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Revision as of 18:00, 23 May 2008
, 18:00, 23 May 2008prep figures for planet math
<pre>
<pre>
o-----------------------------------------------------------o
o-----------------------------------------------------------o
| X |
| X . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| o-------------o |
| . . . . . . . . . . .o-------------o . . . . . . . . . . .|
| / \ |
| . . . . . . . . . . / . . . . . . . \ . . . . . . . . . . |
| / \ |
| . . . . . . . . . ./ . . . . . . . . \ . . . . . . . . . .|
| / \ |
| . . . . . . . . . / . . . . . . . . . \ . . . . . . . . . |
| / \ |
| . . . . . . . . ./ . . . . . . . . . . \ . . . . . . . . .|
| / \ |
| . . . . . . . . / . . . . . . . . . . . \ . . . . . . . . |
| o o |
| . . . . . . . .o . . . . . . . . . . . . o . . . . . . . .|
| | U | |
| . . . . . . . .| . . . . . .U . . . . . .| . . . . . . . .|
| | | |
| . . . . . . . .| . . . . . . . . . . . . | . . . . . . . .|
| | | |
| . . . . . . . .| . . . . . . . . . . . . | . . . . . . . .|
| | | |
| . . . . . . . .| . . . . . . . . . . . . | . . . . . . . .|
| | | |
| . . . . . . . .| . . . . . . . . . . . . | . . . . . . . .|
| o--o----------o o----------o--o |
| . . . . . . o--o----------o . o----------o--o . . . . . . |
| / \%%%%%%%%%%\ /%%%%%%%%%%/ \ |
| . . . . . ./ . .\%%%%%%%%%%\ /%%%%%%%%%%/ . .\ . . . . . .|
| / \%%%%%%%%%%o%%%%%%%%%%/ \ |
| . . . . . / . . .\%%%%%%%%%%o%%%%%%%%%%/ . . .\ . . . . . |
| / \%%%%%%%%/%\%%%%%%%%/ \ |
| . . . . ./ . . . .\%%%%%%%%/%\%%%%%%%%/ . . . .\ . . . . .|
| / \%%%%%%/%%%\%%%%%%/ \ |
| . . . . / . . . . .\%%%%%%/%%%\%%%%%%/ . . . . .\ . . . . |
| / \%%%%/%%%%%\%%%%/ \ |
| . . . ./ . . . . . .\%%%%/%%%%%\%%%%/ . . . . . .\ . . . .|
| o o--o-------o--o o |
| . . . o . . . . . . .o--o-------o--o . . . . . . .o . . . |
| | |%%%%%%%| | |
| . . . | . . . . . . . . |%%%%%%%| . . . . . . . . | . . . |
| | |%%%%%%%| | |
| . . . | . . . . . . . . |%%%%%%%| . . . . . . . . | . . . |
| | |%%%%%%%| | |
| . . . | . . . . . . . . |%%%%%%%| . . . . . . . . | . . . |
| | V |%%%%%%%| W | |
| . . . | . . . .V . . . .|%%%%%%%| . . . .W . . . .| . . . |
| | |%%%%%%%| | |
| . . . | . . . . . . . . |%%%%%%%| . . . . . . . . | . . . |
| o o%%%%%%%o o |
| . . . o . . . . . . . . o%%%%%%%o . . . . . . . . o . . . |
| \ \%%%%%/ / |
| . . . .\ . . . . . . . . \%%%%%/ . . . . . . . . / . . . .|
| \ \%%%/ / |
| . . . . \ . . . . . . . . \%%%/ . . . . . . . . / . . . . |
| \ \%/ / |
| . . . . .\ . . . . . . . . \%/ . . . . . . . . / . . . . .|
| \ o / |
| . . . . . \ . . . . . . . . o . . . . . . . . / . . . . . |
| \ / \ / |
| . . . . . .\ . . . . . . . / \ . . . . . . . / . . . . . .|
| o-------------o o-------------o |
| . . . . . . o-------------o . o-------------o . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
o-----------------------------------------------------------o
o-----------------------------------------------------------o
Figure 1. Polymorphous Set Q
Figure 1. Polymorphous Set Q
<pre>
<pre>
o-----------------------------------------------------------o
o-----------------------------------------------------------o
| X |
| X . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| o-------------o |
| . . . . . . . . . . .o-------------o . . . . . . . . . . .|
| / \ |
| . . . . . . . . . . / . . . . . . . \ . . . . . . . . . . |
| / \ |
| . . . . . . . . . ./ . . . . . . . . \ . . . . . . . . . .|
| / \ |
| . . . . . . . . . / . . . . . . . . . \ . . . . . . . . . |
| / \ |
| . . . . . . . . ./ . . . . . . . . . . \ . . . . . . . . .|
| / \ |
| . . . . . . . . / . . . . . . . . . . . \ . . . . . . . . |
| o o |
| . . . . . . . .o . . . . . . . . . . . . o . . . . . . . .|
| | U | |
| . . . . . . . .| . . . . . .U . . . . . .| . . . . . . . .|
| | | |
| . . . . . . . .| . . . . . . . . . . . . | . . . . . . . .|
| | | |
| . . . . . . . .| . . . . . . . . . . . . | . . . . . . . .|
| | | |
| . . . . . . . .| . . . . . . . . . . . . | . . . . . . . .|
| | | |
| . . . . . . . .| . . . . . . . . . . . . | . . . . . . . .|
| o--o----------o o----------o--o |
| . . . . . . o--o----------o . o----------o--o . . . . . . |
| / \%%%%%%%%%%\ /%%%%%%%%%%/ \ |
| . . . . . ./ . .\%%%%%%%%%%\ /%%%%%%%%%%/ . .\ . . . . . .|
| / \%%%%%%%%%%o%%%%%%%%%%/ \ |
| . . . . . / . . .\%%%%%%%%%%o%%%%%%%%%%/ . . .\ . . . . . |
| / \%%%%%%%%/%\%%%%%%%%/ \ |
| . . . . ./ . . . .\%%%%%%%%/%\%%%%%%%%/ . . . .\ . . . . .|
| / \%%%%%%/%%%\%%%%%%/ \ |
| . . . . / . . . . .\%%%%%%/%%%\%%%%%%/ . . . . .\ . . . . |
| / \%%%%/%%%%%\%%%%/ \ |
| . . . ./ . . . . . .\%%%%/%%%%%\%%%%/ . . . . . .\ . . . .|
| o o--o-------o--o o |
| . . . o . . . . . . .o--o-------o--o . . . . . . .o . . . |
| | |%%%%%%%| | |
| . . . | . . . . . . . . |%%%%%%%| . . . . . . . . | . . . |
| | |%%%%%%%| | |
| . . . | . . . . . . . . |%%%%%%%| . . . . . . . . | . . . |
| | |%%%%%%%| | |
| . . . | . . . . . . . . |%%%%%%%| . . . . . . . . | . . . |
| | V |%%%%%%%| W | |
| . . . | . . . .V . . . .|%%%%%%%| . . . .W . . . .| . . . |
| | |%%%%%%%| | |
| . . . | . . . . . . . . |%%%%%%%| . . . . . . . . | . . . |
| o o%%%%%%%o o |
| . . . o . . . . . . . . o%%%%%%%o . . . . . . . . o . . . |
| \ \%%%%%/ / |
| . . . .\ . . . . . . . . \%%%%%/ . . . . . . . . / . . . .|
| \ \%%%/ / |
| . . . . \ . . . . . . . . \%%%/ . . . . . . . . / . . . . |
| \ \%/ / |
| . . . . .\ . . . . . . . . \%/ . . . . . . . . / . . . . .|
| \ o / |
| . . . . . \ . . . . . . . . o . . . . . . . . / . . . . . |
| \ / \ / |
| . . . . . .\ . . . . . . . / \ . . . . . . . / . . . . . .|
| o-------------o o-------------o |
| . . . . . . o-------------o . o-------------o . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
o-----------------------------------------------------------o
o-----------------------------------------------------------o
Figure 1. Polymorphous Set Q
Figure 1. Polymorphous Set Q
</pre>
</pre>
The proposition or the truth-function <math>q\!</math> that describes <math>Q\!</math> is:
The proposition or the truth-function <math>q\!</math> that describes <math>Q\!</math> is:
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| X |
| X . . . . . . . . . . . . . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| o-------------o |
| . . . . . . . . o-------------o . . . . . . . . |
| / \ |
| . . . . . . . ./ . . . . . . . \ . . . . . . . .|
| / \ |
| . . . . . . . / . . . . . . . . \ . . . . . . . |
| / \ |
| . . . . . . ./ . . . . . . . . . \ . . . . . . .|
| / \ |
| . . . . . . / . . . . . . . . . . \ . . . . . . |
| o U o |
| . . . . . .o . . . . . U . . . . . o . . . . . .|
| | | |
| . . . . . .| . . . . . . . . . . . | . . . . . .|
| | | |
| . . . . . .| . . . . . . . . . . . | . . . . . .|
| | | |
| . . . . . .| . . . . . . . . . . . | . . . . . .|
| o---o---------o o---------o---o |
| . . . .o---o---------o . o---------o---o . . . .|
| / \%%%%%%%%%\ /%%%%%%%%%/ \ |
| . . . / . . \%%%%%%%%%\ /%%%%%%%%%/ . . \ . . . |
| / \%%%%%%%%%o%%%%%%%%%/ \ |
| . . ./ . . . \%%%%%%%%%o%%%%%%%%%/ . . . \ . . .|
| / \%%%%%%%/%\%%%%%%%/ \ |
| . . / . . . . \%%%%%%%/%\%%%%%%%/ . . . . \ . . |
| / \%%%%%/%%%\%%%%%/ \ |
| . ./ . . . . . \%%%%%/%%%\%%%%%/ . . . . . \ . .|
| o o---o-----o---o o |
| . o . . . . . . o---o-----o---o . . . . . . o . |
| | |%%%%%| | |
| . | . . . . . . . . |%%%%%| . . . . . . . . | . |
| | V |%%%%%| W | |
| . | . . . .V . . . .|%%%%%| . . . .W . . . .| . |
| | |%%%%%| | |
| . | . . . . . . . . |%%%%%| . . . . . . . . | . |
| o o%%%%%o o |
| . o . . . . . . . . o%%%%%o . . . . . . . . o . |
| \ \%%%/ / |
| . .\ . . . . . . . . \%%%/ . . . . . . . . / . .|
| \ \%/ / |
| . . \ . . . . . . . . \%/ . . . . . . . . / . . |
| \ o / |
| . . .\ . . . . . . . . o . . . . . . . . / . . .|
| \ / \ / |
| . . . \ . . . . . . . / \ . . . . . . . / . . . |
| o-------------o o-------------o |
| . . . .o-------------o . o-------------o . . . .|
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
Figure 1. Polymorphous Set Q
Figure 1. .Polymorphous Set Q
</pre>
</pre>
The proposition or truth-function <math>q : X \to \mathbb{B}</math> that describes <math>Q\!</math> is represented by the following graph and text expressions:
The proposition or truth-function <math>q : X \to \mathbb{B}</math> that describes <math>Q\!</math> is represented by the following graph and text expressions:
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| q |
| q . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| u v u w v w |
| . . . . . . . . u v . u w . v w . . . . . . . . |
| o o o |
| . . . . . . . . . .o . o . o . . . . . . . . . .|
| \ | / |
| . . . . . . . . . . \ .| ./ . . . . . . . . . . |
| \ | / |
| . . . . . . . . . . .\ | / . . . . . . . . . . .|
| \|/ |
| . . . . . . . . . . . \|/ . . . . . . . . . . . |
| o |
| . . . . . . . . . . . .o . . . . . . . . . . . .|
| | |
| . . . . . . . . . . . .| . . . . . . . . . . . .|
| | |
| . . . . . . . . . . . .| . . . . . . . . . . . .|
| | |
| . . . . . . . . . . . .| . . . . . . . . . . . .|
| @ |
| . . . . . . . . . . . .@ . . . . . . . . . . . .|
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| (( u v )( u w )( v w )) |
| . . . . . . (( u v )( u w )( v w )) . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| q.uvw |
| q.uvw . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
| . . . . . . . . . . . . . . . . . . . . . . . . |
| . . . . . . . . u v . u w . v w . . . . . . . . |
| . . . . . . . . . .o . o . o . . . . . . . . . .|
| . . . . . . . . . . \ .| ./ . . . . . . . . . . |
| . . . . . . . . . . .\ | / . . . . . . . . . . .|
| . . . . . . . . . . . \|/ . . . . . . . . . . . |
| . . . . . . . . . . . .o . . . . . . . . . . . .|
| . . . . . . . . . . . .| . . . . . . . . . . . .|
| . . . . . . . . . . . .| . . . . . . . . . . . .|
| . . . . . . . . . . . .| . . . . . . . . . . . .|
| . . . . . . . . . . . .@ u v w . . . . . . . . .|
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . (( u v )( u w )( v w )) u v w . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| Eq |
| Eq . . . . . . . . . . . . . . . . . . . . . . .|
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| u du v dv u du w dw v dv w dw |
| . . .u .du v .dv .u .du w .dw .v .dv w .dw . . .|
| o---o o---o o---o o---o o---o o---o |
| . . .o---o o---o .o---o o---o .o---o o---o . . .|
| \ | | / \ | | / \ | | / |
| . . . \ .| | ./ . .\ .| | ./ . .\ .| | ./ . . . |
| \ | | / \ | | / \ | | / |
| . . . .\ | | / . . .\ | | / . . .\ | | / . . . .|
| \| |/ \| |/ \| |/ |
| . . . . \| |/ . . . .\| |/ . . . .\| |/ . . . . |
| o=o o=o o=o |
| . . . . .o=o . . . . .o=o . . . . .o=o . . . . .|
| \ | / |
| . . . . . . \ . . . . .| . . . . ./ . . . . . . |
| \ | / |
| . . . . . . .\ . . . . | . . . . / . . . . . . .|
| \ | / |
| . . . . . . . \ . . . .| . . . ./ . . . . . . . |
| \ | / |
| . . . . . . . .\ . . . | . . . / . . . . . . . .|
| \ | / |
| . . . . . . . . \ . . .| . . ./ . . . . . . . . |
| \ | / |
| . . . . . . . . .\ . . | . . / . . . . . . . . .|
| \ | / |
| . . . . . . . . . \ . .| . ./ . . . . . . . . . |
| \ | / |
| . . . . . . . . . .\ . | . / . . . . . . . . . .|
| \ | / |
| . . . . . . . . . . \ .| ./ . . . . . . . . . . |
| \ | / |
| . . . . . . . . . . .\ | / . . . . . . . . . . .|
| \|/ |
| . . . . . . . . . . . \|/ . . . . . . . . . . . |
| o |
| . . . . . . . . . . . .o . . . . . . . . . . . .|
| | |
| . . . . . . . . . . . .| . . . . . . . . . . . .|
| | |
| . . . . . . . . . . . .| . . . . . . . . . . . .|
| | |
| . . . . . . . . . . . .| . . . . . . . . . . . .|
| @ |
| . . . . . . . . . . . .@ . . . . . . . . . . . .|
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| (( ( u , du ) ( v , dv ) |
| . . . . .(( .( u , du ) ( v , dv ) . . . . . . .|
| )( ( u , du ) ( w , dw ) |
| . . . . .)( .( u , du ) ( w , dw ) . . . . . . .|
| )( ( v , dv ) ( w , dw ) |
| . . . . .)( .( v , dv ) ( w , dw ) . . . . . . .|
| )) |
| . . . . .)) . . . . . . . . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| Eq.uvw |
| Eq.uvw . . . . . . . . . . . . . . . . . . . . .|
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| u du v dv u du w dw v dv w dw |
| . . .u .du v .dv .u .du w .dw .v .dv w .dw . . .|
| o---o o---o o---o o---o o---o o---o |
| . . .o---o o---o .o---o o---o .o---o o---o . . .|
| \ | | / \ | | / \ | | / |
| . . . \ .| | ./ . .\ .| | ./ . .\ .| | ./ . . . |
| \ | | / \ | | / \ | | / |
| . . . .\ | | / . . .\ | | / . . .\ | | / . . . .|
| \| |/ \| |/ \| |/ |
| . . . . \| |/ . . . .\| |/ . . . .\| |/ . . . . |
| o=o o=o o=o |
| . . . . .o=o . . . . .o=o . . . . .o=o . . . . .|
| \ | / |
| . . . . . . \ . . . . .| . . . . ./ . . . . . . |
| \ | / |
| . . . . . . .\ . . . . | . . . . / . . . . . . .|
| \ | / |
| . . . . . . . \ . . . .| . . . ./ . . . . . . . |
| \ | / |
| . . . . . . . .\ . . . | . . . / . . . . . . . .|
| \ | / |
| . . . . . . . . \ . . .| . . ./ . . . . . . . . |
| \ | / |
| . . . . . . . . .\ . . | . . / . . . . . . . . .|
| \ | / |
| . . . . . . . . . \ . .| . ./ . . . . . . . . . |
| \ | / |
| . . . . . . . . . .\ . | . / . . . . . . . . . .|
| \ | / |
| . . . . . . . . . . \ .| ./ . . . . . . . . . . |
| \ | / |
| . . . . . . . . . . .\ | / . . . . . . . . . . .|
| \|/ |
| . . . . . . . . . . . \|/ . . . . . . . . . . . |
| o |
| . . . . . . . . . . . .o . . . . . . . . . . . .|
| | |
| . . . . . . . . . . . .| . . . . . . . . . . . .|
| | |
| . . . . . . . . . . . .| . . . . . . . . . . . .|
| | |
| . . . . . . . . . . . .| . . . . . . . . . . . .|
| @ u v w |
| . . . . . . . . . . . .@ u v w . . . . . . . . .|
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
| . . . . . . . . . . . . . . . . . . . . . . . . |
| . . . . .(( .( u , du ) ( v , dv ) . . . . . . .|
| . . . . .)( .( u , du ) ( w , dw ) . . . . . . .|
| . . . . .)( .( v , dv ) ( w , dw ) . . . . . . .|
| . . . . .)) . . . . . . . . . . . . . . . . . . |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| . . . . .u v w . . . . . . . . . . . . . . . . .|
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
</pre>
The models of this last expression tell us which combinations of feature changes among the set <math>\{ \operatorname{d}u, \operatorname{d}v, \operatorname{d}w \}</math> will take us from our present interpretation, the center cell expressed by "<math>u\ v\ w</math>", to a true value under the target proposition <code> (( u v )( u w )( v w )) </code>.
