Changes

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 <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.

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'',&nbsp;d''A''}.  For each of the propositions ''f''_''i'' over ''X'', specifically, those whose expression ''e''_''i'' lies in the collection {0,&nbsp;(''A''),&nbsp;''A'',&nbsp;1}, the tacit extension <math>\epsilon</math>''f'' of ''f'' to E''X'' can be phrased as a logical conjunction of two factors, ''f''_''i'' = ''e''_''i''&nbsp;'''·'''&nbsp;<math>\tau</math>&nbsp;, where <math>\tau</math> is a logical tautology that uses all the variables of <font face="lucida calligraphy">Y</font>&nbsp;&ndash;&nbsp;<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&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.

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