Aristotelian Logic

Aristotelian Logic

Aristotle (384-322 B.C.), a student of Plato and tutor of Alexander the Great, invented a system of logic which remained essentially unchanged in Western European philosophy for more than two millennia following his death. Although his predecessors, Socrates and Plato, had placed great emphasis on correct reasoning, Aristotle was the first philosopher to set forth a carefully worked out system, which, he believed, if followed, could eliminate false reasoning. Some have suggested that Aristotle created the "science" of logic, but, to Aristotle, logic was not simply one of the sciences; rather it was the art and method of correct reasoning, a prerequisite to the study of any science. Followers of Aristotle gave the collection of his writings on logic the name Organon, which means "tool." Logic, for Aristotle, was the principal "tool" for reasoning correctly.

At the heart of Aristotle's logic is the syllogism. Perhaps the most famous syllogism is the following: "All men are mortal. Socrates is a man. Therefore, Socrates is mortal." Aristotle introduced variables into logic to emphasize the general form of the syllogism. Thus the above syllogism has the form "All A is B. Some C is A. Therefore, some C is B." In this form, "All A is B" is called the major premise; "Some C is A" is called the minor premise; and "Some C is B" is called the conclusion. Note the use of the word "some" in this syllogistic form distinguishes it from the form "All A is B. All C is A. Therefore, all C is B." This points out the difference between what Aristotle called "particulars" and "universals." So in the first syllogism above, "Socrates" is the name of a particular member of a class, say, Greeks.

This syllogism "proves" only that one Greek, namely Socrates, is mortal; whereas, the syllogism "All men are mortal. All Greeks are men. Therefore, all Greeks are mortal," would be a syllogism purporting to "prove" that all Greeks, not just Socrates, are mortal. From a logical point of view, it does nothing of the sort, because the minor premise is false. Approximately half of all Greeks are not men. (Aristotle would not have made this objection because his use of "man" or "men" was meant to include women.) Using these and several other syllogistic forms, Aristotle thought that all deductive inference could ultimately be stated in terms of syllogisms. Thus he believed that the path to infallible reasoning consisted in stating arguments in one or more of the syllogistic forms and proceed from there according the laws of syllogistic reasoning. Aristotle's development of the syllogistic method of argument formed the basis for the teaching of formal logic until the beginning of the twentieth century when the mathematical logicians Giuseppi Peano (1858-1932), Gottlob Frege (1848-1925), Bertrand Russell (1872-1970), and Kurt Gödel (1906-1978) discovered numerous shortcomings in the Aristotelian system. The work of these men and others led to the establishment of modern symbolic logic as the cornerstone of mathematical deduction. Russell, in particular, criticized Aristotle's overemphasis on the syllogism as the fundamental form of deductive argument, pointing out that deductive proofs in mathematics are rarely given in syllogistic form. Russell also noted that Aristotle was often not careful about the distinction between particulars and universals in the statement of syllogisms, a problem symbolic logic addresses through the use of existential ("There exists an x") and universal ("For all x") quantifiers. Thus "All men are mortal" could be written as "For all x in the class 'men,' x is mortal," whereas "Socrates is mortal" could be expressed as "There exists an x in the class 'men' such that x is mortal and x is Socrates." Now it may seem that such strange statements complicate language rather than simplify it, but it turns out that introducing such innovations as quantifiers and classes was necessary to eliminate some troubling paradoxes in logical reasoning. Another modern criticism of Aristotle's logic is that it gave too much prominence to deduction and not enough to induction, although it should be noted that compared to his mentor, Plato, Aristotle did give strong credence to induction as a method of arriving at the "probable" truth of a premise such as "All men are mortal."

Although modern logicians may see Aristotle's largely syllogistic logic as merely a primitive beginning to their field, they cannot deny the authoritative position that Aristotelian logic held for more than two thousand years. It is difficult to overstate the enormous influence Aristotle's logic had on the generations of philosophers, mathematicians, scientists, and educators who came after him.