Greek, Indian and Arabic Logic

Dov M. Gabbay,John Woods

• Éditeur: North Holland
• Année: 2004
• ISBN:
  9780080532868
• Sujet: Mathématiques
• Sous-sujet: Logique en mathématiques

Greek, Indian and Arabic Logic marks the initial appearance of the multi-volume Handbook of the History of Logic. Additional volumes will be published when ready, rather than in strict chronological order. Soon to appear are The Rise of Modern Logic: From Leibniz to Frege. Also in preparation are Logic From Russell to Gödel, Logic and the Modalities in the Twentieth Century, and The Many-Valued and Non-Monotonic Turn in Logic. Further volumes will follow, including Mediaeval and Renaissance Logic and Logic: A History of its Central.In designing the Handbook of the History of Logic, the Editors have taken the view that the history of logic holds more than an antiquarian interest, and that a knowledge of logic's rich and sophisticated development is, in various respects, relevant to the research programmes of the present day. Ancient logic is no exception. The present volume attests to the distant origins of some of modern logic's most important features, such as can be found in the claim by the authors of the chapter on Aristotle's early logic that, from its infancy, the theory of the syllogism is an example of an intuitionistic, non-monotonic, relevantly paraconsistent logic. Similarly, in addition to its comparative earliness, what is striking about the best of the Megarian and Stoic traditions is their sophistication and originality.Logic is an indispensably important pivot of the Western intellectual tradition. But, as the chapters on Indian and Arabic logic make clear, logic's parentage extends more widely than any direct line from the Greek city states. It is hardly surprising, therefore, that for centuries logic has been an unfetteredly international enterprise, whose research programmes reach to every corner of the learned world.Like its companion volumes, Greek, Indian and Arabic Logic is the result of a design that gives to its distinguished authors as much space as would be needed to produce highly authoritative chapters, rich in detail and interpretative reach. The aim of the Editors is to have placed before the relevant intellectual communities a research tool of indispensable value.Together with the other volumes, Greek, Indian and Arabic Logic, will be essential reading for everyone with a curiosity about logic's long development, especially researchers, graduate and senior undergraduate students in logic in all its forms, argumentation theory, AI and computer science, cognitive psychology and neuroscience, linguistics, forensics, philosophy and the history of philosophy, and the history of ideas.

Logic Before Aristotle: Development or Birth?

Julius Moravcsik

Introduction

“What is the origin of logic as a distinct discipline?” is a complex and partly confusing question. It is confusing because some might misinterpret it as asking for a date at which people discovered the difference between sound and unsound reasoning. But presumably people have been thinking logically, at least in relatively simple contexts, since the origin of humanity. Material elements have behaved “physically” much before the rise of physics as a discipline, and people, at times, have argued logically much before the first system of logic was presented. There is a difference between the two cases. Physics did not start with everything “physical” beginning to think about what normal physical functioning is. Humans, with a certain sense of detachment, started raising that question. Reason had to be applied to natural processes in space and time. In the case of logic, however, reason had to be applied to reason. This application required that people reflect on their own thought processes and that of others. This reflection then had to be coupled with separating the art of logical reasoning from other subjects. However, this separation was not like the separating of two natural sciences, e.g., chemistry and biology. “Separation” in our case has two aspects.¹ We need to separate logic from other disciplines dealing with argumentation and communication, such as rhetoric, advertisement generation, and others. But there is also another sense. For in the case of logic we need to bring our reflection on language to a new, higher level of abstraction. We need to consider language, like mathematics, as an abstract system, and then isolate a higher level abstract quality, namely valid and invalid inference discriminability. Language in practice is a series of sounds. We next abstract from that the phonological, syntactic, and semantic elements. We then consider the grammar and semantics, and attempt to impose on some of this logical structure. So we delineate the valid inference patterns. To justify this effort we need to bring in rules of valid inference. But these do not consist of independent elements. Logic emerges when we can relate our rules of inference and present them as a coherent system. Finally, we reflect on the abstract features of such systems in order to understand what logic itself is. Nobody has come up with an informative and explanatory definition of what logic is. The same holds for mathematics. Logic and mathematics evaluate sequences of elements from a completely detached view. By this we mean that the point of view is not influenced by considerations of utility, pleasure, and other such human interest-relative factors.

