Roots of Logic
People have always
reasoned. Even early civilizations made technical and intellectual
advances that presupposed logical inferences. For example,
the Babylonians discovered and used a theorem (now called
the Pythagorean theorem) that expresses the relation between
the sides and the hypotenuse of right triangles, and the Egyptians
used the formula for calculating the volume of a rectangular
pyramid 'h(a^2 +ab + b^2)/3'. Such results cannot be obtained
without reasoning. However, neither civilization sought formal
proofs for mathematical insights; formulas were regarded only
as recipes or rules of thumb that could be followed to calculate
desired numerical values. In these early cultures, there was
no interest in proofs or arguments and so there was no interest
in general criteria for the correctness of arguments.
532 B.C.), a Greek, was the first historical figure known
to have been interested in the construction of mathematical
proofs. As a young man, Pythagoras traveled widely; he
studied firsthand the mathematical results of the Egyptians
and Babylonians. Later, he established a semireligious brotherhood
whose members were concerned primarily with discovering and
proving mathematical theorems. Members of the brotherhood
probably were the first to prove the Pythagorean theorem,
and they also may have discovered the first proof that the
square root of two is irrational.
The Pythagoreans were not the only Greeks
who were passionately interested in arguments and proofs.
The early Greek philosophers are generally distinguished
from the sages, prophets, and oracles of other early Middle
Eastern civilizations precisely in that they sought to support
their opinions with reasons. An outstanding early example
was Zeno of Elea (ca. 490-430 B.C.), who constructed numerous
brilliant arguments against the philosophical doctrines of
his opponents. Some of Zeno's most interesting arguments were
directed against philosophical beliefs connected with the
mathematical theories of the Pythagoreans. In spite of their
interest in mathematics, the Pythagoreans lacked an efficient
system of numerals--they represented numbers by groups of
dots similar to those found today on dominoes and dice. Use
of this notation was probably connected with their failure
to distinguish sharply between units of arithmetic, geometic
points, and physical atoms. . . .
Another of Zeno's arguments, one that continues
to receive attention even in our own time, is intended to
prove that motion is impossible. As background to this
argument, suppose a runner were to advance from a starting
point S to some goal G. In order to attain G, he must reach
the mid point between S and G, which we can call M. Then,
in order to get from M to the goal G, he must reach the midpoint
between M and G, which we can call N. But to get from N to
G he must reach the midpoint of this segment, which can call
O. This can be repeated endlessly. Therefore, we have an infinite
sequence of intervals, each of which must be traversed for
the runner to reach his goal. . . .
In this second argument, Zeno employs a strategy
similar to that found in the first. He supposes the runner
reaches the goal and shows that this leads to impossible consequences.
From this he concludes that the runner could not reach the
goal. But, since the runner can be taken to represent any
object moving from one point to another, it follows that no
motion is possible.
These arguments are similar in structure:
each begins with a supposition from which we deduce an impossible
consequence (in the first case a contradiction), we then conclude
that the original premise is false. This is a common and
powerful form of reasoning that is important in the writings
of Zeno's successors; Plato used it occasionally in his dialogues.
Here is an argument with essentially the same form as (3:1)
that Origen attributed to the Stoics: "If you know you
are daed, you are dead (since if you know something, it must
be true). If you know you are dead, you are not dead (since
to know something you must be alive). Therefore, you do not
know that you are dead (Benson Mates, Elementary Logic).
From the time of Zeno to the present day,
countless mathematical proofs have been constructed with this
form. Aristotle credited Zeno with the discovery of what
he called "dialectic." Aristotle may have meant
that Zeno was the first to recognize the general form common
to a whole class of arguments. In fact, Zeno is known to have
constructed approximately fifty different arguments, most
of which employ strategies similar to those of (3:1) and (3:2).
Other persons before Zeno probably constructed analogous arguments,
but he may have been the first to become aware of the general
form. Since logic deals with criteria for the correctness
of arguments, and since the correctness of arguments is largely
a function of their form, Zeno's work as a major contribution
in preparing the way for the study of logic.
Another group of persons who helped initiate
the study of logic was the Sophists. These men sustained
themselves throughout ancient Greece by teaching for pay.
They were individualistic, skeptical, and critical of traditional
values. To attract students, many engaged in public debates;
some resorted to theatrical tricks and to exaggerated claims.
Plato and Aristotle disliked their commercialism and deception,
but many Greeks regarded them favorably. Their opinions were
sought in ethics, law, and politics.
(Carter, Codell K. Excerpts from The
Hottest Logic Book on Earth. Pages 113-118)