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.