Changes

For the purposes of this discussion, let it be supposed that each set <math>Q,\!</math> that comprises a subject of interest in a particular discussion or that constitutes a topic of interest in a particular moment of discussion, is a subset of a set <math>X,\!</math> one that is sufficiently universal relative to that discussion or big enough to cover everything that is being talked about in that moment.  In a setting like this it is possible to make a number of useful definitions, to which we now turn.

For the purposes of this discussion, let it be supposed that each set <math>Q,\!</math> that comprises a subject of interest in a particular discussion or that constitutes a topic of interest in a particular moment of discussion, is a subset of a set <math>X,\!</math> one that is sufficiently universal relative to that discussion or big enough to cover everything that is being talked about in that moment.  In a setting like this it is possible to make a number of useful definitions, to which we now turn.

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} \, \underline{(} s \underline{)} \, ^{\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> 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:

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:

X\!-\!Q

X\!-\!Q

& = &

& = &

\{ \, x \in X : \texttt{(} x \in Q \texttt{)} \, \}.

\{ \, x \in X : \underline{(} x \in Q \underline{)} \, \}.

\\

\\

\end{array}</math>

\end{array}</math>

Q\!-\!P

Q\!-\!P

& = &

& = &

\{ \, x \in X : x \in Q ~\operatorname{and}~ \texttt{(} x \in P \texttt{)} \, \}.

\{ \, x \in X : x \in Q ~\operatorname{and}~ \underline{(} x \in P \underline{)} \, \}.

\\

\\

\end{array}</math>

\end{array}</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 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} \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.

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.

<br>

<br>

|- style="background:whitesmoke"

|- style="background:whitesmoke"

| <math>x\!</math>

| <math>x\!</math>

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

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

|-

|-

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

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

<br>

<br>

For sentences, the signs of equality (<math>=\!</math>) and inequality (<math>\ne\!</math>) are reserved to signify the syntactic identity and non-identity, respectively, of the literal strings of characters that make up the sentences in question, while the signs of equivalence (<math>\Leftrightarrow</math>) and inequivalence (<math>\not\Leftrightarrow</math>) refer to the logical values, if any, of these strings, and thus they signify the equality and inequality, respectively, of their conceivable boolean values.  For the logical values themselves, the two pairs of symbols collapse in their senses to a single pair, signifying a single form of coincidence or a single form of distinction, respectively, between the boolean values of the entities involved.

For sentences, the signs of equality <math>(=)\!</math> and inequality <math>(\ne)\!</math> are reserved to signify the syntactic identity and non-identity, respectively, of the literal strings of characters that make up the sentences in question, while the signs of equivalence <math>(\Leftrightarrow)</math>) and inequivalence <math>(\not\Leftrightarrow)</math> refer to the logical values, if any, of these strings, and thus they signify the equality and inequality, respectively, of their conceivable boolean values.  For the logical values themselves, the two pairs of symbols collapse in their senses to a single pair, signifying a single form of coincidence or a single form of distinction, respectively, between the boolean values of the entities involved.

In logical studies, one tends to be interested in all of the operations or all of the functions of a given type, at least, to the extent that their totalities and their individualities can be comprehended, and not just the specialized collections that define particular algebraic structures.  Although the remainder of the dyadic operations on boolean values, in other words, the rest of the sixteen functions of the form <math>f : \underline\mathbb{B} \times \underline\mathbb{B} \to \underline\mathbb{B},</math> could be presented in the same way as the multiplication and addition tables, it is better to look for a more efficient style of representation, one that is able to express all of the boolean functions on the same number of variables on a roughly equal basis, and with a bit of luck, affords us with a calculus for computing with these functions.

In logical studies, one tends to be interested in all of the operations or all of the functions of a given type, at least, to the extent that their totalities and their individualities can be comprehended, and not just the specialized collections that define particular algebraic structures.  Although the remainder of the dyadic operations on boolean values, in other words, the rest of the sixteen functions of the form <math>f : \underline\mathbb{B} \times \underline\mathbb{B} \to \underline\mathbb{B},</math> could be presented in the same way as the multiplication and addition tables, it is better to look for a more efficient style of representation, one that is able to express all of the boolean functions on the same number of variables on a roughly equal basis, and with a bit of luck, affords us with a calculus for computing with these functions.