Logical Reasoning: Syllogisms and Arguments
CSAT verbal logic — syllogisms with Venn diagrams, if–then statements and contrapositives, necessary vs sufficient conditions, assumptions, conclusions and courses of action.
SyllabusCSAT · Logical reasoning and analytical ability
Prerequisites: Logical Reasoning — overview · Reading Comprehension: Question Types and Strategy
The Hook: The Admit Card That Did Not Get You In
Every examination hall in India carries some version of this notice: “Only candidates carrying a valid admit card will be allowed to enter.” A candidate arrives with an admit card but without a photo identity document — and is turned away. Was the notice broken?
No. The notice said the admit card is necessary, not that it is sufficient. “Only if” statements, “unless” clauses and “all/some/none” claims are exactly what CSAT tests in its verbal-logic questions. The logic is simple; the language is the trap.
The Core Concept
1. Syllogisms — the four statement forms
A syllogism gives you two or more statements (assume them true, however odd they sound) and asks which conclusions must follow.
| Form | Meaning | Venn picture | Valid conversion |
|---|---|---|---|
| All A are B | Every A is inside B | A circle wholly inside B | Some B are A |
| No A is B | A and B never overlap | Two separate circles | No B is A |
| Some A are B | At least one A is a B | Overlapping circles | Some B are A |
| Some A are not B | At least one A lies outside B | Part of A outside B | No conversion possible |
Method (Venn, minimum-overlap): draw the least the statements force. A conclusion follows only if it is true in every diagram consistent with the statements. If you can draw even one valid diagram where the conclusion fails, it does not follow.
Standard pairings (with B as the middle term):
| Statement 1 | Statement 2 | Definite conclusion |
|---|---|---|
| All A are B | All B are C | All A are C |
| All A are B | No B is C | No A is C |
| Some A are B | All B are C | Some A are C |
| Some A are B | No B is C | Some A are not C |
| All A are B | Some B are C | None between A and C |
| Some A are B | Some B are C | None between A and C |
| No A is B | No B is C | None between A and C |
Worked example 1
Statements: All teachers are readers. All readers are thinkers. Conclusions: I. All teachers are thinkers. II. Some thinkers are teachers.
Teachers ⊂ readers ⊂ thinkers, so I follows. Since every teacher is a thinker (and teachers exist), some thinkers are teachers — II follows by conversion. Both follow.
Worked example 2
Statements: Some phones are cameras. No camera is a radio. Conclusions: I. Some phones are not radios. II. No phone is a radio.
The phones that are cameras cannot be radios, so I follows. But nothing stops other phones from being radios — draw the phone circle also touching the radio circle and II fails. Only I follows.
Worked example 3 — possibility questions
Statements: All roses are flowers. Some flowers are red. Conclusions: I. Some roses are red. II. All roses being red is a possibility.
I does not follow: the red flowers may all be non-roses. II follows: nothing in the statements rules out the red region covering the whole rose circle. Rule: a “possibility” conclusion follows unless it contradicts a definite statement.
2. Conditional (“if–then”) logic
For “If P, then Q”:
| Form | Statement | Valid? |
|---|---|---|
| Original | If P then Q | Given |
| Contrapositive | If not Q then not P | Always valid |
| Converse | If Q then P | Not valid (affirming the consequent) |
| Inverse | If not P then not Q | Not valid (denying the antecedent) |
Example: “If the monsoon fails, food prices rise.” Prices did rise — you cannot conclude the monsoon failed (prices may rise for other reasons). Prices did not rise — you can conclude the monsoon did not fail.
graph LR
P[P happens] -->|guarantees| Q[Q happens]
NQ[Q does not happen] -->|guarantees| NP[P did not happen]
Q -.->|tells you nothing about| P3. Necessary versus sufficient
| Language | Translation | Example |
|---|---|---|
| “If P, then Q” | P is sufficient for Q; Q is necessary for P | If you score 66, you qualify |
| “P only if Q” | If P then Q (Q is necessary) | You enter only if you have an admit card → entry implies admit card |
| “P if and only if Q” | P and Q go together | Both directions hold |
| “P unless Q” | If not Q, then P | “The match will be played unless it rains” → no rain implies the match is played |
| “No P without Q” | If P then Q | “No entry without a pass” → entry implies a pass |
4. Statement-based argument questions
| Question type | What to check | Typical wrong option |
|---|---|---|
| Assumption (implicit) | An unstated belief the speaker must hold for the statement to make sense | Something desirable but not needed by the statement |
| Conclusion (inference) | Must follow from the statement alone | Plausible, but needs outside facts |
| Course of action | Practical, addresses the problem, proportionate, within the actor’s power | Extreme (“ban all…”), unrelated, or impractical |
| Cause and effect | Is one event a plausible cause of the other, in time order? | Reversed order, or both effects of a third cause |
| Strengthen / weaken | New fact that makes the conclusion more or less likely | Restates the argument, or is irrelevant |
Worked example 4 — assumption
Statement: “A coaching institute advertises: Join us to clear the examination in your first attempt.” Assumptions: I. Many aspirants want to clear the examination in their first attempt. II. Aspirants read or see advertisements.
