Changes

The ''negation'' of a sentence <math>s\!</math>, written as <math>^{\backprime\backprime} \, \texttt{(} s \texttt{)} \, ^{\prime\prime}</math> and read as <math>^{\backprime\backprime} \, \operatorname{not}\ s \, ^{\prime\prime}</math>, is a sentence that is true when <math>s\!</math> is false and false when <math>s\!</math> is true.

The ''negation'' of a sentence <math>s\!</math>, written as <math>^{\backprime\backprime} \, \texttt{(} s \texttt{)} \, ^{\prime\prime}</math> and read as <math>^{\backprime\backprime} \, \operatorname{not}\ s \, ^{\prime\prime}</math>, is a sentence that is true when <math>s\!</math> is false and false when <math>s\!</math> is true.

The ''complement'' of a set <math>Q\!</math> with respect to the universe <math>X\!</math> is denoted by <math>^{\backprime\backprime} \, X\!-\!Q \, ^{\prime\prime}</math>, or simply by <math>^{\backprime\backprime} \, \tilde{Q} \, ^{\prime\prime}</math> when the universe <math>X\!</math> is determinate, and is defined as the set of elements in <math>X\!</math> that do not belong to <math>Q,\!</math> that is:

The ''complement'' of a set <math>Q\!</math> with respect to the universe <math>X\!</math> is denoted by <math>^{\backprime\backprime} \, X\!-\!Q \, ^{\prime\prime}</math> and is defined as the set of elements in <math>X\!</math> that do not belong to <math>Q\!</math>.  When the universe <math>X\!</math> is fixed throughout a given discussion, the complement <math>X\!-\!Q</math> may be denoted either by <math>^{\backprime\backprime} \thicksim \! Q \, ^{\prime\prime}</math> or by <math>^{\backprime\backprime} \, \tilde{Q} \, ^{\prime\prime}</math>.  Thus we have the following series of equivalences:

{| align="center" cellpadding="8" width="90%"

{| align="center" cellpadding="8" width="90%"

|

|

<math>\begin{array}{lllll}

<math>\begin{array}{lllllll}

\tilde{Q}

\tilde{Q}

& = &

& = &

X\!-\!Q

X\!-\!Q

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 and 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.

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 and 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} \texttt{(} x \texttt{)} ^{\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>\texttt{(} x \texttt{)} \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.

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|- style="background:whitesmoke"

|- style="background:whitesmoke"

| <math>x\!</math>

| <math>x\!</math>

| <math>\underline{(} x \underline{)}</math>

| <math>\texttt{(} x \texttt{)}</math>

|-

|-

| <math>\underline{0}</math>

| <math>\underline{0}</math>