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In the
final section we present the mathematical structure of three valued logic
and explain how it is used to attain a logical understanding of the
revelation of Atzmus in the world and the coexistence of opposites that it
entails. Real world applications to computers and physics are described.
Part IV:
Three Valued Logic and the third Redemption
Earlier
we discussed confrontations between emunah and intellect. Regarding the
world order of the Geulah a similar question arises: We have been speaking
of phenomena that are not only beyond the limits of nature (such as
miracles) but also beyond the limits of logic - the combination of the
natural with the miraculous, the coexistence and simultaneous functioning
of opposites etc. Must we rely on emunah to accept these matters or can we
also understand it with our intellect? In other words, must we accept it on
faith or can we open our eyes and see it?
In fact,
the coexistence of opposites in the Era of Moshiach applies not only to the
order of nature but also to intellect and logic itself as the Rebbe, Melech
HaMoshiach says, "The intellect itself will be permeated and filled
with the knowledge of the level of G-dliness which is higher than the
limitations of intellect," [1] so that the paradoxical coexistence of
the natural and the miraculous (the infinite and the finite etc.) will not
only be experienced in the Era of Moshiach but will also be clearly
understood by the intellect. Atzmus itself will be comprehended:
"Within the realm of understanding and comprehension, there will be
the comprehension of the essense of the infinite Atzmus."
Earlier,
we discussed the Sicha of the Rebbe, Melech HaMoshiach, which explains how
the revelations in the sciences since the year 5600 (1840 c.e.) prepare the
world for the revelations of the Era of Moshiach. So it is natural to ask:
Has anything been discovered in mathematics or science that prepares us for
an intellectual understanding of the concept of the coexistence of
opposites? The answer, quite surprisingly, is "Yes!"
Three
Valued Logic
The story
starts shortly after the fountains of wisdom from below opened up. In the
year 1847, around the time that Likutei Torah was published, [1a] a British
mathematician named George Boole published a paper explaining that logic
has an algebraic structure. Thus began the development of a new branch of
mathematics known as mathematical logic. [2] In the same year, a
mathematician named De Morgan, who was working on mathematical logic,
announced that he felt that a third "truth value" should be added
to the structure of logic. This means that in addition to the usual two
truth values "true" and "false" (i.e. the
characterization of statements as being either true or false), which
classical logic had been limited to, there should be a third truth value
characterizing a third state. However, he did not know exactly how to develop
this idea and it remained a vague concept until the 1920's.
In the
early 1920's a Polish mathematician named Lukasiewicz developed this
concept in a very precise mathematical way. The classical truth values
"true" and "false" were represented by the numbers 1
and 0, respectively, and the third truth value, which could represent
"true" and "false" simultaneously, was represented by
1/2. This new system retained the ability to represent all the information
that classical logic did while having the new capability of representing
also the coexistence of opposites, contradictions and paradoxes. It became
known a Three Valued Logic (3VL).
It is
interesting to note that around the same time the New Quantum Mechanics was
being developed and it was found that classical logic was no longer adequte
to deal with physical phenomena. Specifically, it was found that in quantum
systems the distributive law of logic no longer held. This stems from the
fact that in a quantum system, observables are represented aby self-adjoint
operators instead of functions. The failure of the distributive law of
classical logic to hold is a result of the noncommutativity of matrix
multiplication. A new Quantum Logic was thus developed for Quantum
Mechanics. [3]
It is no
accident that in this era not only were new theories of physics being
developed but also new systems of logic were atually being developed! We
already know that all the developments in the sciences since 1840 are
intimately connected with and actually derive from corresponding
developments in Chassidus which prepare the world for the knowledge of
Hashem in the Era Moshiach. An extraordinary event must have occured in the
World of Chassidus making a breakthrough in preparing the world for the
Geulah of Moshiach in order for there to have been the parallel event of
the development of new systems of logic in the world of science. This event
was, of course, the Geulah of the Previous Rebbe on Gimmel Tamuz and 12-13
Tamuz, 5687 (1927). As the Rebbe, Melech HaMoshiach, writes concerning this
Geulah, "It was a revealed miracle higher than nature. Nevertheless,
the miracle was clothed in nature...in a way that nature did not hide the
miracle....Since in this Geulah there was the revelation of the infinite
Atzmus...the Geulah was in a manner that the miracle and nature fused
together." [4] "Through this, the barrier between the lower and
the higher [worlds] was nullified." [5] "Through the fusion of
the miraculous and the natural that occurred in the Geulah, the power was
given to all those who follow the ways of the one who was redeemed [the
Previous Rebbe] that the Emunah which is higher than intellect (the
miraculous) be drawn down also into intellect (the natural), in a manner
that the intellect will not hide the Emunah." [4] "Through this
the power was then given to every Jew to nullify the barrier between his
dealing in worldly matters and the observance of Torah and Mitzvos."
