Syllogism – Study Notes for Competitive Exams
(IIT‑JEE, UPSC, CAT, GMAT, GRE, etc.)
Syllogism is a form of deductive reasoning that uses two premises to deduce a conclusion.
The classical structure deals with categorical propositions (statements that assert a relationship between two classes).
1. Basic Structure
| Term | Symbol | Meaning |
|---|---|---|
| Major Term (M) | (P) | Predicate of the conclusion |
| Minor Term (m) | (S) | Subject of the conclusion |
| Middle Term (M) | (Q) | Appears in both premises, not in the conclusion |
A standard syllogism has three categorical propositions:
- Major Premise – relates Middle Term to Major Term
- Minor Premise – relates Minor Term to Middle Term
- Conclusion – relates Minor Term to Major Term
2. Forms of Categorical Propositions
| Symbol | Form | Example | Truth Conditions |
|---|---|---|---|
| A | Universal affirmative | All (S) are (P) | ∀x(Sx → Px) |
| E | Universal negative | No (S) are (P) | ∀x(Sx → ¬Px) |
| I | Particular affirmative | Some (S) are (P) | ∃x(Sx ∧ Px) |
| O | Particular negative | Some (S) are not (P) | ∃x(Sx ∧ ¬Px) |
Mood = sequence of the three symbols (e.g., AEI).
Figure = position of the middle term in the premises (1–4).
3. Rules of Validity
| Rule | Explanation | Example |
|---|---|---|
| 1. Distribution | A term is distributed if it refers to all members of its class. | In All S are P, S is distributed, P is not. |
| 2. Middle Term Must Be Distributed | The middle term must be distributed at least once in the premises. | All S are M (M distributed) & All M are P → valid. |
| 3. No Contradiction | The premises cannot contradict each other. | All S are P & No S are P → invalid. |
| 4. Conclusion Cannot Be More Specific | The conclusion cannot assert a property that the premises don’t support. | Some M are P & All M are S → cannot conclude Some S are P. |
| 5. Figure‑Specific Constraints | Different figures have different admissible moods (Aristotelian rules). | In Figure 1, EAE is invalid; AEE is valid. |
4. Deriving the Conclusion
- Identify the middle term and its distribution in each premise.
- Apply the rule of distribution to check validity.
- Use logical equivalences (implication, contraposition) if needed.
- Check for hidden contradictions (e.g., Some S are P vs. No S are P).
- Confirm the conclusion’s mood is one of the valid moods for that figure.
Example
Major Premise (Figure 1): All M are P (A)
Minor Premise: All S are M (A)
Conclusion: All S are P (A)
- Middle term M is distributed in the major premise.
- No contradictions.
- Valid mood: AAI in Figure 1 → Valid.
5. Common Mistakes & Tricky Points
| Mistake | Why it’s wrong | How to avoid |
|---|---|---|
| Middle term not distributed | Leads to an undistributed middle fallacy. | Check each premise for the middle term’s coverage. |
| Assuming “All” → “Some” | All S are P does NOT guarantee Some S are P (unless we know S is non‑empty). | Remember: All ≠ Some unless existence is given. |
| Misreading “No” | No S are P ≠ All S are ¬P logically? (They are equivalent). | Use the equivalence: ¬∃x(Sx ∧ Px) ⇔ ∀x(Sx → ¬Px). |
| Ignoring the figure | Certain moods are valid only in specific figures. | Memorize the 24 valid moods per figure (or use a cheat‑sheet). |
| Reversing the conclusion | E.g., concluding Some P are S from All S are P. | The conclusion must preserve the order of terms. |
6. Quick Reference: Valid Moods by Figure
| Figure | Valid Moods (Major → Minor → Conclusion) | Example |
|---|---|---|
| 1 | AAI, EAE, AII, EIO, AIO, EII, IIO | All S are M; All M are P; → All S are P |
| 2 | EAE, IAE, IAA, EIO, AIO, EII, IIO | No S are M; All M are P; → No S are P |
| 3 | EAE, AAE, AII, EIO, AIO, IIO | No M are S; All M are P; → No S are P |
| 4 | EAE, AAE, IAE, AII, EIO, AIO, EII, IIO | *No M are S; No M are P; → No S |