>I salute all the great thinkers in Logic who taught me to think more
>logical and not merely classical.
Dear At,
I have spent the last nights (I still have to work during daytime) to
rethink the relation between mathematics and logic.
There have been many attempts to interpret special mathematical systems as
logic. I will call such attempts "logistics". You refer to one of them,
may be called the calculus of statements (in german: "Aussagenkalkuel").
Others are set theory or lately category theory.
Which authority can decide whether such an attempt is successful, valid,
really the same as logic? After a special logistic has set up an
interpretation (e.g. p,q are statements, T, F are truth values, and
functions are called for example "IMPLY"), I miss questioning whether such
an interpretation makes (common) sense. Instead of questioning, the
logistic follows its business and start to find various kinds of logics or
even start to prove, where logic fails - as if that interpretation "My
calculus is logic, or superior to "classical" logic" can be taken for
granted.
There is an authority, which can decide on the validity of such an
interpretation. This is, what I called pure logic. Pure logic is logic
before any mathematics. It is the ground without which mathematics would
not even be thinkable. I called that pure logic RTL - relation of terms
logic. It is build solely on the identity-diversity-relations of terms.
Let's understand what "imply" means in RTL. RTL is really easy in its
basics, although it can become very complex.
We can logically distinguish 4 possible relations of terms (try to find
more!). Let M and P be terms (I skip to explain what a term is and how to
define them. The examples should make the point sufficiently clear):
1.) All S are P. Name: a-relation. Short: S a P (Universal positive relation)
Example: All companies are organizations. All birds are animals.
All Mathematic is Logic.
(companies and organizations are a-related;
If company then organization)
2.) Some S are P. Name: i-relation. Short: S i P (Particular positive relation)
This relation is symmetrical: Some P are S expresses the same.
It is always allowed that some could be all. If one knew, one would
have an a-relation
Example: Some service organizations are learning organizations.
Some birds are black
3.) All S are not P. Name: e-relation; S e P (Universal negative relation)
This relation is also symmetrical
Example: All organizations are not airplanes. All birds are not cats.
4.) Some S are not P. Name: o-relation; S o P (Particular negative relation)
Example: Some organizations are not learning organizations.
Some birds are not black.
Now one may want to examine, what one can know about the relation between
an S and P, if one knows the relation between S and another middle term M
and also the relation between P and M. Answer: Nothing! As long as if we
do not make use of some fundamental knowledge of relations. These are not
axioms in the mathematical sense. A mathematical axiom can be changed or
added or ommitted and what you get is another system which can be examined
mathematically. Without the following fundamentals, mathematics is not
even thinkable.
Fundamental 1: All S are S. (S a S)
Example: all birds are birds
Fundamental 2: All S are not Not-S. (S e Not-S)
Example: all birds are not something else
Fundamental 3: S a M; M a P therefore S e P is forbidden
All blackbirds are birds; all birds are animals therefore
All blackbirds are not animals is forbidden
S a M; M e P therefore S a P is forbidden as well
All blackbirds are birds; All birds are not cats therefore
All blackbirds are cats is forbidden.
One is for sure in At's usage of the IMPLY, STROKE, AND... for any two or
three terms used, these fundamentals are unquestioned. Take for example T
and F (in fact, regardsless of their "meaning" of true and false):
The three fundamentals read as:
1.: T a T; 2.: T e F
3.: T a T - T a T therefore T e T is forbidden and
T a T - T e F therefore T a F is forbidden too.
Now we can derive exactly 18 basic (using only one middle term M)
inferences. All 18 have a name for easier memorization with three vocals
a,i,e and o. The first vocal indicate the relation between P and M, the
second between S and M and the third the infered relation between S and P.
The easiest one is "barbara":
All M are P
All S are M therefore
All S are P
In short: barbara = M a P; S a M => S a P
In our example: All blackbirds are birds; all birds are animals =>
blackbirds are animals.
[Host's Note: "Barbara" is the name classically given to this form of
syllogism; if we had been born in the previous century, we would have
learned "Barbara" by rote. ...Rick]
You may find the other 17 for yourself as a riddle if you want. I give a
few further as examples. You may check whether you can agree on them:
darapti: M a P; M a S => S i P
datisi : M a P; M i S => S i P
disamis: M i P; M a S => S i P
calemes: M a P; M e S => S e P
felapton: P e M; M a S => S o P
It's really as simple as this.
>From here, also any of the logical fallacies, which John Gunkler mentioned
and all other possible fallacies can be derrived:
1.) From some combination of the two initial relations, no relation
between S and P can be infered (for example: P a M; S a M. Birds are
animals. There are for all relations to birds a possible S, which is also
an animal)
2.) Changing the vocal in the infered relation, except a -> i and e -> o,
because if a relation is valid for all of one kind, it is also valid for
some of the same kind.
Now about the meaning of "IMPLY". Where is it? We cannot answer this
question logically! From the viewpoint of logic, we first have to define
the term "imply", what is "imply" considered to mean? The definition must
be given by giving all a,i,e,o-relations to all other terms related to it.
Then I can check whether what I have defined is RTL-consistent. If it is
not, I have a problem in my web of terms, I try to include one of the
fundamentally forbidden relations somewhere - a defect of Msys of my
system of terms. It is a defect in RTL, which I think is related to the
essentiality sureness - something in my trying to establish categorical
identity in all what I think and mean does not work.
Interpreting any mathematical calculus as logic is as dangerous as any
interpretations of mathematical systems, like differential equations as
models of real systems. The positive with mathematics is that it is
RTL-consistent. But the interpretations have more to do with fruitfulness,
which is in my eyes related to lateral thinking logic. At, you have shown
what you have learnt by such connections and how powerful they are. But
your parable fits to your path: a meandering through the marsh. Whatever
you disregard as classical logic does not touch RTL. You refer to a part
of the logistics you have learnt to be "classical logic". With the term
"classical logic" we have the same situation as with "imply". It depends
on how you define it. You have done it by pointing to how it is part of
your logistics: "classical logic" a "Aussagenkalkuel". I have defined:
"Aussagenkalkuel" a "Mathematics"; "Mathematics" a "RTL". I am not going
to argue definitions. And any definition can lead, if followed
consequently, to profound insights.
But with my presentation here of RTL, I hope that I was able to promote
your concepts on creative learning by strengthening the essentiality
sureness. I suspect that pure logic, free of any ontology or epistemology,
has never been presented to you like this.
In my eyes, the relation between mathematics and logic is well discribed
in the powerful narrative of the lost son and his return to his fathers
house. Mathematics being the lost son, logic the father. The way in which
the son leaves home, the way he lives in the foreign country, the way he
discover his poorness, the way the father is happy and regards him as the
most valuable he has when he returned discribes wonderfully the relation
between mathematics and logic in the creative course of time.
For the end, I have one question left for you: I hope, I could make clear
that there is one and only one pure logic. Yet I stated that we have at
least two more logics to consider: CEL (cause-effect-logic) for liveness
and LFS (logic of front structure) for wholeness. How does this fit
together?
Liebe Gruesse,
Winfried
--"Winfried Dressler" <winfried.dressler@voith.de>
Learning-org -- Hosted by Rick Karash <rkarash@karash.com> Public Dialog on Learning Organizations -- <http://www.learning-org.com>