ARISTOTLE'S LOGIC

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Near the end of a work now called Sophistical Refutations, Aristotle apparently claims to have created the subject of logic. It seems probable that Aristotle's claim is true, although it is, nevertheless, possible for the historian to find all kinds of hints and anticipations of it in the works of earlier thinkers. For example, Plato makes the following statement in the Republic, "The same thing cannot ever act or be acted upon in two opposite ways, or be two opposite things, at the same time, in respect of the same part of itself, and in relation to the same object". Aristotle claims that the most certain of all principles is that "the same attribute cannot at the same time belong and not belong to the same subject and in the same respect". This latter principle is Aristotle's formulation of the law of non-contradiction, and it is tempting to say that Aristotle received not only this law, but many of his ideas on logic from his predecessors. Nevertheless, one should resist this temptation because Plato makes this remark only in passing and there is no evidence that he or anyone else before Aristotle, attempted to codify the rules of correct inference. Thus, we may accept Aristotle's claim and ask what led him to create the subject of logic.

"All men by nature desire to know," Aristotle tells us in the famous opening sentence of the Metaphysics. Both he and Plato believed that philosophy begins in wonder, and there can be little doubt that this motive was strong in Aristotle's logical investigations. Yet it does not seem that this was the only or even the most pressing motive. Rather, two other related but more practical aims were involved, one having to do with maths and the other with sophisms. Maths developed in a number of significant ways between the time of the achievements of the early Pythagoreans and the time of Aristotle. However, from a logical point of view, nothing really new was added to the proof and disproof procedure of the early Pythagoreans. But maths was not the only area which stimulated the development of logic; arguments in philosophy and the law courts did so.

Ancient Athens of Aristotle's time provided a great variety of view points and thinkers. Some of the thinkers were from Athens, but many were from Greek colonies; either they would visit Athens or reports of their views would be brought by their disciples. Further, there were extant writings or oral traditions of a philosophical heritage that even then was well over 200 years old. In order to refute  the arguments of the sundry sophists and philosophers whose conclusions Aristotle found either false or paradoxical, he tried to devise a set of principles by which one could determine whether any given argument is a good one. Zeno of Elea was a typical example of a pre-Socratic whose arguments Aristotle tried to refute. According to Plato, Zeno "has an art of speaking by which he makes the same thing appear to his hearers like and unlike, one and many, at rest and in motion". We have no extant writings of Zeno, and it is even possible (although unlikely) that he wrote nothing at all. Nevertheless, it is clear Zeno devised a good number of puzzles which have philosophical interest. Commentary and criticisms of these paradoxes appeared early and the literature is still growing at a good pace.

However, another play form developed in which there was a game whose object was to defeat one's opponent by words. Perhaps, Zeno has an important influence in the development of it. We do know that there existed a class of teachers who came to be known as sophists. These sophists would travel much like wandering minstrels, and for a fee would teach their students how to speak persuasively on many different kinds of topics. Sophists were also prepared to defeat any opponent in a public argument. The competitiveness of such a spectacle must have been very keen and the arguments often dramatic, so that we can understand why the arrival of an important sophist in town was the occasion of much excitement and why sophists were often able to command large fees.

We may now summarize Aristotle's motivation in inventing logic. First, there is the desire to know the truth about the nature of argument, an intellectual curiosity which needs no further account or justification. Second, there is the desire to know the conditions under which something is proved. This question was perhaps most clearly focused in the case of geometry: How are we going to decide when a math relationship really holds? But the problem was wider and was also, in metaphysical questions, acute. Zeno's arguments provide a good example of the latter. Third, there is the desire to refute opponents. Here there is perhaps an analogy with the invention of probability theory, which was initiated when Chevalier asked Pascal to solve certain problems having to do with odds in gambling. But probability theory is vastly more comprehensive, applying to many, many different areas, including all physical and social sciences. Similarly, logic is vastly more comprehensive and useful than merely a device which may be used to show that an opponent is wrong. Yet we should not overlook the egoism and spirit of competitiveness which marked its origin. Aristotle (384-322 B.C.) must have been a man of almost boundless energy; the range of his intellectuality is astounding. He contributed in important ways to biology, physics, astronomy, political theory, and ethics. His achievement in logic was just one of many others, and even if it were all wrong, we would, nevertheless, consider him an intellectual giant. But just because of the great diversity of his interests, as well as his conviction that logic is an art or a tool, it is unlikely that he would approve of our considering just this aspect of his thought. Indeed, his logical theory is embedded in a number of other related methaphysical doctrines which, is to be considered comprehensively, would have to be discussed in detail.

