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Revision as of 13:42, 28 July 2011
Note 1
| The most fundamental concept in cybernetics is that of "difference",
| either that two things are recognisably different or that one thing
| has changed with time.
|
| William Ross Ashby,
|'An Introduction to Cybernetics',
| Chapman & Hall, London, UK, 1956,
| Methuen & Company, London, UK, 1964.
Linear Topics. The Differential Theory of Qualitative Equations
This chapter is titled "Linear Topics" because that is the heading
under which the derivatives and the differentials of any functions
usually come up in mathematics, namely, in relation to the problem
of computing "locally linear approximations" to the more arbitrary,
unrestricted brands of functions that one finds in a given setting.
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 = <x_1, x_2, x_3>,
y = <y_1, y_2, y_3>,
z = <z_1, z_2, z_3>.
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 our 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 the operator F to f, we write g = Ff.
The first operator, E, associates with a function f : X -> Y
another function Ef, where Ef : X x X -> Y is defined by the
following equation:
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.
Thus, E tells us the absolute effect on f that is obtained by changing its
argument from x by an amount that is equal to y.
Historical Note. The protean "shift operator" E was originally called
the "enlargement operator", hence the initial "E" of the usual notation.
The next operator, D, associates with a function f : X -> Y
another function Df, where Df : X x X -> Y is defined by the
following equation:
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 tells us about 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 : X x X -> Y,
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 : X x X -> Y,
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 : X x X -> Y,
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.
That's the basic idea. The next order of business is to develop
the logical side of the analogy a bit more fully, and to take up
the elaboration of some moderately simple applications of these
ideas to a selection of relatively concrete examples.
Note 2
Linear Topics (cont.)
Example 1. A Polymorphous Concept
I start with an example that is simple enough that it will allow us to compare
the representations of propositions by venn diagrams, truth tables, and my own
favorite version of the syntax for propositional calculus all in a relatively
short space. To enliven the exercise, I borrow an example from a book with
several independent dimensions of interest, 'Topobiology' by Gerald Edelman.
One finds discussed there the notion of a "polymorphous set". Such a set
is defined in a universe of discourse whose elements can be described in
terms of a fixed number k of logical features. A "polymorphous set" is
one that can be defined in terms of sets whose elements have a fixed
number j of the k features.
As a rule in the following discussion, I will use upper case letters as names
for concepts and sets, lower case letters as names for features and functions.
The example that Edelman gives (1988, Fig. 10.5, p. 194) involves sets of
stimulus patterns that can be described in terms of the three features
"round" 'u', "doubly outlined" 'v', and "centrally dark" 'w'. We may
regard these simple features as logical propositions u, v, w : X -> B.
The target concept Q is one whose extension is a polymorphous set Q,
the subset Q of the universe X where the complex feature q : X -> B
holds true. The Q in question is defined by the requirement:
"Having at least 2 of the 3 features in the set {u, v, w}".
Taking the symbols u = "round", v = "doubly outlined", w = "centrally dark",
and using the corresponding capitals to label the circles of a venn diagram,
we get a picture of the target set Q as the shaded region in Figure 1. Using
these symbols as "sentence letters" in a truth table, let the truth function q
mean the very same thing as the expression "{u and v} or {u and w} or {v and w}".
o-----------------------------------------------------------o
| X |
| |
| o-------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| | U | |
| | | |
| | | |
| | | |
| | | |
| 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
In other words, the proposition q is a truth-function of the 3 logical variables
u, v, w, and it may be evaluated according to the "truth table" scheme that is
shown in Table 2. In this representation the polymorphous set Q appears in
the guise of what some people call the "pre-image" or the "fiber of truth"
under the function q. More precisely, the 3-tuples for which q evaluates
to true are in an obvious correspondence with the shaded cells of the
venn diagram. No matter how we get down to the level of actual
information, it's all pretty much the same stuff.