The result of applying the difference operator <math>\operatorname{D}</math> to the initial proposition <math>q\!</math>, conjoined with a query on the center cell, yields:
<pre>
o-------------------------------------------------o
| Dq.uvw . . . . . . . . . . . . . . . . . . . . .|
o-------------------------------------------------o
| . . . . . . . . . . . . . . . . . . . . . . . . |
| . .u .du v .dv .u .du w .dw .v .dv w .dw . . . .|
| . .o---o o---o .o---o o---o .o---o o---o . . . .|
| . . \ .| | ./ . .\ .| | ./ . .\ .| | ./ . . . . |
| . . .\ | | / . . .\ | | / . . .\ | | / . . . . .|
| . . . \| |/ . . . .\| |/ . . . .\| |/ . . . . . |
| . . . .o=o . . . . .o=o . . . . .o=o . . . . . .|
| . . . . . \ . . . . .| . . . . ./ . . . . . . . |
| . . . . . .\ . . . . | . . . . / . . . . . . . .|
| . . . . . . \ . . . .| . . . ./ . . . . . . . . |
| . . . . . . .\ . . . | . . . / . . . . . . . . .|
| . . . . . . . \ . . .| . . ./ . . . . . . . . . |
| . . . . . . . .\ . . | . . / . . . . . . . . . .|
| . . . . . . . . \ . .| . ./ . .u v .u w .v w . .|
| . . . . . . . . .\ . | . / . . . o . o . o . . .|
| . . . . . . . . . \ .| ./ . . . . \ .| ./ . . . |
| . . . . . . . . . .\ | / . . . . . \ | / . . . .|
| . . . . . . . . . . \|/ . . . . . . \|/ . . . . |
| . . . . . . . . . . .o . . . . . . . o . . . . .|
| . . . . . . . . . . .| . . . . . . . | . . . . .|
| . . . . . . . . . . .| . . . . . . . | . . . . .|
| . . . . . . . . . . .| . . . . . . . | . . . . .|
| . . . . . . . . . . .o---------------o . . . . .|
| . . . . . . . . . . . \ . . . . . . / . . . . . |
| . . . . . . . . . . . .\ . . . . . / . . . . . .|
| . . . . . . . . . . . . \ . . . . / . . . . . . |
| . . . . . . . . . . . . .\ . . . / . . . . . . .|
| . . . . . . . . . . . . . \ . . / . . . . . . . |
| . . . . . . . . . . . . . .\ . / . . . . . . . .|
| . . . . . . . . . . . . . . \ / . . . . . . . . |
| . . . . . . . . . . . . . . .@ u v w . . . . . .|
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| (( ( u , du ) ( v , dv ) |
| . . . ( . . . . . . . . . . . . . . . . . . . . |
| )( ( u , du ) ( w , dw ) |
| . . . . .(( .( u , du ) ( v , dv ) . . . . . . .|
| )( ( v , dv ) ( w , dw ) |
| . . . . .)( .( u , du ) ( w , dw ) . . . . . . .|
| )) |
| . . . . .)( .( v , dv ) ( w , dw ) . . . . . . .|
| |
| . . . . .)) . . . . . . . . . . . . . . . . . . |
| u v w |
| . . . , . . . . . . . . . . . . . . . . . . . . |
| |
| . . . . .(( .u v . . . . . . . . . . . . . . . .|
| . . . . .)( .u w . . . . . . . . . . . . . . . .|
| . . . . .)( .v w . . . . . . . . . . . . . . . .|
| . . . . .)) . . . . . . . . . . . . . . . . . . |
| . . . ) . . . . . . . . . . . . . . . . . . . . |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| . . . u v w . . . . . . . . . . . . . . . . . . |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
The models of this last expression tell us which combinations of feature changes among the set <math>\{ \operatorname{d}u, \operatorname{d}v, \operatorname{d}w \}</math> will take us from our present interpretation, the center cell expressed by "<math>u\ v\ w</math>", to a true value under the target proposition <code> (( u v )( u w )( v w )) </code>.
The models of this last proposition are:
<code>
1. u v w du dv dw
2. u v w du dv (dw)
3. u v w du (dv) dw
4. u v w (du) dv dw
</code>
This tells us that changing any two or more of the features <math>u, v, w\!</math> will take us from the center cell, as described by the conjunctive expression "<math>u\ v\ w</math>", to a cell outside the shaded region for the set <math>Q\!</math>.
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| Dq.uvw |
| X . . . . . . . . . . . . . . . . . . . . . . . |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| . . . . . . . . o-------------o . . . . . . . . |
| . . . . . . . ./ . . . . . . . \ . . . . . . . .|
| . . . . . . . / . . . .U . . . .\ . . . . . . . |
| . . . . . . ./ . . . . . . . . . \ . . . . . . .|
| . . . . . . / . . . . . . . . . . \ . . . . . . |
| . . . . . .o . . . . . . . . .@ . .o . . . . . .|
| . . . . . .| . . . . . . . . .^ . .| . . . . . .|
| . . . . . .| . . . . . . . . .|dw .| . . . . . .|
| . . . . . .| . . . . . . . . .| . .| . . . . @ .|
| . . . .o---o---------o . o----|----o---o . .^ . |
| . . . / . . \%%%%%%%%%\ /%%%%%|%%%/ . . \ ./dw .|
| . . ./ . .du \%%%%%dw%%o%%dv%%|%%/ . . . \/ . . |
| . . / .@<-----\-o<----/+\---->o%/ . . . ./\ . . |
| . ./ . . . . . \%%%%%/%|%\%%%%%/ . . . ./ .\ . .|
| . o . . . . . . o---o--|--o---o . . . ./ . .o . |
| . | . . . . . . . . |%%|%%| . . . . . / . . | . |
| . | .V . . . . . . .|%du%%| . . . . ./ . W .| . |
| . | . . . . . . . . |% |%%| . . . . / . . . | . |
| . o . . . . . . . . o%%v%%o . dv . / . . . .o . |
| . .\ . . . . . . . . \%o-/------->@ . . . ./ . .|
| . . \ . . . . . . . . \%/ . . . . . . . . / . . |
| . . .\ . . . . . . . . o . . . . . . . . / . . .|
| . . . \ . . . . . . . / \ . . . . . . . / . . . |
| . . . .o-------------o . o-------------o . . . .|
| . . . . . . . . . . . . . . . . . . . . . . . . |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
Figure 3. Effect of the Difference Operator D
Acting on a Polymorphous Function q
</pre>
| \ | | / \ | | / \ | | / |
Figure 3 shows one way to picture this kind of a situation, by superimposing the paths of indicated feature changes on the venn diagram of the underlying proposition. Here, the models, or the satisfying interpretations, of the relevant ''difference proposition'' <math>\operatorname{D}q</math> are marked with "<code>@</code>" signs, and the boundary crossings along each path are marked with the corresponding ''differential features'' among the collection <math>\{ \operatorname{d}u, \operatorname{d}v, \operatorname{d}w \}</math>. In sum, starting from the cell <math>uvw\!</math>, we have the following four paths:
<pre>
1. du dv dw => Change u, v, w.
2. du dv (dw) => Change u and v.
3. du (dv) dw => Change u and w.
| \ | / |
4. (du) dv dw => Change v and w.
| \ | / |
</pre>
Next I will discuss several applications of logical differentials, developing along the way their logical and practical implications.
| \ | / o o o |
| \ | / \ | / |
We have come to the point of making a connection, at a very primitive level, between propositional logic and the classes of mathematical structures that are employed in mathematical systems theory to model dynamical systems of very general sorts.
| \ | / \ | / |
| \|/ \|/ |
===Recapitulation===
| o o |
| | | |
Here is a flash montage of what has gone before, retrospectively touching on just the highpoints, and highlighting mostly just Figures and Tables, all directed toward the aim of ending up with a novel style of pictorial diagram, one that will serve us well in the future, as I have found it readily adaptable and steadily more trustworthy in my previous investigations, whenever we have to illustrate these very basic sorts of dynamic scenarios to ourselves, to others, to computers.
| | | |
| | | |
We typically start out with a proposition of interest, for example, the proposition <math>q : X \to \mathbb{B}</math> depicted here:
| o---------------o |
<pre>
o-------------------------------------------------o
| q . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
| . . . . . . . . . . . . . . . . . . . . . . . . |
| . . . . . . . . u v . u w . v w . . . . . . . . |
| . . . . . . . . . .o . o . o . . . . . . . . . .|
| . . . . . . . . . . \ .| ./ . . . . . . . . . . |
| |
| . . . . . . . . . . .\ | / . . . . . . . . . . .|
| . . . . . . . . . . . \|/ . . . . . . . . . . . |
| . . . . . . . . . . . .o . . . . . . . . . . . .|
| . . . . . . . . . . . .| . . . . . . . . . . . .|
| . . . . . . . . . . . .| . . . . . . . . . . . .|
| . . . . . . . . . . . .| . . . . . . . . . . . .|
| . . . . . . . . . . . .@ . . . . . . . . . . . .|
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . (( u v )( u w )( v w )) . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
The models of this last proposition are:
The proposition <math>q\!</math> is properly considered as an ''[[abstract object]]'', in some acceptation of those very bedevilled and egging-on terms, but it enjoys an interpretation as a function of a suitable type, and all we have to do in order to enjoy the utility of this type of representation is to observe a decent respect for what befits.
I will skip over the details of how to do this for right now. .I started to write them out in full, and it all became even more tedious than my usual standard, and besides, I think that everyone more or less knows how to do this already.
Once we have survived the big leap of re-interpreting these abstract names as the names of relatively concrete dimensions of variation, we can begin to lay out all of the familiar sorts of mathematical models and pictorial diagrams that go with these modest dimensions, the functions that can be formed on them, and the transformations that can be entertained among this whole crew.
Here is the venn diagram for the proposition <math>q\!</math>.
<pre>
<pre>
o-------------------------------------------------o
o-----------------------------------------------------------o
| X |
| X . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| o-------------o |
| . . . . . . . . . . .o-------------o . . . . . . . . . . .|
| / \ |
| . . . . . . . . . . / . . . . . . . \ . . . . . . . . . . |
| / U \ |
| . . . . . . . . . ./ . . . . . . . . \ . . . . . . . . . .|
| / \ |
| . . . . . . . . . / . . . . . . . . . \ . . . . . . . . . |
| / \ |
| . . . . . . . . ./ . . . . . . . . . . \ . . . . . . . . .|
| o @ o |
| . . . . . . . . / . . . . . . . . . . . \ . . . . . . . . |
| | ^ | |
| . . . . . . . .o . . . . . . . . . . . . o . . . . . . . .|
| | |dw | |
| . . . . . . . .| . . . . . .U . . . . . .| . . . . . . . .|
| | | | @ |
| . . . . . . . .| . . . . . . . . . . . . | . . . . . . . .|
| o---o---------o o----|----o---o ^ |
| . . . . . . . .| . . . . . . . . . . . . | . . . . . . . .|
| / \`````````\ /`````|```/ \ /dw |
| . . . . . . . .| . . . . . . . . . . . . | . . . . . . . .|
| / du \`````dw``o``dv``|``/ \/ |
| . . . . . . . .| . . . . . . . . . . . . | . . . . . . . .|
| / @<-----\-o<----/+\---->o`/ /\ |
| . . . . . . o--o----------o . o----------o--o . . . . . . |
| / \`````/`|`\`````/ / \ |
| . . . . . ./ . .\%%%%%%%%%%\ /%%%%%%%%%%/ . .\ . . . . . .|
| o o---o--|--o---o / o |
| . . . . . / . . .\%%%%%%%%%%o%%%%%%%%%%/ . . .\ . . . . . |
| | |``|``| / | |
| . . . . ./ . . . .\%%%%%%%%/%\%%%%%%%%/ . . . .\ . . . . .|
| | V |`du``| / W | |
| . . . . / . . . . .\%%%%%%/%%%\%%%%%%/ . . . . .\ . . . . |
| | |` |``| / | |
| . . . ./ . . . . . .\%%%%/%%%%%\%%%%/ . . . . . .\ . . . .|
| o o``v``o dv / o |
| . . . o . . . . . . .o--o-------o--o . . . . . . .o . . . |
| \ \`o-/------->@ / |
| . . . | . . . . . . . . |%%%%%%%| . . . . . . . . | . . . |
| \ \`/ / |
| . . . | . . . . . . . . |%%%%%%%| . . . . . . . . | . . . |
| \ o / |
| . . . | . . . . . . . . |%%%%%%%| . . . . . . . . | . . . |
| \ / \ / |
| . . . | . . . .V . . . .|%%%%%%%| . . . .W . . . .| . . . |
| o-------------o o-------------o |
| . . . | . . . . . . . . |%%%%%%%| . . . . . . . . | . . . |
| |
| . . . o . . . . . . . . o%%%%%%%o . . . . . . . . o . . . |
| |
| . . . .\ . . . . . . . . \%%%%%/ . . . . . . . . / . . . .|
o-------------------------------------------------o
| . . . . \ . . . . . . . . \%%%/ . . . . . . . . / . . . . |
Figure 3. Effect of the Difference Operator D
| . . . . .\ . . . . . . . . \%/ . . . . . . . . / . . . . .|
| . . . . . \ . . . . . . . . o . . . . . . . . / . . . . . |
| . . . . . .\ . . . . . . . / \ . . . . . . . / . . . . . .|
| . . . . . . o-------------o . o-------------o . . . . . . |
| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
o-----------------------------------------------------------o
Figure 1. .Venn Diagram for the Proposition q
</pre>
</pre>
By way of excuse, if not yet a full justification, I probably ought to give an account of the reasons why I continue to hang onto these primitive styles of depiction, even though I can hardly recommend that anybody actually try to draw them, at least, not once the number of variables climbs much higher than three or four or five at the utmost. . One of the reasons would have to be this: .that in the relationship between their continuous aspect and their discrete aspect, venn diagrams constitute a form of "iconic" reminder of a very important fact about all ''finite information depictions'' (FID's) of the larger world of reality, and that is the hard fact that we deceive ourselves to a degree if we imagine that the lines and the distinctions that we draw in our imagination are all there is to reality, and thus, that as we practice to categorize, we also manage to discretize, and thus, to distort, to reduce, and to truncate the richness of what there is to the poverty of what we can sieve and sift through our senses, or what we can draw in the tangled webs of our own very tenuous and tinctured distinctions.
