Syllogism – Study Notes
(Reasoning Ability – Competitive Exams – IIT‑JEE, JEE‑Main, JEE‑Advanced, GRE, SAT, etc.)
Syllogism is a deductive inference that contains two premises (major & minor) and a conclusion.
It is formally a categorical argument using the four standard categorical propositions:
A – All S are P, E – No S are P, I – Some S are P, O – Some S are not P.
1. Formal Structure
| Part | Symbol | Meaning |
|---|---|---|
| Major Premise | P | Universal statement about P (the predicate) |
| Minor Premise | Q | Universal statement about Q (the subject) |
| Conclusion | R | Deduction that links Q and P |
Example
Major: All birds are animals (A)
Minor: All sparrows are birds (A)
Concl.: All sparrows are animals (A)
2. The Four Standard Forms
| Form | Symbol | Statement | Example |
|---|---|---|---|
| A | All S are P | Universal affirmative | All A are B |
| E | No S are P | Universal negative | No A are B |
| I | Some S are P | Particular affirmative | Some A are B |
| O | Some S are not P | Particular negative | Some A are not B |
- S – subject term
- P – predicate term
3. Mood & Figure
| Figure | Arrangement of Terms | Example (A‑A‑A) |
|---|---|---|
| 1 | P – S – P (both premises share the predicate) | All B are C; All A are B; All A are C |
| 2 | S – P – S (both premises share the subject) | All B are C; All B are A; All A are C |
| 3 | P – S – S (predicate in major, subject in minor) | All B are C; All B are A; All A are C |
| 4 | S – P – P (subject in major, predicate in minor) | All B are C; All A are C; All A are B |
Mood = (type of major) – (type of minor) – (type of conclusion).
Example: AAA (all premises and conclusion are universal affirmative) in Figure‑1 is a valid syllogism (Barbara).
4. Valid Syllogisms (Aristotelian)
| Mood | Figure | Valid? | Name |
|---|---|---|---|
| AAA | 1 | ✔ | Barbara |
| EAE | 1 | ✔ | Celarent |
| AII | 1 | ✔ | Darii |
| EIO | 1 | ✔ | Ferio |
| EIE | 2 | ✔ | Camestres |
| AOO | 2 | ✔ | Bocardo |
| EIO | 2 | ✔ | Ferio |
| AII | 3 | ✔ | Darii |
| EIO | 3 | ✔ | Ferio |
| AII | 4 | ✔ | Darii |
Note – The list above is not exhaustive.
A valid syllogism must satisfy the following three rules:
- All terms must be used (the subject and predicate of the conclusion must each appear in at least one premise).
- No term may appear twice in the same premise (unless the syllogism is of the hypothetical type, which is not considered here).
- At least one premise must be universal (or the conclusion must be universal).
5. Common Tricky Points
| Tricky Point | Explanation | Quick Check |
|---|---|---|
| Existential Fallacy | A conclusion claiming existence from a universal premise (e.g., “All A are B” → “Some A are B”) is invalid in classical logic. | If I or O conclusion with universal premises → invalid. |
| Distribution | A term is distributed in a premise if it refers to all members of the set. | A & E premises distribute the subject; I & O premises distribute the predicate. |
| Undistributed Middle | The middle term must be distributed at least once. | If middle term appears only in I/O premises → invalid. |
| Figure‑3 & 4 | These figures are rarely used because the middle term is not distributed in the major premise. | Avoid unless you know the exact mood. |
| Mixed Figures | Some syllogisms use a mixed figure (e.g., AAA in Figure‑4). | Check that the middle term is distributed in at least one premise. |
6. Step‑by‑Step Approach
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