Katalin Havas
In this paper I am not trying to give a definite answer to the question wether modern logic is the perfection of the Aristotelian logic or there is some other relationship between the two. I only wish to pose some questions related to this problem which are connected with the more general problem of the philosophy of science. Namely, is logic a cumulatively developing discipline or are the paradigms changing and consequently are the systems based on different paradigms mutually untranslatable or is there a progress in the course of the history of logic, but this progress is somehow different from the cumulatively developing processes?
A few years ago I visited the theater in Epidaurus, built in the fourth century B. C. Of course, like other tourists, I tried out the acoustics of the theater, and confirmed that a whisper, or the sound made by the lighting of a match on the central stage could be heard on each of the 55 rows of seats of the auditorium accommodating an audience of 14 thousand people. All this had been achieved without the use of complex electronic equipment. Can we then state -- I was asking myself -- that the sound systems employed in our contemporary theaters would be more advanced using them in the theater in Epidaurus? More advanced because for the same purpose we employ more complex means? Or is such new technology more advanced not in order to achieve the same purpose in the Epidaurus theater, but rather to achieve other results? (E.g. in sending sound over longer distances.)
A similar question can be raised in connection with logical theories. In which sense is modern logic more advanced than Aristotelian logic? In the first period of the modern logic the representatives of the cumulative theory -- which was the ruling theory in that time -- considered the history of logic a succession that was started by Aristotle, supplemented by the results of some mediaeval logicians, and given its full-blown form by the birth of the Frege-Russell type mathematical-logical calculi. To illustrate how the relationship between Aristotelian and modern logic was described -- according to this view -- let me quote A. N. Whitehead. In his Foreword to Quine's early work «A System of Logistics» (1934) Whitehead wrote: «In the modern development of Logic, the traditional Aristotelian Logic takes its place as a simplification of the problem presented by the subject. In this there is an analogy to arithmetic of primitive tribes compared to modern mathematics.»
To give another example, I would like to mention how Tarski evaluated the whole of traditional logic, including Aristotelian logic: «The new logic surpasses the old in many respects, -- not only because of the solidity of its foundations and the perfection of the methods employed in its development, but mainly on account of the wealth of concepts and theorems that have been established. Fundamentally, the old traditional logic forms only a fragment of the new, a fragment moreover which, from the point of view of the requirements of other sciences, and of mathematics in particular, is entirely insignificant.»<59>Foot note 3_1
On another page of the same book Tarski wrote: «The whole of the old traditional logic can almost entirely be reduced to the theory of the fundamental relations among classes, that is, to a small fragment of the entire theory of classes.»<60>Foot note 3_2
Can the «perfection of the methods» be used as one of the arguments to prove that the new logic surpasses the old logic? Do a theory surpass another if the results are the same and only the methods are different (let me add: more complicated)? Of course the «perfection of the methods» was not the only argument that Tarski used. He mentioned the wealth of concepts and theorems in the new logic. Because of this he thought that the old logic was only a fragment of the new. But, are really the results of Aristotelian syllogistic a fragment of the logic of classes? It is true that in the logic of classes the validity of some deductions are provable which one cannot prove within the framework of Aristotelian syllogistic. But is the Aristotelian theory of syllogism really a fragment of the modern logic of classes? Does this interpretation not alter the Aristotelian theory at least as much as even the best microphone will alter the characteristic of sound traveling in open air? A vast literature is devoted to the subject of the possible interpretations of the Aristotelian theory of syllogism. For example, M. and W. Kneale specify seven possible types of interpretation and prove that none of them fulfills all the conditions given by Aristotle.<61>Foot note 3_3
I will mention here only two of them because they suffice to show why Aristotelian syllogistic cannot be fully interpreted in the logic of classes.
1. The first requirement -- mentioned by the Kneales -- is that it must be natural within the Aristotelian theory to regard singular and general statements as co-ordinate species of a genus. The copula and the predicate should have the same function in both cases and the kinds differ only in the nature of the subject-term.
If however, in the formula Every A is B «A» and «B» are taken as names of classes and the copula is meant to express the relation «is included in» then «A» cannot be replaced by a singular term. If A is replaced by a singular term the relation «is included in» has to be changed to the relation «is an element of». So, in this case the above mentioned requirement is not fulfilled. The copula is not the same and that is why the singular and the general statements are not co-ordinate species of a genus.
2. Another requirement is that every general term must be capable of occurring either as subject or as predicate without change of meaning.
