Changes

Even more simply, the same result is reached by matching up the propositional coefficients of <math>\epsilon</math>''J'' and E''J'' along the cells of [d''u'',&nbsp;d''v''] and adding the pairs under boolean sums (that is, "mod 2", where 1&nbsp;+&nbsp;1&nbsp;=&nbsp;0), as shown in Table&nbsp;43.

Even more simply, the same result is reached by matching up the propositional coefficients of <math>\epsilon</math>''J'' and E''J'' along the cells of [d''u'',&nbsp;d''v''] and adding the pairs under boolean sums (that is, "mod 2", where 1&nbsp;+&nbsp;1&nbsp;=&nbsp;0), as shown in Table&nbsp;43.

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<font face="courier new">

Table 43.  Computation of DJ (Method 3)

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|+ Table 43.  Computation of D''J'' (Method 3)

|                                                                               |

|  DJ  =  !e!J          +   EJ                                                |

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

|                                                                               |

| width="6%"  | D''J''

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

| width="3%" | =

|                                                                               |

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

| width="3%" | +

|                                                                               |

| width="20%" | E''J''

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

| width="48%" | &nbsp;

|                                                                               |

|}

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

|-

|                                                                               |

| DJ  =   0 . (du)(dv) +   u . (du) dv  +     v . du (dv) + ((u, v)) du dv |

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

|                                                                               |

| width="6%"  | <math>\epsilon</math>''J''

| width="23%" | =&nbsp;''u''&nbsp;''v''&nbsp;(d''u'')(d''v'')

</pre>

| width="23%" | +&nbsp;''u''&nbsp;&nbsp;''v''&nbsp;(d''u'')&nbsp;d''v''

| width="23%" | +&nbsp;&nbsp;''u''&nbsp;&nbsp;''v''&nbsp;d''u''&nbsp;(d''v'')

| width="25%" | +&nbsp;&nbsp;''u''&nbsp;&nbsp;''v''&nbsp;&nbsp;d''u''&nbsp;d''v''

|-

| width="6%"  | E''J''

| width="23%" | = ''u'' ''v'' (d''u'')(d''v'')

| width="23%" | + ''u'' (''v'')(d''u'') d''v''

| width="23%" | + (''u'') ''v'' d''u'' (d''v'')

| width="25%" | + (''u'')(''v'') d''u'' d''v''

|}

|-

|

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

| width="6%"  | D''J''

| width="23%" | = 0 <math>\cdot</math> (d''u'')(d''v'')

| width="23%" | + ''u'' <math>\cdot</math> (d''u'') d''v''

| width="23%" | + ''v'' <math>\cdot</math> d''u'' (d''v'')

| width="25%" | + ((''u'', ''v'')) d''u'' d''v''

|}

|}

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The difference map D''J'' can also be given a ''dispositional'' interpretation.  First, recall that <math>\epsilon</math>''J'' exhibits the dispositions to change from anywhere in ''J'' to anywhere at all, and E''J'' enumerates the dispositions to change from anywhere at all to anywhere in ''J''.  Next, observe that each of these classes of dispositions may be divided in accordance with the case of ''J'' versus (''J'') that applies to their points of departure and destination, as shown below.  Then, since the dispositions corresponding to <math>\epsilon</math>''J'' and E''J'' have in common the dispositions to preserve ''J'', their symmetric difference (<math>\epsilon</math>''J'',&nbsp;E''J'') is made up of all the remaining dispositions, which are in fact disposed to cross the boundary of ''J'' in one direction or the other.  In other words, we may conclude that D''J'' expresses the collective disposition to make a definite change with respect to ''J'', no matter what value it holds in the current state of affairs.

The difference map D''J'' can also be given a ''dispositional'' interpretation.  First, recall that <math>\epsilon</math>''J'' exhibits the dispositions to change from anywhere in ''J'' to anywhere at all, and E''J'' enumerates the dispositions to change from anywhere at all to anywhere in ''J''.  Next, observe that each of these classes of dispositions may be divided in accordance with the case of ''J'' versus (''J'') that applies to their points of departure and destination, as shown below.  Then, since the dispositions corresponding to <math>\epsilon</math>''J'' and E''J'' have in common the dispositions to preserve ''J'', their symmetric difference (<math>\epsilon</math>''J'',&nbsp;E''J'') is made up of all the remaining dispositions, which are in fact disposed to cross the boundary of ''J'' in one direction or the other.  In other words, we may conclude that D''J'' expresses the collective disposition to make a definite change with respect to ''J'', no matter what value it holds in the current state of affairs.