[5] This was the formal end to the conflict between emunah and intellect -
between Torah and science - that we have been discussing! The Rebbe, Melech
HaMoshiach, concludes, "This becomes an immediate preparation for the
fulfillment of the prophecy that 'The glory of Hashem will be revealed and
all flesh will see uniformly that the mouth of Hashem has spoken,' that the
flesh will see G-dliness (not just because of the revelation of Hashem's
glory, but also) from the flesh itself." [4]
Three
valued logic provides for us a mathematical-scientific framework within
which to describe and discuss phenomena of the Geulah that result from the
revelation of Atzmus and thus involve the coexistence of opposites. The
first thing to note is that - now that the barrier between the lower and
the higher worlds is removed - the coexistence of opposites that we have
been discussing is a standard part of scince and mathematics. Most
recently, as mentioned above, it was realized that the very stability of
matter involves the coexistence of opposites, the infinite power of Hashem
in the finite atom. This was always part of the world order even though it
was unknown. Now that it's known, it's also part of science i.e part of
intellect. In the next section we outline the structure of three valued
logic and describe one of its most common everyday applications -
computers.
A Short
Course in Three Valued Logic
In
mathematical logic statements are represented and analized as follows: We
consider first an initial set of statements, called atomic sentences. This
is the collection of the basic statements from which more complex statements
are constructed. The atomic sentences are represented by symbols A, B, C
etc. Algebraic operations are then defined on these symbols forming new,
more complex sentences. The operations are defined in such a way as to
correspond to common logical reasoning. We will define four operations
which will correspond to the concepts of negation ("not"),
conjunction ("and"), disjunction ("or"), and
implication ("if...then").
For
example, let's consider two atomic sentences: "The light is on"
represented by the symbol A, and "The bell is ringing"
represented by the symbol B. The four operations and their meanings are
listed below:
Negation:
-A "The light is not on"
Conjunction:
A/\b "The light is on and the bell is ringing"
Disjunction:
A\/B "The light is on or the bell is ringing"
Implication:
A-->B "If the light is on then the bell is ringing"
Now, any
of these statements may be either true, in which case it is assigned the
value 1, or false, in which case it's assigned the value 0. The truth value
(1 or 0) of a complex statement will depend on the truth values of the
atomic sentences. We must now define how the truth value of a complex
statement is determined based on the truth values of the atomic sentences.
For example, consider the statement above "The light is on and the
bell is ringing." If it is true that "The light is on" but
false that "The bell is ringing" then the compound sentence
"The light is on and the bell is ringing" is false. Symbolically
we are saying that if A is true and B is false then A/\B ("A and
B") is false. Stated mathematically, when A has the truth value 1, and
B has the truth value 0, the conjunction A/\B has the truth value 0.
It should
be clear that this is a property of the abstract structure of these
operations and is independent of the statements which the symbols represent
i.e. A being true and B being false will result in A/\B being false no
matter what sentences A and B represent. The truth values for statements
formed by negation, conjunction, disjunction and implication are usually
summarized in tables called "truth tables". (See figure 1).
A | -A A
| B | A/\B A| B | A\/B A | B | A --> B 1 | 0 1 | 1 | 1 1| 1 | 1 1 | 1 |
1 0 | 1 1 | 0 | 0 1| 0 | 1 1 | 0 | 0 0 | 1 | 0 0 | 1 | 1 0 | 1 | 1 0 | 0 |
0 0 | 0 | 0 0 | 0 | 1
figure 1
We note
that there are certain statements that are always true such as A\/(-A)
which means either A is true or the negation of A is true, and -(A/\-A)
which means that it is not true that both A and its negation are true. Here
too, it makes no difference what statement A represents. These statements
are called tautologies.
