For what end? LO23733

From: AM de Lange (
Date: 01/07/00

Replying to LO23472 --

Dear Organlearners,

Skip this long contribution rather than criticising me later on for its
supposed verbage.

Winfried Dressler" <> writes:

>once again, I have difficulties in understanding your writing about
>the law of excluded middle (LEM).

Greetings Winfried,

Thanks for pointing out that my writings on LEM (Law of Excluded Middle)
are still obscure. I will try to articulate my own understanding better.
Were it not for the fact that LEM is used extensively in learning, I would
have suggested that we bury this topic for good.

LEM is a pattern which was first formalised in logic. There it is called
an axiom. Every system of logic is based on some axioms. An axiom is a
simple statement (proposition) which is true for all possible
circumstances. Initially no logician questioned the LEM as an axiom for
logic. But since the days of Lutzen Brouwer early this century more and
more logicians become dissatisfied with LEM.

A statement (or proposition) is a sentence which has one of two possible
truth values -- either T (true) or F (false). When we invoke LEM as an
axiom, it means that a statement cannot be both T (true) and F (false).
When we now apply the negation NOT to a statement, its negated statement
will have the value F (false) if the statement itself had the value T
(true), or the negated statement will have the value T (true) if the
statement itself had the value F (false).

Applying the NOT a second time in succession switches the truth values of
the negated statement (now a double negated statement) back to the truth
value of the original statement.

Let the symbol P be the statement "the moon is made of cheese".
The statement P has the truth value F (false). The single negated
statement NOT P will then have the truth value T (true). The double
negated statement NOT NOT P will again have the truth value F
(false). Thus
because they have the same truth value.

Writing the symbolic expression NOT P back into an ordinary
sentence requires some grammatical modification. The interlinear
substitution of P into NOT P gives:
"NOT the moon is made of cheese".
We have to chance this into
"the moon is not made of cheese"
to get grammatically a better sentence in English.

Writing the symbolic expression NOT NOT P back into an ordinary
sentence requires even more grammatical modification. The
interlinear substitution of P into NOT NOT P gives:
"NOT NOT the moon is made of cheese"
which can be imbettered as before into
"NOT the moon is not made of cheese". But the modification of
this sentence into better English clearly illustrates how English
(and most other languages) do not easily accomodate LEM. The
best we can come up with is:
"IT IS not THE CASE THAT the moon is not made of cheese"
The words in capital letters were needed to introduce the second
NOT and thus the manifestation of LEM into English.

(My own mother tongue Afrikaans use the double negation as a
bracketing for the single negation. It can also be done in French
and some Chinese language, but it is not done on such a regular
basis. Such a practice affords powerful ways to express standard
rather than bracketing double negations.)

Children simplify this repeated negations in their arguments by
leaving out these additional words in capital letters. Thus they
"it is"
"it is not"
"it is" (rather than "IT IS not THE CASE THAT it is not")
"it is not" (rather than "IT IS not THE CASE THAT THE CASE IS NOT it
"it is" (rather than .................)
"it is not" (rather than ................)

To avoid this indefinite repition of negations in debate among adults, one
of the rules is to allow somebody to respond to a statement with a
negation and then to allow the original proposer to respond to this denial
with a double negation, but to cut off any subsequent negations.

The interesting thing is that LEM (or rather its pattern) goes beyond
logic into, for example, ethics. In ethics the values of statements
become G (good) or B (bad). For example, let Q symbolise the
statement "the manager can take whatever he likes". This statement
has the ethical value B (bad). Thus the statement NOT Q has the
ethical value G (good). Translated back into ordinary language it
becomes "NOT the manager can take whatever he likes" which is
grammatically modified into "the manager cannot take whatever he
The double negation is NOT NOT Q. In terms of our example it means
"NOT NOT the manager can take whatever he likes", first modified
into "NOT the manager cannot take whatever he likes" and then
modified into "IT IS not THE CASE THAT the manger cannot take
whatever he likes".

This extension of LEM into ethics is the basis of stewardship. However,
stewards have to command and to my greatest surprise after having
succeeded in formalisng imperative logic (the logic of commands) LEM has
no standing in it. The simple reason is that commans has three possible
outcomes rather than the usual two values of declarative logic.

