[Arisbe] Re: Differential Logic & Dynamic Systems
Jon Awbrey
arisbe@stderr.org
Fri, 24 Aug 2001 00:12:00 -0400
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Da Capo, Al Fine, One More Time!
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
situations to ourselves and to others.
| Notes On A Few Changes In Notation
|
| In the process of editing this uncolorized version,
| I have tried to fix several problems with notation
| that are begining to become a bother in this plain
| courier context, like the excessive "busy-ness" of
| the letter "B" being used not only for the boolean
| domain B = {0, 1}, but also for a logical variable.
|
| I am also switching to the more usual mathematical
| convention of designating the more global space of
| one's immediate interest with the letter "X", thus
| leaving the letters "U", "V", "W", and so on, free
| to designate the more local and transitory patches
| of spaces that will come to occupy one's attention
| in the moment to moment concern with the situation
| that is given as the site of one's manifold senses.
|
| To save capital letters for sets and spaces, in so
| far as it may be convenient, I will default to the
| use of lower case letters for variables or indices
| whenever possible, especially in abstract examples.
|
| Just as an experiment, as I am not too sure of how
| it will work out if pursued in a persistent manner,
| whenever a sign like "u" serves as a logical index,
| by which I mean the index of a logical proposition,
| in which case you may notice that it is very usual
| to treat "u" as being subject to the extra reading
| by which it can also denote a function, that is to
| say, a proposition of the form u : X -> B = {0, 1},
| then I will invest the associated capital with the
| meaning of the antecedent pre-image of 1 under the
| function, what some call the "fiber of truth" in u,
| in summary, accordingly, writing "U = (u^(-1))(1)".
|
| Finally, notice that we have two ways of referring
| to particular points of a logical space X, what we
| may on divers occasions elect to treat as cells in
| a venn diagram, as boolean vectors in a coordinate
| space B^k, or as rows of values in a truth tableau,
| namely, we can indicate such a particular point as
| the vector space sum, x = <x1, x2, ..., xk>, or as
| the conjunctive product x = x1 x2 ... xk, and so
| I will quite freely switch between these two forms.
Let us now run through the typical scenario, or the generic sequence
of stages, in the rudimentary differential analysis of a proposition.
We typically start out with a target proposition q, for example:
| u v u w v w
| o o o
| \ | /
| \ | /
| \|/
| o
| |
| |
| |
| q = @
|
| q = (( u v )( u w )( v w ))
The proposition q is a properly regarded 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 |
| | U | |
| | o o | |
| | /%\ /%\ | |
| | /%%%\ /%%%\ | |
| | /%%%%%.%%%%%\ | |
| | /%%%%%/%\%%%%%\ | |
| | /%%%%%/%Q%\%%%%%\ | |
| | /%%%%%/%%%%%\%%%%%\ | |
| o-----------------------o |
| / /%%%%%%%%%\ \ |
| o o%%%%%%%%%%%o o |
| \ \%%%%%%%%%/ / |
| \ V \%%%%%%%/ W / |
| \ \%%%%%/ / |
| \ \%%%/ / |
| \ \%/ / |
| \ . / |
| \ / \ / |
| \ / \ / |
| o o |
| |
o-----------------------------------o
Figure 1. Venn Diagram For The Proposition q
By way of an excuse, if not yet a full justification, I probably
ought to explain a couple of the reasons why I hang onto to these
primitive styles of depiction, even though I hardly recommend that
anybody actually try to draw them, at least, not once the number of
variables gets much higher than three or four or five at the utmost.
1. 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 are all there
is to reality, and thus, that as we practice to categorize,
we also manage to discretize, to distort, to reduce, and to
truncate the richness of what is to the poverty of what we can
sieve and sift through our senses and what we can draw into the
tangled webs of our own very tenuous and tinctured distinctions.
2. I have temporarily forgotten the other reason. Will amend later.
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 |
|-------------|-----------|-----------|-----------|-------|
| | | | | |
| 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 space X with coordinates <u, v, w> =
<0, 1, 1> or <1, 0, 1> or <1, 1, 0> or <1, 1, 1>.
2. The points of space X with conjunctive depictions:
(u)vw, u(v)w, uv(w), uvw, where "(x)" is "not x".
The next thing that we typically do is consider the effects of
various operators on our target proposition -- and, by the way,
not too coincidentally, I will need to shift our paradigm from
the psychologist's to what is more akin to the mathematician's
way of using the terms "source" and "target" -- and so I shall
begin to speak about the "operand" or the "source" proposition,
instead of the "target" proposition as I have been up till now.
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,
namely, those in the set {u, v, w, du, dv, dw}. Note that this
does not change the expression of any proposition, like q, that
does not mention the extra variables, it only changes how it is
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.
Jon Awbrey
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