Changes

Second, the difference map (or the chordal transformation) D''G'' = ‹D''G''₁, D''G''₂› : E''U''^ • → E''X''^ • is defined in component-wise fashion as the boolean sum of the initial proposition ''G''_''i'' and the enlarged proposition E''G''_''i'', for ''i'' = 1, 2, according to the following set of equations:

Second, the difference map (or the chordal transformation) D''G'' = ‹D''G''₁, D''G''₂› : E''U''^ • → E''X''^ • is defined in component-wise fashion as the boolean sum of the initial proposition ''G''_''i'' and the enlarged proposition E''G''_''i'', for ''i'' = 1, 2, according to the following set of equations:

<pre>

<br><font face="courier new">

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

{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"

|                                                 |

|   DG_i = G_i <u, v> + EG_i <u, v, du, dv>  |

{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"

|                                                 |

| width="8%"  | D''G''_''i''

|         =  G_i <u, v> + G_i <u + du, v + dv>  |

| width="4%" | =

|                                                 |

| width="20%" | ''G''_''i''‹''u'', ''v''›

| width="4%" | +

</pre>

| width="64%" | E''G''_''i''‹''u'', ''v'', d''u'', d''v''›

|-

| width="8%"  | &nbsp;

| width="4%" | =

| width="20%" | ''G''_''i''‹''u'', ''v''›

| width="4%" | +

| width="64%" | ''G''_''i''‹''u'' + d''u'', ''v'' + d''v''›

|}

|}

</font><br>

Maintaining a strict analogy with ordinary difference calculus would perhaps have us write D''G''_''i''&nbsp;=&nbsp;E''G''_''i''&nbsp;&ndash;&nbsp;''G''_''i'', but the sum and difference operations are the same thing in boolean arithmetic.  It is more often natural in the logical context to consider an initial proposition ''q'', then to compute the enlargement E''q'', and finally to determine the difference D''q''&nbsp;=&nbsp;''q''&nbsp;+&nbsp;E''q'', so we let the variant order of terms reflect this sequence of considerations.

Maintaining a strict analogy with ordinary difference calculus would perhaps have us write D''G''_''i''&nbsp;=&nbsp;E''G''_''i''&nbsp;&ndash;&nbsp;''G''_''i'', but the sum and difference operations are the same thing in boolean arithmetic.  It is more often natural in the logical context to consider an initial proposition ''q'', then to compute the enlargement E''q'', and finally to determine the difference D''q''&nbsp;=&nbsp;''q''&nbsp;+&nbsp;E''q'', so we let the variant order of terms reflect this sequence of considerations.