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DLOG. Note B4
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| It is one of the rules of my system of general harmony,
| 'that the present is big with the future', and that he
| who sees all sees in that which is that which shall be.
|
| Leibniz, 'Theodicy'
|
| Gottfried Wilhelm, Freiherr von Leibniz,
|'Theodicy: Essays on the Goodness of God,
| The Freedom of Man, & The Origin of Evil',
| Edited with an Introduction by Austin Farrer,
| Translated by E.M. Huggard from C.J. Gerhardt's
| Edition of the 'Collected Philosophical Works',
| 1875-90; Routledge & Kegan Paul, London, UK, 1951;
| Open Court, La Salle, IL, 1985. Paragraph 360, Page 341.
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?
Consider the polymorphous set Q of Example 1 and focus on the central cell,
described by the conjunction of logical features in the expression "u v w".
o-------------------------------------------------o
| X |
| |
| o-------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| o U o |
| | | |
| | | |
| | | |
| 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 truth-function q : X -> B that
describes Q is represented by the following graph
and text expressions:
o-------------------------------------------------o
| q |
o-------------------------------------------------o
| |
| u v u w v w |
| o o o |
| \ | / |
| \ | / |
| \|/ |
| o |
| | |
| | |
| | |
| @ |
| |
o-------------------------------------------------o
| (( u v )( u w )( v w )) |
o-------------------------------------------------o
Conjoining the query that specifies the center cell gives:
o-------------------------------------------------o
| q.uvw |
o-------------------------------------------------o
| |
| u v u w v w |
| o o o |
| \ | / |
| \ | / |
| \|/ |
| o |
| | |
| | |
| | |
| @ u v w |
| |
o-------------------------------------------------o
| (( u v )( u w )( v w )) u v w |
o-------------------------------------------------o
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:
o-------------------------------------------------o
| Eq |
o-------------------------------------------------o
| |
| u du v dv u du w dw v dv w dw |
| o---o o---o o---o o---o o---o o---o |
| \ | | / \ | | / \ | | / |
| \ | | / \ | | / \ | | / |
| \| |/ \| |/ \| |/ |
| o=o o=o o=o |
| \ | / |
| \ | / |
| \ | / |
| \ | / |
| \ | / |
| \ | / |
| \ | / |
| \ | / |
| \ | / |
| \ | / |
| \|/ |
| o |
| | |
| | |
| | |
| @ |
| |
o-------------------------------------------------o
| |
| (( ( u , du ) ( v , dv ) |
| )( ( u , du ) ( w , dw ) |
| )( ( v , dv ) ( w , dw ) |
| )) |
| |
o-------------------------------------------------o
Conjoining a query on the center cell yields:
o-------------------------------------------------o
| Eq.uvw |
o-------------------------------------------------o
| |
| u du v dv u du w dw v dv w dw |
| o---o o---o o---o o---o o---o o---o |
| \ | | / \ | | / \ | | / |
| \ | | / \ | | / \ | | / |
| \| |/ \| |/ \| |/ |
| o=o o=o o=o |
| \ | / |
| \ | / |
| \ | / |
| \ | / |
| \ | / |
| \ | / |
| \ | / |
| \ | / |
| \ | / |
| \ | / |
| \|/ |
| o |
| | |
| | |
| | |
| @ u v w |
| |
o-------------------------------------------------o
| |
| (( ( u , du ) ( v , dv ) |
| )( ( u , du ) ( w , dw ) |
| )( ( v , dv ) ( w , dw ) |
| )) |
| |
| u v w |
| |
o-------------------------------------------------o
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:
o-------------------------------------------------o
| Dq.uvw |
o-------------------------------------------------o
| |
| u du v dv u du w dw v dv w dw |
| o---o o---o o---o o---o o---o o---o |
| \ | | / \ | | / \ | | / |
| \ | | / \ | | / \ | | / |
| \| |/ \| |/ \| |/ |
| o=o o=o o=o |
| \ | / |
| \ | / |
| \ | / |
| \ | / |
| \ | / |
| \ | / |
| \ | / u v u w v w |
| \ | / o o o |
| \ | / \ | / |
| \ | / \ | / |
| \|/ \|/ |
| o o |
| | | |
| | | |
| | | |
| o---------------o |
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| @ u v w |
| |
o-------------------------------------------------o
| |
| ( |
| (( ( u , du ) ( v , dv ) |
| )( ( u , du ) ( w , dw ) |
| )( ( v , dv ) ( w , dw ) |
| )) |
| , |
| (( u v |
| )( u w |
| )( v w |
| )) |
| ) |
| |
| u v w |
| |
o-------------------------------------------------o
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,
as described by the conjunctive expression "u v w",
to a cell outside the shaded region for the set Q.
o-------------------------------------------------o
| X |
| |
| o-------------o |
| / \ |
| / U \ |
| / \ |
| / \ |
| o @ o |
| | ^ | |
| | |dw | |
| | | | @ |
| o---o---------o o----|----o---o ^ |
| / \`````````\ /`````|```/ \ /dw |
| / du \`````dw``o``dv``|``/ \/ |
| / @<-----\-o<----/+\---->o`/ /\ |
| / \`````/`|`\`````/ / \ |
| o o---o--|--o---o / o |
| | |``|``| / | |
| | V |`du``| / W | |
| | |` |``| / | |
| o o``v``o dv / o |
| \ \`o-/------->@ / |
| \ \`/ / |
| \ o / |
| \ / \ / |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
Figure 3. Effect of the Difference Operator D
Acting on a Polymorphous Function q
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" Dq are marked with "@" signs, and
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:
1. du dv dw => Change u, v, w.
2. du dv (dw) => Change u and v.
3. du (dv) dw => Change u and w.
4. (du) dv dw => Change v and w.
Next I will discuss several applications of logical differentials,
developing along the way their logical and practical implications.
Jon Awbrey
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http://www.cs.bsu.edu/homepages/mighty/history.html
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