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	Name	Meaning
ἀξίωμα	—	axiōma	Assertible; the bearer of truth, which may change with time
συνημμένον	—	synēmmenon	Conditional, "if p, q"
συμπεπλεγμένον	—	sympeplegmenon	Conjunction, "both p and q"
διεζευγμένον	—	diezeugmenon	Disjunction, read exclusively and exhaustively
ἀποφατικόν	—	apophatikon	Negation, prefixed to the whole assertible
ἀναπόδεικτος	—	anapodeiktos	Indemonstrable; one of the five primitive schemata
θέμα	—	thema	Reduction meta-rule; four are reported, two attested
ἀνάλυσις	—	analysis	Reduction of a valid argument to indemonstrable steps
κυριεύων	—	kyrieuōn	The Master Argument, constraining the modal notions
p, q	—	Schematic letters	The modern stand-ins for the Stoic ordinals "the first", "the second"
1–5	—	The five indemonstrables	The primitive schemata the themata reduce arguments to
→	U+2192	Rightwards arrow / Implication	The modern symbol; the Philonian conditional is exactly this

Metatheory. The system is rule-based rather than axiomatic: the indemonstrables are inference schemata and the themata are meta-rules that reduce any valid argument to a chain of them, which makes analysis a reduction procedure and gives the Stoics an explicit completeness ambition for propositional consequence — the earliest on record. Bobzien (1996) reconstructed themata 2 and 4 from the surviving reports of thema 1 and thema 3 and showed the resulting calculus is deductively complete for the Stoic argument forms, so the ambition was in principle met. The Philonian conditional is exactly material implication, which makes Philo's account the first statement of a truth table for →; Diodorus required truth at all times and Chrysippus required that the contradictory of the consequent conflict with the antecedent, so the Chrysippean conditional is not truth-functional at all. Mates (1953) is the modern reconstruction that established these identifications; Łukasiewicz had already argued in the 1930s that the Stoic system is propositional and not a defective term logic, against the reading inherited from Prantl. The Master Argument, reported by Epictetus, sets out three modal claims that cannot jointly hold, and the Hellenistic schools divide by which one they surrender, Chrysippus and Cleanthes rejecting different premises.

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

Limitations. The system is propositional only, so nothing is said about quantification, relations or subject-predicate structure, and the Aristotelian syllogisms are outside it rather than derivable in it. It is also not truth-functional as Chrysippus set it up: his conditional turns on a conflict relation between antecedent and consequent, and the disjunction is exclusive and exhaustive with a modal component, so the connectives cannot be given truth tables and the semantics has no compositional account beyond the Philonian case. Assertibles change truth value with time, so there is no fixed valuation over which validity can be defined, and the reports do not say whether an argument valid now must be valid later. Negation attaches only to a whole assertible, so scope distinctions inside a compound cannot be drawn. And the schemata are stated on ordinal placeholders rather than variables, so no substitution rule is formulated and the generality of a schema has to be read off examples. What is transmitted is fragmentary — nothing survives from Chrysippus directly, and themata 2 and 4 are reconstructions — so the calculus whose completeness is discussed is in part a modern one.