SUO: Re: VOFIOTI ADO
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| The present is big with the future.
|
| ~~ Leibniz
Da Capo, Re Typo, Al Fine --
I will repeat what has gone before,
silently correcting a few minor typos,
and then continue with the presentation.
This time we delve into subject matters
that are more specifically logical in
the character of their interpretation.
Notational Note. The so-called "shift operator" E
was originally called the "enlargement operator",
hence the initial "E" of the customary notation.
¤~~~~~~~~~¤~~~~~~~~~¤~~APPENDICES~~¤~~~~~~~~~¤~~~~~~~~~¤
Chapter 3. Linear Topics
The Differential Theory of Qualitative Equations
To denote lists of propositions and to detail their components,
we use notations like:
#a# = <a, b, c>, #p# = <p, q, r>, #x# = <x, y, z>,
or, in more complicated situations:
x = <x1, x2, x3>, y = <y1, y2, y3>, z = <z1, z2, z3>.
In a universe where some region is ruled by a proposition,
it is natural to ask whether we can change the value of that
proposition by changing the features of a current state.
Given a venn diagram with a shaded region and starting from
any cell in that universe, what sequences of feature changes,
what traverses of cell walls, will take us from shaded to
unshaded areas, or the reverse?
In order to discuss questions of this type, it is useful
to define several "operators" on functions. An operator
is nothing more than a function between sets that happen
to have functions as members.
A typical operator F takes us from thinking about a given function f
to thinking about another function g. To express the fact that g can
be obtained by applying F to f, we write g = Ff.
The first operator, E, associates with a function f : A -> B
another function Ef, where Ef : AxA -> B is defined as follows:
Ef(x, y) = f(x + y).
E is called a "shift operator" because it takes us from
contemplating the value of f at a place x to considering
the value of f at a shift of y away. In effect, it tells
us the absolute effect on f of changing its argument from x
by an amount that is equal to y.
The second operator, D, associates with a function f : A -> B
another function Df, where Df : AxA -> B is defined as follows:
Df(x, y) = Ef(x, y) - f(x),
or, equivalently,
Df(x, y) = f(x + y) - f(x).
D is called a "difference operator" because it informs us of the
relative change in the value of f along the shift from x to x + y.
In practice, one of the variables, x or y, is often
considered to be "less variable" than the other one,
being fixed in the context of a concrete discussion.
Thus, we might find any one of the following idioms:
1. Df : AxA -> B,
Df(c, x) = f(c + x) - f(c).
Here, c is held constant and Df(c, x) is regarded
mainly as a function of the second variable x,
giving the relative change in f at various
distances x from the center c.
2. Df : AxA -> B,
Df(x, h) = f(x + h) - f(x).
Here, h is either a constant (usually 1), in discrete contexts,
or a variably "small" amount (near to 0) over which a limit is
being taken, as in continuous contexts. Df(x, h) is regarded
mainly as a function of the first variable x, in effect, giving
the differences in the value of f between x and a neighbor that
is a distance of h away, all the while that x itself ranges over
its various possible locations.
3. Df : AxA -> B,
Df(x, dx) = f(x + dx) - f(x).
This is yet another variant of the previous form,
with dx denoting small changes contemplated in x.
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Imagine that we are sitting in one of the cells of a venn diagram,
contemplating the walls. There are N of them, one for each positive
feature x1, ..., xN in our universe of discourse. Our particular cell
is described by a concatenation of N signed assertions, positive or negative,
regarding each of these features. But are we locked into this interpretation?
With respect to each edge x of the cell we consider a test proposition dx,
to determine our decision to differ on x. If dx is true we decide to cross
over edge x at some point in the future. To reckon the effect of several such
decisions on our current interpretation (or the value of the reigning proposition),
we transform that proposition by making the following set of substitutions everywhere
in its expression:
| Substitute "( x1 , dx1 )" for "x1",
|
| Substitute "( x2 , dx2 )" for "x2",
|
| ...,
|
| Substitute "( xN , dxN )" for "xN".
For concreteness, consider the polymorphous set T' of Example 1
and focus on the central cell, as described by the conjunction
of logical features in the expression "A B C".
The proposition or the truth-function T that describes T' is:
(( A B )( A C )( B C ))
Conjoining the query that specifies the center cell gives:
(( A B )( A C )( B C )) A B C
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 T yields:
| (( ( A , dA )( B , dB )
| )( ( A , dA )( C , dC )
| )( ( B , dB )( C , dC )
| ))
Conjoining a query on the center cell yields:
| (( ( A , dA )( B , dB )
| )( ( A , dA )( C , dC )
| )( ( B , dB )( C , dC )
| ))
|
| A B C
The models of this last expression tell us which combinations of
feature changes among the set {dA, dB, dC} will take us from our
present interpretation, the center cell expressed by "A B C", to
a true value under the target proposition (( A B )( A C )( B C )).
The result of applying the difference operator D
to the initial proposition T, conjoined with
a query on the center cell, yields:
| (
| (( ( A , dA )( B , dB )
| )( ( A , dA )( C , dC )
| )( ( B , dB )( C , dC )
| ))
| ,
| (( A B
| )( A C
| )( B C
| ))
| )
|
| A B C
The models of this last proposition are:
1. A B C dA dB dC
2. A B C dA dB (dC)
3. A B C dA (dB) dC
4. A B C (dA) dB dC
This tells us that changing any two or more of the
features A, B, C will take us from the center cell
to a cell outside the shaded region for the set T'.
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¤~~~~~~~~~¤~~~~~~~~~¤~~SECIDNEPPA~~¤~~~~~~~~~¤~~~~~~~~~¤
E(Nuff)!
Have a good weekend ...
Jon Awbrey
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