Systematical Patterns in Boolean Logic LO25063 [complex]

From: AM de Lange (
Date: 07/12/00

Replying to LO25047 --

Dear Organleaners,

Greetings to you all.

In my reply to the topic Efficiency and Emergence LO25047, I did
somethings so as to initiate this contribution. I wrote:

>This latter sentence has an important logical undertone.
>Since Gavin says I am stuck in logic, I have to elucidate
>this logical undertone so that fellow learners can know
>exactly what I am doing and where I am stuck. The logical
>undertone here was first formally introduced by the American
>philospher CS Peirce as "inclusive denials". .... It is often
>called among logicians as Peirce's "dagger" and can be
>symbolised by "|". It can be defined as
>a | b == (NOT a) AND (NOT b)
>This "dagger" seems to be a self-destruction of truth in statements.

This "handle" gives me the opportunity to say something more on the
relationship between logic and Learning Organisations.

I specifically avoid calling this contribution "Systems Thinking and
Boolean Logic". This is to give you full opportunity to discover self the
connections between Systems Thinking and Boolean Logic. If you persist to
the very end of this contribution, you may perhaps get a novel insight as
to how the LO-dialogue happens.

First, allow me to paint the bigger picture in which Boolean Logic is
contained. This context is necessary so that we do not over emphasise
Boolean Logic and thus get a distorted picture of it.

We will restrict ourselves to "declarative truth logic". It concerns
sentences which are declarations (propositions, statements), but not
commands and not questions. Furthermore, the sentences concern truth based
on the values "true" and "false", but not ethics based on the values
"good" and "bad" and not morals based on the values "correct" and "wrong".
We also exclude modal qualifiers such as "some", "probable", "pretty",

Boolean Logic is somewhat of a misnomer. It should have been called the
Boolean Model of Logic. In this model every proposition can have one of
several truth values. In the Binary Bollean Model any proposition has one
of two values, namely T (symbol for true) and F (symbol for false).

The qualification "model" above is important because we also have the
Proof Theory of Logic. In Proof Theory we work only with true
propositions, i.e never with false propositions. It is metaphorically like
walking through a marsh on stones. Stepping off a stone into the bog of
the marsh is like working with a false proposition. The stepping stones at
the place of departure are called axioms. The mode of striding from one
stepping stone to another is called an inference rule. Every stone reached
after the stones at the departure is called a theorem. At first the
stepping stones are close to each other and the strides are one stone at a
time. But as we reach stepping stones (theorems) further away, the more we
develop strides (inference rules) to allow us to make vast jumps or even
acrobatic jumps from one stepping stone to another.

Kurt Goedel proved his famous "incompleteness theorem". It says that
whatever our point of departure and its stepping stones there, there are
some stepping stones in the marsh that we will not be able to reach no
matter how much we meander through the marsh. Thus the idea with the
Binary Boolean Model is to extend the meandering with Proof Theory by
walking on stepping stones as well as walking in the the bog surrounding
them trying to reach stepping stones which we cannot reach by Proof
Theory. Hopefully we will not be sucked into the bog a little later on ;-)

Lutzen Brouwer was one of the few logicians who appreciated the fact that
logicians knew about some stepping stones (theorems) before they have
even reached them first by logical means. In his days the post WWII
concept "tacit knowledge" (formulated by Michael Polanyi) was not yet
known. So he suspected that "intuition" had very much to do with knowing
stepping stones in advance. Also in his days the post WWII concept
"creativity" has been used less than a dozen times in all literature. So
he suspected that "constructivity" had very much to do with creating
intuitive knowledge in advance on stepping stones laying beyond. For many
years Brouwer was a lone voice in the desert. However, eventually one of
his students (Arend Heyting) managed to create a model for this
"intuitionistic, constructivist" logic of Brouwer. It is now called the
Heyting Model of Logic. It provided logicians with the insight that the
Boolean Model and the Heyting Model are complementary duals of each other,
although some would love to demarcate them into dialectical duals.

