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Directory:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
Revision as of 16:58, 5 May 2008
, 16:58, 5 May 2008→Note 3: png format & code -> pre
Applying the enlargement operator <math>\operatorname{E}</math> to the initial proposition <math>q\!</math> yields:
Applying the enlargement operator <math>\operatorname{E}</math> to the initial proposition <math>q\!</math> yields:
<code>
<pre>
(( ( u , du )( v , dv )
(( ( u , du )( v , dv )
)( ( u , du )( w , dw )
)( ( u , du )( w , dw )
)( ( v , dv )( w , dw )
)( ( v , dv )( w , dw )
))
))
</code>
</pre>
Conjoining a query on the center cell yields:
Conjoining a query on the center cell yields:
<code>
<pre>
(( ( u , du )( v , dv )
(( ( u , du )( v , dv )
)( ( u , du )( w , dw )
)( ( u , du )( w , dw )
u v w
u v w
</code>
</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 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>\operatorname{q}</math>, conjoined with a query on the center cell, yields:
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:
<code>
<pre>
(
(
(( ( u , du )( v , dv )
(( ( u , du )( v , dv )
u v w
u v w
</code>
</pre>
The models of this last proposition are:
The models of this last proposition are:
<code>
<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
</code>
</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 to a cell outside the shaded region for the set <math>Q\!</math>.
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>
===Note 4===
===Note 4===