Table 2. Polymorphous Function q
o---------------o-----------o-----------o-----------o-------o
| u v w | u & v | u & w | v & w | q |
o---------------o-----------o-----------o-----------o-------o
| | | | | |
| 0 0 0 | 0 | 0 | 0 | 0 |
| | | | | |
| 0 0 1 | 0 | 0 | 0 | 0 |
| | | | | |
| 0 1 0 | 0 | 0 | 0 | 0 |
| | | | | |
| 0 1 1 | 0 | 0 | 1 | 1 |
| | | | | |
| 1 0 0 | 0 | 0 | 0 | 0 |
| | | | | |
| 1 0 1 | 0 | 1 | 0 | 1 |
| | | | | |
| 1 1 0 | 1 | 0 | 0 | 1 |
| | | | | |
| 1 1 1 | 1 | 1 | 1 | 1 |
| | | | | |
o---------------o-----------o-----------o-----------o-------o
With the pictures of the venn diagram and the truth table before us,
we have come to the verge of seeing how the word "model" is used in
logic, namely, to distinguish whatever things satisfy a description.
In the venn diagram presentation, to be a model of some conceptual
description !F! is to be a point x in the corresponding region F
of the universe of discourse X.
In the truth table representation, to be a model of a logical
proposition f is to be a data-vector !x! (a row of the table)
on which a function f evaluates to true.
This manner of speaking makes sense to those who consider the ultimate meaning of
a sentence to be not the logical proposition that it denotes but its truth value
instead. From the point of view, one says that any data-vector of this type
(k-tuples of truth values) may be regarded as an "interpretation" of the
proposition with k variables. An interpretation that yields a value
of true is then called a "model".
For the most threadbare kind of logical system that we find residing
in propositional calculus, this notion of model is almost too simple
to deserve the name, yet it can be of service to fashion some form
of continuity between the simple and the complex.
| Reference:
|
| Edelman, Gerald M.,
|'Topobiology: An Introduction to Molecular Embryology',
| Basic Books, New York, NY, 1988.
Note 3
Linear Topics (cont.)
| The present is big with the future.
|
| ~~ Leibniz
Here I now delve into subject matters
that are more specifically logical in
the character of their interpretation.
Imagine that we are sitting in one of the cells of a venn diagram,
contemplating the walls. There are k of them, one for each positive
feature x_1, ..., x_k in our universe of discourse. Our particular cell
is described by a concatenation of k signed assertions, positive or negative,
regarding each of these features, and this description of our position amounts
to what is called an "interpretation" of whatever proposition may rule the space,
or reign on the universe of discourse. But are we locked into this interpretation?
With regard to each edge x of the cell we consider a test proposition dx
that determines our decision whether or not we will make a difference in
how we stand with regard to x. If dx is true then it marks our decision,
intention, or plan to cross over the edge x at some point in the purview
of the contemplated plan.
To reckon the effect of several such decisions on our current interpretation,
or the value of the reigning proposition, we transform that position or that
proposition by making the following array of substitutions everywhere in its
expression:
1. Substitute (x_1, dx_1) for x_1,
2. Substitute (x_2, dx_2) for x_2,
3. Substitute (x_3, dx_3) for x_3,
...
k. Substitute (x_k, dx_k) for x_k.
For concreteness, consider the polymorphous set Q of Example 1
and focus on the central cell, specifically, the cell described
by the conjunction of logical features in the expression "u v w".
o-----------------------------------------------------------o
| X |
| |
| o-------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| | U | |
| | | |
| | | |
| | | |
| | | |
| 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 the truth-function q that describes Q is:
(( u v )( u w )( v w ))
Conjoining the query that specifies the center cell gives:
(( u v )( u w )( v w )) u v w
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:
(( ( u , du )( v , dv )
)( ( u , du )( w , dw )
)( ( v , dv )( w , dw )
))
Conjoining a query on the center cell yields:
(( ( u , du )( v , dv )
)( ( u , du )( w , dw )
)( ( v , dv )( w , dw )
))
u v w
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:
(
(( ( 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
to a cell outside the shaded region for the set Q.
Note 4
Linear Topics (cont.)
| 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 o |
| | U | |
| | | |
| | | |
| | | |
| | | |
| 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.