Without I, the pitch makes no sense; without II, there is no point advertising. Both are implicit.
Worked example 5 — course of action
Statement: Several cases of food poisoning were reported in a village after a community feast. Courses of action: I. The health department should send a team to treat the affected and test food samples. II. All community feasts in the district should be banned.
I is proportionate and addresses the problem. II is extreme and punishes everyone for one incident. Only I follows.
Memory Anchors
“Some is not none, and all is not only.”
- “Some A are B” does not rule out “All A are B”.
- “All A are B” does not mean “only A are B” — B may contain other things.
Contrapositive in one line — “Flip and negate”: If P then Q ⟹ If not Q then not P.
Conversions — “ASSN”: All → Some (reverse); Some → Some (reverse); No → No (reverse); “Some… not” → nothing.
Middle-term rule: if the common (middle) term is not fully covered by an “All” or “No” statement in at least one premise, no definite conclusion links the end terms.
For Prelims
- Treat the statements as true even if they contradict common sense (“All fish are birds”).
- A conclusion follows only if it is true in every diagram consistent with the statements.
- “Some A are not B” cannot be converted.
- Two “Some” premises, or two negative premises, never give a definite conclusion.
- “Either I or II follows” is correct only when I and II form a complementary pair about the same terms (e.g., “Some A are B” and “No A is B”), and neither follows on its own.
- The contrapositive is the only always-valid rewrite of an if–then statement.
Speed Strategy
- Draw, don’t imagine. Two quick circles take five seconds and prevent most errors.
- Test the conclusion, not the statements. Try to build one diagram where the conclusion fails. If you can, eliminate it.
- Translate words into arrows. Rewrite every “only if”, “unless” and “no… without” as “If… then…” before answering.
- Assumption questions: apply the negation test from comprehension — if negating the option makes the statement pointless, it is an assumption.
- What to skip: four- or five-statement syllogisms with multiple “possibility” conclusions can take three minutes; attempt them after the shorter questions.
Common Traps
Practice
Prelims (CSAT)
Statements:
- All doctors are graduates.
- Some graduates are artists.
Conclusions:
- I. Some doctors are artists.
- II. Some artists are graduates.
Which of the conclusions logically follow(s)?
Solution
The middle term “graduates” links doctors and artists only through “Some graduates are artists” — the artist-graduates may all be non-doctors, so I does not follow. II is the conversion of “Some graduates are artists”, so II follows.
Answer: B
“If it rains, the match is cancelled.” Which one of the following must be true?
Solution
The contrapositive — “if not Q then not P” — is always valid: if the match is not cancelled, it did not rain (D).
- A is the converse (the match may be cancelled for other reasons).
- B is the inverse (not raining does not guarantee the match goes ahead).
- C says rain is the only cause — not stated.
Answer: D
Statement: The government has decided to send weather alerts to farmers as SMS messages on their mobile phones.
Assumptions:
- I. A substantial number of farmers have access to mobile phones.
- II. Weather forecasts are always accurate.
Which of the assumptions is/are implicit in the statement?
Solution
Negate I: if few farmers had mobile phones, SMS alerts would be pointless — so I is implicit. II is extreme: forecasts can be useful without being always accurate, so the decision does not need II.
Answer: A
Statements:
- All pens are boxes.
- No box is a chair.
Conclusions:
- I. No pen is a chair.
- II. Some boxes are pens.
Which of the conclusions logically follow(s)?
Solution
Pens lie wholly inside boxes, and boxes never touch chairs, so pens never touch chairs — I follows (All A are B + No B is C → No A is C). “All pens are boxes” converts to “Some boxes are pens” — II follows.
Answer: C
“Only those who have passed the written test will be called for the interview.” Ravi has been called for the interview. Which one of the following must be true?
Solution
“Called only if passed” means: called → passed. Ravi was called, so he passed (B).
- A goes beyond the statement — passing is not topping.
- C is the converse — passing is necessary, not sufficient, for a call.
- D is about selection, which the statement does not address.
Answer: B
Quick Revision
| Point | One-liner |
|---|---|
| All A are B | Converts to “Some B are A” |
| No A is B | Converts to “No B is A” |
| Some A are not B | No conversion |
| All + All | All |
| All + No | No |
| Some + All | Some |
| Some + No | Some… not |
| Some + Some, No + No | No definite conclusion |
| If P then Q | Only the contrapositive (not Q → not P) is valid |
| P only if Q | P → Q |
| P unless Q | not Q → P |
| Course of action | Practical, proportionate, addresses the problem |
Key Takeaways
- A conclusion follows only if no valid diagram can contradict it.
- “Some” means “at least one, possibly all”.
- Rewrite “only if”, “unless” and “no… without” as if–then before answering.
- The contrapositive is the only safe rewrite of a conditional.
- Assumptions are unstated and necessary; courses of action are practical and proportionate.
Related Topics
- Logical Reasoning: Puzzles and Arrangements
- Logical Reasoning — quick revision
- Logical Reasoning — practice questions
- Reading Comprehension: Question Types and Strategy
- Decision Making and Problem Solving