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Directory:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
Revision as of 03:06, 29 April 2008
, 03:06, 29 April 2008→Note 4: markup
4. u v w (du) dv dw
4. u v w (du) dv dw
</code>
</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
| X |
| X |
Figure 3. Effect of the Difference Operator D
Figure 3. Effect of the Difference Operator D
Acting on a Polymorphous Function q
Acting on a Polymorphous Function q
</pre>
Figure 3 shows one way to picture this kind of a situation,
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:
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
the boundary crossings along each path are marked with the
corresponding "differential features" among the collection
{du, dv, dw}. In sum, starting from the cell uvw, we have
the following four paths:
<pre>
1. du dv dw => Change u, v, w.
1. du dv dw => Change u, v, w.
2. du dv (dw) => Change u and v.
2. du dv (dw) => Change u and v.
3. du (dv) dw => Change u and w.
3. du (dv) dw => Change u and w.
4. (du) dv dw => Change v and w.
4. (du) dv dw => Change v and w.
</pre>
Next I will discuss several applications of logical differentials,
Next I will discuss several applications of logical differentials, developing along the way their logical and practical implications.
developing along the way their logical and practical implications.
===Note 5===
===Note 5===