \underline\mathbb{B},</math> the product corresponds to the logical operation of ''conjunction'', written <math>^{\backprime\backprime} x \land y ^{\prime\prime}</math> or <math>^{\backprime\backprime} x\!\And\!y ^{\prime\prime}</math> and read as <math>^{\backprime\backprime} x ~\operatorname{and}~ y ^{\prime\prime}.</math> In accord with common practice, the multiplication sign is often omitted from written expressions.

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<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"

|+ '''Table 8.  Product Operation for the Boolean Domain'''

|+ '''Table 9.  Product Operation for the Boolean Domain'''

|- style="background:whitesmoke"

|- style="background:whitesmoke"

| <math>\cdot\!</math>

| <math>\cdot\!</math>

<br>

<br>

The ''sum'' of <math>x\!</math> and <math>y,\!</math> for values <math>x, y \in \underline\mathbb{B},</math> is presented in Table&nbsp;9.  Viewed as a function of logical values, <math>+ : \underline\mathbb{B} \times \underline\mathbb{B} \to \underline\mathbb{B},</math> this corresponds to the logical operation that is commonly called ''exclusive disjunction'' and otherwise read as "<math>x\!</math> or <math>y\!</math> but not both". Depending on the context, other signs and readings that invoke this operation are:  <math>^{\backprime\backprime} x \ne y ^{\prime\prime},</math> read as "<math>x\!</math> is not equal to <math>y\!</math>" or as "exactly one of <math>x\!</math> and <math>y\!</math> is true", and <math>^{\backprime\backprime} x \not\equiv y ^{\prime\prime},</math> read as "<math>x\!</math> is not equivalent to <math>y\!</math>".

The ''sum'' <math>x + y\!</math> of two values <math>x\!</math> and <math>y\!</math> in <math>\underline\mathbb{B}</math> is given in Table&nbsp;10.  As a dyadic operation on boolean values, that is, a function of the form <math>f : \underline\mathbb{B} \times \underline\mathbb{B} \to \underline\mathbb{B},</math> the sum corresponds to the logical operation of ''exclusive disjunction'', usually read as <math>^{\backprime\backprime} x ~\text{or}~ y ~\text{but not both} ^{\prime\prime}.</math>  Depending on the context, other signs and readings that invoke this operation are:  <math>^{\backprime\backprime} x \ne y ^{\prime\prime}</math> or <math>^{\backprime\backprime} x \not\equiv y ^{\prime\prime},</math> read as <math>^{\backprime\backprime} x ~\text{is not equal to}~ y ^{\prime\prime},</math> <math>^{\backprime\backprime} x ~\text{is not equivalent to}~ y ^{\prime\prime},</math> or <math>^{\backprime\backprime} \text{exactly one of}~ x, y ~\text{is true} ^{\prime\prime}.</math>

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<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"

|+ '''Table 9.  Sum Operation for the Boolean Domain'''

|+ '''Table 10.  Sum Operation for the Boolean Domain'''

|- style="background:whitesmoke"

|- style="background:whitesmoke"

| <math>+\!</math>

| <math>+\!</math>