It would be misleading to characterize the emergence of the first system of logic as either an invention or a discovery. It is not an invention like artifacts, such as wheel, car, or less concrete, such as an alphabet. On the other hand, it is not like the discovery of a mountain nobody knew about, or a chemical element. There are elements of both invention and discovery in the formulation of a logic. We discover necessary relationships between abstract elements that can be characterized linguistically or conceptually. We have, however, options as to how to characterize these relationships. At that point, inventiveness enters the scene. We cannot credit the discoverer of a river we did not know, with originality. Nor do we credit a logician with originality when he first presents the law of non-contradiction. But we do credit him with originality in view of the particular ways in which he shapes rules for deduction and proofs.

If we think — as we should — of logic as a system of justifiable rules of inference that demarcates the valid from the invalid ones, then we should say that Aristotle was the first creator of logic in Western culture, and that this achievement came spontaneously. As Carl Hempel used to say, it was “a free creation of the human mind”. It took a special leap in cognition to arrive at this high level of abstractness, and to formulate the laws of syllogisms the way Aristotle did. On the basis of these considerations one can say that there was no logic in the West before Aristotle, and that emergence was spontaneous, not a matter of gradual development.

Nevertheless, we should not think of the rise of logic as taking place in an intellectual vacuum. Logic presupposes a set of concepts that provide its back ground and elements. These concepts are interrelated.² They do not surface isolated from each other. Their respective developments need not be interrelated, but the final products must be linked that way.

In the following we shall trace the developments of some of these concepts. We must not think that once these concepts are parts of a culture, that logic must emerge. The conceptual background we will trace constitutes a necessary but not also sufficient background for the kind of ingenious work Aristotle did.

Thus we divide the question: “was there logic before Aristotle?” into two sub-questions. First: “did anyone produce a system of logic comparable to that of Aristotle prior to the work of the Stagirite?” and “what, in any, jumps in level of abstraction, and development of new concepts were required for the rise of logic?” The answer to the first question is negative. To the second question we respond by showing what levels of abstraction were needed, and how concepts developed that formed the needed background for logic. In the next section we turn to a brief sketch of this conceptual background.

1 Developing the Conceptual Foundations

The conceptual foundation has two aspects. One of these is the development of the needed detachment and objectivity. One can assess arguments in a number of ways. For example, how persuasive these are, how colorfully they are presented, how easy it is to understand them and so on. We need to eliminate such considerations when we assess something as valid or not. Thus we should trace the development of the notions of generality and objectivity. Detailed work in this area is yet to be done. We need to trace the move from “this must be right because the gods say so” to “this must be a sound deduction because each step can be justified by the rules of inference.” In general terms, the tracing of this development was started by Bruno Snell, with the felicitous title to one of his chapters, “from myth to logic”.³

Before we start on the details, it is worth clarifying where we should look for relevant evidence. For example, the Kneales suggests that we should look primarily at texts of geometry and the natural sciences that started around that time.⁴ They suggest that literary sources are less likely to contain logically sound texts. But it seems that if we regard the use of logic as a universal human practice that is to be codified, then we should look at literary texts as well There are many arguments in Homer’s Iliad. The problem lies not so much with the deductive links but with the premisses; clearly a matter outside the province of logic. Many would not accept the premiss that everything that is loved by the gods is necessarily good, but whether the other beliefs that surround this one are consistent with it or not is a matter of logic. Some discussions involve sketching alternatives among which the characters must choose. For example, who should have more prestige and power the tribe, the one of royal blood, or the most successful warrior in the tribe?⁵ Some notion of what argument supports what — apart from what one would like to be the case — seems implicit in the exchanges. Such material can be seen as leading to the development of the notion of consistency. Some might be tempted to compare agreements, and see something more general and abstract that these have in common, even if this cannot be yet articulated as logical consistency. For the latter concept to be used, one needs the notion of logical form, more abstract than anything we need in other disciplines except mathematics. We have no idea what enables humans to discern the same “form” in so many arguments with different vocabularies and dealing with different topics, different domains. Some suddenly found the similarities, on this abstract level, puzzling. This puzzlement need be resolved; hence the notion of logical form. To take these conceptual steps may seem to many today easy. This would blind us to the difficulty with which different cultures sooner or later managed to form the right notion. A good comparison would be that with grammar. Logic needs some sort of a grammar for the internal analysis of sentences. Yet the first formal complete logic was formulated only in 120 B.C., quite a bit later than Aristotle’s activity.⁶ Yet Aristotle does rely on grammatical distinctions and composition rules.

