Changes

Earlier I defined the tacit extension operators <math>\epsilon</math>&nbsp;:&nbsp;''X''^&nbsp;&bull;&nbsp;&rarr;&nbsp;''Y''^&nbsp;&bull; as maps embedding each proposition of a given universe ''X''^&nbsp;&bull; in a more generously given universe ''Y''^&nbsp;&bull; containing ''X''^&nbsp;&bull;.  Of immediate interest are the tacit extensions <math>\epsilon</math>&nbsp;:&nbsp;''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;E''U''^&nbsp;&bull;, that locate each proposition of ''U''^&nbsp;&bull; in the enlarged context of E''U''^&nbsp;&bull;.  In its application to the propositional conjunction ''J''&nbsp;=&nbsp;''u''&nbsp;''v'' in [''u'',&nbsp;''v''], the tacit extension operator <math>\epsilon</math> produces the proposition <math>\epsilon</math>''J'' in E''U''^&nbsp;&bull;&nbsp;=&nbsp;[''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''].  The extended proposition <math>\epsilon</math>''J'' may be computed according to the scheme in Table&nbsp;36, in effect, doing nothing more than conjoining a tautology of [d''u'',&nbsp;d''v''] to ''J'' in ''U''^&nbsp;&bull;.

Earlier I defined the tacit extension operators <math>\epsilon</math>&nbsp;:&nbsp;''X''^&nbsp;&bull;&nbsp;&rarr;&nbsp;''Y''^&nbsp;&bull; as maps embedding each proposition of a given universe ''X''^&nbsp;&bull; in a more generously given universe ''Y''^&nbsp;&bull; containing ''X''^&nbsp;&bull;.  Of immediate interest are the tacit extensions <math>\epsilon</math>&nbsp;:&nbsp;''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;E''U''^&nbsp;&bull;, that locate each proposition of ''U''^&nbsp;&bull; in the enlarged context of E''U''^&nbsp;&bull;.  In its application to the propositional conjunction ''J''&nbsp;=&nbsp;''u''&nbsp;''v'' in [''u'',&nbsp;''v''], the tacit extension operator <math>\epsilon</math> produces the proposition <math>\epsilon</math>''J'' in E''U''^&nbsp;&bull;&nbsp;=&nbsp;[''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''].  The extended proposition <math>\epsilon</math>''J'' may be computed according to the scheme in Table&nbsp;36, in effect, doing nothing more than conjoining a tautology of [d''u'',&nbsp;d''v''] to ''J'' in ''U''^&nbsp;&bull;.

<pre>

<font face="courier new">

Table 36.  Computation of !e!J

o---------------------------------------------------------------------o

|+ Table 36.  Computation of <math>\epsilon</math>''J''

|                                                                     |

| !e!J  = J<u, v>                                                    |

{| align="left" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"

|                                                                     |

| width="8%" | <math>\epsilon</math>''J''

|       = u v                                                       |

| width="4%" | =

|                                                                     |

| ''J''‹''u'', ''v''›

|       = u v (du)(dv) + u v (du) dv + u v du (dv)  + u v du dv |

|-

|                                                                     |

| width="8%" | &nbsp;

o---------------------------------------------------------------------o

| width="4%" | =

|                                                                     |

| ''u'' ''v''

| !e!J = u v (du)(dv)  +                                           |

|-

|         u v (du) dv  +                                           |

| width="8%" | &nbsp;

|         u v du (dv)  +                                           |

| width="4%" | =

|         u v  du  dv                                                |

| ''u'' ''v'' (d''u'')(d''v'') || +

|                                                                     |

| ''u'' ''v'' (d''u'') d''v'' || +

| ''u'' ''v'' d''u'' (d''v'')  || +

</pre>

| ''u'' ''v'' d''u'' d''v''

|}

|-

|

{| align="left" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"

| width="8%"  | <math>\epsilon</math>''J''

| width="4%" | =

| width="24%" | ''u'' ''v''&nbsp;(d''u'')(d''v'')

| width="4%" | +

| width="60%" | &nbsp;

| width="8%"  | &nbsp;

| width="24%" | ''u'' ''v''&nbsp;(d''u'')&nbsp;d''v''

| width="4%"  | +

| width="60%" | &nbsp;

| width="24%" | ''u'' ''v''&nbsp;&nbsp;d''u''&nbsp;(d''v'')

| width="4%" | +

| width="60%" | &nbsp;

|-

| width="24%" | ''u'' ''v''&nbsp;&nbsp;d''u''&nbsp;&nbsp;d''v''

| width="4%" | &nbsp;

| width="60%" | &nbsp;

</font><br>

The lower portion of the Table contains the dispositional features of <math>\epsilon</math>''J'' arranged in such a way that the variety of ordinary features spreads across the rows and the variety of differential features runs through the columns.  This organization serves to facilitate pattern matching in the remainder of our computations.  Again, the tacit extension is usually so trivial a concern that we do not always bother to make an explicit note of it, taking it for granted that any function ''F'' that is being employed in a differential context is equivalent to <math>\epsilon</math>''F'', for a suitable <math>\epsilon</math>.

The lower portion of the Table contains the dispositional features of <math>\epsilon</math>''J'' arranged in such a way that the variety of ordinary features spreads across the rows and the variety of differential features runs through the columns.  This organization serves to facilitate pattern matching in the remainder of our computations.  Again, the tacit extension is usually so trivial a concern that we do not always bother to make an explicit note of it, taking it for granted that any function ''F'' that is being employed in a differential context is equivalent to <math>\epsilon</math>''F'', for a suitable <math>\epsilon</math>.