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To round out the presentation of the "Polymorphous" Example 1, I will go through what has gone before and lay in the graphic forms of all of the propositional expressions. These graphs, whose official botanical designation makes them out to be a species of "painted and rooted cacti" (PARC's), are not too far from the actual graph-theoretic data-structures that result from parsing the Cactus string expressions, the "painted and rooted cactus expressions" (PARCE's). Finally, I will add a couple of venn diagrams that will serve to illustrate the "difference opus" Dq. If you apply an operator to an operand you must arrive at either an opus or an opera, no?
To round out the presentation of the Polymorphous Example 1, I will go through what has gone before and lay in the graphic forms of all of the propositional expressions. These graphs, whose official botanical designation makes them out to be a species of ''painted and rooted cacti'' (PARC's), are not too far from the actual graph-theoretic data-structures that result from parsing the cactus string expressions, the ''painted and rooted cactus expressions'' (PARCE's). Finally, I will add a couple of venn diagrams that will serve to illustrate the ''difference opus'' <math>\operatorname{D}q</math>. If you apply an operator to an operand you must arrive at either an opus or an opera, no?
Consider the polymorphous set <math>Q\!</math> of Example 1 and focus on the central cell, described by the conjunction of logical features in the expression "<math>u\ v\ w\!</math>".
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| X |
| X |
o-------------------------------------------------o
o-------------------------------------------------o
Figure 1. Polymorphous Set Q
Figure 1. Polymorphous Set Q
</pre>
The proposition or truth-function q : X -> B that
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:
describes Q is represented by the following graph
and text expressions:
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| q |
| q |
| (( u v )( u w )( v w )) |
| (( u v )( u w )( v w )) |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
Conjoining the query that specifies the center cell gives:
Conjoining the query that specifies the center cell gives:
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| q.uvw |
| q.uvw |
| (( u v )( u w )( v w )) u v w |
| (( u v )( u w )( v w )) u v w |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
And we know the value of the interpretation by
And we know the value of the interpretation by whether this last expression issues in a model.
whether this last expression issues in a model.
Applying the enlargement operator E
Applying the enlargement operator <math>\operatorname{E}</math> to the initial proposition <math>q\!</math> yields:
to the initial proposition q yields:
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| Eq |
| Eq |
| |
| |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
Conjoining a query on the center cell yields:
Conjoining a query on the center cell yields:
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| Eq.uvw |
| Eq.uvw |
| |
| |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
The models of this last expression tell us which combinations of
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>.
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>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:
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| Dq.uvw |
| Dq.uvw |
| |
| |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
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>
<pre>
This tells us that changing any two or more of the
This tells us that changing any two or more of the
features u, v, w will take us from the center cell,
features u, v, w will take us from the center cell,