To get an idea how the Boolean and Heyting models are related to each
other, think again of the marsh as our metaphor. Do not think of an evenly
distributed bog underneath the water, but of deep potholes and trenches in
it. To step into a hole or trench of the bog, we need stilts in order not
to submerge and drown. Walking with stilts along the bog is easy, but to
walk with stilts on the stepping stones too requires advanced acrobatic
qualities. It is as if the Heyting Model is sort of a mirror image of the
Boolean Model.

What kind of acrobatics do we need when we combine the Boolean Model,
Heying Model and Proof Theory? A seemingly "strange logic" which began to
develop since the late seventies with the name "toposlogic" (Greek
"topos"=place). (Do not confuse "toposlogic" with "topology" because
"Toposlogic" is the context for "topology".) However, "toposlogic" is not
that strange. It is very much the kind of logic which chemical systems and
biological systems seem to follow. In chemical systems we would rather
speak of it as empirical structure and reactivity. In biological systems
it is better known as empirical morphology and physiology. In other words,
the intuitive idea of "becoming-being" becomes most important to

Secondly, allow me to paint the rest of picture on Boolean Logic itself.
It is possible to paint a rich picture using several hunderds of screens.
I will not paint that rich, yet I will try not to paint the picture too

As has been said above, in Binary Boolean Logic (BBL) every
proposition gets the value either T (true) or F (false), but never
"both T and F" and never "none T and F". This is how the Law
of the Excluded Middle (LEM) is firmly entrenched in the truth
values. Let us symbolise any proposition by the variable (symbol)
"p". A proposition can be a sentence like "The cat has six legs."
Then we can illustrate this LEM by a so-called truth table.
. p
1 T
2 F
There are two cases possible. In the first case p has the value T.
In the second case p has the value F. Should LEM not apply to
the truth values, then we would have had four cases in the truth
. p
1 T
2 F
3 TF
4 --
In the third case p is both true-false and in the fourth case p has
none of T and F as value.

LEM is used above to bring two ideas into BBL, namely "spareness"
and "sureness". The limiting makeup of "spareness" is to restrict
the values to T and F and not also allow other combinations of T
and F like "both" (TF) and "none" (--). The quantitative makeup of
"spareness" is that we restrict ourselves to two cases as in the
first truth table and not four cases as in the second truth table. The
categorical makeup of "sureness" is the "binary", i.e only the two
values T and F are possible. The identity makeup of sureness is
that it is only one of them for each case. In the case of a Ternary
Boolean Logic, sureness will entail one of three possible values,
namely T, F and a third member which we might indicate by U.
Thus the truth table for a Ternary Boolean Logic will be
. p
1 T
2 F
3 U
We will not even take a peep into Ternary Boolean Logic or
Multivalued Boolean Logic.

Two other ideas are very important to BBL. The one is "fruitfulness", i.e.
to make effective connections between two (or more) propositions like p
and q. The other one is "otherness", i.e to exhaust all possibilities of a
particular kind of connection. But let us first make sure what we mean by
"p" and "q". The "p" may be a sentence like "At types on the keyboard" and
the "q" like "At looks at the screen". Now, if "p" corresponds to the fact
that I am typing on the keyboard, then it gets the value T. But if "p"
corresponds to the fact that I rub my hands together, then "p" gets the
value F.

Here is an effective connection between two propositions "p" and
"q". It is symbolised by pANDq. It will be a composite sentence
like "At types on the keyboard AND At looks at the screen". Only
when I am doing both simultaneously, the composite sentence
pANDq will be true. Should I be typing on the keyboard, but not
looking at the screen, then pANDq will be false. Should I not be
typing on the keyboard, but still looking at the screen, then pANDq
will also be false. Should I neither be typing on the keyboard nor
be looking at the screen, the pANDq will also be false. This
effective connection called "AND" can be illustrated by a truth table
as follows
. p q pANDq
1 T T T
2 T F F
3 F T F
4 F F F

We ought to oberve that in the latter truth table some attempt is made to
wave away a strict application of LEM. This is done by allowing cases 2
and 3 where we combine T and F. Only cases 1 (connection in only T) and 4
(connection in only F) would survive should we apply LEM once again.