The
innovation in three valued logic is the introduction of a third truth
value, 1/2, to represent the notion of being both true and false
simultaneously. For this concept to be logical, truth values must be
assigned to complex sentences in a consistent manner and in such a way that
the system of three valued logic is an extension of two valued logic. This
means that in a case where the third value 1/2 is not used, the three
valued system gives the same results as the two valued system. This is in
fact the case as can be seen from the truth tables for three valued logic
given in Figure 2.
A/\B A\/B
\ B | 1 |
1/2 | 0 \ B | 1 | 1/2 | 0 A \ | | | . A \ | | | . 1 | 1 | 1/2 | 0 1 | 1 | 1
| 1 1/2 | 1/2| 1/2 | 0 1/2 | 1 | 1/2 | 1/2 0 | 0 | 0 | 0 0 | 1 | 1/2 | 0
-A A
--> B
A | -A \
B | 1 | 1/2 | 0 1 | 0 A \ | | | . 1/2| 1/2 1 | 1 | 1/2 | 0 0 | 1 1/2 | 1 |
1 | 1/2 0 | 1 | 1 | 1
figure 2
What
about tautologies in 3VL? This is where things get interesting and the
results are surprising. Firstly lets consider the tautologies of the two
valued system and see if they are still tautologies in the three valued
system. Remember that to be a tautology it must always be true i.e. its
truth value must always be 1. As Figure 3 shows, when A is assigned the
value 1/2, the statement A\/(-A) has the value 1/2 so it is not a
tautology. Thus in this system it is not valid to say that either A or its negation
will always be true.
A | -A ||
A\/-A | A/\-A | -(A/\-A) 1 | 0 || 1 | 0 | 1 1/2 | 1/2 || 1/2 | 1/2 | 1/2 0
| 1 || 1 | 0 | 1
figure 3
Figure 3
also shows that when A has the truth value 1/2, the statement -(A/\-A) has
the value 1/2 so this statement is also not a tautology. In this case we
notice that the contradictory statement A/\-A also has the value 1/2 which
shows that contradictions are not rejected in this logical system!
Now it
gets even better. We show that there is a contradictory statement in 3VL
that actually has the truth value 1, i.e. it is a tautology. Consider the
statement (A --> -A)/\(-A --> A). This means that "A implies the
negation of A and the negation of A implies A," or, to use the
terminology of logic, "A is true if and only if the negation of A is
true," a very contradictory statement. Yet the truth table in Figure 4
shows that when A has the value 1/2, this statement has the truth value 1,
meaning true!
A | -A ||
A --> -A | -A --> A | (A --> -A)/\(-A --> A) 1 | 0 || 0 | 1 | 0
1/2 | 1/2 || 1 | 1 | 1 0 | 1 || 1 | 0 | 0
figure 4
Now,
let's symbolize this contradictory statement by the symbol C (for
contradiction). We now show that there is a tautology in the three valued
system which is built on C. The statement A\/(-A)\/C is a tautology as
shown in Figure 5.
A | -A ||
C | A\/-A\/C 1 | 0 || 0 | 1 1/2 | 1/2 || 1 | 1 0 | 1 || 0 | 1
figure 5
We
already saw that the statement A\/-A, a tautology of two valued logic, is
not a tautology in three valued logic. Now we see that by combining it with
the contradictory statement C, we have a tautology, a statement which is
always true! This tautology says that in three valued logic it is always
true that either A or the negation of A or the contradictory statement C is
true.
We went through
the details of this to show that three valued logic is in fact logical and
consistent. We now go even further and show that this is not just an
abstract concept but actually has down to earth applications every day in
computers.