First the easy part of the understanding:-

We have to understand that one of the requirements for LEM is
EXACTLY two OPPOSITE values. Typical examples are the
complementary duality T (true) and F (false), the complementary
duality G (good) and B (bad) or the complementary duality W (being)
and J (becoming). As soon as we have any other number of values
than only TWO values, LEM fails. Say, for example, we have a
property P with one of three possible values X, Y and Z. When P
has the value X, then NOT P will have the value Y or Z, but not both.
Say NOT P has the value Y. Then NOT NOT P has the value X or
Y. Because Y is also a possibility for NOT NOT P whereas X is
the only possibility for P, we cannot invoker LEM as

This is also what your illustration boils down to.

>Let me give a simple example - not wholeness but a little black
>blue bird = NOT black bird
>NOT blue bird = NOT NOT black bird
>The LEM states that
>NOT NOT black bird = black bird
>but obviously
>NOT blue bird can also be a red bird.
>The problem is solved by noticing that
> blue bird = NOT black bird (apartheid = NOT wholeness)
>is wrong. NOT black bird means: ALL BUT black birds.
>Blue birds are only a subset of those 'all but black birds':

Your example is fine with respect to the colour of birds since
we have more than two possible colours.

But you assumption that it also models the case
(apartheid = NOT wholeness)
is wrong because wholeness is unique. There is but ONE
wholeness which has only ONE value, namely whole. Wholeness
does not have two or more different values.

I will answer more fully on the identity
apartheid = NOT wholeness
in my reply to John Gunkler.

You also write:

>LEM works perfectly. The mistake was to take something
>specific (be it blue bird or apartheid) for the general 'ALL BUT'
>(be it NOT black bird or NOT wholeness).
>This mistake is really very common, and I think, this is what
>you wanted to point out: If one is against something bad, people
>expect the outcome to be something good. This is of course a fallacy:
>NOT a specific bad = ALL BUT this specific bad (including all good,
>but also all other bad things).
>But the whole issue does not scratch LEM, if it is properly applied.

Yes, but how do we know when to apply LEM properly?

This brings us to the difficult part of the understanding:-

We also have to understand that the other requirement for LEM is
that there shall be no evolution (the "one-to-many-mapping" of
"entropy production") in the possible values of a property.

For example, without evolution false statements remain false forever.
They never grow gradually into true statements (by a process which
we often articulate as modification or rectification). If this is
really true,
then we have no hope for truth to evolve. Also the third step
(falsify) of
scientific thinking (observation => speculation => falsification)
becomes a hoax. But worse of all, learning cannot happen.

However, what about the "many value" rather than "two value"
cases. It seems as if LEM is excluded because we are dealing
with many values. However, "underneath" these "many values"
we may find a "two value" substrate which brings LEM into the
picture after all. But together with it comes "entropy production"
which is the only other quantitiy connected forever with time like
the velocity of light. Unfortunately for us because of our traditional
thinking, whereas the velocity of light is constant, "entropy
production" has a dance which we have to focus on. This dance
is the source of all complexity.

As an introduction I want you to consider the following line of
thinking. An adult is a human. A child is not an adult. Hence some
people infer incorrectly that a child is not a human. As a teacher I
had countless battles against people who consider a child not as a
human, or to put it more mildly, people who consider a child less of
an human than an adult. With this example we can pick out many
other examples, even in the domain of organisational management.
A typical one is that a person in a lower order of an hirarchy is not
(or less of) a manager like the person in the highest order of the
hirarchy. This fallacy is often pointed out in a study of predicate
(first order) logic. But there is something deeper to its

A property (predicate) evolves through a whole spectrum of values
in a very definite, IRREVERSIBLE ("entropy producing") order. (In the
case of true/false the evolution sweeps through only two values in one
step.) At every given instant of time the property has only one of all
possible values. When we compare its value at one instant of time with
a different value at a later instant of time, the later value negate
former value. This negation is in a positive sense because it does
NOT REVERSE the evolution.

For example
P = person ### is 4 years old
N+O+T P = person ### is 5 years old = person ### is not 4 years old
NOT N+O+T P = it is not that person ### is not 4 years old
= person ### is 4 years old
= P
(Please read N+O+T as "the postive or constructive kind of

But when we compare its value at one instant of time with a different
value at a former (earlier) instant of time which belongs to the past,
this negation is in a negative (destructive) sense because it is not
possible for evolution. Evolution is not a reversible process. A
biologist with the name Dollo first had this profound insight into
Only many years afterwards Prigogine succeeded in showing why --
physical evolution is caused by irreversible entropy production
he would say)

P = person ### is 4 years old
N-O-T P = person ### is 3 years old = person ### is not 4 years old
This negative N-O-T P is completely different to the positive N+O+T P
above. The LEM may be applied to the positive denial N+O+T P, but not
to this negative denial N-O-T P. The reason is that a double negation
cannot be constructed on a single negation which cannot happen. In
other words, we cannot make sound inferences in terms of