In terms of "fruitfulness", the construction pANDq is one possible
effective connection between p and q. Another well known possible
construction to create which use also only two propositions (variables) is
pORq. It can be illustrated by the following truth table:
. p q pORq
1 T T T
2 T F T
3 F T T
4 F F F
Should the "p" be a sentence like "At types on the keyboard" and
the "q" be like "At looks at the screen", then our use of OR in
common language is that only when in fact "neither is At typing
on the keyboard and nor is At looking at the screen" will pORq be
false (case 4). I may be typing on the keyboard and yet not be
looking at the screen for which pORq will be true (case 2). I may
also not be typing on the keyboard and yet be looking at the
screen for which pORq will be true (case 3).

LEM has been brought into BBL at the very low level of truth
values. Thus it can surface in BBL at the level of theorems. To
see how, consider for the expression NOT(pAND(NOTp)) its
truth table:
The first line of pAND(NOTp) correspond to line 2 of pANDq
The second line of pAND(NOTp) correspond to line 3 of pANDq

Whenever we have only T's in the column of the suspected theorem (like the
least column above), it is indeed a theorem. The expression
NOT(pAND(NOTp)) is nothing else that the refomulation of LEM as a theorem.

Many people claim that logic is reductionistic. However, consider
for the expression (NOTp)OR(pORq) its truth table:
p q NOTp pORq (NOTp)OR(pORq)
Since all the truth values in the column of (NOTp)OR(pORq) are
T's, it must be a theorem. This theorem tells us how we can
indeed EXPAND an argument containing a proposition p to
include another proposition q too. If logic was reductionistic, this
expansion would not have been possible. In fact, we then would
have had a column of F's underneath (NOTp)OR(pORq).

Please observe that pOR(pORq) is not a theorem as is shown by
p q pORq pOR(pORq)
The fourth line has a F. The fact that (NOTp)OR(pORq) is a
theorem whereas pOR(pORq) has an important bearing on the
way in which people participate in a discussion. Those intuitively
aware of this distinction between (NOTp)OR(pORq) and pOR(pORq)
will take pains first to refute with NOTp the other person's claim p
so as to introduce their own claim q by (NOTp)OR(pORq).

It is not logic which is per se reductionistic. It is rather the
underlying creativity of the mind which is impaired destructively into
reductionism. This happens when ideas like sureness, fruitfulness,
spareness and otherness get neglected or even ignored.

We have now looked at two possible effective connections AND and OR. In
the light of the idea of "otherness" what are all the possible effective
connections between two propositions (variables)? Well, there are 16
(2^2^2) of them of which pANDq and pORq are two possibilities. (In a three
valued logic there will be 3^3^3 possibilities.) For the sake of reference
purposes, we will count them by giving each also a name f[##](p, q) where
the ## in the tag or label [##] is a number ranging from 01, 02, ..., 15,
16. The (p,q) part in f[##](p,q) is just a way of indicating that f[##]
is a two variable function. A three variable function would look like
f(p,q, r) and a four variable function like f(p,q,r,s).

Here are all 16 truth tables for connecting two propositions
        f[01](p,q) f[02](p,q) f[03](p,q) f[04](p,q)

        f[05](p,q) f[06](p,q) f[07](p,q) f[08](p,q)

        f[09](p,q) f[10](p,q) f[11](p,q) f[12](p,q)

         f[13](p,q) f[14](p,q) f[15](p,q) f[16](p,q)

These 16 connectives (functions) are astounding. Let us compare some of
them to get the feeling.

The most common connections we make use of in every day conversations, are
pANDq (the f[02]) as well as pORq (the f[12]). We have discussed them
before introducing the 16 connections. There is a real danger in making
excessively use of the one or the other. Excessive use of one of them (or
any of the 14 other connections) leads to a peculiar kind of linear
thinking. Excessive use of pANDq gives the impression of "hard" (strict)
reasoning while excessive use of pORq gives the impression of "soft" (lax)

Let us now compare f[01] as pFALSEq with f[16] as pTRUEq. Let us consider
any two propositions p and q. Someone who persists making the connection
between the two propositions false irrespective of whether each
proposition is true or false, uses the pFALSEq connection. Such a person
defy logic in the "hard" sense that all arguments will always be false,
thus making logic irrelevant. The opposite of this is the "nice guy" who
persistently claim that the connection is true whatever the truth values
of its constituents. Such a person defy logic in the "soft" sense that all
arguments will always be true, making logic superficial. Sometimes a
person switch from pFALSEq to pTRUEq depending on the situation at hand.
In "hard" situations the person will use pFALSEq while in "soft"
situations the person will use pTRUEq. Such a preson is somewhat like a
"loose cannon" on a ship.

