The Megarians and the Stoics were different philosophical schools, and also presented internal variations in a number of respects. Their logics, however, are interrelated enough to be usefully outlined together.1 To indicate their differences and to identify historical connections between the two, I use biographical data, on the one hand, and continuities in logical developments, on the other.

Several ancient philosophers and philosophical schools address issues about terms and propositions. The most important contributions are offered by Plato, Aristotle, and the Stoics. In the Sophist, Plato distinguishes names and verbs (which roughly correspond to subject-expressions and predicate-expressions), he claims that truth and falsehood qualify only speeches (which roughly correspond to complete sentences), and he sketches accounts of truth and falsehood for speeches of the simplest sort. In De Interpretatione, Aristotle picks up Plato’s distinction between names and verbs and identifies the bearers of truth and falsehood with sentences of a special sort, namely, declarative sentences. In the Prior Analytics, he develops a theory of inferences constructed from propositions and terms, but he ignores the distinction between names and verbs. With the Stoics, a contrast analogous to that between terms and propositions is found at the level (not of speech, but) of sayables, incorporeal items signified by utterances. The Stoics single out a special type of sayable, the statable, as the bearer of truth and falsehood. Of the four sections of this chapter, the first is dedicated to Plato, the second to Aristotle’s views in De Interpretatione, the third to his position in the Prior Analytics, and the fourth to the Stoics.

Greek mathematics may not have been unique, among the ancient mathematical traditions, in its very use of proof (in the last generation, Høyrup and Chemla have shown the role of proof in the Babylonian and Chinese mathematical traditions, respectively1). But in no other ancient mathematical tradition was proof so explicit and foregrounded. Other mathematical civilisations proved: Greeks did so self-consciously.

The notion of validity and the systematic codification of valid forms of deductive argument are central to the discipline of logic. This chapter will reconstruct how, in antiquity, Aristotle and the Stoics constructed two different deductive systems meant to capture and codify an especially important subset of valid inferences, which they called ‘syllogisms’.1 In the process, we will also emphasise some key similarities and differences between the two systems, and between both of them and modern conceptions of validity and formal logic. Because of space limitations we will not include in our presentation other interesting and for the most part subsequent developments in the classification of valid forms of inference.2

In late antiquity interpreters of Plato’s philosophy insisted that the whole of logic was already present in his dialogues. All kinds of syllogisms were used by Socrates and his interlocutors, and it was left to Aristotle and his successors only to name, classify and formalise them.1 This approach remained popular among interpreters until the first half of the twentieth century.2 More recent historians of logic have protested that in order to ‘discover’ or ‘invent’ logic it is not sufficient to reason according to certain valid patterns, or to represent someone acting in this way in a fictional dialogue. But there is a sense in which Plato did play a key role in the birth and development of ancient logic, a role which is often underplayed in histories of logic. In his dialogues Plato identified and explored a number of central philosophical issues to which logical concepts and methods offered powerful responses, if not definitive solutions. In this way, he was an essential catalyst for the birth of logic: if ancient logic was the promised land, Plato was its Moses. He never set foot in it, but enabled others to see the destination. Of course, when setting this agenda, Plato was not operating in a philosophical vacuum; often he was engaging in original ways with problems raised or foreshadowed by some of his predecessors and contemporaries (on the ‘prehistory’ of logic see Chapter 1 – Denyer).

Logic didn’t develop much in the period from the early Stoics to Boethius. Certainly, there was nothing to match the ground-breaking discoveries of Aristotle and Chrysippus. Aristotle found for us the figures of the syllogism, and in the course of proving those forms to be correct, uncovered and exploited numerous logical laws. He even managed to make steps in the right direction with his modal syllogistic. The early Stoics found for us the indemonstrables, the method of analysis, and the themata. There are few, if any, comparable discoveries in the later period. The closest anyone came was Galen with his relational syllogisms, a ‘third class’ of syllogisms that Galen argued was a necessary supplement to Aristotelian and Stoic syllogistic. Moreover, it is also hard to deny that there were some steps backwards taken by some of the later logicians, by which I mean there were some serious misunderstandings of the logical theory of their predecessors.

The rejection of the medieval scholastic tradition that characterised the logic of the Renaissance did not imply the rejection of ancient logic. On the contrary, the philological expertise of humanist scholars made it possible to read the writings of ‘practically all classical Greek authors’.1 In particular, Aristotle’s logical writings and many commentaries on them, together with new Latin translations and revised scholastic ones, became widely available not only in manuscript form but also in print.2 This favoured a lively dialogue with ancient logical literature, even when it was tinged with criticism.