It is in this sense that
NOT (apartheid is the law) = NOT (N-O-T wholeness is essential)
and thus
NOT (N-O-T wholeness is essential) = wholeness is essential

Winfried, please do some reading on the constructionist and intuitionist
movements in the evolution of logics and mathematics, spearheaded by the
courageous Brouwer. You will then understand that the application of LEM
is not as simple as humans have assumed for a couple of millenia,
beginning with Aristoteles. So when "deep creativity" enters the scene, we
have to prepare ourselves for a complex understanding otherwise we will
find the understanding far too difficult to comprehend.

This does not mean that through the ages people did not have the slightest
clue what to do. The Bible in many places warns us not to judge unless we
are ready to be judged ourselves by a much stricter law than the one which
we use in our own judgements. This warning evolves clearly through the
Bible, culminating in the teachings of Jesus. The kind of judgement which
the Bible warns us against is the NOT N-O-T kind -- denying (the NOT)
people the possibility to become free from destructive creativity (the
N-O-T). (By the way, the New Testament consistently use the Greek words
"kritikos" and "kritiko" for the judgement and judging which it warns us

I have explained many times how destructive creativity happens. It happens
when one or more of the seven essentialities are "impaired" -- the N-O-T.
(By "impaired" I mean than that some of them are far less developed than
the others and what the system's level of development requires). Should a
person in judgement now willfully deny these essentialities -- the NOT --
that person is effectively arguing that the NOT N-O-T is good enough for
the evolution of the system.

We are now entering a new millenium while we are experiencing one of the
greatest paradigm shifts in the history of humankind. I think that each of
us has to try and learn just how destructive any critique is when it
denies wholeness (or any of the other six essentialities for the same
reason). We need the whole picture and nothing less before we should even
think of possibly making a judgement. Any fragmented judgement is far
worse than that issue which is judged irrespective of how bad or false the
issue is. Should we make such a fragmented judgement, we are jeopardising
the whole of evolution far more than the situation which we are judging
has done. What happens is that we let "entropy production" become our
master rather than becoming masters in "entropy production". Are the many
millenia of it in the past not enough? Do we really have to continue with
it it in this new millenia?

Here is an example of what I mean. The heating of the earth's atmosphere
due to the combustion of fossil fuels is a situation for great concern.
But some people argue that such a concern is needless because we do not
have reliable temperature meaurements of the earth's atmosphere going back
even fifty years. Temperature measurements are direct evidence. Alowing
only them and not any indirect evidence in terms of the effects caused by
an increasing temperature is a serious fragmentation.

I can show anyone of you fellow learners in the Namib how the succulent
plant species Euphorbia venenata has responded to this heating the past 30
years compared to the previous 170 years. Every year a branch of any E
venenata ought to grow a bit, expanding and lengthening as it grows. By
the end of the season it gets constricted again. By counting the number of
contrictions along a branch, we can determine the age of the branch. The
lowest branches may reach an age of circa 200 years before they die off.
The distance between two consecutive constrictions gives us an indication
how favourable or severe the climate was in a particular season. By
comparing the constrictions of, say 20-21 years ago on statistically
sufficient plants and from several localities we will get a powerful
indication of the severity of the climate in the summer of 1980.

The advance of earthquakes are clearly indicated by seismographs. People
who denounce such seismographic evidence are foollish and often have paid
dearly for such foolishness. On the other hand, vulcanologists have made
the most extraordinary movies of erupting vulcanoes because they were
there at the right time, making wisely use of seismographic information.
Euphorbia venenata is one of several "climatographs" we can make use of if
we want to. What "climatographic" information do they afford us? Come with
me into the Namgorab desert of Namibia and I will show you how these
plants frantically try to tell us that something is seriously wrong with
our climate -- something which is unprecedented for the past 200 years.
The old farmers like my friend Lourens Cornelision will also tell you
that, but who still believe farmers -;)

Now for a spiritual (immaterial) rather than a physical (material)

My former colleague Herman Potgieter and I had some extraordinary results
in teaching engineering students chemistry 1. In all their subjects as in
the rest of the university in its awesome curriculum, students have to
build up a "semester mark" in each subject based on the average of all the
tests during that semester on the subject. Then they write an examination
to obtain an "examination mark". Their performance is eventually based on
the average of these two marks. Failing them for achieving less than 50%
is nothing else than a NOT N-O-T situation -- the death of evolution in

I convinced Herman to try a different strategy -- the NOT N+O+T. Three
weeks before the examination we tell the students that there will be a
"pre-examination" in a week's time. It is not compulsory. Should they
take the "pre-examination", we will "forget" all their "sins" (the
N-O-T's) during all their tests the entire semester. Should they do
better in this "pre-examination", we will use its mark rather than the
customary "semester mark".

