LOGIKĒ · IV · the logic of whole statements

Chrysippus & the Stoics

Chrysippus of Soli · c. 279 – c. 206 BCE · "if there had been no Chrysippus, there would have been no Stoa"

Aristotle reasoned about categories — all men, some Greeks, no mortals. The Stoics did something the gate would need and he could not give: they reasoned about whole statements — true-or-false propositions joined by connectives: and, or, not, if… then. A proposition is true or false — that is the bit. The connectives are the operators. And Chrysippus's rules of inference are how proofs compute. This is propositional logic, born in Athens, and it — not Aristotle's — is the logic your processor runs on.

✓ STRONG

They built propositional logic. Connectives, a truth-functional conditional, and inference rules — recognised by modern logicians (Łukasiewicz, Mates, Bobzien) as the genuine ancestor of Boolean logic and the gate.

◐ RECONSTRUCTED

Almost all of it is lost. Chrysippus wrote 700+ books; none survive whole. We rebuild Stoic logic from fragments and hostile critics (Sextus Empiricus, Diogenes Laertius).

◔ VINDICATED LATE

Dismissed for 2,000 years. Aristotle's logic was held superior; only in the 20th century was it seen that the Stoics had the deeper, computing-relevant system first.

I · The connectives — and the first truth table

Set two propositions P and Q to true or false, and watch the connectives compute their value. The star is the conditional: Philo of Megara taught that "if P then Q" is false in exactly one case — P true, Q false — and true otherwise. That is the material conditional, the modern truth table, written down around 300 BCE. Every box below is a logic gate that wouldn't be built in silicon for 2,200 years.

P =

TRUE

Q =

FALSE

— click to toggle

The Stoics called the four cases the only thing that matters about a connective — its value in every row. They had, in all but the name, truth-functional semantics: the meaning of "and / or / if" is its table. Frege and Wittgenstein would re-derive this; Shannon would weld it to circuits.

II · The five indemonstrables

From those connectives Chrysippus distilled five basic, self-evident inference patterns — the indemonstrables — out of which all valid argument is built. They are the first natural-deduction rules in history, and you'll recognise them: they're how a proof, or a circuit, moves from premises to a conclusion. With the P, Q above set, the ones whose premises currently hold light up.

"The first indemonstrable: if the first, then the second; but the first; therefore the second." — Sextus Empiricus, reporting Chrysippus (modus ponens)

III · Lost, dismissed, vindicated

Chrysippus was the second founder of Stoicism and, by reputation, the greatest logician of antiquity — Diogenes Laertius credits him with 705 books. They are all gone. What we know of the most consequential logic before Frege survives only because critics quoted it to argue against it.

Gate kept on. For two thousand years the verdict was that Aristotle's term logic was real logic and the Stoic stuff was a lesser cousin — so it was neglected, and the texts perished. Only in the 20th century, when propositional logic became the spine of mathematics and then of computing, did logicians (Łukasiewicz, Benson Mates, Susanne Bobzien) realise the Stoics had got there first and got there deeper. So treat the precise truth-functional reading with care: it's a faithful but partly modern reconstruction from fragments, not a surviving Stoic textbook. The honest shape of this one is a first that was forgotten — and turned out to be the more important first.