Aplications
of Three Valued Logic in the Real World
When
computers were put to the task of dealing with very large amounts of data,
it was found that the way in which the data was organized, or
"modeled", was an issue of great importance which would affect
the efficiency and accuracy of the processing of the data. The best and the
most popular model, called the "Relational Model", was introduced
by the mathematician E. F. Codd in 1970. It organized data into related
tables. It had a very solid foundation in theoretical mathematics so that
any question regarding the model could be dealt with rigorously i.e. by
making mathematical statements which could be proved or disproved. [6] Then
a language had to be developed to interact with this model. A language
based on mathematical logic, called the "Structured Query
Language" or SQL, was developed at IBM in the 1970's. SQL is still the
most important tool for defining and manipulating relational databases. [7]
SQL is
based on three valued logic. The third truth value, instead of being used
to represent contradictory information, is used to represent unknown
information. <8>For example, suppose we are dealing with a database of
employees of a company and the computer program must evaluate the truth
value of the statement: "(profession is engineer)/\(age is greater
than 65)." If both parts of the statement are true then the whole
statement is true; if one part is true and the other is false then the
whole statement is false in accordance with the rules of /\
("and") in both two valued and three valued logic. But what if
part of the information is unknown? What if it is known that the profession
is engineer but the age is unknown? Since SQL is based on three valued
logic, the unknown information "(age is greater than 65)" is assigned
the value 1/2. Then, in accordance with the rules for /\ ("and")
in three valued logic (in figure 2) the whole statement is assigned the
value 1/2 and the program continues. It is an amazing fact that the
computers we use every day for our information needs use three valued logic
to process the data! But when we consider the fact that we are now in the
Era of Moshiach when Atzmus is out in the open, it is not so amazing that
three valued logic is part of our daily lives.
Another
application of three valued logic in the real world is switching theory
(electronic circuits). However, we will not discuss the details of it. We
conclude this discussion with a fascinating example from Quantum Mechanics.
Actually,
Quantum Mechanics has shown that at the foundation of physical reality - at
the microscopic level of atoms and subatomic particles - paradoxes and
contradictions are commonplace. Now, in the Era of Moshiach, the
paradoxical aspects of this hidden level of reality are now coming out in
the open.
Much of
the paradoxical nature of the world apparant at the microscopic level is
due to the fact that at this level, all information about a particle cannot
be given precisely. This is expressed by the well known "Uncertainty
Principle." The information must be given in terms of probabilities
and the determination of which of the possible events occurs is made by the
observer of the event! The relationship between this concept and the
teachings of Chassidus has been analyzed and written about extensively by
Professor Yirmiyahu Branover. [9]
What is
the nature of the reality, i.e. what is the state of the particles
involved, before the observer views the event? This question was debated
openly by two Jewish physicists, Albert Einstein and Neils Bohr, in the
early years of the development of Quantum Mechanics. The opinion that
prevailed, that of Neils Bohr of Copenhagen, came to be known as the
"Copenhagen Interpretation" of Quantum Mechanics. According to
this view, two contradictory mutually exclusive possibilities can exist
simultaneously in reality until the observer makes his decision as to how
he will observe the event (eg. as a particle or a wave, spin up or spin
down, etc.). [10]
Within
the past year this level of reality has been brought out into the open. A
team of physicists working with Dr. C. Monroe at the National Institute of
Standards and Technology in Boulder, Colorado, succeeded in getting an
entire atom to exist simultaneously in two widely separated places. As
reported in the New York Times, [11] "The real significance of the
institute's feat, Dr. Monroe said in an interview, is that the two states
of the same atom were not only pulled apart but were separated by a
relatively enormous distance - a distance large enough to represent a
transition from the domain of quantum mechanics to the everyday
world...." But for us the significance of the event is the appearance
in the everyday world of an aspect of physical reality involving a
contradiction (and thus requiring three valued logic to deal with it) - an
event in line with the current Era of Moshiach when Atzmus (and coexistence
of opposites) is out in the open.
We see
from the above examples that the mathematical structure that serves as a
model for the revelation of Atzmus in the world, while not so long ago it
was a mystery even to the mathematicians, now, in the Era of Moshiach, it
is part of everyday math, science and technology.
We have
presented a logical structure within which we can understand the presence
of Atzmus in the world. But any logical structure is finite so its
usefulness as a model for the presence of the infinite Atzmus in the world
is limited. As the Previous Lubavitcher Rebbe wrote, [12] "The
greatest understanding has a limit. But...belief is a feeling without a
limit. Melech HaMoshiach will enable us to understand the greatness of the
sincere, earnest service of Hashem."
Where
logic leaves off emunah (belief) takes over. In fact emunah must be the
foundation upon which any logical structure must be built. Rabbi Levi
Yitzchok Schneerson, the father of the Rebbe Melech HaMoshiach, explains
this in a letter to the Rebbe Melech HaMoshiach and Rebbitzin Chaya Mushka.
