SUO: Re: VOFIOTI ADO
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Before going on -- in order to keep alive the will to go on! --
it would probably be a good idea to remind ourselves of just
why we are going through with this exercise. It is to unify
the world of change, for which aspect or regime of the world
I occasionally evoke the eponymous figures of Prometheus and
Heraclitus, and the world of logic, for which facet or realm
of the world I periodically recur to the prototypical shades
of Epimetheus and Parmenides, at least, that is, to state it
more carefully, to encompass the antics and the escapades of
these all too manifestly strife-born twins within the scopes
of our thoughts and within the charts of our theories, which
is most likely the only places where they will ever, for the
moment and as long as it lasts, be seen or be heard together.
With that intermezzo, with all of its echoes of the opening overture,
over and done, let us now return to that droller drama, already fast
in progress, the differential disentanglements, hopefully toward the
end of a grandly enlightening denouement, of the ever-polymorphous q.
The next transformation of the source proposition q, that we are
typically aiming to contemplate in the process of carrying out a
"differential analysis" of its "dynamic" effects or implications,
is the yield of the so-called "difference" or "delta" operator D.
The resultant "difference proposition" Dq is defined in terms of
the source proposition q and the "shifted proposition" Eq thusly:
| Dq = Eq - q = Eq - eq.
|
| Since "+" and "-" signify the same operation over B, we have:
|
| Dq = Eq + q = Eq + eq.
|
| Since "+" = "exclusive-or", RefLog syntax expresses this as:
|
| Eq q Eq eq
| o---o o---o
| \ / \ /
| Dq = @ = @
|
| Dq = ( Eq , q ) = ( Eq , eq ).
|
| Recall that the k-place bracket "(x1, x2, ..., xk)"
| is interpreted (in the "existential" interpretation)
| to mean "Exactly one of the xj is false", and so the
| two-place bracket is equivalent to the exclusive-or.
The result of applying the difference operator D to the source
proposition q, conjoined with a query on the center cell c, is:
| u v u w v w
| o o o o o o
| / \ / \ / \ / \ / \ / \
| o---o---o o---o---o o---o---o
| du \ dv du | dw dv / dw
| \ | /
| \ | /
| \ | /
| \ | / u v u w v w
| \ | / o o o
| \ | / \ | /
| \ | / \ | /
| \|/ \|/
| o o
| | |
| | |
| | |
| o-------------------o
| \ /
| \ /
| \ /
| \ /
| \ /
| \ /
| \ /
| \ /
| \ /
| @ u v w
|
| (
| (( ( u , du )( v , dv )
| )( ( u , du )( w , dw )
| )( ( v , dv )( w , dw )
| ))
| ,
| (( u v
| )( u w
| )( v w
| ))
| )
|
| u v w
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
that is marked by the conjunctive expression "uvw",
to a cell outside the shaded region for the area Q.
o-----------------------------------o
| X |
| o-----------------------o |
| | U | |
| | o o | |
| | /`\ /`\ * | |
| | /```\ /```\ / | |
| | /`````.`````dw | |
| | /`````/`\````/\ | |
| | /``@-dw-@-dv-@``\ | |
| | /``/``/``|``\`````\ | |
| o---du-----du-----------o |
| / / /````|````\ \ |
| o * o`````|`````o o |
| \ \````@````/ / |
| \ V \````\``/ W / |
| \ \````dv / |
| \ \```/\ / |
| \ \`/ * / |
| \ . dw |
| \ / \ /\ |
| \ / \ / * |
| o o |
| |
o-----------------------------------o
Figure 4. Extended Venn Diagram For Dq.c
Figure 4 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 stars (*), 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.
That sums up, but rather more carefully, the material that
I ran through just a bit too quickly the first time around.
Next time, I will begin to develop an alternative style of
diagram for depicting these types of differential settings.
Until Then,
Jon Awbrey
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