Although Aristotle did not give a definition of argument, it is clear that he meant by it what we mean; namely, a set of propositions of which one is claimed to follow from the others. That is, the one is claimed to be true if the other proposition or propositions are also true. The proposition which follows from the otner proposition or propositions is called a conclusion that from which it follows is the premise ox premises. Now a valid argument is one in which if the premises are true, the conclusion must necessarily also be true. An invalid argument is any argument which is not valid. The validity of an argument is in general independent of the truth or falsehood of the premises. It is perfectly possible for a valid argument to have a false conclusion and for an invalid argument to have a true conclusion.

The distinction between valid and true and invalid and false was understood by Aristotle, but its first clear and accurate formulation was made by the Stoics. A similar statement must be made about the distinction between a sentence and a proposition. If one refers to the fifth postulate of Euclid, one does not generally mean something in the Greek language. Consider an English sentence. Translate it into a dozen languages. We could then have thirteen sentences, but only one proposition. A proposition is what is expressed by a sentence. Aristotle was the first to use symbols in logic. In geometry it was the triangle qua triangle that was important and the geometer had to abstract from the trian¬gle he drew by ignoring any of its particular conditions. It can be easily seen that it is not the content of a proposition that is important in logic but the form. Apparently, early in his career Aristotle followed a similar procedure in logic. At some time, however, he came to see that he could use a letter to stand for any term. This practice was, perhaps, suggested by the use of letters for points, lines, triangles, etc. in geometry. In any case, this procedure not only reveals the structure of the proposition more clearly but it is also more convenient to use. Thus Aristotle began using what are now called propositional forms: "Every В is P", "No S is /"', etc. The term "formal logic" arose to designate the study of the various kinds of relationships which these forms might have.

The kind of inference of Aristotle came to be called immediate presumably because the inference depended on only one premise, and thus the conclusion followed immediately without the introduction of any other premise. Aristotle's main contribution to logic was not this theory of the immediate inference, but rather the theory of the syllogism, which involves mediate inference, that is, an inference depending on more than one premise. At first Aristotle understood a syllogism to be any argument, but later he took it to mean an argument with two premises and a conclusion. He concentrated most of his attention on what later came to be known as categorical syllogisms.

There are two aspects of Aristotle's logic which should be mentioned, if only in passing. The first is that Aristotle developed a theory of modal logic in connection with his theory of the syllogism. Modal logic concerns inferences involving such notions as necessity, contingency, and possibility. Aristotle's theory of mo¬dal logic is quite complex and much of it is wrong. It represents, however, an attempt to deal with notions that any logic which aims at completeness must encompass. The other aspect of Aristotle's theory is his theory of definition. "A definition," Aristotle tells us, "is a phrase signifying a thing's essence." Objects have properties that are both essential and accidental, but only the former enter into a definition of an object. A definition, as Aristotle pointed out in a number of places, does not assert the existence of the thing defined. This must be either assumed or proved. All in all, Aristotle's logic is a magnificent achievement; he started with virtually no predecessors and invented a theory which today is considered in many respects right and even complete. If from today's vantage point it also seems limited, it must be remembered that the discovery of its limitations is a rather recent achievement, and that it was 2000 years before anyone besides the Stoics made substantial progress in formal logic. There are really only two further developments in Greek maths which turned out to be important for logic:

The first is the systematization of geometrical knowledge, the second is the formulation of geometrical problems which the Greeks themselves could not solve. Ancient Greece developed another tradition in logic besides the Aristotelian. This was the Stoic logic, which evolved from the philosophy of Parmenides and Zeno. Stoic school of logic formulated numerous paradoxes, only one of which is of interest today. However, this one was destined to play an important part in the development of math logic. It is called the Liar paradox, the version of which is the following:
 1. Cretans always lie [uttered by a Cretan].
 2. Whoever says "I lie" lies and speaks the truth at the same time.
 3. This proposition is not true.
The difficulty with the Liar paradox is that it leads to a contradiction, whether the proposition involved is considered true or false. Stoic logic was surely very rich. For example, as we have seen, Aristotle was aware of the distinction between true and valid, but the first explicit and clear statement of it was made in Stoic writings. Nevertheless, since none of the modern innovators of math logic knew anything of Stoic logic, it does not form part of the historical development of math logic. Indeed, it was not until 1927 that the Polish logician Lukasiewicz showed that in many ways Stoic logic was the unknown forerunner of contemporary logic. Thus, much of what had been thought to be recent discoveries were really rediscoveries. Stoic logic is fundamentally simpler than Aristotelian logic, which cannot even be systematically presented without the equivalent of the Stoic logic of propositions.