There are four "hard" connections, yet not as very "hard" as pFALSEq, (the
f[01]). They are f[02] as pANDq, f[03] as pNOIMPLYq, f[04] as pFITq and
f[05] as pNOORq. In logic the pNOORq is usually written as pNORq, dropping
the one O. We retain the O here to indicate that the truth table of pNOORq
is exactly the opposite of pORq. The pNOORq has T whenever pORq has F and
vice versa. Hence the NO in NOOR means that the OR has been denied. It
seems that the pNOORq has been first used by CS Peirce to elucidate
certain strange patterns of thinking. It is also called the DAGGER
operation because it scrutinizes truth in a very "hard" way, almost like a
distrusting non-conformist will do. It claims that whenever there is a
possibility for one or both the constituent propositions to be true, the
connection have to be considered as false until it has been re-evalued by
other means. It kinds of invite logics to be supplemented by other means
like empirics (exemplar-exploring), art-expressing, problem-solving and

The connection f[04] as pFITq is some times also described as a
"non-converse implication". It means the fitting or adapting of a second
proposition q despite its truth value provided the initial proposition p
has the truth value false. Its intention is to include the second
proposition by way of admitting that the first proposition is false. The
connection then becomes like the second proposition. But observe that
this is a very "hard" fitting because when the first proposition is true
(the first two lines), the fitting connection is excluded by making it
false. It is also used in competitive debates in the sense of "just prove
that my claim is false and I will concede to your claim whether it is true
or false".

The connection f[03] as pNOIMPLYq is occasionaly also described as a
"non-conditional implication". It is frequently used in sarcastic
discourses in which biting gibes, cutting taunts and scorning ironies
abound like "Yes, the heaven is blue and we will all wear blue caps when
it falls". What happens is that when the person is very sure that the
first proposition p is true, the person then will make this connection
which is in truth the exact opposite of the second proposition q so as to
denigrate the truth perception of the other person. Furthermore, the NO in
NOIMPLY means that it is also denying the connection f[14] as pIMPLYq by
having exactly the opposite truth values.

There are six connections which are intermediate between "hard" and
"soft" in truth. They are f[06] as pFIRSTq, f[07] as pLASTq, f[08] as
pEQUIVq, f[09] as pEXORq, f[10] as pNOLASTq and f[11] as pNOFIRSTq. We
will soon see that they can not only be grouped into complementary duals,
but that each of them points to much logical inertia.

The first two connections pFIRSTq and pLASTq complement each other. In
pFIRSTq the effective connection is forced to have exactly the same truth
values as the first proposition p whatever the values of the second
proposition q. The pFIRSTq is often used by people who begin a discussion
and invite other opinions. However, they firmly stay with their own
opinions whatever course the argument takes. It seems as if they have a
massive resistance to change in logical thinking. It is the same with
pLASTq. In this case the person usually waits until the discourse is
completed and then as an afterthought force all previous previous
propositons to abide by the claim in the last proposition.

The next two connections pEQUIVq and pEXORq are also complementary to each
other. The EQUIV is a shortening of the word "equivalent". It means that p
and q, although they may concern different things, are equal in their
truth values. This effective connection is often used in conversation by
people who want to equalise the thoughts of other participants because of
being preoccupied by equality. They use it as an instrument for logical
conformation. Obviously, its dual is pEXORq which is better known as
pXORq, the "exclusive or" (exlusive disjunction). The idea with using
pEXORq is to force the two propositions to have opposite truth values.
Note that by excluding the first line, this connection is the same as pORq
( f[12] ). A preoccupation with "non-conformism" as well as LEM usually
leads to excessive use of pEXORq.

