Stoic Logic

Origin. Stoa, 3rd c. BCE. Chrysippus of Soli, building on Diodorus Cronus and Philo of Megara. The first systematic propositional logic. Reconstructed from Sextus Empiricus, Diogenes Laertius, Galen. Foundation of Western sentential inference, rediscovered by 20th-century logicians.

Models. Reasoning about whole propositions (axiōmata), not terms. Where Aristotle analyzes subject-predicate form, the Stoics take complete assertibles as primitive and study connectives and inference schemata. Core question: which argument forms are valid in virtue of their propositional structure?

Formalism.

Assertibles (axiōmata): Simple: atomic propositions, bearers of truth at a time. Non-simple: built with connectives. Truth is temporal — an assertible may change truth value.

Connectives: Conditional (synēmmenon): "if p, q." Philonian (material) vs Diodorean (necessitated) readings. Conjunction (sympeplegmenon): "both p and q." Disjunction (diezeugmenon): "either p or q" — exclusive and exhaustive. Negation (apophatikon): "not: p" — external negation.

The five indemonstrables (anapodeiktoi):

If p then q; p; therefore q. (modus ponens)
If p then q; not q; therefore not p. (modus tollens)
Not both p and q; p; therefore not q.
Either p or q; p; therefore not q.
Either p or q; not p; therefore q.

Themata: Four meta-rules reducing complex arguments to the indemonstrables. Thema 1 (a form of antilogism) and Thema 3 are attested; 2 and 4 reconstructed. Analysis (analysis) = reduction of any valid argument to indemonstrable form.

Diodorean modality: Possible = is or will be true. Necessary = is and always will be true. The Master Argument (kyrieuōn) constrains the modal notions.

Symbols.

Symbol	Unicode	Greek	Meaning
axiōma	—	ἀξίωμα	assertible / proposition
synēmmenon	—	συνημμένον	conditional
sympeplegmenon	—	συμπεπλεγμένον	conjunction
diezeugmenon	—	διεζευγμένον	(exclusive) disjunction
anapodeiktos	—	ἀναπόδεικτος	indemonstrable
thema	—	θέμα	reduction meta-rule

Metatheory. The indemonstrables are inference schemata, not axioms; the themata give a reduction procedure so that every valid syllogism analyzes into indemonstrable steps — an early completeness ambition for propositional consequence. The Philonian conditional is exactly material implication. The Diodorean/Philonian dispute is the first recorded debate over the truth conditions of the conditional.

Applies to. History of logic. Propositional consequence. Theory of the conditional. Modal logic (Master Argument, Diodorean modality). Comparative logic against Aristotelian term logic.

Limitations. Fragmentary transmission; no surviving primary text by Chrysippus. Themata 2 and 4 are reconstructions. No variables in the modern sense — schemata are stated on ordinal placeholders. Quantificational and relational reasoning fall outside the propositional core.