Chrysippus · Philosophy
How Chrysippus invented a logic of whole sentences, centuries before its time.
Aristotle had founded logic a generation before the Stoa, and his system - the theory of the syllogism - analysed reasoning in terms of the relations between terms. A classic Aristotelian inference runs: all men are mortal; Socrates is a man; therefore Socrates is mortal. The logic tracks how the terms ‘man,’ ‘mortal,’ and ‘Socrates’ relate. Chrysippus did something different and, in the long run, equally fundamental. He built a logic not of terms but of whole propositions - complete statements that are true or false - and of the ways they combine using the words we now call logical connectives: if … then, and, or, not.
A Stoic inference looks like this: If it is day, then it is light; it is day; therefore it is light. Notice that here the basic units are not terms but entire statements - ‘it is day,’ ‘it is light’ - that get linked by ‘if … then.’ This is propositional logic, and Chrysippus was its founder. For over two thousand years it was overshadowed by Aristotle’s term-logic, dismissed or misunderstood. Only with the rise of modern mathematical logic in the nineteenth and twentieth centuries did scholars realise that the Stoics had been doing, with remarkable sophistication, exactly the kind of logic that lies at the foundation of mathematics and computing.
Before he could build a logic of propositions, Chrysippus had to say what a proposition is - and here the Stoics made a subtle and important distinction. When you say ‘it is day,’ there are, the Stoics noted, three different things in play. There is the sound you make (mere noise, which even a parrot can produce); there is the state of affairs in the world (the actual daylight); and there is, in between, the meaning - what you are saying, the thought expressed. This meaning the Stoics called a lekton, a ‘sayable.’ The sayable is not the sound and not the physical fact; it is the content, the thing that is true or false.
This was a remarkable move. The Stoics had identified what later philosophers would call propositions or meanings as a distinct kind of item, neither word nor thing - strikingly close to what the logician Gottlob Frege, two thousand years later, would call ‘senses’ or ‘thoughts.’ A complete sayable that can be asserted, that is true or false, the Stoics called an axioma - an assertible, the Stoic equivalent of a proposition. Logic, for Chrysippus, is the study of how assertibles relate: which ones follow from which, which combinations are consistent, which inferences are valid. By locating logic at the level of meanings rather than mere words, Chrysippus gave it a rigour and generality that pure grammar could never supply.
The crown of Chrysippus’ logic was his system of indemonstrables - five basic argument-forms so self-evidently valid that they need no proof, and from which (with rules of analysis) every other valid inference can be derived. In modern terms they are the foundational rules of propositional reasoning. The five, stated schematically, are: (1) If the first, then the second; but the first; therefore the second. (2) If the first, then the second; but not the second; therefore not the first. (3) Not both the first and the second; but the first; therefore not the second. (4) Either the first or the second; but the first; therefore not the second. (5) Either the first or the second; but not the second; therefore the first.
Filled in with content, the first runs: ‘If it is day, it is light; it is day; therefore it is light.’ The second: ‘If it is day, it is light; it is not light; therefore it is not day.’ These are exactly the forms a modern logician calls modus ponens and modus tollens, plus rules for conjunction and disjunction. Chrysippus’ brilliant claim was that these five forms are complete in a sense - that every valid propositional argument can be reduced to a chain of them by definite rules. This is a genuinely systematic vision: not a grab-bag of valid arguments, but a small set of axioms generating the whole. It is one of the great intellectual achievements of antiquity, and the world had to wait until the nineteenth century to fully appreciate it.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
Chrysippus founded propositional logic: a system reasoning about whole assertibles (the Stoic ‘sayables,’ or lekta ) connected by ‘if,’ ‘and,’ ‘or.’ He laid down five basic indemonstrable argument-forms from which all valid inference could be derived - a system so powerful the ancients said the gods would reason his w…
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