The last two connections among the six, namely pNOLASTq and pNOFIRSTq are
also complementary duals. The NO in the name of each indicates that they
are also denials of pLASTq and pFIRSTq by having opposite truth values.
They are often used by people who abstain from actual participation in the
discussion when trying to prevent falling into the traps of pFIRST q and
pLASTq. These people (called "lurkers") usually waits for the discussion
to repeat itself at a latter opportunity so as to participate. But when
that opportunity arrives, they still act according to pNOLASTq and

Finally there are four connections which each is decidedly "soft" in
truth. They are f[12] as pORq, f[13] as pNOFITq, f[14] as pIMPLYq and
f[15] as pNOANDq. We have discussed pORq sufficiently. The pNOFITq is the
denial of pFITq. It is sometimes also called the "converse implication".
In this sense it means that only when the second q as T leads to the first
p as F, it makes the connection false. The "converse" means that from the
second q the first p is implied. Thus it works in the converse direction
as pIMPLYq.

Looking at pIMPLYq, the basic idea here is to prevent making progress from
the first p as T to a second q as F false (the second line). Should the
argument then begin with an axiom or theorem p which is T, the conclusion
q will then also be T (the first line). The fact that the third and fourth
lines allow any conclusion q, true T or false F, as true. seems to worry
many. But should these two lines also be made false, we end up with pANDq.
Thus it is wise to view pANDq as a very "hard" inference rule and pIMPLYq
as its very "soft" counterpart in going from p to q. The "medium
hard-soft" counterparts of pANDq and pIMPLYq would then be pLASTq and
pEQUIVq which have their own peculiar drawbacks. The pIMPLYq is frequently
reformulated in language as IFpTHENq. The last two lines of pIMPLYq, i.e.
IFpTHENq play a powerful role in metaphoric speach. For exmple, the
sentence "If I were a cat, then you are the mouse which should flee"
refers to the forth line.

In the connection f[15] as pNOANDq the O is usually dropped to give the
better known "pNANDq gate" in digital technology. Much of digital
technology is based on hooking vast arrays of NOR and NAND gates together
to give LSI (Large Scale Integration) chips used in modern computers. The
pNOANDq connection was first elucidated in logics by W F Quine. Later on J
Nicod managed to show that all of Proof Theory which relies on pIMPLYq as
its initial rule of inference (stride to step from one stone to another)
can be reformulated by using pNOANDq in one initial inference rule and one
axiom (initial theorem). This reformulation of Nicod shows how complex
Proof Theory is when beginning with only one axiom and one inference rule.
Its almost like the complexity when arguing that all evolution can be
explained by beginning with LEC (Law of Energy Conservation) as "axiom"
and LEP (Law of Entropy Production) as "inference rule".

I want to invite fellow learners to draw the connnecting lines to the
following "lattice points". At the top is the most "hard" point f[01] as
pFALSEq. Then in the next level follow four "hard" points with f[02] as
pANDq, f[03] as pNOIMPLYq, f[04] as pFITq and f[05] as pNOORq. In the
middle level follow six "medium hard-soft" points as f[06] as pFIRSTq,
f[07] as pLASTq, f[08] as pEQUIVq, f[09] as pEXORq, f[10] as pNOLASTq and
f[11] as pNOFIRSTq. In the second lowest level are the four "soft" points
with f[12] as pORq, f[13] as pNOFITq, f[14] as pIMPLYq and f[15] as
pNOANDq. Down at the bottom level is the most "soft" point f[16] as
pTRUEq. The pattern of "lattice points" looks as follows:

. f[01]

. f[02] f[03] f[04] f[05]

f[06] f[07] f[08] f[09] f[10] f[11]

. f[12] f[13] f[14] f[15]

. f[16]

Make a screen-copy of it on a sheet of paper. Then make the
following connections to let the lattice diagram of the BBL crystal