.
Another common scheme for description and evaluation of a proposition is the so-called ''truth table'' or the ''semantic tableau'', for example:
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
|+ '''Table 2. .Truth Table for the Proposition ''q'' '''
|- style="background:paleturquoise"
! style="width:20%" | ''u v w''
! style="width:20%" | ''u'' ∧ ''v''
! style="width:20%" | ''u'' ∧ ''w''
! style="width:20%" | ''v'' ∧ ''w''
! style="width:20%" | ''q''
|-
| 0 0 0 || 0 || 0 || 0 || 0
|-
| 0 0 1 || 0 || 0 || 0 || 0
|-
| 0 1 0 || 0 || 0 || 0 || 0
|-
| 0 1 1 || 0 || 0 || 1 || 1
|-
| 1 0 0 || 0 || 0 || 0 || 0
|-
| 1 0 1 || 0 || 1 || 0 || 1
|-
| 1 1 0 || 1 || 0 || 0 || 1
|-
| 1 1 1 || 1 || 1 || 1 || 1
|}
Reading off the shaded cells of the venn diagram or the rows of the truth table that have a "1" in the q column, we see that the ''models'', or satisfying interpretations, of the proposition <math>q\!</math> are the four that can be expressed, in either the ''additive'' or the ''multiplicative'' manner, as follows:
# The points of the space <math>X\!</math> that are assigned the coordinates:<br><math>(u, v, w)\!</math> = <math>(0, 1, 1)\!</math> or <math>(1, 0, 1)\!</math> or <math>(1, 1, 0)\!</math> or <math>(1, 1, 1)\!</math>.
# The points of the space <math>X\!</math> that have the conjunctive descriptions:<br><code>(u) v w</code> or <code>u (v) w</code> or <code>u v (w)</code> or <code>u v w</code>, where "<code>(x)</code>" is "not <code>x</code>".
===Recapitulation===
The next thing that one typically does is to consider the effects of various ''operators'' on the proposition of interest, which may be called the ''operand'' or the ''source'' proposition, leaving the corresponding terms ''opus'' or ''target'' as names for the result.
In our initial consideration of the proposition <math>q\!</math>, we naturally interpret it as a function of the three variables that it wears on its sleeve, as it were, namely, those that we find contained in the basis <math>\{ u, v, w \}</math>. As we begin to regard this proposition from the standpoint of a differential analysis, however, we may need to regard it as ''tacitly embedded'' in any number of higher dimensional spaces. Just by way of starting out, our immediate interest is with the ''first order differential analysis'' (FODA), and this requires us to regard all of the propositions in sight as functions of the variables in the first order extended basis, specifically, those in the set <math>\{ u, v, w, \operatorname{d}u, \operatorname{d}v, \operatorname{d}w \}</math>. Now this does not change the expression of any proposition, like <math>q\!</math>, that does not mention the extra variables, only changing how it gets interpreted as a function. A level of interpretive flexibility of this order is very useful, and it is quite common throughout mathematics. In this discussion, I will invoke its application under the name of the ''[[tacit extension]]'' of a proposition to any universe of discourse based on a superset of its original basis.
I think that we finally have enough of the preliminary set-ups and warm-ups out of the way that we can begin to tackle the differential analysis proper of the sample proposition, the truth-function <math>q(u, v, w)\!</math> that is given by the following expression:
<blockquote><code>
(( u v )( u w )( v w ))
</code></blockquote>
When <math>X\!</math> is the type of space that is generated by <math>\{ u, v, w \}\!</math>, let <math>\operatorname{d}X</math> be the type of space that is generated by <math>\{ \operatorname{d}u, \operatorname{d}v, \operatorname{d}w \}</math>, and let <math>X \times \operatorname{d}X</math> be the type of space that is generated by the extended set of boolean basis elements <math>\{ u, v, w, \operatorname{d}u, \operatorname{d}v, \operatorname{d}w \}</math>. For convenience, define a notation "<math>\operatorname{E}X</math>" so that <math>\operatorname{E}X = X \times \operatorname{d}X</math>. Even though the differential variables are in some abstract sense no different than other boolean variables, it usually helps to mark their distinctive roles and their differential interpretation by means of the distinguishing domain name "<math>\operatorname{d}\mathbb{B}</math>". Using these designations of logical spaces, the propositions over them can be assigned both abstract and concrete types.
For instance, consider the proposition <math>q(u, v, w)\!</math>, as before, and then consider its tacit extension <math>q(u, v, w, \operatorname{d}u, \operatorname{d}v, \operatorname{d}w)\!</math>, the latter of which may be indicated more explicitly as "<math>\operatorname{e}q\!</math>".
#<p>Proposition <math>q\!</math> is abstractly typed as <math>q : \mathbb{B}^3 \to \mathbb{B}.</math></p><p>Proposition <math>q\!</math> is concretely typed as <math>q : X \to \mathbb{B}.</math></p>
#<p>Proposition <math>\operatorname{e}q\!</math> is abstractly typed as <math>\operatorname{e}q : \mathbb{B}^3 \times \operatorname{d}\mathbb{B}^3 \to \mathbb{B}.</math></p><p>Proposition <math>\operatorname{e}q\!</math> is concretely typed as <math>\operatorname{e}q : X \times \operatorname{d}X \to \mathbb{B}.</math></p><p>Succinctly, <math>\operatorname{e}q : \operatorname{E}X \to \mathbb{B}.</math></p>
We now return to our consideration of the effects of various differential operators on propositions. This time around we have enough exact terminology that we shall be able to explain what is actually going on here in a rather more articulate fashion.
The first transformation of the source proposition <math>q\!</math> that we may wish to stop and examine, though it is not unusual to skip right over this stage of analysis, frequently regarding it as a purely intermediary stage, holding scarcely even so much as the passing interest, is the work of the ''enlargement'' or ''shift'' operator <math>\operatorname{E}.</math>
Applying the operator <math>\operatorname{E}</math> to the operand proposition <math>q\!</math> yields:
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| q |
| Eq . . . . . . . . . . . . . . . . . . . . . . .|
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| u v u w v w |
| . . .u .du v .dv .u .du w .dw .v .dv w .dw . . .|
| o o o |
| . . .o---o o---o .o---o o---o .o---o o---o . . .|
| \ | / |
| . . . \ .| | ./ . .\ .| | ./ . .\ .| | ./ . . . |
| \ | / |
| . . . .\ | | / . . .\ | | / . . .\ | | / . . . .|
| \|/ |
| . . . . \| |/ . . . .\| |/ . . . .\| |/ . . . . |
| o |
| . . . . .o=o . . . . .o=o . . . . .o=o . . . . .|
| | |
| . . . . . . \ . . . . .| . . . . ./ . . . . . . |
| | |
| . . . . . . .\ . . . . | . . . . / . . . . . . .|
| | |
| . . . . . . . \ . . . .| . . . ./ . . . . . . . |
| @ |
| . . . . . . . .\ . . . | . . . / . . . . . . . .|
| |
| . . . . . . . . \ . . .| . . ./ . . . . . . . . |
| . . . . . . . . .\ . . | . . / . . . . . . . . .|
| . . . . . . . . . \ . .| . ./ . . . . . . . . . |
| . . . . . . . . . .\ . | . / . . . . . . . . . .|
| . . . . . . . . . . \ .| ./ . . . . . . . . . . |
| . . . . . . . . . . .\ | / . . . . . . . . . . .|
| . . . . . . . . . . . \|/ . . . . . . . . . . . |
| . . . . . . . . . . . .o . . . . . . . . . . . .|
| . . . . . . . . . . . .| . . . . . . . . . . . .|
| . . . . . . . . . . . .| . . . . . . . . . . . .|
| . . . . . . . . . . . .| . . . . . . . . . . . .|
| . . . . . . . . . . . .@ . . . . . . . . . . . .|
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| (( u v )( u w )( v w )) |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| . . . . .(( .( u , du ) ( v , dv ) . . . . . . .|
| . . . . .)( .( u , du ) ( w , dw ) . . . . . . .|
| . . . . .)( .( v , dv ) ( w , dw ) . . . . . . .|
| . . . . .)) . . . . . . . . . . . . . . . . . . |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
The proposition <math>q\!</math> is properly considered as an ''[[abstract object]]'', in some acceptation of those very bedevilled and egging-on terms, but it enjoys an interpretation as a function of a suitable type, and all we have to do in order to enjoy the utility of this type of representation is to observe a decent respect for what befits.
The enlarged proposition <math>\operatorname{E}q</math> is minimally interpretable as a function on the six variables of <math>\{ u, v, w, \operatorname{d}u, \operatorname{d}v, \operatorname{d}w \}.</math> .In other words, <math>\operatorname{E}q : \operatorname{E}X \to \mathbb{B},</math> or <math>\operatorname{E}q : X \times \operatorname{d}X \to \mathbb{B}.</math>
Conjoining a query on the center cell, <math>c = u\ v\ w\!</math>, yields:
<pre>
o-------------------------------------------------o
| Eq.c . . . . . . . . . . . . . . . . . . . . . .|
o-------------------------------------------------o
| . . . . . . . . . . . . . . . . . . . . . . . . |
| . . .u .du v .dv .u .du w .dw .v .dv w .dw . . .|
| . . .o---o o---o .o---o o---o .o---o o---o . . .|
| . . . \ .| | ./ . .\ .| | ./ . .\ .| | ./ . . . |
| . . . .\ | | / . . .\ | | / . . .\ | | / . . . .|
| . . . . \| |/ . . . .\| |/ . . . .\| |/ . . . . |
| . . . . .o=o . . . . .o=o . . . . .o=o . . . . .|
| . . . . . . \ . . . . .| . . . . ./ . . . . . . |
| . . . . . . .\ . . . . | . . . . / . . . . . . .|
| . . . . . . . \ . . . .| . . . ./ . . . . . . . |
| . . . . . . . .\ . . . | . . . / . . . . . . . .|
| . . . . . . . . \ . . .| . . ./ . . . . . . . . |
| . . . . . . . . .\ . . | . . / . . . . . . . . .|
| . . . . . . . . . \ . .| . ./ . . . . . . . . . |
| . . . . . . . . . .\ . | . / . . . . . . . . . .|
| . . . . . . . . . . \ .| ./ . . . . . . . . . . |
| . . . . . . . . . . .\ | / . . . . . . . . . . .|
| . . . . . . . . . . . \|/ . . . . . . . . . . . |
| . . . . . . . . . . . .o . . . . . . . . . . . .|
| . . . . . . . . . . . .| . . . . . . . . . . . .|
| . . . . . . . . . . . .| . . . . . . . . . . . .|
| . . . . . . . . . . . .| . . . . . . . . . . . .|
| . . . . . . . . . . . .@ u v w . . . . . . . . .|
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
| . . . . . . . . . . . . . . . . . . . . . . . . |
| . . . . .(( .( u , du ) ( v , dv ) . . . . . . .|
| . . . . .)( .( u , du ) ( w , dw ) . . . . . . .|
| . . . . .)( .( v , dv ) ( w , dw ) . . . . . . .|
| . . . . .)) . . . . . . . . . . . . . . . . . . |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| . . . . .u v w . . . . . . . . . . . . . . . . .|
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
</pre>
The models of this last expression tell us which combinations of feature changes among the set <math>\{ \operatorname{d}u, \operatorname{d}v, \operatorname{d}w \}</math> will take us from our present interpretation, the center cell expressed by "<code>u v w</code>", to a true value under the given proposition <code>(( u v )( u w )( v w ))</code>.
The models of <math>\operatorname{E}q \cdot c</math> can be described in the usual ways as follows:
* The points of the space <math>\operatorname{E}X</math> that have the following coordinate descriptions:
<pre>
<u, v, w, du, dv, dw> =
<1, 1, 1, 0, 0, 0>,
<1, 1, 1, 0, 0, 1>,
<1, 1, 1, 0, 1, 0>,
<1, 1, 1, 1, 0, 0>.
</pre>
* The points of the space <math>\operatorname{E}X</math> that have the following conjunctive expressions:
<pre>
<pre>
u v w (du)(dv)(dw),
u v w (du)(dv) dw ,
u v w (du) dv (dw),
u v w du (dv)(dw).
</pre>
</pre>
In summary, <math>\operatorname{E}q \cdot c</math> informs us that we can get from <math>c\!</math> to a model of <math>q\!</math> by changing our position with respect to <math>u, v, w\!</math> according to the following description:
<blockquote>
Change none or just one among <math>\{ u, v, w \}.\!</math>
</blockquote>
I think that it would be worth our time to diagram the models of the ''enlarged'' or ''shifted'' proposition, <math>\operatorname{E}q,</math> at least, the selection of them that we find issuing from the center cell <math>c.\!</math>
! style="width:20%" | ''u'' ∧ ''v''
Figure 4 is an extended venn diagram for the proposition <math>\operatorname{E}q \cdot c,</math> where the shaded area gives the models of <math>q\!</math> and the "<code>@</code>" signs mark the terminal points of the requisite feature alterations.
<pre>
o-------------------------------------------------o
| X |
| |
| o-------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| o U o |
| | | |
| | | |
| | | |
| o---o---------o o---------o---o |
| / \%%%%%%%%%\ /%%%%%%%%%/ \ |
| / \%%%%%dw%%o%%dv%%%%%/ \ |
| / \%@<----/@\---->@%/ \ |
| / \%%%%%/%|%\%%%%%/ \ |
| o o---o--|--o---o o |
| | |%%|%%| | |
| | V |%du%%| W | |
| | |% |%%| | |
| o o%%v%%o o |
| \ \%@%/ / |
| \ \%/ / |
| \ o / |
| \ / \ / |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
Figure 4. Effect of the Enlargement Operator E
On the Proposition q, Evaluated at c
</pre>
One more piece of notation will save us a few bytes in the length of many of our schematic formulations.
Let <math>\mathcal{X} = \{ x_1, \ldots, x_k \}</math> be a finite set of variables, regarded as a formal alphabet of formal symbols but listed here without quotation marks. Starting from this initial alphabet, the following items may then be defined:
#<p>The "(first order) differential alphabet",</p><p><math>\operatorname{d}\mathcal{X} = \{ \operatorname{d}x_1, \ldots, \operatorname{d}x_k \}.</math></p>
#<p>The "(first order) extended alphabet",</p><p><math>\operatorname{E}\mathcal{X} = \mathcal{X} \cup \operatorname{d}\mathcal{X},</math></p><p><math>\operatorname{E}\mathcal{X} = \{ x_1, \dots, x_k, \operatorname{d}x_1, \ldots, \operatorname{d}x_k \}.</math></p>
Before we continue with the differential analysis of the source proposition <math>q\!</math>, we need to pause and take another look at just how it shapes up in the light of the extended universe <math>\operatorname{E}X,</math> in other words, to examine in detail its tacit extension <math>\operatorname{e}q.\!</math>
The models of <math>\operatorname{e}q\!</math> in <math>\operatorname{E}X\!</math> can be comprehended as follows:
*<p>Working in the ''summary coefficient'' form of representation, if the coordinate list <math>\mathbf{x}\!</math> is a model of <math>q\!</math> in <math>X,\!</math> then one can construct a coordinate list <math>\operatorname{e}\mathbf{x}\!</math> as a model for <math>\operatorname{e}q\!</math> in <math>\operatorname{E}X\!</math> just by appending any combination of values for the differential variables in <math>\operatorname{d}\mathcal{X}.</math></p><p>For example, to focus once again on the center cell <math>c,\!</math> which happens to be a model of the proposition <math>q\!</math> in <math>X,\!</math> one can extend <math>c\!</math> in eight different ways into <math>\operatorname{E}X,\!</math> and thus get eight models of the tacit extension <math>\operatorname{e}q\!</math> in <math>\operatorname{E}X.\!</math></p><p>It is a trivial exercise to write these out, but it is useful to do so at least once in order to see the patterns of data involved.</p><p>The tacit extensions of <math>c\!</math> that are models of <math>\operatorname{e}q\!</math> in <math>\operatorname{E}X\!</math> are as follows:</p>
<pre>
<u, v, w, du, dv, dw> =
<pre>
<1, 1, 1, 0, 0, 0>,
<1, 1, 1, 0, 0, 1>,
<1, 1, 1, 0, 1, 0>,
<1, 1, 1, 0, 1, 1>,
<1, 1, 1, 1, 0, 0>,
<1, 1, 1, 1, 0, 1>,
<1, 1, 1, 1, 1, 0>,
<1, 1, 1, 1, 1, 1>.