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 "1" 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.
o-----------------------------------------------------------o
| X |
| |
| o-------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| | U 1 | |
| | ^ | |
| | | | |
| | |dw | |
| | | | |
| o--o----------o o----------o--o |
| / \ \ / | / \ |
| / \ o | / \ |
| / du \ dw / \ dv | / \ |
| / 1<-----\--0<--/-0-\-->0 / \ |
| / \ / | \ / \ |
| o o--o-------o--o o |
| | | | | | |
| | | du | | |
| | | | | | |
| | V | v | W | |
| | | 0 | | |
| o o \ o o |
| \ \ \/ / |
| \ \ /\dv / |
| \ \ / \ / dw |
| \ o 1------------/----->1 |
| \ / \ / |
| o-------------o o-------------o |
| |
| |
o-----------------------------------------------------------o
Figure 3. Effect of the Difference Operator D
Acting on a Polymorphous Function q
Next I will discuss several applications of logical differentials,
developing along the way their logical and practical implications.
Note 5
Linear Topics (cont.)
We have come to the point of making a connection,
at a very primitive level, between propositional
logic and the classes of mathematical structures
that are employed in mathematical systems theory
to model dynamical systems of very general sorts.
Here is a flash montage of what has gone before,
retrospectively touching on just the highpoints,
and highlighting mostly just Figures and Tables,
all directed toward the aim of ending up with a
novel style of pictorial diagram, one that will
serve us well in the future, as I have found it
readily adaptable and steadily more trustworthy
in my previous investigations, whenever we have
to illustrate these very basic sorts of dynamic
scenarios to ourselves, to others, to computers.
We typically start out with a proposition of interest,
for example, the proposition q : X -> B depicted here:
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
The proposition q is properly considered as an "abstract object",
in some acceptation of those very bedevilled and egging-on terms,
but it enjoys an interpretation as a function of a suitable type,
and all we have to do in order to enjoy the utility of this type
of representation is to observe a decent respect for what befits.
I will skip over the details of how to do this for right now.
I started to write them out in full, and it all became even
more tedious than my usual standard, and besides, I think
that everyone more or less knows how to do this already.
Once we have survived the big leap of re-interpreting these
abstract names as the names of relatively concrete dimensions
of variation, we can begin to lay out all of the familiar sorts
of mathematical models and pictorial diagrams that go with these
modest dimensions, the functions that can be formed on them, and
the transformations that can be entertained among this whole crew.
Here is the venn diagram for the proposition q.
o-----------------------------------------------------------o
| X |
| |
| o-------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| | U | |
| | | |
| | | |
| | | |
| | | |
| 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. Venn Diagram for the Proposition q
By way of excuse, if not yet a full justification, I probably ought to give
an account of the reasons why I continue to hang onto these primitive styles
of depiction, even though I can hardly recommend that anybody actually try to
draw them, at least, not once the number of variables climbs much higher than
three or four or five at the utmost. One of the reasons would have to be this:
that in the relationship between their continuous aspect and their discrete aspect,
venn diagrams constitute a form of "iconic" reminder of a very important fact about
all "finite information depictions" (FID's) of the larger world of reality, and that
is the hard fact that we deceive ourselves to a degree if we imagine that the lines
and the distinctions that we draw in our imagination are all there is to reality,
and thus, that as we practice to categorize, we also manage to discretize, and
thus, to distort, to reduce, and to truncate the richness of what there is to
the poverty of what we can sieve and sift through our senses, or what we can
draw in the tangled webs of our own very tenuous and tinctured distinctions.
Another common scheme for description and evaluation of a proposition
is the so-called "truth table" or the "semantic tableau", for example:
Table 2. Truth Table for the Proposition q
o---------------o-----------o-----------o-----------o-------o
| u v w | u & v | u & w | v & w | q |
o---------------o-----------o-----------o-----------o-------o
| | | | | |
| 0 0 0 | 0 | 0 | 0 | 0 |
| | | | | |
| 0 0 1 | 0 | 0 | 0 | 0 |
| | | | | |
| 0 1 0 | 0 | 0 | 0 | 0 |
| | | | | |
| 0 1 1 | 0 | 0 | 1 | 1 |
| | | | | |
| 1 0 0 | 0 | 0 | 0 | 0 |
| | | | | |
| 1 0 1 | 0 | 1 | 0 | 1 |
| | | | | |
| 1 1 0 | 1 | 0 | 0 | 1 |
| | | | | |
| 1 1 1 | 1 | 1 | 1 | 1 |
| | | | | |
o---------------o-----------o-----------o-----------o-------o
Reading off the shaded cells of the venn diagram or the
rows of the truth table that have a "1" in the q column,
we see that the "models", or satisfying interpretations,
of the proposition q are the four that can be expressed,
in either the "additive" or the "multiplicative" manner,
as follows:
1. The points of the space X that are assigned the coordinates:
<u, v, w> = <0, 1, 1> or <1, 0, 1> or <1, 1, 0> or <1, 1, 1>.
