Syllogism – Advanced University‑Level Study Notes
(Navy SSR – Reasoning Ability – Syllogism)
Goal: Build a deep, rigorous understanding of categorical syllogisms from first principles, connect them to modern logic, explore edge cases, and challenge you with a hard problem.
Prerequisites:
- Basic set theory & propositional logic
- Familiarity with predicate logic (first‑order logic)
- Comfort with mathematical proofs
Structure
- Foundations – What a syllogism is and why it matters
- Categorical Propositions – Grammar & semantics
- Syllogistic Structure – Figures, moods, and the laws of inference
- Formalization – From ancient syllogisms to FOL proofs
- Edge Cases & Advanced Topics – Existential import, conversion, non‑standard terms, and beyond
- Challenging Problem – Test your mastery
1. Foundations
1.1 Definition
A syllogism is a two‑premise argument of the form:
Premise 1: All M are P
Premise 2: All S are M
Conclusion: Therefore, all S are P
More generally, it is a categorical argument with three terms:
- Major term (
P) – predicate of the conclusion - Minor term (
S) – subject of the conclusion - Middle term (
M) – appears in both premises but not the conclusion
1.2 Historical Context
- Aristotle (4th century BC): Catophrastic logic; introduced the four figures (arrangements of terms) and six moods (combinations of qualities).
- Boole (19th century): Algebraic logic; formalized categorical statements as Boolean equations.
- Modern Logic: Syllogisms are a subset of first‑order quantification; they can be expressed and proved in FOL.
2. Categorical Propositions
| Symbol | Quality | Quantity | Form | Example | Logical Form (FOL) |
|---|---|---|---|---|---|
| A | Universal affirmative | All | ∀x (M(x) → P(x)) |
All M are P | ∀x (M(x) → P(x)) |
| E | Universal negative | No | ∀x (M(x) → ¬P(x)) |
No M are P | ∀x (M(x) → ¬P(x)) |
| I | Particular affirmative | Some | ∃x (M(x) ∧ P(x)) |
Some M are P | ∃x (M(x) ∧ P(x)) |
| O | Particular negative | Some not | ∃x (M(x) ∧ ¬P(x)) |
Some M are not P | ∃x (M(x) ∧ ¬P(x)) |
2.1 Terminology
- Quantity: Universal (
All/No) vs. Particular (Some/Such that) - Quality: Affirmative vs. Negative
- Distribution: A term is distributed if it refers to all members of its class (
All,No).
Rule of Distribution
- If a term occurs universal in a premise, it is distributed.
- In a particular statement, the subject is distributed, the predicate is not.
3. Syllogistic Structure
3.1 Figures
The figure is determined by the position of the middle term:
| Figure | Premise 1 (Major) | Premise 2 (Minor) | Middle term | Example |
|---|---|---|---|---|
| 1 | All M are P |
All S are M |
1st in major, 2nd in minor | Barbara |
| 2 | All P are M |
All S are M |
2nd in major, 2nd in minor | Celarent |
| 3 | All M are P |
All M are S |
1st in major, 1st in minor | Darii |
| 4 | All P are M |
All M are S |