In a passage from his New Essays on Human Understanding (4.5.3–11) Leibniz distinguishes between three kinds of truth: propositional, moral, and metaphysical. Propositional truth belongs to true affirmations and negations, and consists in ‘correspondence of the propositions which are in the intellect with the things they are about’. Moral truth or truthfulness consists in ‘talking about things in accordance with the belief of our spirit’. Finally, metaphysical truth ‘is the real existence of things, in conformity with the ideas which we have about them’ and ‘is typically interpreted by metaphysicians as an attribute of being’. In ancient Greek we can find a similar tripartition of the meanings of the noun alētheia and the adjective alēthēs, which at least from the classical age (fifth–fourth century BCE) prevailed over the other terms adopted in the rich Homeric vocabulary for truth. Therefore, in ancient Greek there are three possible meanings of alētheia: (1) truth as opposed to falsehood (pseudos), (2) truthfulness as opposed to lying, and (3) reality as opposed to appearance. The first meaning is what I will call ‘logical truth’ (obviously not in the sense of ‘logical tautology’) and is an attribute of declarative sentences and of the beliefs that they express. The second meaning, moral truth, applies in particular to people, but also to oracles or dreams; in the case of people it is the ethical virtue of those who are sincere in their discourses (en tois logois),2 namely, of those who say what they believe without hiding anything, and is opposed to the moral vice of lying, which belongs to those ‘who hide something in themselves and declare something else’.3 Finally, there is reality as opposed to appearance, which Leibniz calls ‘metaphysical truth’ and which I prefer to call, faute de mieux, ‘ontological truth’; I distinguish this both from Leibniz’s definition and from the definition of it as an attribute of being given by the metaphysicians of Leibniz’s time and later by Heidegger, which Leibniz considered to be ‘a useless and almost senseless attribute’. By ‘ontological truth’ I will thus mean the attributive use of the adjective ‘true’, as applied ‘to each object, if one wants to express that it really is what it should be according to the name given to it’.4

The Greeks invented the philosophical discipline known as ‘logic’, whose core is the study and classification of valid forms of inference. Since its inception Greek logical inquiry was motivated by the need to establish the standards of correctness for philosophical reasoning and argument. Throughout antiquity this inquiry also focused on the identification, diagnosis, and classification of forms of argument that are invalid, unsound, or otherwise problematic. Within these, special attention was devoted to those forms of argument that, despite their deficiency, somehow appear to be valid, and thus can easily induce us in error, or can be exploited ‘sophistically’ to mislead others. To be able to defend oneself, by detecting the fallacies in someone else’s reasoning, was a valuable skill in a context in which philosophical discourse developed in a dialectical setting, and one’s opponents could use, whether consciously or inadvertently, fallacious arguments to (apparently) refute one’s side of the argument and to win the debate. In addition, the study of fallacies was deemed important to avoid errors in one’s own reasoning, which was construed by Plato as a sort of inner, silent dialogue which one entertains with oneself.1

Aristotle is history’s first great logician and Chrysippus is the second. We know more of Aristotle’s work than Chrysippus’ (whose works have been almost entirely lost), but we have enough at hand to identify the principal achievements of each. Aristotle’s logical particles of the syllogistic were ‘all’, ‘no’, ‘some’, and ‘non-’. Chrysippus’ were ‘if-then’, ‘it is not the case’, and ‘or’. This inclines the modern reader to see in Aristotle’s term-logic a precursor of predicate logic, and in Chrysippus’ logic the precursor of propositional logic. Because space is limited, I shall take the ancient logic of this chapter to be Aristotelian and Chrysippean logics.

Early Greek rhetoricians dealt with a wide range of persuasive techniques: emotional appeals, stylistic ornamentation, slander, and eristic tricks were all part of their repertoire. Notwithstanding this variety, there was a basic understanding that any technique of persuasion has to incorporate elements of genuine argumentation. This is why even early systems of rhetoric came to acknowledge the relevance of notions like proof, sign, probability, contrariety, etc. Still, Aristotle was the first to conceive rhetoric as an endeavour that essentially relies on arguments and, thus, requires some expertise in logic. He strikingly requires the rhetorician, who has to deal with rhetorical proofs or arguments, to be an expert in all sorts of sullogismos, or deductive argument. At the time Aristotle wrote the core of his Art of Rhetoric, he seems to have taken for granted that it is the dialectician who is the expert on all sorts of sullogismoi. The logic that he thus adopts for his account of rhetorical arguments seems to be the same logic that underlies his dialectic, which is unfolded in his Topics and Sophistical Refutations. The underlying logic of these works includes a clear understanding of deductive arguments, the difference between deductive and inductive arguments, and the role of premises, conclusions, and refutations. Most notably, it introduces the so-called topoi, argumentative schemes that enable the dialectician to construct premises from which he can derive the intended conclusion.1 These are the most important tools that Aristotle uses for reinterpreting the terminology of the traditional rhetorical manuals. For example, he takes over the traditional notion of an enthymeme, which had previously been used for condensed and antithetical formulations (see Section II below), and redefines it as a sullogismos that is used for rhetorical purposes. He also tries to adapt the sullogismos as it was defined and used in dialectic to the peculiar circumstances of a public speech, taking into account, for example, that the mostly contingent and variable subject matter of rhetorical arguments seldom allows for necessary proofs. This insight brings the role of the sullogismos, in certain cases, close to the role of the likelihoods and probability arguments, which were prominent in early rhetoric; but, unlike his predecessors, Aristotle is able to clearly distinguish between the modal quality of a premise and the (logical) necessity of a conclusion. Equipped with these logical distinctions he reinterprets the use of likelihoods in traditional rhetoric as a modal modification of the premises of a rhetorical sullogismos.

Aristotelian logic was the basis for various traditions of medieval logic: Greek, Armenian, Syriac, Latin, Arabic, and Hebrew. This chapter will concentrate on just two, the Latin and the Arabic. They are certainly the two great traditions of logic in the Middle Ages and the very different ways in which they use the ancient heritage makes a fascinating comparison. A fuller discussion would certainly include the other four traditions: they are omitted here, not just due to considerations of space, but also because scholarship in these areas has not yet reached the stage to make a survey possible.