ONT Re: Differential Logic -- Series B
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DLOG. Note B5
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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 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 |
| / \ |
| / \ |
| / \ |
| / \ |
| / \ |
| 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.
Jon Awbrey
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http://www.cs.bsu.edu/homepages/mighty/history.html
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