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Strictly speaking, however, there is a subtle distinction in type between the function ''f''<sub>''i''</sub> : ''X'' → '''B''' and the corresponding function ''g''<sub>''j''</sub> : E''X'' → '''B''', even though they share the same logical expression. Being human, we insist on preserving all the aesthetic delights afforded by the abstractly unified form of the "cake" while giving up none of the diverse contents that its substantive consummation can provide. In short, we want to maintain the logical equivalence of expressions that represent the same proposition, while appreciating the full diversity of that proposition's functional and typical representatives. Both perspectives, and all the levels of abstraction extending through them, have their reasons, as will develop in time.
Strictly speaking, however, there is a subtle distinction in type between the function <math>f_i : X \to \mathbb{B}</math> and the corresponding function <math>g_j : \operatorname{E}X \to \mathbb{B},</math> even though they share the same logical expression. Naturally, we want to maintain the logical equivalence of expressions that represent the same proposition while appreciating the full diversity of that proposition's functional and typical representatives. Both perspectives, and all the levels of abstraction extending through them, have their reasons, as will develop in time.
Because this special circumstance points up an important general theme, it is a good idea to discuss it more carefully. Whenever there arises a situation like this, where one alphabet <font face="lucida calligraphy">X</font> is a subset of another alphabet <font face="lucida calligraphy">Y</font>, then we say that any proposition ''f'' : 〈<font face="lucida calligraphy">X</font>〉 → ''B'' has a ''tacit extension'' to a proposition <math>\epsilon</math>''f'' : 〈<font face="lucida calligraphy">Y</font>〉 → '''B''', and that the space (〈<font face="lucida calligraphy">X</font>〉 → '''B''') has an ''automatic embedding'' within the space (〈<font face="lucida calligraphy">Y</font>〉 → '''B'''). The extension is defined in such a way that <math>\epsilon</math>''f'' puts the same constraint on the variables of <font face="lucida calligraphy">X</font> that are contained in <font face="lucida calligraphy">Y</font> as the proposition ''f'' initially did, while it puts no constraint on the variables of <font face="lucida calligraphy">Y</font> outside of <font face="lucida calligraphy">X</font>, in effect, conjoining the two constraints.
Because this special circumstance points up an important general theme, it is a good idea to discuss it more carefully. Whenever there arises a situation like this, where one alphabet <math>\mathcal{X}</math> is a subset of another alphabet <math>\mathcal{Y},</math> then we say that any proposition <math>f : \langle \mathcal{X} \rangle \to \mathbb{B}</math> has a ''tacit extension'' to a proposition <math>\epsilon f : \langle \mathcal{Y} \rangle \to \mathbb{B},</math> and that the space <math>(\langle \mathcal{X} \rangle \to \mathbb{B})</math> has an ''automatic embedding'' within the space <math>(\langle \mathcal{Y} \rangle \to \mathbb{B}).</math> The extension is defined in such a way that <math>\epsilon f\!</math> puts the same constraint on the variables of <math>\mathcal{X}</math> that are contained in <math>\mathcal{Y}</math> as the proposition <math>f\!</math> initially did, while it puts no constraint on the variables of <math>\mathcal{Y}</math> outside of <math>\mathcal{X},</math> in effect, conjoining the two constraints.
If the variables in question are indexed as <font face="lucida calligraphy">X</font> = {''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>} and <font face="lucida calligraphy">Y</font> = {''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>, …, ''x''<sub>''n''+''k''</sub>}, then the definition of the tacit extension from <font face="lucida calligraphy">X</font> to <font face="lucida calligraphy">Y</font> may be expressed in the form of an equation:
If the variables in question are indexed as <math>\mathcal{X} = \{ x_1, \ldots, x_n \}</math> and <math>\mathcal{Y} = \{ x_1, \ldots, x_n, \ldots, x_{n+k} \},</math> then the definition of the tacit extension from <math>\mathcal{X}</math> to <math>\mathcal{Y}</math> may be expressed in the form of an equation:
: <math>\epsilon</math>''f''(''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>, …, ''x''<sub>''n''+''k''</sub>) = ''f''(''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>).
: <math>\epsilon f(x_1, \ldots, x_n, \ldots, x_{n+k}) = f(x_1, \ldots, x_n).</math>
On formal occasions, such as the present context of definition, the tacit extension from <font face="lucida calligraphy">X</font> to <font face="lucida calligraphy">Y</font> is explicitly symbolized by the operator <math>\epsilon</math> : (〈<font face="lucida calligraphy">X</font>〉 → '''B''') → (〈<font face="lucida calligraphy">Y</font>〉 → '''B'''), where the appropriate alphabets <font face="lucida calligraphy">X</font> and <font face="lucida calligraphy">Y</font> are understood from context, but normally one may leave the "<math>\epsilon</math>" silent.
