Aristotelian Syllogistics

Origin. Aristotle (4th c. BCE). Prior Analytics. Formal deduction from premises. Foundation of Western logic for two millennia.

Models. Categorical propositions. Subject-predicate form. Four figures. Valid and invalid moods. Term logic.

Formalism.

Categorical propositions: A: All S are P (universal affirmative). E: No S are P (universal negative). I: Some S are P (particular affirmative). O: Some S are not P (particular negative).

Syllogism structure: Two premises, one conclusion. Three terms: major, minor, middle. Middle term links premises. Conclusion relates major and minor.

Four figures: Figure I: M-P, S-M ⊢ S-P. Figure II: P-M, S-M ⊢ S-P. Figure III: M-P, M-S ⊢ S-P. Figure IV: P-M, M-S ⊢ S-P.

Valid moods (examples): Barbara (AAA-1): All M are P, All S are M ⊢ All S are P. Celarent (EAE-1): No M are P, All S are M ⊢ No S are P. Darii (AII-1): All M are P, Some S are M ⊢ Some S are P. Names encode mood and figure.

Square of opposition: A-E: contraries (not both true). I-O: subcontraries (not both false). A-O, E-I: contradictories. A-I, E-O: subalternation.

Conversion: E converts simply: No S are P ↔ No P are S. I converts simply: Some S are P ↔ Some P are S. A converts per accidens: All S are P → Some P are S.

Symbols.

Symbol	Unicode	Name	Meaning
A	—	Universal affirmative	All S are P
E	—	Universal negative	No S are P
I	—	Particular affirmative	Some S are P
O	—	Particular negative	Some S are not P
S	—	Minor term	Subject of the conclusion
P	—	Major term	Predicate of the conclusion
M	—	Middle term	In both premises, not in the conclusion
⊢	U+22A2	Turnstile	Premises yield the conclusion
↔	U+2194	Biconditional	Simple conversion, valid both ways
→	U+2192	Conditional	Conversion per accidens, one way only
Barbara	—	Mood name	Vowels encode the mood; AAA in the first figure

Metatheory. Syllogistic is a decidable system: its sentences translate into the monadic predicate calculus, whose satisfiability Löwenheim (1915) proved decidable, and validity for a given mood is settled by inspecting finitely many cases. Aristotle's own metatheorem is the reduction — the moods of the second and third figures are reduced to the perfect moods of the first by conversion and subalternation, with Baroco and Bocardo requiring reductio per impossibile because their premises do not convert; the fourth figure is post-Aristotelian, traditionally credited to Galen, and its moods, which Aristotle treats only indirectly as additions to the first figure, were reduced by the later tradition. That is a completeness proof for a proof system relative to a semantics, twenty-three centuries early. Łukasiewicz (1951) reaxiomatized syllogistic as a theory on top of the propositional calculus, with Barbara and Datisi as axioms alongside "All A is A" and "Some A is A", and proved it complete and decidable. Corcoran (1972) and Smiley (1973) instead reconstructed it as a natural deduction system whose derivations are Aristotle's reductions, and proved soundness and completeness for set-theoretic models with non-empty extensions. On the traditional reading, which grants existential import to universal sentences, exactly 24 moods are valid; on the modern reading, which does not, only 15 are.

Applies to. Term logic. Natural language reasoning. Historical foundation. Scholastic logic.

Limitations. The language has only monadic terms and no relations, so relational inference is unformalizable — De Morgan's "every head of a horse is the head of an animal" has no syllogistic derivation. There are no propositional connectives: premises are categorical sentences, so no argument with a conditional or disjunctive premise is expressible and the Stoic indemonstrables fall outside the system entirely. Quantifiers do not nest, so "every boy loves some girl" cannot be stated. Validity depends on existential import: Darapti, Felapton, Bramantip and Fesapo, together with the subalternate moods and the subalternation relation itself, become invalid once terms may have empty extensions, and the square of opposition retains only its contradictory diagonals. Singular sentences have no categorical form, so treating "Socrates is mortal" as a universal affirmative is a stipulation rather than a translation. And the system is properly weaker than the monadic predicate calculus, having no term negation and no way to compound terms.