However, if for example -- corresponding to Tarski's interpretation --, the formula Every A is B is interpreted as a form where A refers to certain individuals that are separated from other individuals by properties which they have in common and, furthermore, if the copula is the sign of predication and B expresses a property ascribed to individuals A, then it is impossible to interchange subject and predicate without change of meaning of A and of B.
Are these not sufficient argument to support the assumption that the objects of Aristotelian syllogistic and the logic of classes are different? Hence, these two theories do not speak of the same objects and consequently, both Aristotelian syllogistic and the logic of classes are fragments of the whole of logic in the sense that they are different parts of it.
The objects of the Aristotelian theory of syllogism are the general terms of the natural languages of everyday conversation and science. Aristotle was aware of the dual logical function in which the general term is used in natural languages. That is to say, in the role of logical subject its function is to denominate an individual or to refer to an individual and as logical predicate its function is to indicate what belongs or does not belong to an individual. Peter Geach mentioned in his book Logic Matters that in modern logic «we do not have such a formal theory that recognizes the name-status of general terms without eclipsing the difference between name and predicate.»<62>Foot note 3_4 The objects of the logic of classes as well as of the predicate logic were constructed by taking out only one function -- and abstracted from the other functions -- of general terms. That is why the doubly-functioning general terms of natural languages are only indirect objects of these modern logics. Can we evaluate the theories created by the segregation of functions, as unquestionable progress in the development of theories? Or does this question contain its answer -- like in the case of the theater at Epidaurus -- depending on the universe of discourse? Are we looking at it from the viewpoint of the area of objects within the Aristotelian theory, or from some other area? Do we wish to speak of the logical relationship expressed in natural language, do we wish to explore the rules of argumentation? Or do we think that the logical relationships expressed in natural language -- which was a subject matter of the Aristotle's investigations -- are entirely insignificant «from the point of view of the requirements of sciences, and of mathematics in particular»?
The answers to those questions are closely linked with our way of defining the task of logic.
It is well known that Aristotle did not use the word «logic» for his works which later in the first century B. C. was collected under the name «Organon». The topics such as Aristotle discussed in the works contained in the Organon were what in later centuries most people have called logic. However, within the Organon, Aristotle is not content with merely providing the axiomatic theory of syllogism. The range of means offered by Aristotle for the victorious conduct of arguments in discussions is much broader than that. Thus it is evident, that in subsequent centuries, based on the Organon, logic contains much more than the theory of formal analysis of deduction or the theory of some abstract objects. On the basis of the Organon, such «aids of thinking» were born -- under the collective name of logical theory -- as discuss the role played in cognition by various kinds of concepts, and by various kinds of statements, as well as numerous problems in methodology. With the emergence of modern logic, that is with the appearance of the Frege-Russell type of so-called classical symbolic logic a great advance undoubtedly was made in the logic whose objects are special kinds of abstract objects. At the same time the philosophical spirit of logic was almost entirely lost. It become removed from what Aristotelian, Stoic and Scholastic logic had set for itself as an important task: to explore the features of argumentation in ordinary language and to establish the rules of correct inference in order to improve the methods of cognition.
But today, when we speak of contemporary logic, we cannot mean exclusively that part of modern logic which is called classical symbolic logic. Contemporary logic contains the non-classical logics as well (intuitionist, relevant, paraconsistent logics, etc.). Contemporary logic provides more than formal study of deducibility. Making use of the results of formal studies and not divorced from them, it examines problems with philosophical content, some of them already occurring in Aristotle's work and only later removed from logic by the members of the early neopositivistic movement during the initial stage of modern logic.
To resort again to an analogy, let me mention the fact that most of the ancient Greek statues were originally colored. In the course of centuries they become soiled. When they were found and people tried to clean them, they lost their coloring. That was one of the reasons why uncolored sculpture became fashionable. Aristotelian logic also lost its colors through the centuries. But the time of rediscovering the beauty of its colors has returned. It is the right time to return to Aristotle's idea according to which logic has a double aim: it is concerned with apodeixis, and at the same time it is an episteme. To realize this idea of course is not possible in one logic, which would be «the true logic» and which would outdo Aristotelian logic in every respect. Hopefully, though, with different logical theories -- built up with different aims and with different methods -- logic as a whole, while retaining the Aristotelian ideas, will at the same time surpass them.
Katalin G. Havas
International Academy of the Philosophy of Science
and Institute of Philosophyof the Hungarian Academy of Science
Address: K£tv”lgyi £t 41.
1125 Budapest, Hungary