</pre>
</pre>
*<p>Working in the ''conjunctive product'' form of representation, if the conjunctive proposition <math>x\!</math> is a model of <math>q\!</math> in <math>X,\!</math> then one can construct a conjunctive proposition <math>\operatorname{e}x\!</math> as a model for <math>\operatorname{e}q\!</math> in <math>\operatorname{E}X\!</math> just by appending any combination of values for the differential variables in <math>\operatorname{d}\mathcal{X}.</math></p><p>The tacit extensions of <math>c\!</math> that are models of <math>\operatorname{e}q\!</math> in <math>\operatorname{E}X\!</math> are as follows:</p>
<pre>
u v w (du)(dv)(dw),
u v w (du)(dv) dw ,
u v w (du) dv (dw),
u v w (du) dv dw ,
u v w du (dv)(dw),
u v w du (dv) dw ,
u v w du dv (dw),
u v w du dv dw .
</pre>
</pre>
In short, <math>\operatorname{e}q \cdot c</math> just enumerates all of the possible changes in <math>\operatorname{E}X\!</math> that ''derive from'', ''issue from'', or ''stem from'' the cell <math>c\!</math> in <math>X.\!</math>
That was pretty tedious, and I know that it all appears to be totally trivial, which is precisely why we usually just leave it "tacit" in the first place, but hard experience, and a real acquaintance with the confusion that can beset us when we do not render these implicit grounds explicit, have taught me that it will ultimately be necessary to get clear about it, and by this ''clear'' to say ''marked'', not merely ''transparent''.
Before going on, it would probably be a good idea to remind ourselves of just why we are going through with this exercise. It is to unify the world of change, for which aspect or regime of the world I occasionally evoke the eponymous figures of Prometheus and Heraclitus, and the world of logic, for which facet or realm of the world I periodically recur to the prototypical shades of Epimetheus and Parmenides, at least, that is, to state it more carefully, to encompass the antics and the escapades of these all too manifestly strife-born twins within the scopes of our thoughts and within the charts of our theories, as it is most likely the only places where ever they will, for the moment and as long as it lasts, be seen or be heard together.
With that intermezzo, with all of its echoes of the opening overture, over and done, let us now return to that droller drama, already fast in progress, the differential disentanglements, hopefully toward the end of a grandly enlightening denouement, of the ever-polymorphous <math>Q.\!</math>
</pre>
The next transformation of the source proposition <math>q,\!</math> that we are typically aiming to contemplate in the process of carrying out a ''differential analysis'' of its ''dynamic'' effects or implications, is the yield of the so-called ''difference'' or ''delta'' operator <math>\operatorname{D}.</math> The resultant ''difference proposition'' <math>\operatorname{D}q</math> is defined in terms of the source proposition <math>q\!</math> and the ''shifted proposition'' <math>\operatorname{E}q</math> thusly:
<pre>
: <math>\operatorname{D}q = \operatorname{E}q - q = \operatorname{E}q - \operatorname{e}q.</math>
</pre>
: Since "+" and "-" signify the same operation over <math>\mathbb{B},</math> we have:
: <math>\operatorname{D}q = \operatorname{E}q + q = \operatorname{E}q + \operatorname{e}q.</math>
: Since "+" = "exclusive-or", cactus syntax expresses this as:
<pre>
<pre>
o-------------------------------------------------o
Eq q Eq eq
o---o o---o
\ / \ /
Dq = @ = @
Dq = ( Eq , q ) = ( Eq , eq ).
</pre>
</pre>
Recall that a <math>k</math>-place bracket "<math>(x_1, x_2, \ldots, x_k)\!</math>" is interpreted (in the ''existential interpretation'') to mean "Exactly one of the <math>x_j\!</math> is false", thus the two-place bracket is equivalent to the exclusive-or.
The result of applying the difference operator <math>\operatorname{D}</math> to the source proposition <math>q,\!</math> conjoined with a query on the center cell <math>c,\!</math> is:
<pre>
o-------------------------------------------------o
| Dq.uvw |
o-------------------------------------------------o
| |
| u du v dv u du w dw v dv w dw |
| o---o o---o o---o o---o o---o o---o |
| \ | | / \ | | / \ | | / |
| \ | | / \ | | / \ | | / |
| \| |/ \| |/ \| |/ |
| o=o o=o o=o |
| \ | / |
| \ | / |
| \ | / |
| \ | / |
| \ | / |
| \ | / |
| \ | / u v u w v w |
| \ | / o o o |
| \ | / \ | / |
</pre>
| \ | / \ | / |
| \|/ \|/ |
| o o |
| | | |
| | | |
| | | |
| o---------------o |
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| @ u v w |
| |
o-------------------------------------------------o
| |
| ( |
| (( ( u , du ) ( v , dv ) |
| )( ( u , du ) ( w , dw ) |
| )( ( v , dv ) ( w , dw ) |
| )) |
| , |
| (( u v |
| )( u w |
| )( v w |
| )) |
| ) |
| |
| u v w |
| |
o-------------------------------------------------o
</pre>
The models of the difference proposition <math>\operatorname{D}q \cdot uvw\!</math> are:
<pre>
<pre>
1. u v w du dv dw
2. u v w du dv (dw)
3. u v w du (dv) dw
4. u v w (du) dv dw
</pre>
</pre>
This tells us that changing any two or more of the features <math>u, v, w\!</math> will take us from the center cell that is marked by the conjunctive expression "<math>u\ v\ w</math>" to a cell outside the shaded region for the area <math>Q.\!</math>
<pre>
<pre>
o-------------------------------------------------o
| X |
| |
| o-------------o |
| / \ |
| / U \ |
| / \ |
| / \ |
| o @ o |
| | ^ | |
| | |dw | |
| | | | @ |
| o---o---------o o----|----o---o ^ |
| / \%%%%%%%%%\ /%%%%%|%%%/ \ /dw |
| / du \%%%%%dw%%o%%dv%%|%%/ \/ |
| / @<-----\-o<----/+\---->o%/ /\ |
| / \%%%%%/%|%\%%%%%/ / \ |
| o o---o--|--o---o / o |
| | |%%|%%| / | |
| | V |%du%%| / W | |
| | |% |%%| / | |
| o o%%v%%o dv / o |
| \ \%o-/------->@ / |
| \ \%/ / |
| \ o / |
| \ / \ / |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
Figure 3. Effect of the Difference Operator D
Acting on a Polymorphous Function q
</pre>
</pre>
Figure 3 shows one way to picture this kind of a situation, by superimposing the paths of indicated feature changes on the venn diagram of the underlying proposition. Here, the models, or the satisfying interpretations, of the relevant ''difference proposition'' <math>\operatorname{D}q</math> are marked with "<code>@</code>" signs, and the boundary crossings along each path are marked with the corresponding ''differential features'' among the collection <math>\{ \operatorname{d}u, \operatorname{d}v, \operatorname{d}w \}</math>. In sum, starting from the cell <math>u\ v\ w,</math> we have the following four paths:
<pre>
1. du dv dw = Change u, v, w.
2. du dv (dw) = Change u and v.
<pre>
3. du (dv) dw = Change u and w.
o-------------------------------------------------o
4. (du) dv dw = Change v and w.
</pre>
| |
| u du v dv u du w dw v dv w dw |
That sums up, but rather more carefully, the material that I ran through just a bit too quickly the first time around. Next time, I will begin to develop an alternative style of diagram for depicting these types of differential settings.
| o---o o---o o---o o---o o---o o---o |
| \ | | / \ | | / \ | | / |
Another way of looking at this situation is by letting the (first order) differential features <math>\operatorname{d}u, \operatorname{d}v, \operatorname{d}w</math> be viewed as the features of another universe of discourse, called the ''tangent universe'' to <math>X\!</math> with respect to the interpretation <math>c\!</math> and represented as <math>\operatorname{d}X \cdot c</math> . In this setting, <math>\operatorname{D}q \cdot c</math> , the ''difference proposition'' of <math>q\!</math> at the interpretation <math>c\!</math> , where <math>c = u\ v\ w</math> , is marked by the shaded region in Figure 4.
| \ | | / \ | | / \ | | / |
| \| |/ \| |/ \| |/ |
<pre>
| o=o o=o o=o |
o-----------------------------------------------------------o
| \ | / |
| dX.c |
| \ | / |
| |
| \ | / |
| o-------------o |
| \ | / |
| / \ |
| / \ |
| \ | / |
| / \ |
| / \ |
| \ | / o o o |
| / \ |
| \ | / \ | / |
| o o |
| \ | / \ | / |
| | dU | |
| \|/ \|/ |
| | | |
| o o |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
| o--o----------o o----------o--o |
| o---------------o |
| / \%%%%%%%%%%\ /%%%%%%%%%%/ \ |
| \ / |
| / \%%%%2%%%%%o%%%%%3%%%%/ \ |
| \ / |
| / \%%%%%%%%/%\%%%%%%%%/ \ |
| \ / |
| / \%%%%%%/%%%\%%%%%%/ \ |
| \ / |
| / \%%%%/%%1%%\%%%%/ \ |
| \ / |
| o o--o-------o--o o |
| | |%%%%%%%| | |
| | |%%%%%%%| | |
| | |%%%%%%%| | |
| |
| | dV |%%%4%%%| dW | |
o-------------------------------------------------o
| | |%%%%%%%| | |
| o o%%%%%%%o o |
| \ \%%%%%/ / |
| \ \%%%/ / |
| \ \%/ / |
| \ o / |
| \ / \ / |
| o-------------o o-------------o |
| |
| |
o-----------------------------------------------------------o
Figure 4. Tangent Venn Diagram for Dq.c
</pre>
</pre>
Taken in the context of the tangent universe to <math>X\!</math> at <math>c = u\ v\ w</math> , written <math>\operatorname{d}X \cdot c</math> or <math>\operatorname{d}X \cdot u\ v\ w</math> , the shaded area of Figure 4 indicates the models of the difference proposition <math>\operatorname{d}q \cdot u\ v\ w</math> , specifically:
<pre>
<pre>
1. u v w du dv dw
1. u v w du dv dw
2. u v w du dv (dw)
2. u v w du dv (dw)
3. u v w du (dv) dw
3. u v w du (dv) dw
4. u v w (du) dv dw
4. u v w (du) dv dw
</pre>
</pre>
===Example 2. Jets and Sharks===
'''Reference.''' Awbrey, J., and Awbrey, S. (1989), "Theme One : A Program of Inquiry", unpublished manuscript, 09 Aug 1989.
The propositional calculus that is based on the boundary operator can be interpreted in a way that resembles the logic of activation states and competition constraints in certain neural network models. One way to do this is by interpreting the blank or unmarked state as the resting state of a neural pool, the bound or marked state as its activated state, and by representing a mutually inhibitory pool of neurons <math>A, B, C\!</math> in the expression "<math>(A, B, C)\!</math>". To illustrate this possibility, we transcribe a well-known example from the parallel distributed processing literature (McClelland and Rumelhart, 1988) and work through two of the associated exercises as portrayed in Existential Graph format.
<pre>
<pre>
o-------------------------------------------------o
File "jas.log". Jets and Sharks Example
o-----------------------------------------------------------o
| |
| (( art ),( al ),( sam ),( clyde ),( mike ), |
| / \ |
| ( jim ),( greg ),( john ),( doug ),( lance ), |
| / U \ |
| ( george ),( pete ),( fred ),( gene ),( ralph ), |
| / \ |
| ( phil ),( ike ),( nick ),( don ),( ned ), |
| ( karl ),( ken ),( earl ),( rick ),( ol ), |
| ( neal ),( dave )) |
| |
| ( jets , sharks ) |
| |
| o---o---------o o----|----o---o ^ |
| ( jets , |
| ( art ),( al ),( sam ),( clyde ),( mike ), |
| / du \`````dw``o``dv``|``/ \/ |
| ( jim ),( greg ),( john ),( doug ),( lance ), |
| ( george ),( pete ),( fred ),( gene ),( ralph )) |
| |
| o o---o--|--o---o / o |
| ( sharks , |
| | |``|``| / | |
| ( phil ),( ike ),( nick ),( don ),( ned ),( karl ), |
| | V |`du``| / W | |
| ( ken ),( earl ),( rick ),( ol ),( neal ),( dave )) |
| | |` |``| / | |
| |
| o o``v``o dv / o |
| (( 20's ),( 30's ),( 40's )) |
| \ \`o-/------->@ / |
| \ \`/ / |
| \ / \ / |
| o-------------o o-------------o |
| |
| |
| |
| o-------------o |
| ( 20's , |
| ( sam ),( jim ),( greg ),( john ),( lance ), |
| ( george ),( pete ),( fred ),( gene ),( ken )) |
| |
| ( 30's , |
| ( al ),( mike ),( doug ),( ralph ),( phil ), |
| ( ike ),( nick ),( don ),( ned ),( rick ), |
| ( ol ),( neal ),( dave )) |
| |
| ( 40's , |
| ( art ),( clyde ),( karl ),( earl )) |
| |
| (( junior_high ),( high_school ),( college )) |
| / \``````````\ /``````````/ \ |
| | |```````| | |
| | |```````| | |
| | |```````| | |
| | dV |```4```| dW | |
| | |```````| | |
| o o```````o o |
| \ \`````/ / |
| \ \```/ / |
| \ \`/ / |
| \ o / |
| \ / \ / |
| |
| |
| ( junior_high , |
| ( art ),( al ),( clyde ),( mike ),( jim ), |
| ( john ),( lance ),( george ),( ralph ),( ike )) |
| |
| |
| ( high_school , |
| ( greg ),( doug ),( pete ),( fred ), |
| ( nick ),( karl ),( ken ),( earl ), |
| ( rick ),( neal ),( dave )) |
| |
| |
| (( art ),( al ),( sam ),( clyde ),( mike ), |
| ( college , |
| ( jim ),( greg ),( john ),( doug ),( lance ), |
| ( sam ),( gene ),( phil ),( don ),( ned ),( ol )) |
| |
| |
| ( jets , sharks ) |
| (( single ),( married ),( divorced )) |
| |
| |
| ( jets , |
| ( single , |
| ( art ),( al ),( sam ),( clyde ),( mike ), |
| ( art ),( sam ),( clyde ),( mike ),( doug ), |
| ( jim ),( greg ),( john ),( doug ),( lance ), |
| ( pete ),( fred ),( gene ),( ralph ),( ike ), |
| ( george ),( pete ),( fred ),( gene ),( ralph )) |
| ( nick ),( ken ),( neal )) |
| |
| |
| ( sharks , |
| ( married , |
| ( phil ),( ike ),( nick ),( don ),( ned ),( karl ), |
| ( al ),( greg ),( john ),( lance ),( phil ), |
| ( ken ),( earl ),( rick ),( ol ),( neal ),( dave )) |
| ( don ),( ned ),( karl ),( earl ),( ol )) |
| |
| |
| (( 20's ),( 30's ),( 40's )) |
| ( divorced , |
| ( jim ),( george ),( rick ),( dave )) |
| |
| |
| ( 20's , |
| (( bookie ),( burglar ),( pusher )) |
| |
| |
| ( 30's , |
| ( bookie , |
| ( al ),( mike ),( doug ),( ralph ),( phil ), |
| ( sam ),( clyde ),( mike ),( doug ), |
| ( ike ),( nick ),( don ),( ned ),( rick ), |
| ( pete ),( ike ),( ned ),( karl ),( neal )) |
| |
| |
| ( 40's , |
| ( burglar , |
| ( art ),( clyde ),( karl ),( earl )) |
| ( al ),( jim ),( john ),( lance ), |
| ( george ),( don ),( ken ),( earl ),( rick )) |
| |
| |
| (( junior_high ),( high_school ),( college )) |
| ( pusher , |
| ( art ),( greg ),( fred ),( gene ), |
| ( ralph ),( phil ),( nick ),( ol ),( dave )) |
| |
| |
o-----------------------------------------------------------o
</pre>
We now apply the ''Study'' tool to the proposition defining the Jets and Sharks data base.