The logical form of a sentence explains some of the semantic features of that unit. And yet the relation between logical deduction and explanatory power is complex. Part of the problem is that there is no complete agreement on what explanatory power in general is. Some think that this can be explained within the formalisms of today’s logic,⁷ while others deny that.⁸ So one could point to the understanding of a mathematical proof as a holistic cognitive phenomenon, going beyond mere understanding of logical relations.

It is safe to say that not all deductive patterns have explanatory power. Our drive towards logic is not based solely on finding sound patterns for explanation. Logic is also involved in human efforts to prove something, to sustain an argument in debate, and to find an unassailable stand which nobody can legitimately attack. There is no one drive for logic. The motivation is pluralistic.

In Aristotle’s way of thinking there is clearly a strong link between explanation and demonstration and hence also deduction. All men are mortal, all Greeks are men, therefore all Greeks are mortal. The term “men” carries the key explanatory power in this. configuration. It is through being human, of having human nature? that the Greeks have their mortality.

Yet Aristotle himself points out that the explanatory power does not come from the form alone. As he says, one must select the “middle term” — “human” in our example, very carefully.⁹ There will be a number of co-extensive middle term candidates; so how do we choose the “right one”? Aristotle thinks that this is a matter of insight; we need intuitive understanding of what is puzzling us in a context to see what “really” explains.

So explanation and deduction overlap. Some explanations are not deductions, and some deductions are not explanations. But the overlap is important, or at least it seemed to Aristotle the most important part of our rational activities. His predecessors did not link the two, but Aristotle saw a way in which — he hoped — one could.

One of the expressions the development of which led to the notions of evidence and thus also to premiss, is “signs”. Signs show in Homer what the gods want, signs suggest happiness or tragedy, signs signal to Priam that he can trust, at the end, Achilles.

Once we trust regularities in nature, signs need not be arbitrary. We use these in order to predict storms and other weather conditions. Interestingly, both of these uses indicate something that will give us — or so they thought — certainty, but for quite different reasons. In one case the certainty is derived from religious thinking. The source of the certain, the assurance, is beyond typical rational understanding. In the other case the source is the regular observation of general truths in nature. In modern times we link the predictability of rain, storm and other such phenomena to probabilistic reasoning, but it was not construed that way in ancient times. This turned out well in a way, for in this development kinds of certainty paved the path for forging the necessity — and correlated One might think that the sciences that promise only probability developed first, and the ones bringing certainty only later. But exactly the opposite is the case. Mathematics and geometry had their early flowering and the “natural’ sciences only later.

One might think that the development was “from probability to certainty”, but in fact the reverse took place. Humans reach for what promises certainty and settle for the probable only when the methodology and selection of the right domain of objects is not at hand.

We shall now look into the developments, using Bruno Snell’s work, the title of which we cited already. In the light of the last paragraph we can see now myth and logic not only as terminus ab quo and terminus ad quem, but also as sharing an important characteristic, namely the promise of certainty.

There is not enough evidence to trace out the separation between the two sources of certainty. In many countries the two live side by side, and in Plato’s Timaeus the rough equivalent to some of what are today natural sciences is introduced within a religious framework. One can speculate about how questioning of a well known sort, namely, going from the particular to the general helped to sort out different kinds of certainty. In the one case, people presumably started to ask questions about the reliability of specific divine commands and alleged forecasts. But after that they could ask questions about the nature of these pronouncements in general. For example, an important critic, Plato, treats them as a group, but here generality is not available. Divine orders, as also divinities, remain particular. On the other hand, after we agree on specific rules of deductive inference for some patterns, we can ask for more general justifications, and a system of logic will provide this for us.

In our tracing...

Table des matières

• Cover image
• Title page
• Table of Contents
• Front Matter
• Front Matter
• Copyright page
• Preface
• Contributors
• Chapter 1: Logic Before Aristotle: Development or Birth?
• Chapter 2: Aristotle’S Early Logic
• Chapter 3: Aristotle’s Underlying Logic
• Chapter 4: Aristotle’s Modal Syllogisms
• Chapter 5: Indian Logic
• Chapter 6: The Megarians and the Stoics
• Chapter 7: Arabic Logic
• Chapter 8: The Translation of Arabic Works on Logic into Latin in the Middle Ages and the Renaissance
• Index