The Historical 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.

Pythagoras (ca. 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)