With a query on the name "ken" we obtain all of the propositional features associated with Ken, as shown in the following output.
<pre>
File "ken.sen". Output of Query on "ken"
o-----------------------------------------------------------o
| |
| |
| ( high_school , |
| ken |
| ( greg ),( doug ),( pete ),( fred ), |
| sharks |
| ( nick ),( karl ),( ken ),( earl ), |
| 20's |
| ( rick ),( neal ),( dave )) |
| high_school |
| single |
| burglar |
| |
| |
o-----------------------------------------------------------o
</pre>
With a query on the two features "college" and "sharks" we obtain the following outline of all features satisfying these constraints.
<pre>
File "cos.sen". Output of Query on "college" and "sharks"
o-----------------------------------------------------------o
| |
| |
| (( single ),( married ),( divorced )) |
| college |
| sharks |
| 30's |
| ( art ),( sam ),( clyde ),( mike ),( doug ), |
| married |
| ( pete ),( fred ),( gene ),( ralph ),( ike ), |
| bookie |
| ned |
| |
| burglar |
| don |
| ( al ),( greg ),( john ),( lance ),( phil ), |
| pusher |
| phil |
| ol |
| ( sam ),( clyde ),( mike ),( doug ), |
| |
| ( al ),( jim ),( john ),( lance ), |
| |
| ( art ),( greg ),( fred ),( gene ), |
| ( ralph ),( phil ),( nick ),( ol ),( dave )) |
| |
| |
o-----------------------------------------------------------o
o-----------------------------------------------------------o
</pre>
</pre>
From this we discover that all college Sharks are 30-something and married. Further, we have a complete listing of their names broken down by occupation, as no doubt all of them will be, eventually.
Those who already know the tune,<br>
Be at liberty to sing out of it.
<pre>
===Interlude===
<blockquote>
<p>"The burden of genius is undeliverable"</p>
<p>From a poster, as I once misread it,<br>Marlboro, Vermont, ''c.'' 1976</p>
</blockquote>
How does Cosmo, and by this I mean my pet personification of cosmic order in the universe, not to be too tautologous about it, preserve a memory like that, a goodly fraction of a century later, whether localized to this body that's kept going by this heart, and whether by common assumption still more localized to the spongey fibres of this brain, or not?
It strikes me, as it has struck others, that it's terribly unlikely to be stored in persistent patterns of activation, for ''activation'' and ''persistent'' are nigh a contradiction in terms, as even the author, Cosmo, of the ''I Ching'' knew.
But that was then, this is now, so let me try to say it planar.
===Notes on Cactus Language===
I happened on the graphical syntax for propositional calculus that I now call the ''cactus language'' while exploring the confluence of three streams of thought. There was C.S. Peirce's use of operator variables in logical forms and the operational representations of logical concepts, there was George Spencer Brown's explanation of a variable as the contemplated presence or absence of a constant, and then there was the graph theory and group theory that I had been picking up, bit by bit, since I first encountered them in tandem in Frank Harary's foundations of math course, ''c.'' 1970.
More on that later, as the memories unthaw, but for the moment I want very much to take care of some long-unfinished business, and give a more detailed explanation of how I used this syntax to represent a popular exercise from the PDP literature of the late 1980's, McClelland's and Rumelhart's "Jets and Sharks".
The knowledge base of the case can be expressed as a single proposition. The following display presents it in the corresponding text file format.
<pre>
<pre>
File "cos.sen". Output of Query on "college" and "sharks"
File "jas.log". Jets and Sharks Example
o-----------------------------------------------------------o
o-----------------------------------------------------------o
| |
| |
| college |
| (( art ),( al ),( sam ),( clyde ),( mike ), |
| sharks |
| ( jim ),( greg ),( john ),( doug ),( lance ), |
| 30's |
| ( george ),( pete ),( fred ),( gene ),( ralph ), |
| married |
| ( phil ),( ike ),( nick ),( don ),( ned ), |
| bookie |
| ( karl ),( ken ),( earl ),( rick ),( ol ), |
| ned |
| ( neal ),( dave )) |
| burglar |
| |
| don |
| ( jets , sharks ) |
| pusher |
| |
| phil |
| ( jets , |
| ol |
| ( art ),( al ),( sam ),( clyde ),( mike ), |
| ( jim ),( greg ),( john ),( doug ),( lance ), |
| ( george ),( pete ),( fred ),( gene ),( ralph )) |
| |
| ( sharks , |
| ( phil ),( ike ),( nick ),( don ),( ned ),( karl ), |
| ( ken ),( earl ),( rick ),( ol ),( neal ),( dave )) |
| |
| |
| (( 20's ),( 30's ),( 40's )) |
| |
| |
| (( art ),( al ),( sam ),( clyde ),( mike ), |
| ( 20's , |
| ( jim ),( greg ),( john ),( doug ),( lance ), |
| ( sam ),( jim ),( greg ),( john ),( lance ), |
| ( george ),( pete ),( fred ),( gene ),( ralph ), |
| ( george ),( pete ),( fred ),( gene ),( ken )) |
| |
| |
| ( 30's , |
| ( al ),( mike ),( doug ),( ralph ),( phil ), |
| ( 30's , |
| ( al ),( mike ),( doug ),( ralph ),( phil ), |
| ( ike ),( nick ),( don ),( ned ),( rick ), |
| ( ike ),( nick ),( don ),( ned ),( rick ), |
| ( ol ),( neal ),( dave )) |
| ( ol ),( neal ),( dave )) |
| |
| |
| o-------------o |
| o-------------o |
| /```````````````\ |
| /%%%%%%%%%%%%%%%\ |
| /`````````````````\ |
| /%%%%%%%%%%%%%%%%%\ |
| /```````````````````\ |
| /%%%%%%%%%%%%%%%%%%%\ |
| /`````````````````````\ |
| /%%%%%%%%%%%%%%%%%%%%%\ |
| /```````````````````````\ |
| /%%%%%%%%%%%%%%%%%%%%%%%\ |
| o`````````````````````````o |
| o%%%%%%%%%%%%%%%%%%%%%%%%%o |
| |``````````` X ```````````| |
| |%%%%%%%%%%% X %%%%%%%%%%%| |
| |`````````````````````````| |
| |%%%%%%%%%%%%%%%%%%%%%%%%%| |
| |`````````````````````````| |
| |%%%%%%%%%%%%%%%%%%%%%%%%%| |
| |`````````````````````````| |
| |%%%%%%%%%%%%%%%%%%%%%%%%%| |
| |`````````````````````````| |
| |%%%%%%%%%%%%%%%%%%%%%%%%%| |
| o--o----------o```o----------o--o |
| o--o----------o%%%o----------o--o |
| /````\ \`/ /````\ |
| /%%%%\ \%/ /%%%%\ |
| /``````\ o /``````\ |
| /%%%%%%\ o /%%%%%%\ |
| /````````\ / \ /````````\ |
| /%%%%%%%%\ / \ /%%%%%%%%\ |
| /``````````\ / \ /``````````\ |
| /%%%%%%%%%%\ / \ /%%%%%%%%%%\ |
| /````````````\ / \ /````````````\ |
| /%%%%%%%%%%%%\ / \ /%%%%%%%%%%%%\ |
| o``````````````o--o-------o--o``````````````o |
| o%%%%%%%%%%%%%%o--o-------o--o%%%%%%%%%%%%%%o |
| |`````````````````| |`````````````````| |
| |%%%%%%%%%%%%%%%%%| |%%%%%%%%%%%%%%%%%| |
| |`````````````````| |`````````````````| |
| |%%%%%%%%%%%%%%%%%| |%%%%%%%%%%%%%%%%%| |
| |`````````````````| |`````````````````| |
| |%%%%%%%%%%%%%%%%%| |%%%%%%%%%%%%%%%%%| |
| |``````` Y ```````| |`````` Z ````````| |
| |%%%%%%% Y %%%%%%%| |%%%%%% Z %%%%%%%%| |
| |`````````````````| |`````````````````| |
| |%%%%%%%%%%%%%%%%%| |%%%%%%%%%%%%%%%%%| |
| o`````````````````o o`````````````````o |
| o%%%%%%%%%%%%%%%%%o o%%%%%%%%%%%%%%%%%o |
| \`````````````````\ /`````````````````/ |
| \%%%%%%%%%%%%%%%%%\ /%%%%%%%%%%%%%%%%%/ |
| \`````````````````\ /`````````````````/ |
| \%%%%%%%%%%%%%%%%%\ /%%%%%%%%%%%%%%%%%/ |
| \`````````````````\ /`````````````````/ |
| \%%%%%%%%%%%%%%%%%\ /%%%%%%%%%%%%%%%%%/ |
| \`````````````````o`````````````````/ |
| \%%%%%%%%%%%%%%%%%o%%%%%%%%%%%%%%%%%/ |
| \```````````````/ \```````````````/ |
| \%%%%%%%%%%%%%%%/ \%%%%%%%%%%%%%%%/ |
| o-------------o o-------------o |
| o-------------o o-------------o |
| |
| |
| |
| |
| o-------------o |
| o-------------o |
| /```````````````\ |
| /%%%%%%%%%%%%%%%\ |
| /`````````````````\ |
| /%%%%%%%%%%%%%%%%%\ |
| /```````````````````\ |
| /%%%%%%%%%%%%%%%%%%%\ |
| /`````````````````````\ |
| /%%%%%%%%%%%%%%%%%%%%%\ |
| /```````````````````````\ |
| /%%%%%%%%%%%%%%%%%%%%%%%\ |
| o`````````````````````````o |
| o%%%%%%%%%%%%%%%%%%%%%%%%%o |
| |``````````` X ```````````| |
| |%%%%%%%%%%% X %%%%%%%%%%%| |
| |`````````````````````````| |
| |%%%%%%%%%%%%%%%%%%%%%%%%%| |
| |`````````````````````````| |
| |%%%%%%%%%%%%%%%%%%%%%%%%%| |
| |`````````````````````````| |
| |%%%%%%%%%%%%%%%%%%%%%%%%%| |
| |`````````````````````````| |
| |%%%%%%%%%%%%%%%%%%%%%%%%%| |
| o--o----------o```o----------o--o |
| o--o----------o%%%o----------o--o |
| /````\ \`/ /````\ |
| /%%%%\ \%/ /%%%%\ |
| /``````\ o /``````\ |
| /%%%%%%\ o /%%%%%%\ |
| /````````\ / \ /````````\ |
| /%%%%%%%%\ / \ /%%%%%%%%\ |
| /``````````\ / \ /``````````\ |
| /%%%%%%%%%%\ / \ /%%%%%%%%%%\ |
| /````````````\ / \ /````````````\ |
| /%%%%%%%%%%%%\ / \ /%%%%%%%%%%%%\ |
| o``````````````o--o-------o--o``````````````o |
| o%%%%%%%%%%%%%%o--o-------o--o%%%%%%%%%%%%%%o |
| |`````````````````| |`````````````````| |
| |%%%%%%%%%%%%%%%%%| |%%%%%%%%%%%%%%%%%| |
| |`````````````````| |`````````````````| |
| |%%%%%%%%%%%%%%%%%| |%%%%%%%%%%%%%%%%%| |
| |`````````````````| |`````````````````| |
| |%%%%%%%%%%%%%%%%%| |%%%%%%%%%%%%%%%%%| |
| |``````` Y ```````| |`````` Z ````````| |
| |%%%%%%% Y %%%%%%%| |%%%%%% Z %%%%%%%%| |
| |`````````````````| |`````````````````| |
| |%%%%%%%%%%%%%%%%%| |%%%%%%%%%%%%%%%%%| |
| o`````````````````o o`````````````````o |
| o%%%%%%%%%%%%%%%%%o o%%%%%%%%%%%%%%%%%o |
| \`````````````````\ /`````````````````/ |
| \%%%%%%%%%%%%%%%%%\ /%%%%%%%%%%%%%%%%%/ |
| \`````````````````\ /`````````````````/ |
| \%%%%%%%%%%%%%%%%%\ /%%%%%%%%%%%%%%%%%/ |
| \`````````````````\ /`````````````````/ |
| \%%%%%%%%%%%%%%%%%\ /%%%%%%%%%%%%%%%%%/ |
| \`````````````````o`````````````````/ |
| \%%%%%%%%%%%%%%%%%o%%%%%%%%%%%%%%%%%/ |
| \```````````````/ \```````````````/ |
| \%%%%%%%%%%%%%%%/ \%%%%%%%%%%%%%%%/ |
| o-------------o o-------------o |
| o-------------o o-------------o |
| |
| |
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| x y |
| . . . . . . . . . . . x y . . . . . . . . . . . |
| @ |
| . . . . . . . . . . . .@. . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| x and y |
| . . . . . . . . . . x and y . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| @ |
| . . . . . . . . . . . .@. . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| true |
| . . . . . . . . . . .true . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| o |
| . . . . . . . . . . . .o. . . . . . . . . . . . |
| | |
| . . . . . . . . . . . .|. . . . . . . . . . . . |
| @ |
| . . . . . . . . . . . .@. . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| false |
| . . . . . . . . . . .false . . . . . . . . . . .|
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| o-----------o o-----------o |
| . . . . .o-----------o. .o-----------o. . . . . |
| / \ / \ |
| . . . . / . . . . . . \ / . . . . . . \ . . . . |
| / o \ |
| . . . ./. . . . . . . .o. . . . . . . .\. . . . |
| / /%\ \ |
| . . . / . . . . . . . /%\ . . . . . . . \ . . . |
| / /%%%\ \ |
| . . ./. . . . . . . ./%%%\. . . . . . . .\. . . |
| o o%%%%%o o |
| . . o . . . . . . . o%%%%%o . . . . . . . o . . |
| | |%%%%%| | |
| . . | . . . . . . . |%%%%%| . . . . . . . | . . |
| | |%%%%%| | |
| . . | . . . . . . . |%%%%%| . . . . . . . | . . |
| | x |%%%%%| y | |
| . . | . . . x . . . |%%%%%| . . . y . . . | . . |
| | |%%%%%| | |
| . . | . . . . . . . |%%%%%| . . . . . . . | . . |
| | |%%%%%| | |
| . . | . . . . . . . |%%%%%| . . . . . . . | . . |
| o o%%%%%o o |
| . . o . . . . . . . o%%%%%o . . . . . . . o . . |
| \ \%%%/ / |
| . . .\. . . . . . . .\%%%/. . . . . . . ./. . . |
| \ \%/ / |
| . . . \ . . . . . . . \%/ . . . . . . . / . . . |
| \ o / |
| . . . .\. . . . . . . .o. . . . . . . ./. . . . |
| \ / \ / |
| . . . . \ . . . . . . / \ . . . . . . / . . . . |
| o-----------o o-----------o |
| . . . . .o-----------o. .o-----------o. . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
Now ask yourself: What is the value of the proposition ''xy'' at a distance of ''dx'' and ''dy'' from the cell ''xy'' where you are standing?
