ONT Re: Differential Logic -- Series B
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DLOG. Note B3
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| The present is big with the future.
|
| ~~ Leibniz
Here I now delve into subject matters
that are more specifically logical in
the character of their interpretation.
Imagine that we are sitting in one of the cells of a venn diagram,
contemplating the walls. There are k of them, one for each positive
feature x_1, ..., x_k in our universe of discourse. Our particular cell
is described by a concatenation of k signed assertions, positive or negative,
regarding each of these features, and this description of our position amounts
to what is called an "interpretation" of whatever proposition may rule the space,
or reign on the universe of discourse. But are we locked into this interpretation?
With respect to each edge x of the cell we consider a test proposition dx
that determines our decision whether or not we will make a difference in
how we stand regarding to x. If dx is true then it marks our decision,
intention, or plan to cross over the edge x at some point within the
purview of the contemplated plan.
To reckon the effect of several such decisions on our current interpretation,
or the value of the reigning proposition, we transform that position or that
proposition by making the following array of substitutions everywhere in its
expression:
1. Substitute "( x_1 , dx_1 )" for "x_1"
2. Substitute "( x_2 , dx_2 )" for "x_2"
3. Substitute "( x_3 , dx_3 )" for "x_3"
...
k. Substitute "( x_k , dx_k )" for "x_k"
For concreteness, consider the polymorphous set Q of Example 1
and focus on the central cell, specifically, the cell described
by the conjunction of logical features in the expression "u v w".
o-----------------------------------------------------------o
| X |
| |
| o-------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| | U | |
| | | |
| | | |
| | | |
| | | |
| o--o----------o o----------o--o |
| / \%%%%%%%%%%\ /%%%%%%%%%%/ \ |
| / \%%%%%%%%%%o%%%%%%%%%%/ \ |
| / \%%%%%%%%/%\%%%%%%%%/ \ |
| / \%%%%%%/%%%\%%%%%%/ \ |
| / \%%%%/%%%%%\%%%%/ \ |
| o o--o-------o--o o |
| | |%%%%%%%| | |
| | |%%%%%%%| | |
| | |%%%%%%%| | |
| | V |%%%%%%%| W | |
| | |%%%%%%%| | |
| o o%%%%%%%o o |
| \ \%%%%%/ / |
| \ \%%%/ / |
| \ \%/ / |
| \ o / |
| \ / \ / |
| o-------------o o-------------o |
| |
| |
o-----------------------------------------------------------o
Figure 1. Polymorphous Set Q
The proposition or the truth-function q that describes Q is:
(( u v )( u w )( v w ))
Conjoining the query that specifies the center cell gives:
(( u v )( u w )( v w )) u v w
And we know the value of the interpretation by
whether this last expression issues in a model.
Applying the enlargement operator E
to the initial proposition q yields:
(( ( u , du )( v , dv )
)( ( u , du )( w , dw )
)( ( v , dv )( w , dw )
))
Conjoining a query on the center cell yields:
(( ( u , du )( v , dv )
)( ( u , du )( w , dw )
)( ( v , dv )( w , dw )
))
u v w
The models of this last expression tell us which combinations of
feature changes among the set {du, dv, dw} will take us from our
present interpretation, the center cell expressed by "u v w", to
a true value under the target proposition (( u v )( u w )( v w )).
The result of applying the difference operator D
to the initial proposition q, conjoined with
a query on the center cell, yields:
(
(( ( u , du )( v , dv )
)( ( u , du )( w , dw )
)( ( v , dv )( w , dw )
))
,
(( u v
)( u w
)( v w
))
)
u v w
The models of this last proposition are:
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
This tells us that changing any two or more of the
features u, v, w will take us from the center cell
to a cell outside the shaded region for the set Q.
Jon Awbrey
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http://www.cs.bsu.edu/homepages/mighty/history.html
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