Syllogisms
Syllogisms for CGPSC CSAT: A, E, I, O statements, distribution, conversion, Venn diagrams, validity rules, either-or and possibility conclusions.
Prerequisites
- No mathematics is needed — only the ability to draw overlapping circles (Venn diagrams).
- This is the first topic of Logical Reasoning and Analytical Ability. The “read only what is given” habit you build here carries straight into Blood Relations and Seating Arrangement.
- Inference questions in Reading Comprehension use the same discipline: a conclusion must follow from the passage, not from your general knowledge.
The Hook: “Ramu Gets Rice, So Ramu Has a Ration Card?”
A notice at a fair price shop (उचित मूल्य दुकान) in a village near Kanker reads: “All ration card holders of this panchayat receive rice at this shop.” A villager watches Ramu collect a bag of rice and concludes, “So Ramu must hold a ration card of this panchayat.”
Sounds reasonable — but it does not follow. The notice says every card holder gets rice; it never says that only card holders do. Ramu may be buying under some other arrangement. The notice puts all card holders inside the circle of rice receivers; it says nothing about who else is in that circle.
The question this page answers: given a few statements, how do you decide — mechanically and without trusting your instinct — which conclusions must be true, which only might be true, and which cannot be true?
Why it matters for CGPSC: syllogism (न्याय-वाक्य) questions are a staple of reasoning sections in CSAT-type papers, and the same skill decides inference and assumption questions in comprehension passages. The statements are often deliberately absurd (“All pens are trees”) so that your real-world knowledge cannot help you — only the logic can.
The Four Standard Statements
Every syllogism statement is one of four types. The letters come from Latin logic: A and I from AffIrmo (I affirm), E and O from nEgO (I deny).
| Type | Form | Quantity | Quality | Example |
|---|---|---|---|---|
| A | All S are P | Universal | Affirmative | All officers are graduates. |
| E | No S is P | Universal | Negative | No officer is a farmer. |
| I | Some S are P | Particular | Affirmative | Some officers are farmers. |
| O | Some S are not P | Particular | Negative | Some officers are not farmers. |
S is the subject and P is the predicate. In logic, “some” means at least one, possibly all. So “Some officers are graduates” is still true if every officer is a graduate.
- “Every / each / any S is P” → A
- “None of the S is P”, “S is never P” → E
- “Many / a few / most / at least one S is P” → I
- “Not all S are P” → O: Some S are not P
“All S are not P” is ambiguous in ordinary English (it can mean “No S is P” or “Not all S are P”). Well-set questions avoid it; if you meet it, read the options to see which sense the setter intends.
The Venn diagram of each type
Draw S and P as circles. Shaded or overlapping regions show what the statement guarantees:
| Type | Basic diagram | What it guarantees |
|---|---|---|
| A: All S are P | S circle fully inside P | No part of S lies outside P |
| E: No S is P | S and P separate | No overlap at all |
| I: Some S are P | S and P overlap | At least one element in the overlap |
| O: Some S are not P | Part of S outside P | At least one S lies outside P |
An A statement can also be drawn with S and P as the same circle (all S are P and all P are S). Since “All S are P” does not rule this out, it remains a possible case — one of the reasons possibility questions exist.
Distribution: Which Term Is “Fully Covered”?
A term is distributed when the statement says something about every member of it.
| Type | Subject distributed? | Predicate distributed? |
|---|---|---|
| A — All S are P | Yes | No |
| E — No S is P | Yes | Yes |
| I — Some S are P | No | No |
| O — Some S are not P | No | Yes |
“All officers are graduates” tells you about every officer but not about every graduate, so only the subject is distributed. “No officer is a farmer” tells you about every officer and every farmer (none of them is an officer), so both are distributed.
Conversion (Immediate Inference)
Conversion swaps subject and predicate. Only some conversions are valid:
| Statement | Valid conversion | Note |
|---|---|---|
| All S are P (A) | Some P are S (I) | Never “All P are S” |
| No S is P (E) | No P is S (E) | Fully reversible |
| Some S are P (I) | Some P are S (I) | Fully reversible |
| Some S are not P (O) | None | O cannot be converted |
Combining Two Statements
When two statements share a middle term (the term that appears in both but not in the conclusion), write them so the middle term sits at the end of the first and the start of the second: X–Y, Y–Z. Then read the conclusion from this table. A “+” means “first statement, then second”.
| First + Second (aligned X–Y, Y–Z) | Definite conclusion |
|---|---|
| A + A | All X are Z (A) |
| A + E | No X is Z (E) |
| E + A | Some Z are not X (O, reversed) |
| E + I | Some Z are not X (O, reversed) |
| I + A | Some X are Z (I) |
| I + E | Some X are not Z (O) |
| A + I, I + I, E + E, any pair with O | No definite conclusion between X and Z |
Each conclusion can then be converted using the conversion table: from “All X are Z” you also get “Some Z are X”; from “No X is Z”, “No Z is X”.