Now ask yourself: .What is the value of the proposition ''xy'' at a distance of ''dx'' and ''dy'' from the cell ''xy'' where you are standing?
Don't think about it -- just compute:
Don't think about it -- just compute:
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| dx o o dy |
| . . . . . . . . . dx o. .o dy . . . . . . . . . |
| / \ / \ |
| . . . . . . . . . . / \ / \ . . . . . . . . . . |
| x o---@---o y |
| . . . . . . . . .x o---@---o y. . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| (x + dx) and (y + dy) |
| . . . . . . .(x + dx) and (y + dy). . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| x dx y dy |
| . . . . . . . . . x .dx y .dy . . . . . . . . . |
| o---o o---o |
| . . . . . . . . . o---o o---o . . . . . . . . . |
| \ | | / |
| . . . . . . . . . .\. | | ./. . . . . . . . . . |
| \ | | / |
| . . . . . . . . . . \ | | / . . . . . . . . . . |
| \| |/ |
| . . . . . . . . . . .\| |/. . . . . . . . . . . |
| @=@ |
| . . . . . . . . . . . @=@ . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| (x + dx) and (y + dy) |
| . . . . . . .(x + dx) and (y + dy). . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| x dx |
| . . . . . . . . . . x . .dx . . . . . . . . . . |
| o---o |
| . . . . . . . . . . .o---o. . . . . . . . . . . |
| \ / |
| . . . . . . . . . . . \ / . . . . . . . . . . . |
| @ |
| . . . . . . . . . . . .@. . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| x + dx |
| . . . . . . . . . . .x + dx . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
Next question: What is the difference between the value of the proposition ''xy'' "over there" and the value of the proposition ''xy'' where you are, all expressed as general formula, of course? Here 'tis:
Next question: .What is the difference between the value of the proposition ''xy'' "over there" and the value of the proposition ''xy'' where you are, all expressed as general formula, of course? .Here 'tis:
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| x dx y dy |
| . . . . . . x .dx y .dy . . . . . . . . . . . . |
| o---o o---o |
| . . . . . . o---o o---o . . . . . . . . . . . . |
| \ | | / |
| . . . . . . .\. | | ./. . . . . . . . . . . . . |
| \ | | / |
| . . . . . . . \ | | / . . . . . . . . . . . . . |
| \| |/ x y |
| . . . . . . . .\| |/. . . . .x y. . . . . . . . |
| o=o-----------o |
| . . . . . . . . o=o-----------o . . . . . . . . |
| \ / |
| . . . . . . . . .\. . . . . ./. . . . . . . . . |
| \ / |
| . . . . . . . . . \ . . . . / . . . . . . . . . |
| \ / |
| . . . . . . . . . .\. . . ./. . . . . . . . . . |
| \ / |
| . . . . . . . . . . \ . . / . . . . . . . . . . |
| \ / |
| . . . . . . . . . . .\. ./. . . . . . . . . . . |
| \ / |
| . . . . . . . . . . . \ / . . . . . . . . . . . |
| @ |
| . . . . . . . . . . . .@. . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| ((x + dx) & (y + dy)) - xy |
| . . . . . ((x + dx) & (y + dy)) - xy. . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
Oh, I forgot to mention: Computed over '''B''', plus and minus are the very same operation. This will make the relationship between the differential and the integral parts of the resulting calculus slightly stranger than usual, but never mind that now.
Oh, I forgot to mention: .Computed over '''B''', plus and minus are the very same operation. .This will make the relationship between the differential and the integral parts of the resulting calculus slightly stranger than usual, but never mind that now.
Last question, for now: What is the value of this expression from your current standpoint, that is, evaluated at the point where ''xy'' is true? Well, substituting 1 for ''x'' and 1 for ''y'' in the graph amounts to the same thing as erasing those labels:
Last question, for now: .What is the value of this expression from your current standpoint, that is, evaluated at the point where ''xy'' is true? .Well, substituting 1 for ''x'' and 1 for ''y'' in the graph amounts to the same thing as erasing those labels:
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| dx dy |
| . . . . . .dx . .dy . . . . . . . . . . . . . . |
| o---o o---o |
| . . . . . . o---o o---o . . . . . . . . . . . . |
| \ | | / |
| . . . . . . .\. | | ./. . . . . . . . . . . . . |
| \ | | / |
| . . . . . . . \ | | / . . . . . . . . . . . . . |
| \| |/ |
| . . . . . . . .\| |/. . . . . . . . . . . . . . |
| o=o-----------o |
| . . . . . . . . o=o-----------o . . . . . . . . |
| \ / |
| . . . . . . . . .\. . . . . ./. . . . . . . . . |
| \ / |
| . . . . . . . . . \ . . . . / . . . . . . . . . |
| \ / |
| . . . . . . . . . .\. . . ./. . . . . . . . . . |
| \ / |
| . . . . . . . . . . \ . . / . . . . . . . . . . |
| \ / |
| . . . . . . . . . . .\. ./. . . . . . . . . . . |
| \ / |
| . . . . . . . . . . . \ / . . . . . . . . . . . |
| @ |
| . . . . . . . . . . . .@. . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| ((1 + dx) & (1 + dy)) - 1&1 |
| . . . . . ((1 + dx) & (1 + dy)) - 1&1 . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| dx dy |
| . . . . . . . . . . dx. .dy . . . . . . . . . . |
| o o |
| . . . . . . . . . . .o. .o. . . . . . . . . . . |
| \ / |
| . . . . . . . . . . . \ / . . . . . . . . . . . |
| o |
| . . . . . . . . . . . .o. . . . . . . . . . . . |
| | |
| . . . . . . . . . . . .|. . . . . . . . . . . . |
| @ |
| . . . . . . . . . . . .@. . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| dx or dy |
| . . . . . . . . . .dx or dy . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
We have just met with the fact that the differential of the "and" is the "or" of the differentials.
We have just met with the fact that the differential of the "and" is the "or" of the differentials.
: ''x'' and ''y'' --Diff--> ''dx'' or ''dy''.
: ''x'' and ''y'' .--Diff--> .''dx'' or ''dy''.
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| dx dy |
| . . . . . . . . . . . . . . . . . .dx . dy. . . |
| o o |
| . . . . . . . . . . . . . . . . . . o . o . . . |
| \ / |
| . . . . . . . . . . . . . . . . . . .\ /. . . . |
| o |
| . . . . . . . . . . . . . . . . . . . o . . . . |
| x y | |
| . . . .x y. . . . . . . . . . . . . . | . . . . |
| @ --Diff--> @ |
| . . . . @ . . . . .--Diff-->. . . . . @ . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| x y --Diff--> ((dx) (dy)) |
| . . . .x y. . . . .--Diff-->. . .((dx) (dy)). . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
It will be necessary to develop a more refined analysis of this statement directly, but that is roughly the nub of it.
It will be necessary to develop a more refined analysis of this statement directly, but that is roughly the nub of it.
If the form of the above statement reminds you of De Morgan's rule, it is no accident, as differentiation and negation turn out to be closely related operations. Indeed, one can find discussions of logical difference calculus in the Boole-De Morgan correspondence and [[C.S. Peirce]] also made use of differential operators in a logical context, but the exploration of these ideas has been hampered by a number of factors, not the least of which being a syntax adequate to handle the complexity of expressions that evolve.
If the form of the above statement reminds you of De Morgan's rule, it is no accident, as differentiation and negation turn out to be closely related operations. .Indeed, one can find discussions of logical difference calculus in the Boole-De Morgan correspondence and [[C.S. Peirce]] also made use of differential operators in a logical context, but the exploration of these ideas has been hampered by a number of factors, not the least of which being a syntax adequate to handle the complexity of expressions that evolve.
For my part, it was definitely a case of the calculus being smarter than the calculator thereof. The graphical pictures were catalytic in their power over my thinking process, leading me so quickly past so many obstructions that I did not have time to think about all of the difficulties that would otherwise have inhibited the derivation. It did eventually became necessary to write all this up in a linear script, and to deal with the various problems of interpretation and justification that I could imagine, but that took another 120 pages, and so, if you don't like this intuitive approach, then let that be your sufficient notice.
For my part, it was definitely a case of the calculus being smarter than the calculator thereof. The graphical pictures were catalytic in their power over my thinking process, leading me so quickly past so many obstructions that I did not have time to think about all of the difficulties that would otherwise have inhibited the derivation. .It did eventually became necessary to write all this up in a linear script, and to deal with the various problems of interpretation and justification that I could imagine, but that took another 120 pages, and so, if you don't like this intuitive approach, then let that be your sufficient notice.
Let us run through the initial example again, this time attempting to interpret the formulas that develop at each stage along the way.
Let us run through the initial example again, this time attempting to interpret the formulas that develop at each stage along the way.
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| o-----------o o-----------o |
| . . . . .o-----------o. .o-----------o. . . . . |
| / \ / \ |
| . . . . / . . . . . . \ / . . . . . . \ . . . . |
| / o \ |
| . . . ./. . . . . . . .o. . . . . . . .\. . . . |
| / /`\ \ |
| . . . / . . . . . . . /%\ . . . . . . . \ . . . |
| / /```\ \ |
| . . ./. . . . . . . ./%%%\. . . . . . . .\. . . |
| o o`````o o |
| . . o . . . . . . . o%%%%%o . . . . . . . o . . |
| | |`````| | |
| . . | . . . . . . . |%%%%%| . . . . . . . | . . |
| | |`````| | |
| . . | . . . . . . . |%%%%%| . . . . . . . | . . |
| | x |``f``| y | |
| . . | . . . x . . . |%%f%%| . . . y . . . | . . |
| | |`````| | |
| . . | . . . . . . . |%%%%%| . . . . . . . | . . |
| | |`````| | |
| . . | . . . . . . . |%%%%%| . . . . . . . | . . |
| o o`````o o |
| . . o . . . . . . . o%%%%%o . . . . . . . o . . |
| \ \```/ / |
| . . .\. . . . . . . .\%%%/. . . . . . . ./. . . |
| \ \`/ / |
| . . . \ . . . . . . . \%/ . . . . . . . / . . . |
| \ o / |
| . . . .\. . . . . . . .o. . . . . . . ./. . . . |
| \ / \ / |
| . . . . \ . . . . . . / \ . . . . . . / . . . . |
| o-----------o o-----------o |
| . . . . .o-----------o. .o-----------o. . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| x y |
| . . . . . . . . . . . x y . . . . . . . . . . . |
| @ |
| . . . . . . . . . . . .@. . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| f = x y |
| f = . . . . . . . . . x y . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
A function like this has an abstract type and a concrete type. The abstract type is what we invoke when we write things like ''f'' : '''B''' × '''B''' → '''B''' or ''f'' : '''B'''<sup>2</sup> → '''B'''. The concrete type takes into account the qualitative dimensions or the "units" of the case, which can be explained as follows.
A function like this has an abstract type and a concrete type. .The abstract type is what we invoke when we write things like ''f'' : '''B''' × '''B''' → '''B''' or ''f'' : '''B'''<sup>2</sup> → '''B'''. .The concrete type takes into account the qualitative dimensions or the "units" of the case, which can be explained as follows.
* Let ''X'' be the set of values {(''x''), ''x''} = {not ''x'', ''x''}.
* Let ''X'' be the set of values {(''x''), ''x''} = {not ''x'', ''x''}.
Then interpret the usual propositions about ''x'', ''y'' as functions of the concrete type ''f'' : ''X'' × ''Y'' → '''B'''.
Then interpret the usual propositions about ''x'', ''y'' as functions of the concrete type ''f'' : ''X'' × ''Y'' → '''B'''.
We are going to consider various "operators" on these functions. Here, an operator ''F'' is a function that takes one function ''f'' into another function ''Ff''.
We are going to consider various "operators" on these functions. .Here, an operator ''F'' is a function that takes one function ''f'' into another function ''Ff''.
The first couple of operators that we need to consider are logical analogues of those that occur in the classical "finite difference calculus", namely:
The first couple of operators that we need to consider are logical analogues of those that occur in the classical "finite difference calculus", namely:
These days, ''E'' is more often called the ''shift'' operator.
These days, ''E'' is more often called the ''shift'' operator.
In order to describe the universe in which these operators operate, it will be necessary to enlarge our original universe of discourse. We mount up from the space ''U'' = ''X'' × ''Y'' to its ''differential extension'',
In order to describe the universe in which these operators operate, it will be necessary to enlarge our original universe of discourse. .We mount up from the space ''U'' = ''X'' × ''Y'' to its ''differential extension'',
''EU'' = ''U'' × ''dU'' = ''X'' × ''Y'' × ''dX'' × ''dY'', with ''dX'' = {(''dx''), ''dx''} and ''dY'' = {(''dy''), ''dy''}. The interpretations of these new symbols can be diverse, but the easiest for now is just to say that ''dx'' means "change x" and ''dy'' means "change y". To draw the differential extension ''EU'' of our present universe ''U'' = ''X'' × ''Y'' as a venn diagram, it would take us four logical dimensions ''X'', ''Y'', ''dX'', ''dY'', but we can project a suggestion of what it's about on the universe ''X'' × ''Y'' by drawing arrows that cross designated borders, labeling the arrows as ''dx'' when crossing the border between ''x'' and (''x'') and as ''dy'' when crossing the border between ''y'' and (''y''), in either direction, in either case.
''EU'' = ''U'' × ''dU'' = ''X'' × ''Y'' × ''dX'' × ''dY'', with ''dX'' = {(''dx''), ''dx''} and ''dY'' = {(''dy''), ''dy''}. .The interpretations of these new symbols can be diverse, but the easiest for now is just to say that ''dx'' means "change x" and ''dy'' means "change y". .To draw the differential extension ''EU'' of our present universe ''U'' = ''X'' × ''Y'' as a venn diagram, it would take us four logical dimensions ''X'', ''Y'', ''dX'', ''dY'', but we can project a suggestion of what it's about on the universe ''X'' × ''Y'' by drawing arrows that cross designated borders, labeling the arrows as ''dx'' when crossing the border between ''x'' and (''x'') and as ''dy'' when crossing the border between ''y'' and (''y''), in either direction, in either case.