Connect f[01], f[03], f[06], f[02] and f[01] to form a "diamond face".
Connect f[01], f[04], f[08], f[03] and f[01] to form a "diamond face".
Connect f[01], f[05], f[11], f[04] and f[01] to form a "diamond face".
Connect f[02], f[07], f[12], f[06] and f[02] to form a "diamond face".
Connect f[02], f[09], f[14], f[07] and f[02] to form a "diamond face".
Connect f[03], f[08], f[12], f[06] and f[03] to form a "diamond face".
Connect f[03], f[10], f[15], f[08] and f[03] to form a "diamond face".
Connect f[04], f[11], f[14], f[07] and f[04] to form a "diamond face".
Connect f[05], f[11], f[15], f[10] and f[05] to form a "diamond face".
Connect f[05], f[11], f[14], f[09] and f[05] to form a "diamond face".
Connect f[06], f[13], f[16], f[12] and f[06] to form a "diamond face".
Connect f[07], f[14], f[16], f[12] and f[07] to form a "diamond face".
Connect f[08], f[15], f[16], f[12] and f[08] to form a "diamond face".
Connect f[09], f[14], f[16], f[13] and f[08] to form a "diamond face".
Connect f[10], f[15], f[16], f[13] and f[10] to form a "diamond face".
Connect f[11], f[15], f[16], f[14] and f[11] to form a "diamond face".
Perhaps you will have to change the relative position of points in
a level to get a better perspective.

The result should form beautiful lattice diagram like that of a crystal.
You can also replace each f[##] with its infix name like pANDq for f[02]
so that the logical quality of each lattice point shines clearer. Observe
how the lattice points change from "hard" to "soft" as we go lower from
level to level. It is almost as if we can distinguish five systematical
levels in the lattice. But this is by far not the only systematical
abstractions possible in Binary Boolean Logic. We will leave the others,
some which are very abstaract and deep, for a later time otherwise this
contribution will become too complex and abstract.

What can we learn with this lattice diagram with respect to the
LO-dialogue? When a LOer stays fixed at a point in the diagram, the
contribution of that LOer tends to force the LO-dialogue into a
discussion. The more a LOer works in the middel level of the diagram, the
more it is forced into a debate and sometimes even a percussion. The
"hard" LOers tend to work in the upper half whereas the "soft" LOers tend
to work in the lower half. They often find it difficult to connect to the
"hard" LOers and vice versa because of having to jump several lattice
points. It is AS IF the "hard" LOers have an affinity for each other (hard
seeks hard) while the "soft" LOers have an affinity for each other (soft
seeks soft). It is AS IF "hard" and "soft" LOers do not have an affinity
for each other. But the AS IF is captalised to indicate fixed positions on
the lattice diagram. The whole LO-dialogue change when LOers are also able
to change from position to position, i.e. allow for becoming in their

In fact, the beauty of the LO-dialogue begins to unfold itself when the
LOers begin to move around in the lattice diagram and not remain fixed at
some lattice point. These movements are more obvious when they go along
the edges of "diamond faces". For example, a LOer may move from f[05] as
pNOORq in the second level (which is explicit distrusting) from the top to
the second level (which is implicit trusting) from the bottom to come
closer to a LOer working from say f[14] as pIMPLYq. It will be the case
when a creative non-conformist who often uses f[05] want to work with
somebody who often uses f[14] like a conformed mathematician. The
vertical movements (along diagonal face edges) between levels is often far
less erratic than horizontal movements (again by means of diagonal faces
edges) within a level.

Can some of you fellow learner see the connection to Soft Systems
Methodology? But this is not all. There is also a profound connection to
"hard-soft" chemistry! This complexifies even into "hard-soft"

I invite all fellow learners to study once again in our LO-archive, for
several topics, all the contributions to each topic. See of you can
picture in your mind how some participating LOers stay at a fixed point in
the lattice diagram while others move from point to point as the topic
requires. The ability to move from any point to any other point affords a
LOer to become aware of the viewpoints of all other LOers in the
LO-dialogue. Read once again Andrew Campbell's beautiful notes on the
LO-dialogue to see how they describe the becoming in the lattice diagram
of Binary Boolean Logic. See Dialogue - Notes LO24825
< >

Sometimes the cutting of a lattice diagram in a LO-dialogue so that all
its "diamond faces" become clear, gives rise to a crying much like that of
cutting faces to a physical diamond. The bigger the diamond, the more
audible the "crying" during the cutting of its faces. To Rick who host The
LO-dailogue, that crying may appear on occasions to be nerve breaking.

With care 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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