In this "pre-exam" we thoroughly tested their understanding of chemistry
by making the use of the seven essentialities as the basis for

Then an extraodinary surge in authentic learning happens. At least 95% of
the students begin to act as students of old. They suspend their social
life. They even skip some of their meals. Obviously, and to the anger of
lecturers in other subjects, they even fail to attend the last lectures in
those subjects. For one week they become most intimate with chemistry in
all its aspects. The results? Whereas 70% fail in their "semester mark",
only 30% fail in their "pre-exam mark". Whereas their average for the
"semester mark" is in the vincinity of 40%, it is in the vincinity of 60%
in the "pre-exam mark". When the final examination comes, they improve
even up to 10% on their "pre-exam mark". Eventually their performance in
chemistry 1 is twice as good as in any of their other subjects. They even
begin to become fond of chemistry ;-)

Herman and I managed to maintain this performance for four years before
the axe of tradition stopped us dead in our tracks. How dare we upset the
system! How dare we advance from the customary bashing of their brains
with a feather!

This topic "for what end?" began as a contemplation on the final outcome
for organisations as a result of the diverse endeavours of all their
members. Let us now contemplate the final outcome of the whole earth as a
result of the diverse endeavours of humankind. I think that the author
Nevil Shute's reflection in his extraordinary novel On The Beach is as
good as the best among any of our contributions. He used the aftermath of
a nuclear war as his scenario. I think we have a much worse "war" to
consider -- humankind trying to evolve in direct opposition to the rest of

Fighting destructive creativity with destructive creativity is the worst
think we can do. (A nuclear war is merely an instantiation of it.) What
we rather should do, is gradually replace destructive creativity with
constructive creativity at every possible opportunity.

In order to do so so, we have to think about "deep creativity", the way
how this Creation evolves towards the future. Such thinking requires an
immense shift from traditional thinking. I need not to elaborate on it
because your experiences make you primarily conscious of such a shift.
Your tacit knowledge will evolve just as mine and eventually you will have
to articulate it just as I have tried to do. May you have far more success
than me.

While in the desert, thinking on how I will answer you, it gradually
dawned on me that most revered educators in the history of humankind (like
Socrates, Comenius, Melanchton, Pestalozzi and especially Rousseau)
stressed that education should be "natural" rather than adhering to the
wims of past thinkers. I could not wait to come back and check this
insight. Yes, each of them changed in major way the practice of education
with this insistence on the "natural way".

What is this "natural way" of teaching? I invite you to read once again
each of them carefully and see how much they have articulated what I am
struggling to do once again with "deep creativity".

>At, I am like you aware that the LEM does not apply to wholeness -
>LEM even distroys wholeness. Because this is as important for me
>as it is for you I tried my best to show that your formalization of
>knowledge failed. This does NOT mean, that the knowledge itself is
>wrong, only that there is need for another approach to formalize.

Winfried, may your articulation be more fruitful to other fellow learners
than mine! Two teachers are better than one!

>So do wholeness and sureness have to be separated for eternity?
>Of course not - the world and we with it emerged as a matter of fact.
>It is only the formalization of wholeness in terms of sureness, which
>doesn't work.

I was not aware that I was doing in it. Perhaps I am caught up in my
own South African circumstances because I cannot understand how
even you misinterpret the identity
    apartheid = NOT wholeness
But I do concede that it is possible for fellow learners to get this
impression in which case I realise that my explanation will fail. So I
have tried to rectify it -- I hope it works this time.

>Now I have to catch my train home for a weekend with family.
>Aren't families the places to learn how wholeness and sureness can
>come together by means of fruitfullness? What a life!

What is a family if it cannot emerge into a LO? Show me any emergence,
physical or spiritual, which circumvented wholeness, sureness,
fruitfulness or any of the other four essentialities and I will concede
that I have made the gravest of all conceptual errors. But obviously,
should we not care for emergences, why should we care for the

With caring and best wishes


At de Lange <> Snailmail: A M de Lange Gold Fields Computer Centre Faculty of Science - University of Pretoria Pretoria 0001 - Rep of South Africa

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