2. The points of the space X that have the conjunctive descriptions:
"(u) v w", "u (v) w", "u v (w)", "u v w", where "(x)" is "not x".
The next thing that one typically does is to consider the effects
of various "operators" on the proposition of interest, which may
be called the "operand" or the "source" proposition, leaving the
corresponding terms "opus" or "target" as names for the result.
In our initial consideration of the proposition q, we naturally
interpret it as a function of the three variables that it wears
on its sleeve, as it were, namely, those that we find contained
in the basis {u, v, w}. As we begin to regard this proposition
from the standpoint of a differential analysis, however, we may
need to regard it as "tacitly embedded" in any number of higher
dimensional spaces. Just by way of starting out, our immediate
interest is with the "first order differential analysis" (FODA),
and this requires us to regard all of the propositions in sight
as functions of the variables in the first order extended basis,
specifically, those in the set {u, v, w, du, dv, dw}. Now this
does not change the expression of any proposition, like q, that
does not mention the extra variables, only changing how it gets
interpreted as a function. A level of interpretive flexibility
of this order is very useful, and it is quite common throughout
mathematics. In this discussion, I will invoke its application
under the name of the "tacit extension" of a proposition to any
universe of discourse based on a superset of its original basis.
Note 6
Linear Topics (cont.)
I think that we finally have enough of the preliminary
set-ups and warm-ups out of the way that we can begin
to tackle the differential analysis proper of our
sample proposition q = (( u v )( u w )( v w )).
When X is the type of space that is generated by {u, v, w},
let dX be the type of space that is generated by (du, dv, dw},
and let X x dX be the type of space that is generated by the
extended set of boolean basis elements {u, v, w, du, dv, dw}.
For convenience, define a notation "EX" so that EX = X x dX.
Even though the differential variables are in some abstract
sense no different than other boolean variables, it usually
helps to mark their distinctive roles and their differential
interpretation by means of the distinguishing domain name "dB".
Using these designations of logical spaces, the propositions
over them can be assigned both abstract and concrete types.
For instance, consider the proposition q<u, v, w>, as before,
and then consider its tacit extension q<u, v, w, du, dv, dw>,
the latter of which may be indicated more explicitly as "eq".
1. Proposition q is abstractly typed as q : B^3 -> B.
Proposition q is concretely typed as q : X -> B.
2. Proposition eq is abstractly typed as eq : B^3 x dB^3 -> B.
Proposition eq is concretely typed as eq : X x dX -> B.
Succinctly, eq : EX -> B.
We now return to our consideration of the effects
of various differential operators on propositions.
This time around we have enough exact terminology
that we shall be able to explain what is actually
going on here in a rather more articulate fashion.
The first transformation of the source proposition q that we may
wish to stop and examine, though it is not unusual to skip right
over this stage of analysis, frequently regarding it as a purely
intermediary stage, holding scarcely even so much as the passing
interest, is the work of the "enlargement" or "shift" operator E.
Applying the operator E to the operand 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
The enlarged proposition Eq is a minimally interpretable as
as a function on the six variables of {u, v, w, du, dv, dw}.
In other words, Eq : EX -> B, or Eq : X x dX -> B.
Conjoining a query on the center cell, c = uvw, yields:
o-----------------------------------------------------------o
| Eq.c |
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 given proposition (( u v )( u w )( v w )).
The models of Eq.c can be described in the usual ways as follows:
1. The points of the space EX that have
the following coordinate descriptions:
<u, v, w, du, dv, dw> =
<1, 1, 1, 0, 0, 0>,
<1, 1, 1, 0, 0, 1>,
<1, 1, 1, 0, 1, 0>,
<1, 1, 1, 1, 0, 0>.
2. The points of the space EX that have
the following conjunctive expressions:
u v w (du)(dv)(dw),
u v w (du)(dv) dw ,
u v w (du) dv (dw),
u v w du (dv)(dw).
In summary, Eq.c informs us that we can get from c to a model of q by
making the following changes in our position with respect to u, v, w,
to wit, "change none or just one among {u, v, w}".