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| o-----------o o-----------o |
| . . . . .o-----------o. .o-----------o. . . . . |
| / \ / \ |
| . . . . / . . . . . . \ / . . . . . . \ . . . . |
| / x o y \ |
| . . . ./. . . .x. . . .o. . . .y. . . .\. . . . |
| / /`\ \ |
| . . . / . . . . . . . /%\ . . . . . . . \ . . . |
| / /```\ \ |
| . . ./. . . . . . . ./%%%\. . . . . . . .\. . . |
| o o`````o o |
| . . o . . . . . . . o%%%%%o . . . . . . . o . . |
| | |`````| | |
| . . | . . . . . . . |%%%%%| . . . . . . . | . . |
| | dy |`````| dx | |
| . . | . . . . dy. . |%%%%%| . .dx . . . . | . . |
| | <---------|--o--|---------> | |
| . . | . . <---------|--o--|---------> . . | . . |
| | |`````| | |
| . . | . . . . . . . |%%%%%| . . . . . . . | . . |
| | |`````| | |
| . . | . . . . . . . |%%%%%| . . . . . . . | . . |
| o o`````o o |
| . . o . . . . . . . o%%%%%o . . . . . . . o . . |
| \ \```/ / |
| . . .\. . . . . . . .\%%%/. . . . . . . ./. . . |
| \ \`/ / |
| . . . \ . . . . . . . \%/ . . . . . . . / . . . |
| \ o / |
| . . . .\. . . . . . . .o. . . . . . . ./. . . . |
| \ / \ / |
| . . . . \ . . . . . . / \ . . . . . . / . . . . |
| o-----------o o-----------o |
| . . . . .o-----------o. .o-----------o. . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| x dx y dy |
| . . . . . . . . . x .dx y .dy . . . . . . . . . |
| o---o o---o |
| . . . . . . . . . o---o o---o . . . . . . . . . |
| \ | | / |
| . . . . . . . . . .\. | | ./. . . . . . . . . . |
| \ | | / |
| . . . . . . . . . . \ | | / . . . . . . . . . . |
| \| |/ |
| . . . . . . . . . . .\| |/. . . . . . . . . . . |
| @=@ |
| . . . . . . . . . . . @=@ . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| Ef = (x, dx) (y, dy) |
| Ef = . . . . . .(x, dx) (y, dy) . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| x dx y dy |
| . . . . . . x .dx y .dy . . . . . . . . . . . . |
| o---o o---o |
| . . . . . . o---o o---o . . . . . . . . . . . . |
| \ | | / |
| . . . . . . .\. | | ./. . . . . . . . . . . . . |
| \ | | / |
| . . . . . . . \ | | / . . . . . . . . . . . . . |
| \| |/ x y |
| . . . . . . . .\| |/. . . . .x y. . . . . . . . |
| o=o-----------o |
| . . . . . . . . o=o-----------o . . . . . . . . |
| \ / |
| . . . . . . . . .\. . . . . ./. . . . . . . . . |
| \ / |
| . . . . . . . . . \ . . . . / . . . . . . . . . |
| \ / |
| . . . . . . . . . .\. . . ./. . . . . . . . . . |
| \ / |
| . . . . . . . . . . \ . . / . . . . . . . . . . |
| \ / |
| . . . . . . . . . . .\. ./. . . . . . . . . . . |
| \ / |
| . . . . . . . . . . . \ / . . . . . . . . . . . |
| @ |
| . . . . . . . . . . . .@. . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| Df = ((x, dx)(y, dy), xy) |
| Df = . . . . . ((x, dx)(y, dy), xy) . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
We did not yet go through the trouble to interpret this (first order) ''difference of conjunction'' fully, but were happy simply to evaluate it with respect to a single location in the universe of discourse, namely, at the point picked out by the singular proposition ''xy'', in as much as if to say, at the place where ''x'' = 1 and ''y'' = 1. This evaluation is written in the form ''Df''|''xy'' or ''Df''|<1, 1>, and we arrived at the locally applicable law that states that ''f'' = ''xy'' = ''x'' & ''y'' ⇒ ''Df''|''xy'' = ((''dx'')(''dy'')) = ''dx'' or ''dy''.
We did not yet go through the trouble to interpret this (first order) ''difference of conjunction'' fully, but were happy simply to evaluate it with respect to a single location in the universe of discourse, namely, at the point picked out by the singular proposition ''xy'', in as much as if to say, at the place where ''x'' = 1 and ''y'' = 1. .This evaluation is written in the form ''Df''|''xy'' or ''Df''|<1, 1>, and we arrived at the locally applicable law that states that ''f'' = ''xy'' = ''x'' & ''y'' ⇒ ''Df''|''xy'' = ((''dx'')(''dy'')) = ''dx'' or ''dy''.
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| o-----------o o-----------o |
| . . . . .o-----------o. .o-----------o. . . . . |
| / \ / \ |
| . . . . / . . . . . . \ / . . . . . . \ . . . . |
| / x o y \ |
| . . . ./. . . .x. . . .o. . . .y. . . .\. . . . |
| / /`\ \ |
| . . . / . . . . . . . /%\ . . . . . . . \ . . . |
| / /```\ \ |
| . . ./. . . . . . . ./%%%\. . . . . . . .\. . . |
| o o`````o o |
| . . o . . . . . . . o%%%%%o . . . . . . . o . . |
| | |`````| | |
| . . | . . . . . . . |%%%%%| . . . . . . . | . . |
| | dy (dx) |`````| dx (dy) | |
| . . | . . .dy (dx). |%%%%%| .dx (dy). . . | . . |
| | o<----------|--o--|---------->o | |
| . . | . o<----------|--o--|---------->o . | . . |
| | |``|``| | |
| . . | . . . . . . . |%%|%%| . . . . . . . | . . |
| | |``|``| | |
| . . | . . . . . . . |%%|%%| . . . . . . . | . . |
| o o``|``o o |
| . . o . . . . . . . o%%|%%o . . . . . . . o . . |
| \ \`|`/ / |
| . . .\. . . . . . . .\%|%/. . . . . . . ./. . . |
| \ \|/ / |
| . . . \ . . . . . . . \|/ . . . . . . . / . . . |
| \ | / |
| . . . .\. . . . . . . .|. . . . . . . ./. . . . |
| \ /|\ / |
| . . . . \ . . . . . . /|\ . . . . . . / . . . . |
| o-----------o | o-----------o |
| . . . . .o-----------o | o-----------o. . . . . |
| | |
| . . . . . . . . . . . .|. . . . . . . . . . . . |
| dx|dy |
| . . . . . . . . . . .dx|dy. . . . . . . . . . . |
| | |
| . . . . . . . . . . . .|. . . . . . . . . . . . |
| v |
| . . . . . . . . . . . .v. . . . . . . . . . . . |
| o |
| . . . . . . . . . . . .o. . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| dx dy |
| . . . . . . . . . . dx. .dy . . . . . . . . . . |
| o o |
| . . . . . . . . . . .o. .o. . . . . . . . . . . |
| \ / |
| . . . . . . . . . . . \ / . . . . . . . . . . . |
| o |
| . . . . . . . . . . . .o. . . . . . . . . . . . |
| | |
| . . . . . . . . . . . .|. . . . . . . . . . . . |
| @ |
| . . . . . . . . . . . .@. . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| Df|xy = ((dx) (dy)) |
| Df|xy = . . . . . ((dx) (dy)) . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
The picture illustrates the analysis of the inclusive disjunction ((''dx'')(''dy'')) into the exclusive disjunction: ''dx''(''dy'') + ''dy''(''dx'') + ''dx dy'', a proposition that may be interpreted to say "change x or change y or both". And this can be recognized as just what you need to do if you happen to find yourself in the center cell and desire a detailed description of ways to depart it.
The picture illustrates the analysis of the inclusive disjunction ((''dx'')(''dy'')) into the exclusive disjunction: .''dx''(''dy'') + ''dy''(''dx'') + ''dx dy'', a proposition that may be interpreted to say "change x or change y or both". .And this can be recognized as just what you need to do if you happen to find yourself in the center cell and desire a detailed description of ways to depart it.
We have just computed what will variously be called the ''difference map'', the ''difference proposition'', or the ''local proposition'' ''Df''<sub>''p''</sub> for the proposition ''f''(''x'', ''y'') = ''xy'' at the point ''p'' where ''x'' = 1 and ''y'' = 1.
We have just computed what will variously be called the ''difference map'', the ''difference proposition'', or the ''local proposition'' ''Df''<sub>''p''</sub> for the proposition ''f''(''x'', ''y'') = ''xy'' at the point ''p'' where ''x'' = 1 and ''y'' = 1.
In the universe ''U'' = ''X'' × ''Y'', the four propositions ''xy'', ''x''(''y''), (''x'')''y'', (''x'')(''y'') that indicate the "cells", or the smallest regions of the venn diagram, are called ''singular propositions''. These serve as an alternative notation for naming the points <1, 1>, <1, 0>, <0, 1>, <0, 0>, respectively.
In the universe ''U'' = ''X'' × ''Y'', the four propositions ''xy'', ''x''(''y''), (''x'')''y'', (''x'')(''y'') that indicate the "cells", or the smallest regions of the venn diagram, are called ''singular propositions''. .These serve as an alternative notation for naming the points <1, 1>, <1, 0>, <0, 1>, <0, 0>, respectively.
Thus, we can write ''Df''<sub>''p''</sub> = ''Df''|''p'' = ''Df''|<1, 1> = ''Df''|''xy'', so long as we know the frame of reference in force.
Thus, we can write ''Df''<sub>''p''</sub> = ''Df''|''p'' = ''Df''|<1, 1> = ''Df''|''xy'', so long as we know the frame of reference in force.
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| x dx y dy |
| . . . . . . x .dx y .dy . . . . . . . . . . . . |
| o---o o---o |
| . . . . . . o---o o---o . . . . . . . . . . . . |
| \ | | / |
| . . . . . . .\. | | ./. . . . . . . . . . . . . |
| \ | | / |
| . . . . . . . \ | | / . . . . . . . . . . . . . |
| \| |/ x y |
| . . . . . . . .\| |/. . . . .x y. . . . . . . . |
| o=o-----------o |
| . . . . . . . . o=o-----------o . . . . . . . . |
| \ / |
| . . . . . . . . .\. . . . . ./. . . . . . . . . |
| \ / |
| . . . . . . . . . \ . . . . / . . . . . . . . . |
| \ / |
| . . . . . . . . . .\. . . ./. . . . . . . . . . |
| \ / |
| . . . . . . . . . . \ . . / . . . . . . . . . . |
| \ / |
| . . . . . . . . . . .\. ./. . . . . . . . . . . |
| \ / |
| . . . . . . . . . . . \ / . . . . . . . . . . . |
| @ |
| . . . . . . . . . . . .@. . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| Df = ((x, dx)(y, dy), xy) |
| Df = . . . .((x, dx)(y, dy), xy). . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| dx dy |
| . . . . . . . .dx . .dy . . . . . . . . . . . . |
| o---o o---o |
| . . . . . . o---o o---o . . . . . . . . . . . . |
| \ | | / |
| . . . . . . .\. | | ./. . . . . . . . . . . . . |
| \ | | / |
| . . . . . . . \ | | / . . . . . . . . . . . . . |
| \| |/ |
| . . . . . . . .\| |/. . . . . . . . . . . . . . |
| o=o-----------o |
| . . . . . . . . o=o-----------o . . . . . . . . |
| \ / |
| . . . . . . . . .\. . . . . ./. . . . . . . . . |
| \ / |
| . . . . . . . . . \ . . . . / . . . . . . . . . |
| \ / |
| . . . . . . . . . .\. . . ./. . . . . . . . . . |
| \ / |
| . . . . . . . . . . \ . . / . . . . . . . . . . |
| \ / |
| . . . . . . . . . . .\. ./. . . . . . . . . . . |
| \ / |
| . . . . . . . . . . . \ / . . . . . . . . . . . |
| @ |
| . . . . . . . . . . . .@. . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| Df|xy = ((dx) (dy)) |
| Df|xy = . . . . . ((dx) (dy)) . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| o |
| . . . . . . . . . o . . . . . . . . . . . . . . |
| dx | dy |
| . . . . . . . .dx | .dy . . . . . . . . . . . . |
| o---o o---o |
| . . . . . . o---o o---o . . . . . . . . . . . . |
| \ | | / |
| . . . . . . .\. | | ./. . . . . . . . . . . . . |
| \ | | / o |
| . . . . . . . \ | | / . . . . o . . . . . . . . |
| \| |/ | |
| . . . . . . . .\| |/. . . . . | . . . . . . . . |
| o=o-----------o |
| . . . . . . . . o=o-----------o . . . . . . . . |
| \ / |
| . . . . . . . . .\. . . . . ./. . . . . . . . . |
| \ / |
| . . . . . . . . . \ . . . . / . . . . . . . . . |
| \ / |
| . . . . . . . . . .\. . . ./. . . . . . . . . . |
| \ / |
| . . . . . . . . . . \ . . / . . . . . . . . . . |
| \ / |
| . . . . . . . . . . .\. ./. . . . . . . . . . . |
| \ / |
| . . . . . . . . . . . \ / . . . . . . . . . . . |
| @ |
| . . . . . . . . . . . .@. . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| Df|x(y) = (dx) dy |
| Df|x(y) = . . . . .(dx) dy. . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| o |
| . . . . . . o . . . . . . . . . . . . . . . . . |
| | dx dy |
| . . . . . . | .dx . .dy . . . . . . . . . . . . |
| o---o o---o |
| . . . . . . o---o o---o . . . . . . . . . . . . |
| \ | | / |
| . . . . . . .\. | | ./. . . . . . . . . . . . . |
| \ | | / o |
| . . . . . . . \ | | / . . . . o . . . . . . . . |
| \| |/ | |
| . . . . . . . .\| |/. . . . . | . . . . . . . . |
| o=o-----------o |
| . . . . . . . . o=o-----------o . . . . . . . . |
| \ / |
| . . . . . . . . .\. . . . . ./. . . . . . . . . |
| \ / |
| . . . . . . . . . \ . . . . / . . . . . . . . . |
| \ / |
| . . . . . . . . . .\. . . ./. . . . . . . . . . |
| \ / |
| . . . . . . . . . . \ . . / . . . . . . . . . . |
| \ / |
| . . . . . . . . . . .\. ./. . . . . . . . . . . |
| \ / |
| . . . . . . . . . . . \ / . . . . . . . . . . . |
| @ |
| . . . . . . . . . . . .@. . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| Df|(x)y = dx (dy) |
| Df|(x)y = . . . . . .dx (dy). . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| o o |
| . . . . . . o . . o . . . . . . . . . . . . . . |
| | dx | dy |
| . . . . . . | .dx | .dy . . . . . . . . . . . . |
| o---o o---o |
| . . . . . . o---o o---o . . . . . . . . . . . . |
| \ | | / |
| . . . . . . .\. | | ./. . . . . . . . . . . . . |
| \ | | / o o |
| . . . . . . . \ | | / . . . o . o . . . . . . . |
| \| |/ \ / |
| . . . . . . . .\| |/. . . . .\./. . . . . . . . |
| o=o-----------o |
| . . . . . . . . o=o-----------o . . . . . . . . |
| \ / |
| . . . . . . . . .\. . . . . ./. . . . . . . . . |
| \ / |
| . . . . . . . . . \ . . . . / . . . . . . . . . |
| \ / |
| . . . . . . . . . .\. . . ./. . . . . . . . . . |
| \ / |
| . . . . . . . . . . \ . . / . . . . . . . . . . |
| \ / |
| . . . . . . . . . . .\. ./. . . . . . . . . . . |
| \ / |
| . . . . . . . . . . . \ / . . . . . . . . . . . |
| @ |
| . . . . . . . . . . . .@. . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| Df|(x)(y) = dx dy |
| Df|(x)(y) = . . . . .dx dy. . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| x |
| .x. . . . . . . . . . . . . . . . . . . . . . . |
| o-o-o-...-o-o-o |
| .o-o-o-...-o-o-o. . . . . . . . . . . . . . . . |
| \ / |
| . \ . . . . . / . . . . . . . . . . . . . . . . |
| \ / |
| . .\. . . . ./. . . . . . . . . . . . . . . . . |
| \ / |
| . . \ . . . / . . . . . . . . . . . . . . . . . |
| \ / x |
| . . .\. . ./. . . . . . . . . . . . . x . . . . |
| \ / o |
| . . . \ . / . . . . . . . . . . . . . o . . . . |
| \ / | |
| . . . .\./. . . . . . . . . . . . . . | . . . . |
| @ = @ |
| . . . . @ . . . . . . .=. . . . . . . @ . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| (x, , ... , , ) = (x) |
| .(x, , ... , , ). . . .=. . . . . . .(x). . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| o |
| . . . . . . . .o. . . . . . . . . . . . . . . . |
| x_1 x_2 x_k | |
| x_1 x_2 . x_k .|. . . . . . . . . . . . . . . . |
| o---o-...-o---o |
| .o---o-...-o---o. . . . . . . . . . . . . . . . |
| \ / |
| . \ . . . . . / . . . . . . . . . . . . . . . . |
| \ / |
| . .\. . . . ./. . . . . . . . . . . . . . . . . |
| \ / |
| . . \ . . . / . . . . . . . . . . . . . . . . . |
| \ / |
| . . .\. . ./. . . . . . . . . . . . . . . . . . |
| \ / |
| . . . \ . / . . . . . . . . . . . . . . . . . . |
| \ / x_1 ... x_k |
| . . . .\./. . . . . . . . . . . .x_1 ... x_k. . |
| @ = @ |
| . . . . @ . . . . . . .=. . . . . . . @ . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| (x_1, ..., x_k, ()) = x_1 ... x_k |
| .(x_1, ..., x_k, ()). .=. . . . .x_1 ... x_k. . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| o |
| . . . . . . . . . . . .o. . . . . . . . . . . . |
| | |
| . . . . . . . . . . . .|. . . . . . . . . . . . |
| dx|dy |
| . . . . . . . . . . .dx|dy. . . . . . . . . . . |
| | |
| . . . . . . . . . . . .|. . . . . . . . . . . . |
| o-----------o | o-----------o |
| . . . . .o-----------o.|.o-----------o. . . . . |
| / \|/ \ |
| . . . . / . . . . . . \|/ . . . . . . \ . . . . |
| / x | y \ |
| . . . ./. . . .x. . . .|. . . .y. . . .\. . . . |
| / /|\ \ |
| . . . / . . . . . . . /|\ . . . . . . . \ . . . |
| / /`|`\ \ |
| . . ./. . . . . . . ./%|%\. . . . . . . .\. . . |
| o o``|``o o |
| . . o . . . . . . . o%%|%%o . . . . . . . o . . |
| | dy (dx) |``v``| dx (dy) | |
| . . | . . .dy (dx). |%%v%%| .dx (dy). . . | . . |
| | o-----------|->o<-|-----------o | |
| . . | . o-----------|->o<-|-----------o . | . . |
| | |`````| | |
| . . | . . . . . . . |%%%%%| . . . . . . . | . . |
| | o<----------|--o--|---------->o | |
| . . | . o<----------|--o--|---------->o . | . . |
| | dy (dx) |``|``| dx (dy) | |
| . . | . . .dy (dx). |%%|%%| .dx (dy). . . | . . |
| o o``|``o o |
| . . o . . . . . . . o%%|%%o . . . . . . . o . . |
| \ \`|`/ / |
| . . .\. . . . . . . .\%|%/. . . . . . . ./. . . |
| \ \|/ / |
| . . . \ . . . . . . . \|/ . . . . . . . / . . . |
| \ | / |
| . . . .\. . . . . . . .|. . . . . . . ./. . . . |
| \ /|\ / |
| . . . . \ . . . . . . /|\ . . . . . . / . . . . |
| o-----------o | o-----------o |
| . . . . .o-----------o.|.o-----------o. . . . . |
| | |
| . . . . . . . . . . . .|. . . . . . . . . . . . |
| dx|dy |
| . . . . . . . . . . .dx|dy. . . . . . . . . . . |
| | |
| . . . . . . . . . . . .|. . . . . . . . . . . . |
| v |
| . . . . . . . . . . . .v. . . . . . . . . . . . |
| o |
| . . . . . . . . . . . .o. . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
By inspection, it is fairly easy to understand ''Df'' as telling you what you have to do from each point of ''U'' in order to change the value borne by ''f''(''x'', ''y'').
