Origins: Quod Erat Demonstrandum

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When did people begin to think logically? There is no answer to this question. Rationality of some kind existed before even the earliest stone tools, since such objects must have been produced in the rational expectation that they would make certain tasks easier. As far back as we can imagine, human beings behaved rationally (which is not to say that they haven’t also behaved irrationally): It has always been “logical” to suppose that the sun would indeed rise again, that a rock thrown upward would fall to the ground, and that briny seawater would not make a very healthy drink.

Systematic or formal logic, on the other hand, is a different matter. It was invented by Aristotle (384–322 B.C.) in Athens, and recorded in a group of writings known as the Organon. In one of these works, the Prior Analytics, Aristotle attempted to provide a complete analysis of the valid forms of reasoning. His attempt was so successful that it remained the dominant strain in logic until the mid-19th century and is still taught in some universities today.

The Organon treats what is known as syllogistic logic (from sullogismos, meaning “deduction”). Aristotle discusses the valid forms of syllogism, of which the following argument is an example: “All humans are mortal, all Greeks are humans, therefore all Greeks are mortal.” This argument is valid in that the conclusion follows from the premises. Aristotle well understood, however, that valid arguments do not always make true statements about reality. The following syllogism, for example, is valid but its conclusion is not true: “All humans are immortal, all Greeks are human, therefore all Greeks are immortal.”

For our understanding of logic itself, it does not matter whether the premises and conclusion are true. Systematic logic is the study of arguments. It aims to explain what distinguishes good arguments from bad. Valid arguments with true premises are good arguments. They are a reliable source of truth because the truth of the premises guarantees that the conclusion is true. It is not up to logic to determine whether the premises are true (whether all humans are indeed mortal is not the responsibility of logic to decide), so logic focuses on the validity of arguments.

Aristotle noticed that a conclusion follows from its premises by virtue of the form of the argument. For example, “No humans are immortal, all Greeks are humans, so no Greeks are immortal” is a valid argument, but “All gods are immortal, no collies are gods, so no collies are immortal” is invalid, even though the conclusion is true. (Consider the parallel argument, “All humans are mortal, no collies are humans, so no collies are mortal,” which has the same form but a false conclusion.) The genius of Aristotle lay in his 008discovery that there are rules determining the validity of syllogisms. He also analyzed arguments involving circular reasoning, reductio ad absurdum and infinite regress; and he showed how to apply his logic to real-life situations, including debates (for example, he noted that a false conclusion cannot follow from true premises, but a true conclusion can follow from false ones).

What led Aristotle to invent this system? We would expect systematic logic to emerge at a time when importance was placed on arguing successfully, and when it made a difference whether an argument was valid. Early fourth-century B.C. Athens was such a place, in a number of ways.

In democratic Athens of the fifth century B.C., both political debate and legal procedures demanded that parties give reasons for their views and rebut opposing views. For the first time, rhetoric became explicitly a subject of instruction, and the teachers, known as Sophists, competed with one another for fee-paying students. One instructional technique required students to argue both sides of an issue, to construct effective arguments and to find holes in opponents’ arguments. In such contexts the goal was often victory rather than truth, and the Sophists were charged with making the weaker argument appear the stronger (a complaint still raised against lawyers). To determine the truth, the ancients needed to learn how to strip away irrelevancies and focus on the nature of proper argumentation. Aristotle tackled central parts of this task in the Topics and Sophistical Refutations, the two concluding works of the Organon.

In the natural sciences, too, it was important to be able to identify good arguments. Such topics as the origins of the world, the basic physical composition of reality and the nature of weather phenomena could not be settled by direct observation; such things could only be explained by forming theories, which would then be subject to scrutiny, criticism and revision. What was needed was an objective way to determine whether claims were true or false.

Philosophy offered some hope. The early fifth-century B.C. philosopher Parmenides, for example, held that reality consists of a single unchanging substance, which has no beginning or end; that the senses are totally unreliable as guides to reality; and that the world we think we know is sheer illusion. Parmenides did not reach this view by inspecting the world he could see and touch; he did so by reasoning. He developed his doctrine by drawing deductions from a principle he judged incontrovertible: That which is, is. And he grasped the essential point that if a valid argument is based on true premises, then its conclusion—no matter how unpalatable or contrary to our expectations and beliefs—must be true. Even today, philosophers debate the meaning of Parmenides’s principle (“That which is, is”) and the validity of his deductions, but the important thing for us is that he faced the question of what to do when reason conflicts with the senses, and gave the only rational answer—reason must prevail. The question then becomes, How do we decide whether an argument is rational?

Finally, mathematics. Although the Greeks were by no means the first people to develop a sophisticated mathematics, they made the important contribution of the concept of proof. At least as early as the fifth century B.C., we find the beginnings of a tendency to demand not only correct results but results (theorems) that are proved correct. A famous example is the theorem that a square’s diagonal is incommensurable with its side (the length of the diagonal equals the length of the side multiplied by the square root of 2, which is an irrational number). This conclusion could not have been reached by observation, only by proof, and in fact more than one ancient proof of this result is preserved.

Not content with a collection of isolated proofs, fifth-century B.C. mathematicians set the foundations of the concept of an axiomatic system in which some true propositions, the principles, are laid down as unprovable and the rest, the theorems, are systematically deduced from them. The most famous example of such a system is Euclid’s Elements (c. 300 B.C.). The successes of mathematics were magnificent in comparison with those of natural science; mathematicians agreed not only about which claims were true and which were false, but also on the need to prove results and on what constitutes proof.

It is plausible, then, that Aristotle, inspired by current work in mathematics, decided to develop such a system for logic. In the Posterior Analytics, the work that immediately follows the Prior Analytics in the traditional arrangement of the Organon, he outlined a model of science as a deductive system based on unprovable principles, strongly resembling the system of geometry developed by Euclid a generation later. To make good this claim, Aristotle needed to give a thorough account of deduction, to show how to distinguish valid from invalid arguments. Thus the birth of formal logic.

When did people begin to think logically? There is no answer to this question. Rationality of some kind existed before even the earliest stone tools, since such objects must have been produced in the rational expectation that they would make certain tasks easier. As far back as we can imagine, human beings behaved rationally (which is not to say that they haven’t also behaved irrationally): It has always been “logical” to suppose that the sun would indeed rise again, that a rock thrown upward would fall to the ground, and that briny seawater would not make a very healthy […]