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	Meaning
A, E, I, O	—	proposition types
S, P, M	—	subject, predicate, middle term
Barbara, etc.	—	mood names

Metatheory. Completeness via reduction. All valid moods reduce to Figure I. Existential import assumed.

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

Limitations. Only monadic predicates. No relations. Existential import issues. Superseded by predicate logic for formal purposes.