Rabbi Levi Yitzchok writes that the Etz HaChaim (Tree of Life) and Etz
HaDaas (Tree of Knowledge) refer to emunah and intellect. Emunah is life as
the Prophet Habakuk says, 'The tzadik lives through his emunah.' "If
the emunah preceeds the knowledge ...then the knowledge is also
life....These two trees must be joined together....This is what it says [in
the prophecy of Yeshayahu] about the Era of Moshiach - that 'Emunah will be
the belt of his body.' [Only] at the end [of the prophecy] does it mention
knowledge - 'The earth will be filled with the knowledge of Hashem.' "
This
gives us guidance and direction in living a life of Geulah. Our foundation
must be the sichos and statements of the Rebbe Melech HaMoshiach that
declare that we are in the Era of Moshiach and that the Geulah is already
here. Based on this foundation we must refine our perception and deepen our
understanding to look at and analize the world in a new way. We realize
that everything that is happening in the world and in our personal lives is
part of the unfolding of the Geulah and we come to understand the world and
ourselves differently. We see Geulah in the world and we develop a Geulah
mentality. In this way we truly live a Geulah life and our "knowledge
is also life."
"Yechi
Adoneinu Moreinu V'Rabeinu Melech HaMoshiach L'Olam Voed!"
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NOTES:
1.
Hadranim on the Rambam and Shaas, p. 33 1a. This was preceeded by the
printing of Torah Or about 10 years earlier. The reader may get the
impression that the "exotic" three valued logic is from Tohu
while the "normal" two valued logic is from Tikun. However, from
a ma'amar in Torah Or (p. 24d) it is evident that the reverse is true. This
is in line with the concept that Tikun has "many kelim" (three
values instead of just two.)
2. See
Pesach (Paul) C. Rosenbloom, The Elements of Mathematical Logic, Dover
Publications, New York, 1950, especially ch.2, sec. 4 on many valued
logics. See also L. Bolc and P. Borowick, Many Valued Logics 1: Theoretical
Foundations, Springer-Verlag, 1992. Our own discussion is based on Richard
L. Epstein, The Semantic Foundations of Logic - Propositional Logics, 2ed,
Oxford University Press, New York, 1995, sec. VIII: Many Valued Logics.
3. P.
Bamberg and S. Sternberg, A Course in Mathematics for Students of Physics,
vol. 2, Cambridge University Press, pp.831-835, "Quantum States and
Quantum Logic". See also the section "Quantum Logic" in Yu.
I. Manin, A Course in Mathematical Logic, Springer, pp. 82-95
4.
Kuntres Chag HaGeulah 12-13 Tamuz 5751 in Sefer HaMa'amorim Melukat vol. 5,
p. 325.
5.
Kuntres Gimmel Tamuz 5749 in Sefer HaMa'amorim Melukat vol. 3, p.207
6. For a
mathematical discussion of the Relational Model see Dan A. Simovici and
Richard Tenney, Relational Database Systems, Academic Press, 1995, ch. 2.
7. ibid.
ch 3. Many books have been written on SQL in recent years. See for example,
J. Hursch, SQL The Structured Query Language, 1st ed, TAB Professional
Books, Blue Ridge Summit, PA, 1988, especially ch. 11, "Logic and
SQL."
8.
Simovici and Tenney, Relational Database Systems, pp. 104-106. The use of
3VL to represent unknown information in SQL is currently a matter of lively
discussion in computer journals. Many articles in the journal
"Database Programming and Design" over the past few years have
dealt with this.
9. See
Rabinowitz and Branover, "The Role of the Observer in Halacha and
Quantum Physics" in Fusion - Absolute Standards in a World of
Relativity, B'Or HaTorah Publications and Feldheim Publishers, Jerusalem,
1990, pp. 91-109.
10. Robert
Eisberg and Robert Resnick, Quantum Physics of Atoms, Molecules, Solids,
Nuclei and Particles, 2nd ed, John Wiley & Sons, New York, 1985, pp.
77-80.
11. The
New York Times, Late Edition - Final, Tue. May 28, 1996, sec. C Science
Desk, p. 1, "Physicists Put Atom in Two Places at Once." See also
The New York Times, Late Edition - Final, Sunday June 2, 1996, sec. 4, Week
in Review Desk, p. 16, and Science News, vol. 149, May 25, 1996, p. 325,
"Two Atoms in One."
12.
Quoted by the Rebbe Melech HaMoshiach in HaYom Yom of Hey Teves. 13.
Likutei Levi Yitzchok, Igros Kodesh (p. 413-14, quoted in Sefer HaSichos
5750, p. 629)
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