I think that it would be worth our time to diagram the models
of the "enlarged" or "shifted" proposition, Eq, at least, the
selection of them that we find issuing from the center cell c.
Figure 4 is an extended venn diagram for the proposition Eq.c,
where the shaded area gives the models of q and the "@" signs
mark the terminal points of the requisite feature alterations.
o-----------------------------------------------------------o
| X |
| |
| o-------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| | U | |
| | | |
| | | |
| | | |
| | | |
| o--o----------o o----------o--o |
| / \ \ / / \ |
| / \ o / \ |
| / \ dw / \ dv / \ |
| / \ 1<--/-1-\-->1 / \ |
| / \ / | \ / \ |
| o o--o-------o--o o |
| | | | | | |
| | | du | | |
| | | | | | |
| | V | v | W | |
| | | 1 | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| o-------------o o-------------o |
| |
| |
o-----------------------------------------------------------o
Figure 4. Effect of the Enlargement Operator E
On the Proposition q, Evaluated at c
Note 7
Linear Topics (cont.)
One more piece of notation will save us a few bytes
in the length of many of our schematic formulations.
Let !X! = {x_1, ..., x_k} be a finite class of variables --
whose names I list, according to the usual custom, without
what seems to my semiotic consciousness like the necessary
quotation marks around their particular characters, though
not without not a little trepidation, or without a worried
cognizance that I may be obligated to reinsert them all to
their rightful places at a subsequent stage of development --
with regard to which we may now define the following items:
1. The "(first order) differential alphabet",
d!X! = {dx_1, ..., dx_k}.
2. The "(first order) extended alphabet",
E!X! = !X! |_| d!X!,
E!X! = {x_1, ..., x_k, dx_1, ..., dx_k}.
Before we continue with the differential analysis
of the source proposition q, we need to pause and
take another look at just how it shapes up in the
light of the extended universe EX, in other words,
to examine in utter detail its tacit extension eq.
The models of eq in EX can be comprehended as follows:
1. Working in the "summary coefficient" form of representation,
if the coordinate list x is a model of q in X, then one can
construct a coordinate list ex as a model for eq in EX just
by appending any combination of values for the differential
variables in d!X!.
For example, to focus once again on the center cell c,
which happens to be a model of the proposition q in X,
one can extend c in eight different ways into EX, and
thus get eight models of the tacit extension eq in EX.
Though it may seem an utter triviality to write these
out, I will do it for the sake of seeing the patterns.
The models of eq in EX that are tacit extensions of c:
<u, v, w, du, dv, dw> =
<1, 1, 1, 0, 0, 0>,
<1, 1, 1, 0, 0, 1>,
<1, 1, 1, 0, 1, 0>,
<1, 1, 1, 0, 1, 1>,
<1, 1, 1, 1, 0, 0>,
<1, 1, 1, 1, 0, 1>,
<1, 1, 1, 1, 1, 0>,
<1, 1, 1, 1, 1, 1>.
2. Working in the "conjunctive product" form of representation,
if the conjunct symbol x is a model of q in X, then one can
construct a conjunct symbol ex as a model for eq in EX just
by appending any combination of values for the differential
variables in d!X!.
The models of eq in EX that are tacit extensions of c:
u v w (du)(dv)(dw),
u v w (du)(dv) dw ,
u v w (du) dv (dw),
u v w (du) dv dw ,
u v w du (dv)(dw),
u v w du (dv) dw ,
u v w du dv (dw),
u v w du dv dw .
In short, eq.c just enumerates all of the possible changes in EX
that "derive from", "issue from", or "stem from" the cell c in X.
Okay, that was pretty tedious, and I know that it all appears
to be totally trivial, which is precisely why we usually just
leave it "tacit" in the first place, but hard experience, and
a real acquaintance with the confusion that can beset us when
we do not render these implicit grounds explicit, have taught
me that it will ultimately be necessary to get clear about it,
and by this "clear" to say "marked", not merely "transparent".
Note 8
Linear Topics (cont.)