- Two particular statements (I, O) → no conclusion.
- Two negative statements (E, O) → no conclusion.
- If one statement is negative, the conclusion must be negative.
- If one statement is particular, the conclusion must be particular.
- The middle term must be distributed in at least one statement.
- A term not distributed in the statements cannot be distributed in the conclusion.
A + I fails rule 5: in “All X are Y” and “Some Y are Z”, Y is distributed in neither statement.
Statements: All teachers are graduates. All graduates are voters. Conclusions: I. All teachers are voters. II. Some voters are teachers.
A + A gives “All teachers are voters” — I follows. Converting it gives “Some voters are teachers” — II follows. Both follow.
Statements: Some farmers are traders. No trader is a clerk. Conclusions: I. Some farmers are not clerks. II. No farmer is a clerk.
I + E gives “Some farmers are not clerks” — I follows. II is not forced: the farmers who are not traders may or may not be clerks. Only I follows.
“Either I or II”: Complementary Pairs
Sometimes neither conclusion follows on its own, yet in every possible diagram at least one of them is true. The answer is then “Either I or II follows”. This happens when all three conditions hold:
- Both conclusions have the same subject and predicate.
- Neither follows individually.
- They form an I–O, A–O or E–I pair.
Statements: All mangoes are fruits. Some fruits are sweet. Conclusions: I. Some mangoes are sweet. II. Some mangoes are not sweet.
A + I gives no definite conclusion, so neither I nor II follows alone. But they share subject and predicate and form an I–O pair: either some mangoes fall in the “sweet” circle, or some fall outside it (or both). Either I or II follows.
“Possibility” Conclusions
Modern question setters often ask whether something “is a possibility”. The test changes:
| Conclusion type | It follows if… |
|---|---|
| Definite (“All A are C”, “Some A are C”) | True in every diagram consistent with the statements |
| Possibility (“All A being C is a possibility”) | True in at least one diagram consistent with the statements |
A possibility conclusion fails only when it would contradict a statement or a definite conclusion.
Statements: All rivers are streams. No stream is a pond. Conclusions: I. No river is a pond. II. Some ponds being rivers is a possibility.
A + E gives “No river is a pond” — I follows definitely. II asks for a diagram where a pond overlaps a river; but every river sits inside “streams”, and streams and ponds never meet. II contradicts a definite conclusion. Only I follows.
The Method in Five Steps
graph TD
A[Read each statement and label A/E/I/O] --> B[Draw the basic Venn diagram]
B --> C[Check each conclusion: true in EVERY diagram?]
C -->|Yes| D[Follows]
C -->|No| E[Possibility conclusion? Can ANY diagram make it true?]
E --> F[Finally test unresolved pairs for Either-Or]Memory Tricks
- “AffIrmo, nEgO”. A and I affirm; E and O deny.
- “US-NP” (think “US News Paper”). Universals (A, E) distribute the Subject; Negatives (E, O) distribute the Predicate.
- Conversion: “A shrinks, E and I swap freely, O stays stuck.” All S are P → Some P are S; E and I reverse as they are; O never converts.
- The six winning pairs: “AA, AE, EA, EI, IA, IE”. Any other pairing (and anything with O) gives no definite conclusion. When E comes first (EA, EI), the conclusion is reversed: Some Z are not X.
- “Some” means “at least one, maybe all”. Say it every time you see “some”.
Don’t Confuse
| This | vs | That |
|---|---|---|
| All S are P → Some P are S (valid) | All S are P → All P are S (invalid) | |
| “Some S are P” — true even if all S are P | Everyday “some” — implies “not all” | |
| Definite conclusion — true in every diagram | Possibility — true in at least one diagram | |
| Either-or needs same subject and predicate | Some A are B / Some B are not A — not a complementary pair | |
| A + E → No X is Z | E + A → Some Z are not X (reversed O) |
Common Mistakes
Practice
Level 1: Foundation
- Classify: “No tiger is a herbivore.”
- Classify: “Some officers are not punctual.”
- Write the valid conversion of “Some teachers are poets.”
- In “All S are P”, which term is distributed?
- What do you get from: All A are B; No B is C?
Answers
- E (universal negative).
- O (particular negative).
- Some poets are teachers.
- Only the subject S.
- No A is C (A + E), and by conversion No C is A.
Level 2: Prelims MCQs
For Q1, Q3 and Q5, take the statements as true even if they seem to differ from common knowledge.
