Syllogism
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🔒 Log in to trackA syllogism gives 2-3 'statements' taken as TRUE, plus conclusions that must be judged: do they follow necessarily? The four statement forms:
- All A are B (A inside B, overlap allowed only as A ⊆ B).
- No A is B (two disjoint circles).
- Some A are B (circles overlap, with at least the overlap populated).
- Some A are not B (A has at least one member outside B).
The bullet-proof method is the minimal-diagram + test method: draw the least-committal diagram that satisfies the statements (e.g. 'Some A are B' = two overlapping circles), then try to violate each conclusion while keeping the statements true. If some valid diagram violates the conclusion, it does NOT follow. A conclusion follows only when every valid diagram keeps it true. The classic error is drawing the diagram you want (drawing A inside B because a conclusion says so) — that proves nothing.
Detailed notes
What is a syllogism?
A syllogism gives you two or three statements that you must accept as true, and a set of conclusions. You must decide which conclusions follow necessarily. The topic of the sentences is irrelevant — if the paper says "All cats are engineers", you accept it and reason from it. Real-world truth never matters here.
The four statement forms
| Statement | Meaning | Least-committal picture |
|---|---|---|
| All A are B | every A is inside B; B may be bigger | A-circle inside B-circle |
| No A is B | the two classes never meet | two separate circles |
| Some A are B | at least one member sits in both | two overlapping circles |
| Some A are not B | at least one member of A sits outside B | A-circle partly outside B |
Notice the asymmetry: All and No fix the whole class, while Some fixes only that at least one member exists. That is why conclusions built on "some" are weak and easy to break.
The minimal-diagram method
- Draw the least-committal diagram that satisfies all statements — never draw what a conclusion wants. "Some A are B" is two overlapping circles, nothing more.
- For each conclusion, try to break it: move the free parts of the diagram while keeping every statement true.
- If even one valid diagram breaks the conclusion, it does not follow. If no valid diagram can break it, it follows.
This is the opposite of how students usually work. You are not collecting evidence for a conclusion; you are acting as its enemy. One successful attack kills it.
Moves that are always legal
- All-chain: All A are B + All B are C → All A are C (and hence Some C are A).
- Conversion of All: All A are B → Some B are A (classes are never empty in these exams).
- Conversion of Some: Some A are B → Some B are A.
- Conversion of No: No A is B → No B is A.
- Some + All chain: Some A are B + All B are C → Some A are C. The shared member travels through.
- No + Some: Some B are C + No A is B → Some C are not A (the shared member inherits B's exile from A).
Moves that look legal but are not
- All A are B does not give All B are A (B may be much bigger).
- Some A are B does not give Some A are not B (possibly all A are B).
- No A is B + No B is C gives nothing about A and C.
- Some A are B + Some B are C gives nothing definite about A and C.
Worked example
Statements: No glass is metal. Some metals are solids. Conclusion I: Some solids are not glass. Conclusion II: All glasses are solids. The metal-that-is-solid cannot be glass, so that solid is "a solid not a glass" — I follows. Glass has no fixed link with solids, so a diagram with all glass inside solids and another with glass outside are both valid — II is breakable, so it fails.
Common traps
- Drawing the conclusion's picture instead of the statements'.
- Rejecting "Some B are A" because it "reverses" the statement — that reversal is always valid.
- Judging a conclusion from the first diagram you drew instead of asking whether a second diagram could break it.
Quick revision
- Accept statements as true; ignore real life.
- Draw the least-committal diagram, then hunt one counter-diagram per conclusion.
- Legal: All-chains, All→Some, Some/No conversions, Some+All travel, No+Some exile.
- Illegal: reverse-All, Some→Some-not, No+No chains, Some+Some chains.
- Follows = true in every valid diagram. Break it once = reject.
Types of questions asked
Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.
Type 1: Some + All chain (shared member travels)very common2 practice Q
A 'Some A are B' statement followed by 'All B are C' (or the reverse order), with a conclusion about A and C.
- Spot the term B that appears in both statements.
- The guaranteed A∩B member is a B, so it is also a C — the conclusion 'Some A are C' is forced.
- Check any second conclusion independently (it usually reverses the All, which is illegal).
Why it works: one real member travels through both statements; the rest of the classes stay free.
Example: Some chairs are tables. All tables are desks. Conclusions: I. Some chairs are desks. II. All desks are chairs.
The chair-that-is-a-table is a desk → I follows. II reverses an All → fails. Only I follows.
Type 2: All-chain with a conversion conclusionvery common2 practice Q
Two 'All' statements in a chain, and one conclusion is the reversed 'Some C are A' while the other reverses an All directly.
- Chain the All's: A ⊂ B ⊂ C gives All A are C.
- Convert freely: Some C are A also follows.
- Reject any 'All C are A' — the chain never runs backwards.
Why it works: containment carries forward; the reverse needs new information the statements never give.
Example: All stones are rocks. All rocks are minerals. Conclusions: I. All stones are minerals. II. All minerals are stones.
Stones ⊂ rocks ⊂ minerals → I follows. Minerals stretch beyond stones → II fails. Only I follows.
Type 3: No + Some gives Some-notcommon2 practice Q
A 'No A is B' plus a 'Some B are C' (or 'Some C are B'), conclusion about C and A.
- Find the shared class B.
- The guaranteed B∩C member cannot be an A, so it is 'a C that is not an A' → Some C are not A follows.
- Any stronger conclusion (No C is A, All C are A) is breakable.
Why it works: the shared member inherits B's complete exile from A.
Example: No lizard is a bird. Some birds are black. Conclusion: Some black things are not lizards.
The black bird is a bird, and no bird is a lizard → it is a black thing that is not a lizard. Follows.
Type 4: Conversion-only pair (is the reversal enough?)common2 practice Q
A single statement and two conclusions where one is its exact conversion and the other overreaches it.
- Recall: All→Some converts; Some converts; No converts.
- The conversion conclusion follows.
- The other conclusion adds a new quantifier (All, Some-not, No) that the statement never fixed → reject.
Why it works: conversion only swaps the terms; it never strengthens the claim.
Example: All actors are artists. Conclusions: I. Some artists are actors. II. All artists are actors.
All→Some conversion makes I follow; II is the illegal reverse of an All. Only I follows.
Formulas
'All A are B + All B are C → All A are C' is the only freely chained rule.
Valid (classes are assumed non-empty): 'Some B are A' follows.
Both convert symmetrically.
Shortcut tricks
⚡ Hunt for one counter-diagram
To kill a conclusion, you need just ONE diagram satisfying all statements that breaks it. Ask: 'can the two circles stay apart / can this element sit outside?' If yes, the conclusion fails.
Example: Q. Statements: Some pens are scales. Do 'Some scales are pens' and 'All scales are pens' follow?
Sol. 'Some scales are pens' converts ✓. 'All scales are pens' — draw a scale outside every pen; statements still hold → does NOT follow.
One violating diagram is enough to reject; demanding diagrams can never prove a conclusion.
Where students lose marks
Drawing the conclusion's diagram instead of the statements'.
Rejecting a conclusion only because it 'feels' strong — test it.
Forgetting conversions: Some B are A follows from All A are B.
Practice sets — 14 questions
Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.
Topic test · 10 questions
Suggested time 9 min · wrong answers go to your mistake notebook automatically.