By inspection, it is fairly easy to understand ''Df'' as telling you what you have to do from each point of ''U'' in order to change the value borne by ''f''(''x'', ''y'').
We have been studying the action of the difference operator ''D'', also known as the ''localization operator'', on the proposition ''f'' : ''X'' × ''Y'' → '''B''' that is commonly known as the conjunction ''xy''. We described ''Df'' as a (first order) differential proposition, that is, a proposition of the type ''Df'' : ''X'' × ''Y'' × ''dX'' × ''dY'' → '''B'''. Abstracting from the augmented venn diagram that illustrates how the ''models'', or the ''satisfying interpretations'', of ''Df'' distribute within the extended universe ''EU'' = ''X'' × ''Y'' × ''dX'' × ''dY'', we can depict ''Df'' in the form of a ''digraph'' or ''directed graph'', one whose points are labeled with the elements of ''U'' = ''X'' × ''Y'' and whose arrows are labeled with the elements of ''dU'' = ''dX'' × ''dY''.
We have been studying the action of the difference operator ''D'', also known as the ''localization operator'', on the proposition ''f'' : ''X'' × ''Y'' → '''B''' that is commonly known as the conjunction ''xy''. .We described ''Df'' as a (first order) differential proposition, that is, a proposition of the type ''Df'' : ''X'' × ''Y'' × ''dX'' × ''dY'' → '''B'''. .Abstracting from the augmented venn diagram that illustrates how the ''models'', or the ''satisfying interpretations'', of ''Df'' distribute within the extended universe ''EU'' = ''X'' × ''Y'' × ''dX'' × ''dY'', we can depict ''Df'' in the form of a ''digraph'' or ''directed graph'', one whose points are labeled with the elements of ''U'' = .''X'' × ''Y'' and whose arrows are labeled with the elements of ''dU'' = ''dX'' × ''dY''.
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| f = x y |
| f = . . . . . . . . . x y . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| Df = x y ((dx)(dy)) |
| Df =. . . . . . . x .y. ((dx)(dy)). . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| + x (y) (dx) dy |
| . . . . . + . . . x (y) .(dx) dy. . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| + (x) y dx (dy) |
| . . . . . + . . .(x) y. . dx (dy) . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| + (x)(y) dx dy |
| . . . . . + . . .(x)(y) . dx .dy. . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| x y |
| . . . . . . . . . . . x y . . . . . . . . . . . |
| x (y) o<------------->o<------------->o (x) y |
| .x (y) o<------------->o<------------->o (x) y. |
| (dx) dy ^ dx (dy) |
| . . . . . . (dx) dy . .^. . dx (dy) . . . . . . |
| | |
| . . . . . . . . . . . .|. . . . . . . . . . . . |
| | |
| . . . . . . . . . . . .|. . . . . . . . . . . . |
| dx | dy |
| . . . . . . . . . . dx | dy . . . . . . . . . . |
| | |
| . . . . . . . . . . . .|. . . . . . . . . . . . |
| | |
| . . . . . . . . . . . .|. . . . . . . . . . . . |
| v |
| . . . . . . . . . . . .v. . . . . . . . . . . . |
| o |
| . . . . . . . . . . . .o. . . . . . . . . . . . |
| (x) (y) |
| . . . . . . . . . . (x) (y) . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
Any proposition worth its salt has many equivalent ways to view it, any one of which may reveal some unsuspected aspect of its meaning. We will encounter more and more of these variant readings as we go.
Any proposition worth its salt has many equivalent ways to view it, any one of which may reveal some unsuspected aspect of its meaning. .We will encounter more and more of these variant readings as we go.
The enlargement operator ''E'', also known as the ''shift operator'', has many interesting and very useful properties in its own right, so let us not fail to observe a few of the more salient features that play out on the surface of our simple example, ''f''(''x'', ''y'') = ''xy''.
The enlargement operator ''E'', also known as the ''shift operator'', has many interesting and very useful properties in its own right, so let us not fail to observe a few of the more salient features that play out on the surface of our simple example, ''f''(''x'', ''y'') = ''xy''.
: ''Ef''(''x''<sub>1</sub>, …, ''x''<sub>''k''</sub>, ''dx''<sub>1</sub>, …, ''dx''<sub>''k''</sub>) = ''f''(''x''<sub>1</sub> + ''dx''<sub>1</sub>, …, ''x''<sub>''k''</sub> + ''dx''<sub>''k''</sub>).
: ''Ef''(''x''<sub>1</sub>, …, ''x''<sub>''k''</sub>, ''dx''<sub>1</sub>, …, ''dx''<sub>''k''</sub>) = ''f''(''x''<sub>1</sub> + ''dx''<sub>1</sub>, …, ''x''<sub>''k''</sub> + ''dx''<sub>''k''</sub>).
It should be noted that the so-called ''differential variables'' ''dx''<sub>''j''</sub> are really just the same kind of boolean variables as the other ''x''<sub>''j''</sub>. It is conventional to give the additional variables these brands of inflected names, but whatever extra connotations we might choose to attach to these syntactic conveniences are wholly external to their purely algebraic meanings.
It should be noted that the so-called ''differential variables'' ''dx''<sub>''j''</sub> are really just the same kind of boolean variables as the other ''x''<sub>''j''</sub>. .It is conventional to give the additional variables these brands of inflected names, but whatever extra connotations we might choose to attach to these syntactic conveniences are wholly external to their purely algebraic meanings.
For the example ''f''(''x'', ''y'') = ''xy'', we obtain:
For the example ''f''(''x'', ''y'') = ''xy'', we obtain:
: ''Ef''(''x'', ''y'', ''dx'', ''dy'') = ''x y'' + ''x dy'' + ''y dx'' + ''dx dy''
: ''Ef''(''x'', ''y'', ''dx'', ''dy'') = ''x y'' + ''x dy'' + ''y dx'' + ''dx dy''
To understand what this means in logical terms, for instance, as expressed in a boolean expansion or a ''disjunctive normal form'' (DNF), it is perhaps a little better to go back and analyze the expression the same way that we did for ''Df''. Thus, let us compute the value of the enlarged proposition ''Ef'' at each of the points in the universe of discourse ''U'' = ''X'' × ''Y''.
To understand what this means in logical terms, for instance, as expressed in a boolean expansion or a ''disjunctive normal form'' (DNF), it is perhaps a little better to go back and analyze the expression the same way that we did for ''Df''. .Thus, let us compute the value of the enlarged proposition ''Ef'' at each of the points in the universe of discourse ''U'' = ''X'' × ''Y''.
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| x dx y dy |
| . . . . . . . . . x .dx y .dy . . . . . . . . . |
| o---o o---o |
| . . . . . . . . . o---o o---o . . . . . . . . . |
| \ | | / |
| . . . . . . . . . .\. | | ./. . . . . . . . . . |
| \ | | / |
| . . . . . . . . . . \ | | / . . . . . . . . . . |
| \| |/ |
| . . . . . . . . . . .\| |/. . . . . . . . . . . |
| @=@ |
| . . . . . . . . . . . @=@ . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| Ef = (x, dx) (y, dy) |
| Ef = . . . . . .(x, dx) (y, dy) . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| dx dy |
| . . . . . . . . . . .dx . .dy . . . . . . . . . |
| o---o o---o |
| . . . . . . . . . o---o o---o . . . . . . . . . |
| \ | | / |
| . . . . . . . . . .\. | | ./. . . . . . . . . . |
| \ | | / |
| . . . . . . . . . . \ | | / . . . . . . . . . . |
| \| |/ |
| . . . . . . . . . . .\| |/. . . . . . . . . . . |
| @=@ |
| . . . . . . . . . . . @=@ . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| Ef|xy = (dx) (dy) |
| Ef|xy = . . . . . .(dx) (dy). . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| o |
| . . . . . . . . . . . . o . . . . . . . . . . . |
| dx | dy |
| . . . . . . . . . . .dx | .dy . . . . . . . . . |
| o---o o---o |
| . . . . . . . . . o---o o---o . . . . . . . . . |
| \ | | / |
| . . . . . . . . . .\. | | ./. . . . . . . . . . |
| \ | | / |
| . . . . . . . . . . \ | | / . . . . . . . . . . |
| \| |/ |
| . . . . . . . . . . .\| |/. . . . . . . . . . . |
| @=@ |
| . . . . . . . . . . . @=@ . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| Ef|x(y) = (dx) dy |
| Ef|x(y) = . . . . .(dx) .dy . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| o |
| . . . . . . . . . o . . . . . . . . . . . . . . |
| | dx dy |
| . . . . . . . . . | .dx . .dy . . . . . . . . . |
| o---o o---o |
| . . . . . . . . . o---o o---o . . . . . . . . . |
| \ | | / |
| . . . . . . . . . .\. | | ./. . . . . . . . . . |
| \ | | / |
| . . . . . . . . . . \ | | / . . . . . . . . . . |
| \| |/ |
| . . . . . . . . . . .\| |/. . . . . . . . . . . |
| @=@ |
| . . . . . . . . . . . @=@ . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| Ef|(x)y = dx (dy) |
| Ef|(x)y = . . . . . dx .(dy). . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| o o |
| . . . . . . . . . o . . o . . . . . . . . . . . |
| | dx | dy |
| . . . . . . . . . | .dx | .dy . . . . . . . . . |
| o---o o---o |
| . . . . . . . . . o---o o---o . . . . . . . . . |
| \ | | / |
| . . . . . . . . . .\. | | ./. . . . . . . . . . |
| \ | | / |
| . . . . . . . . . . \ | | / . . . . . . . . . . |
| \| |/ |
| . . . . . . . . . . .\| |/. . . . . . . . . . . |
| @=@ |
| . . . . . . . . . . . @=@ . . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| Ef|(x)(y) = dx dy |
| Ef|(x)(y) = . . . . dx . dy . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| f = x y |
| f = . . . . . . . . . x y . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| Ef = x y (dx)(dy) |
| Ef =. . . . . . . x .y. .(dx)(dy) . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| + x (y) (dx) dy |
| . . . . . + . . . x (y) .(dx) dy. . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| + (x) y dx (dy) |
| . . . . . + . . .(x) y. . dx (dy) . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| + (x)(y) dx dy |
| . . . . . + . . .(x)(y) . dx .dy. . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
| (dx) (dy) |
| . . . . . . . . . .(dx) (dy). . . . . . . . . . |
| .--->---. |
| . . . . . . . . . . --->--- . . . . . . . . . . |
| \ / |
| . . . . . . . . . . \ . . / . . . . . . . . . . |
| \x y/ |
| . . . . . . . . . . .\x y/. . . . . . . . . . . |
| \ / |
| . . . . . . . . . . . \ / . . . . . . . . . . . |
| x (y) o-------------->o<--------------o (x) y |
| .x (y) o-------------->o<--------------o (x) y. |
| (dx) dy ^ dx (dy) |
| . . . . . . (dx) dy . .^. . dx (dy) . . . . . . |
| | |
| . . . . . . . . . . . .|. . . . . . . . . . . . |
| | |
| . . . . . . . . . . . .|. . . . . . . . . . . . |
| dx | dy |
| . . . . . . . . . . dx | dy . . . . . . . . . . |
| | |
| . . . . . . . . . . . .|. . . . . . . . . . . . |
| | |
| . . . . . . . . . . . .|. . . . . . . . . . . . |
| | |
| . . . . . . . . . . . .|. . . . . . . . . . . . |
| o |
| . . . . . . . . . . . .o. . . . . . . . . . . . |
| (x) (y) |
| . . . . . . . . . . (x) (y) . . . . . . . . . . |
| |
| . . . . . . . . . . . . . . . . . . . . . . . . |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
We may understand the enlarged proposition ''Ef'' as telling us all the different ways to reach a model of ''f'' from any point of the universe ''U''.
We may understand the enlarged proposition ''Ef'' as telling us all the different ways to reach a model of ''f'' from any point of the universe ''U''.
To broaden our experience with simple examples, let us now contemplate the sixteen functions of concrete type ''X'' × ''Y'' → '''B''' and abstract type '''B''' × '''B''' → '''B'''. For future reference, I will set here a few tables that detail the actions of ''E'' and ''D'' and on each of these functions, allowing us to view the results in several different ways.
To broaden our experience with simple examples, let us now contemplate the sixteen functions of concrete type ''X'' × ''Y'' → '''B''' and abstract type '''B''' × '''B''' → '''B'''. .For future reference, I will set here a few tables that detail the actions of ''E'' and ''D'' and on each of these functions, allowing us to view the results in several different ways.
By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions in a number of different languages for zeroth order logic.
By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions in a number of different languages for zeroth order logic.
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ '''Table 1. Propositional Forms on Two Variables'''
|+ '''Table 1. .Propositional Forms on Two Variables'''
|- style="background:paleturquoise"
|- style="background:paleturquoise"
! style="width:15%" | L<sub>1</sub>
! style="width:15%" | L<sub>1</sub>
| f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x ← y
| f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x ← y
|-
|-
| f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y || x ∨ y
| f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y .|| x ∨ y
|-
|-
| f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || (( )) || true || 1
| f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || (( )) || true || 1
<br>
<br>
The next four Tables expand the expressions of ''Ef'' and ''Df'' in two different ways, for each of the sixteen functions. Notice that the functions are given in a different order, here being collected into a set of seven natural classes.
The next four Tables expand the expressions of ''Ef'' and ''Df'' in two different ways, for each of the sixteen functions. .Notice that the functions are given in a different order, here being collected into a set of seven natural classes.
<pre>
<pre>
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| | o---------------|->o<-|---------------o | |
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