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Directory:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
Revision as of 17:30, 28 April 2008
, 17:30, 28 April 2008→Note 3: markup
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:
: <code>(( u v )( u w )( v w ))</code>
: <code> (( u v )( u w )( v w )) </code>
Conjoining the query that specifies the center cell gives:
Conjoining the query that specifies the center cell gives:
: <code>(( u v )( u w )( v w )) u v w</code>
: <code> (( u v )( u w )( v w )) u v w </code>
And we know the value of the interpretation by whether this last expression issues in a model.
And we know the value of the interpretation by whether this last expression issues in a model.
</code>
</code>
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 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
The result of applying the difference operator <math>\operatorname{D}</math> to the initial proposition <math>\operatorname{q}</math>, conjoined with a query on the center cell, yields:
to the initial proposition q, conjoined with
a query on the center cell, yields:
<code>
(
(
(( ( u , du )( v , dv )
(( ( u , du )( v , dv )
))
))
)
)
u v w
u v w
</code>
The models of this last proposition are:
The models of this last proposition are:
<code>
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
</code>
This tells us that changing any two or more of the
This tells us that changing any two or more of the features <math>u, v, w</math> will take us from the center cell to a cell outside the shaded region for the set <math>Q\!</math>.
features u, v, w will take us from the center cell
to a cell outside the shaded region for the set Q.
</pre>
===Note 4===
===Note 4===