On formal occasions, such as the present context of definition, the tacit extension from <math>\mathcal{X}</math> to <math>\mathcal{Y}</math> is explicitly symbolized by the operator <math>\epsilon : (\langle \mathcal{X} \rangle \to \mathbb{B}) \to (\langle \mathcal{Y} \rangle \to \mathbb{B}),</math> where the appropriate alphabets <math>\mathcal{X}</math> and <math>\mathcal{Y}</math> are understood from context, but normally one may leave the "<math>\epsilon\!</math>" silent.
Let's explore what this means for the present Example. Here, <font face="lucida calligraphy">X</font> = {''A''} and <font face="lucida calligraphy">Y</font> = E<font face="lucida calligraphy">X</font> = {''A'', d''A''}. For each of the propositions ''f''<sub>''i''</sub> over ''X'', specifically, those whose expression ''e''<sub>''i''</sub> lies in the collection {0, (''A''), ''A'', 1}, the tacit extension <math>\epsilon</math>''f'' of ''f'' to E''X'' can be phrased as a logical conjunction of two factors, ''f''<sub>''i''</sub> = ''e''<sub>''i''</sub> '''·''' <math>\tau</math> , where <math>\tau</math> is a logical tautology that uses all the variables of <font face="lucida calligraphy">Y</font> – <font face="lucida calligraphy">X</font>. Working in these terms, the tacit extensions <math>\epsilon</math>''f'' of ''f'' to E''X'' may be explicated as shown in Table 15.
Let's explore what this means for the present Example. Here, <math>\mathcal{X} = \{ A \}</math> and <math>\mathcal{Y} = \operatorname{E}\mathcal{X} = \{ A, \operatorname{d}A \}.</math> For each of the propositions <math>f_i\!</math> over <math>X\!,</math> specifically, those whose expression <math>e_i\!</math> lies in the collection <math>\{ 0, (A), A, 1 \},\!</math> the tacit extension <math>\epsilon f\!</math> of <math>f\!</math> to <math>\operatorname{E}X</math> can be phrased as a logical conjunction of two factors, <math>f_i = e_i \cdot \tau\ ,</math> where <math>\tau\!</math> is a logical tautology that uses all the variables of <math>\mathcal{Y} - \mathcal{X}.</math> Working in these terms, the tacit extensions <math>\epsilon f\!</math> of <math>f\!</math> to <math>\operatorname{E}X</math> may be explicated as shown in Table 15.
<font face="courier new">
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
|+ '''Table 15. Tacit Extension of [''A''] to [''A'', d''A'']'''
|+ '''Table 15. Tacit Extension of <math>[A]\!</math> to <math>[A, \operatorname{d}A]</math>'''
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|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
|
|
| 0
| <math>0\!</math>
| =
| <math>=\!</math>
| 0
| <math>0\!</math>
| ·
| <math>\cdot\!</math>
| ((d''A''), d''A'')
| <math>((\operatorname{d}A),\ \operatorname{d}A)\!</math>
| =
| <math>=\!</math>
| 0
| <math>0\!</math>
|
|
|-
|-
|
|
| (''A'')
| <math>(A)\!</math>
| =
| <math>=\!</math>
| (''A'')
| <math>(A)\!</math>
| ·
| <math>\cdot\!</math>
| ((d''A''), d''A'')
| <math>((\operatorname{d}A),\ \operatorname{d}A)\!</math>
| =
| <math>=\!</math>
| (''A'')(d''A'') + (''A'') d''A''
| <math>(A)(\operatorname{d}A)\ +\ (A)\ \operatorname{d}A\!</math>
|
|
|-
|-
|
|
| ''A''
| <math>A\!</math>
| =
| <math>=\!</math>
| ''A''
| <math>A\!</math>
| ·
| <math>\cdot\!</math>
| ((d''A''), d''A'')
| <math>((\operatorname{d}A),\ \operatorname{d}A)\!</math>
| =
| <math>=\!</math>
| ''A'' (d''A'') + ''A'' d''A''
| <math>A\ (\operatorname{d}A)\ +\ A\ \operatorname{d}A\!</math>
|
|
|-
|-
|
|
| 1
| <math>1\!</math>
| =
| <math>=\!</math>
| 1
| <math>1\!</math>
| ·
| <math>\cdot\!</math>
| ((d''A''), d''A'')
| <math>((\operatorname{d}A),\ \operatorname{d}A)\!</math>
| =
| <math>=\!</math>
| 1
| <math>1\!</math>
|}
|}
|}
|}
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In its effect on the singular propositions over ''X'', this analysis has an interesting interpretation. The tacit extension takes us from thinking about a particular state, like ''A'' or (''A''), to considering the collection of outcomes, the outgoing changes or the singular dispositions, that spring from that state.
In its effect on the singular propositions over <math>X,\!</math> this analysis has an interesting interpretation. The tacit extension takes us from thinking about a particular state, like <math>A\!</math> or <math>(A),\!</math> to considering the collection of outcomes, the outgoing changes or the singular dispositions, that spring from that state.
===Example 2. Drives and Their Vicissitudes===
===Example 2. Drives and Their Vicissitudes===