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, as it
is most likely the only places where ever they will, 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:
o-----------------------------------------------------------o
| |
| Dq = Eq - q = Eq - eq. |
| |
| Since "+" and "-" signify the same operation |
| over B = GF(2), we have the following equations: |
| |
| Dq = Eq + q = Eq + eq. |
| |
| Since "+" = "exclusive-or", this connective |
| can be expressed in cactus syntax as follows: |
| |
| Eq q Eq eq |
| o---o o---o |
| \ / \ / |
| Dq = @ = @ |
| |
| Dq = (Eq , q) = (Eq , eq). |
| |
| Recall that a k-place bracket "(x_1, x_2, ..., x_k)" |
| is interpreted (in the "existential interpretation") |
| to mean "exactly one of the x_j is false", thus the |
| two-place bracket is equivalent to the exclusive-or. |
| |
o-----------------------------------------------------------o
The result of applying the difference operator D to the source
proposition q, conjoined with a query on the center cell c, is:
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 the difference proposition Dq.uvw 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.
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.
o-----------------------------------------------------------o
| X |
| |
| o-------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| | U 1 | |
| | ^ | |
| | | | |
| | |dw | |
| | | | |
| o--o----------o o----------o--o |
| / \ \ / | / \ |
| / \ o | / \ |
| / du \ dw / \ dv | / \ |
| / 1<-----\--0<--/-0-\-->0 / \ |
| / \ / | \ / \ |
| o o--o-------o--o o |
| | | | | | |
| | | du | | |
| | | | | | |
| | V | v | W | |
| | | 0 | | |
| o o \ o o |
| \ \ \/ / |
| \ \ /\dv / |
| \ \ / \ / dw |
| \ o 1------------/----->1 |
| \ / \ / |
| o-------------o o-------------o |
| |
| |
o-----------------------------------------------------------o
Figure 3. Effect of the Difference Operator D
Acting on a Polymorphous Function q
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.
Note 9
Linear Topics (cont.) Another way of looking at this situation is by letting the (first order) differential features du, dv, dw be viewed as the features of another universe of discourse, called the "tangent universe to X with respect to the interpretation c" and represented as dX.c. In this setting, Dq.c, the "difference proposition of q at the interpretation c", where c = uvw, is marked by the shaded region in Figure 5. o-----------------------------------------------------------o | dX.c | | | | o-------------o | | / \ | | / \ | | / \ | | / \ | | / \ | | o o | | | dU | | | | | | | | | | | | | | | | | | | o--o----------o o----------o--o | | / \%%%%%%%%%%\ /%%%%%%%%%%/ \ | | / \%%% 2 %%%%o%%%% 3 %%%/ \ | | / \%%%%%%%%/%\%%%%%%%%/ \ | | / \%%%%%%/%%%\%%%%%%/ \ | | / \%%%%/% 1 %\%%%%/ \ | | o o--o-------o--o o | | | |%%%%%%%| | | | | |%%%%%%%| | | | | |%%%%%%%| | | | | dV |%% 4 %%| dW | | | | |%%%%%%%| | | | o o%%%%%%%o o | | \ \%%%%%/ / | | \ \%%%/ / | | \ \%/ / | | \ o / | | \ / \ / | | o-------------o o-------------o | | | | | o-----------------------------------------------------------o Figure 5. Tangent Venn Diagram for Dq.c Taken in the context of the tangent universe to X at c = uvw, written dX.c or dX.uvw, the shaded area of Figure 4 indicates the models of the difference proposition Dq.uvw, specifically: 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
Document History
Inquiry List (Nov 2004)
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- http://stderr.org/pipermail/inquiry/2004-November/002041.html
NKS Forum (Nov 2004)
- http://forum.wolframscience.com/archive/topic/632-1.html
- http://forum.wolframscience.com/printthread.php?threadid=632
- http://forum.wolframscience.com/showthread.php?threadid=632
- http://forum.wolframscience.com/showthread.php?postid=2133#post2133
- http://forum.wolframscience.com/showthread.php?postid=2135#post2135
- http://forum.wolframscience.com/showthread.php?postid=2137#post2137
- http://forum.wolframscience.com/showthread.php?postid=2143#post2143
- http://forum.wolframscience.com/showthread.php?postid=2144#post2144
- http://forum.wolframscience.com/showthread.php?postid=2145#post2145
- http://forum.wolframscience.com/showthread.php?postid=2146#post2146
- http://forum.wolframscience.com/showthread.php?postid=2148#post2148
- http://forum.wolframscience.com/showthread.php?postid=2150#post2150
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