Changes

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.

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>&nbsp;=&nbsp;{''x''₁,&nbsp;&hellip;,&nbsp;''x''_''n''} and <font face="lucida calligraphy">Y</font>&nbsp;=&nbsp;{''x''₁,&nbsp;&hellip;,&nbsp;''x''_''n'',&nbsp;&hellip;,&nbsp;''x''_''n''+''k''}, 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''₁,&nbsp;&hellip;,&nbsp;''x''_''n'',&nbsp;&hellip;,&nbsp;''x''_''n''+''k'') = ''f''(''x''₁,&nbsp;&hellip;,&nbsp;''x''_''n'').

: <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>&nbsp;:&nbsp;(〈<font face="lucida calligraphy">X</font>〉&nbsp;&rarr;&nbsp;'''B''')&nbsp;&rarr;&nbsp;(〈<font face="lucida calligraphy">Y</font>〉&nbsp;&rarr;&nbsp;'''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, <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&nbsp;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&nbsp;15.