Q1. Statements: Some officers are graduates. All graduates are voters. Conclusions: I. Some officers are voters. II. Some voters are officers.
(a) Only I follows (b) Only II follows (c) Neither I nor II follows (d) Both I and II follow
Answer: (d) Both I and II follow
Q2. Consider the following statements about conversion:
- “Some A are not B” can be validly converted to “Some B are not A”.
- “No A is B” can be validly converted to “No B is A”.
- “All A are B” can be validly converted to “Some B are A”.
Which of the statements given above is/are correct?
(a) 1 only (b) 2 and 3 only (c) 1 and 3 only (d) 1, 2 and 3
Answer: (b) 2 and 3 only
Q3. Statements: All officers are graduates. Some graduates are farmers. No farmer is lazy. Conclusions: I. Some officers are lazy. II. Some officers are not lazy.
(a) Only I follows (b) Only II follows (c) Either I or II follows (d) Neither I nor II follows
Answer: (c) Either I or II follows
Officers are linked to graduates by A, but graduates link to farmers only by I (A + I gives nothing), so nothing definite connects officers to lazy people. Neither conclusion follows alone. But I and II have the same subject (officers) and predicate (lazy) and form an I–O pair: in any diagram, either some officers are lazy, or some are not (or both). So “Either I or II”.
Note that “Some graduates are not lazy” (I + E) would follow — a likely option in a different version of this question.
Q4. Match the statement types in List I with the terms they distribute in List II:
| List I (Type) | List II (Distributes) |
|---|---|
| A. All S are P | 1. Neither term |
| B. No S is P | 2. Predicate only |
| C. Some S are P | 3. Subject only |
| D. Some S are not P | 4. Both subject and predicate |
(a) A-3, B-4, C-1, D-2 (b) A-3, B-4, C-2, D-1 (c) A-4, B-3, C-1, D-2 (d) A-1, B-4, C-3, D-2
Answer: (a) A-3, B-4, C-1, D-2
Q5. Statements: No pen is a pencil. All pencils are erasers. Conclusions: I. No eraser is a pen. II. Some erasers are not pens.
(a) Only I follows (b) Only II follows (c) Either I or II follows (d) Neither I nor II follows
Answer: (b) Only II follows
Level 3: Speed Drill — Solve in 60 Seconds
Prelims Paper II rewards speed, and the reasoning part of Mains Paper IV (GS-II) expects the same logic with the steps shown — for syllogisms, that means naming the statement types and the rule used. Each item below should take under 60 seconds.
- All A are B; all B are C. Does “Some C are A” follow?
- Some A are B; some B are C. Does “Some A are C” follow?
- No A is B; all B are C. What definite conclusion links A and C?
- All A are B; no B is C. What definite conclusion links A and C?
- Some A are not B. What is its valid conversion?
- Some A are B; no B is C. What definite conclusion links A and C?
Answers with the rule used
- Yes — A + A gives All A are C; convert to Some C are A.
- No — I + I (two particulars) gives no conclusion.
- Some C are not A — E + A gives the reversed O.
- No A is C (and No C is A) — A + E gives E.
- None — an O statement cannot be converted.
- Some A are not C — I + E gives O.
Quick Revision Table
| Idea | Rule |
|---|---|
| A / E / I / O | All S are P / No S is P / Some S are P / Some S are not P |
| “Some” | At least one, possibly all |
| Distribution | Universals → subject; negatives → predicate |
| Conversion | A → I; E → E; I → I; O → none |
| Valid pairs (aligned X–Y, Y–Z) | AA → A; AE → E; EA → O*; EI → O*; IA → I; IE → O (* reversed: Some Z are not X) |
| No conclusion | Two particulars; two negatives; middle term undistributed |
| Either-or | Same subject and predicate; neither follows alone; I–O, A–O or E–I pair |
| Possibility | Follows unless it contradicts a statement or a definite conclusion |
Key Takeaways
- Classify every statement as A, E, I or O before doing anything else.
- A conclusion follows only if it is true in every diagram; a possibility needs just one.
- “All S are P” never reverses into “All P are S” — it converts only to “Some P are S”; O never converts.
- Only six pairings give a definite conclusion: AA, AE, EA, EI, IA, IE.
- Check for “Either I or II” last, and only when neither conclusion follows on its own.
What’s Next
- Coding-Decoding — the next topic: pattern-spotting with letters and numbers.
- Reading Comprehension — inference and assumption questions use the same “only what is given” rule.
- Decision Making and Problem Solving — another CSAT block where reading precisely beats instinct.
- Logical Reasoning — Quick Revision — every rule from this chapter on one page.