The History of Logic: The Logic of Aristotle

It was Aristotle who more than any other philosopher influenced the language of philosophy. Logic, metaphysics, physics can rarely be considered Aristotle's creations. Professor Gianantoni, could you speak to us about Aristotle's logic and the essential role it has had in the history of thought? If we were to trace a history of ancient logic based on the use of the term logic, we would in fact have to exclude Aristotle because he never actually uses this term. The term enters philosophical language later, probably with the Stoics. Aristotle calls the whole body of his research on argument and predication by the name Analytica, analytics, meaning by this term the procedure of analysis, that is the breaking down of a proposition into its component elements and of a proposition based on the premises from which it derives. Nonetheless, Aristotle's analytics is not only part of the history of logic but is also certainly the highest expression of research on this theme in the ancient world. Could you please illustrate Aristotle's theory of logical principles to us, given that Aristotle is credited with the discovery of the most general and important laws of thought? Yes, undoubtedly there is a great deal of truth in this claim, even if, and this is significant in understanding the genesis and history of logical problems, Aristotle discusses logical principles not in a work on logic but in book four of metaphysics. Aristotle in De Interpratione examines at length the relationships that exist between propositions composed of the same subject and of the same predicate. I can say, for example, forming propositions with the terms man and philosopher, all men are philosophers, and this is a universal affirmative statement. I can say some men are philosophers, and this is a particular affirmative statement, and I can say no man is a philosopher, and this is a universal negative statement, and finally I can say some men are not philosophers, and this is a particular negative statement. What are the relationships between these premises? From the square in the picture one can see that the following relationships hold. The universal affirmative and the particular affirmative have a relationship of subordination, in the sense that both propositions can be true. If I say all men are mortal is true, it is also true that some men are mortal, or the particular affirmative can be true even when the universal affirmative is false. This is because it may be false to say that all men are philosophers, but true that some men are philosophers. There is therefore a relationship of subordination. Between the negatives and the positives, on the other hand, there is a relationship of opposition. All men are philosophers, and no man is a philosopher are two opposite or contrary propositions. Finally, there is a third type of relationship, that between the universal affirmative and the particular negative, and that between the universal negative and the particular affirmative, that is, according to the diagonal lines of the square. What characterizes these two propositions? These two propositions are contradictory, that is, they cannot both be true and they cannot both be false, but rather one must necessarily be true and the other necessarily false. In other words, Aristotle distinguishes between the relationship of opposition, or what is contrary, and the relationship of what is contradictory. The difference lies precisely in the fact that only in contradictory relationships must there necessarily be one that is true and the other false. So it is this idea of contradiction that naturally catches Aristotle's attention, because if one of the two contradictory propositions is necessarily true and the other is necessarily false, the consequence is that both cannot apply to the same subject at the same time. This is the genesis of that very famous, that most famous of logical principles, which in the later tradition came to be known by the name of the principle of identity and of non-contradiction. In later treatises on logic, above all in the Middle Ages, and still in the philosophical tradition for the whole 19th century at least, this principle took the formula A is A and is not non-A. Alongside this, treatises on logic included a second logical principle called the law of excluded middle, that is, of the excluded third case. This states that A is either B or is not B. A third hypothesis is not given, tertium non datur, the third or the middle case is excluded. And if we read the text of metaphysics, it emerges very clearly that Aristotle actually does not have in mind the principle of identity and non-contradiction, but rather he has in mind two other different principles, one we could call the principle of determinateness, according to which whatever I think, I think indeed that particular determinate thing. The second is the principle of contradiction, according to which I cannot affirm and negate at the same time, while taking the terms in the same sense, a predicate of a subject. That is, I cannot say simultaneously A is B and A is not B. From this point of view then it is evident that Aristotle's theory of logical principles is strictly linked to his metaphysical theories, since it serves to emphasize the necessary determinateness and identity with itself of every object of my thought. This is against sophist relativism and also against many theories of the pre-Socratics.