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Syllogism

Reasoning · Logical & Analytical Reasoning

6 Part1 worked examples16 practice questions

A syllogism question looks like this: “All mangoes are fruits. Some fruits are sweets.” — these are the statements. After them come two or three conclusions, like “Some mangoes are sweets”. You must decide which conclusion follows from the statements for certain.

The first and most important rule: take the statements as true even if they look strange or wrong in real life. “All chairs are tables” is false in the real world, but in the question it is true. We use only logic, not general knowledge.

What is special here is that we learn it not through “rules to memorise” but through the method of drawing pictures — because a picture never cheats. And the rule tables you will see were made by having a computer check every possible picture, so they cannot hide a mistake.

PartWhat you learn
Part 1The four kinds of statements, their pictures, conclusions from one statement
Part 2Two statements: the Venn method, the rule table, “either-or”, possibility
Part 3Three and four statements and “only a few” statements

How to use this chapter

  1. Look at the picture — each idea comes with a figure; redraw it in your notebook.
  2. Solved example — for each conclusion it shows why it holds in every picture, or the picture where it breaks.
  3. Try it yourself — the solution is hidden. Draw your own picture first, then choose.
  4. Levels — easy questions use letters (A, B, C), hard ones use words and four groups.
1

Foundation — statements, their pictures and conversion

Four statements, a handful of possible pictures

The four kinds of statements and their exact meaning, the possible pictures of each, and the certain conclusions from one statement.

The four kinds of statements

All · no · some · some … not

The four forms of a statement, their exact meaning, what “follows” means and opposite statements.

Every statement tells the relation between two groups (classes). The first is the subject (what is talked about) and the second the predicate (what is said about it). In “All mangoes are fruits”, mangoes is the subject and fruits the predicate. Such statements come in only four forms:

FormStatementMeaningCode
Universal yesAll X are Ynot one member of X lies outside YA
Universal noNo X is YX and Y have no common memberE
Particular yesSome X are Yat least one member is both X and YI
Particular noSome X are not Yat least one X is not in YO

The four letters A, E, I, O are names from old logic; we use them only as a short way to remember.

  • “Some” means “at least one” — maybe all. “Some fruits are sweets” does not rule out that all fruits are sweets.
  • “Some … are not” means “at least one is not” — maybe none are. It does not rule out that none are.
  • In “All X are Y”, Y need not be bigger than X. Y may have other members, and X = Y is also possible.
NoteA standard assumption: every group named in the question has at least one member. That is why “Some mangoes are fruits” follows from “All mangoes are fruits” — accepted in exams.

What “follows” means

WordTest
Definitely followstrue in every possible picture of the statements
Possibilitytrue in at least one possible picture
Does not followsome one possible picture in which it is false

Now opposite statements. Two statements are direct opposites when if one is true the other is certainly false, and if one is false the other is certainly true. The pairs are: “All X are Y” ↔ “Some X are not Y” and “No X is Y” ↔ “Some X are Y”. These pairs will serve us in the “either-or” rule.

BasicTry it yourself first
Which statement is the direct opposite of “No A are B” (false when it is true and true when it is false)?
BasicTry it yourself first
Which statement is the direct opposite of “Some coins are cars” (false when it is true and true when it is false)?
BasicTry it yourself first
Which statement is the direct opposite of “Some books are birds” (false when it is true and true when it is false)?
StandardTry it yourself first
Which statement is the direct opposite of “All clothes are lamps” (false when it is true and true when it is false)?
StandardTry it yourself first
Which statement is the direct opposite of “Some A are not B” (false when it is true and true when it is false)?

The pictures of each statement

One statement, several possible pictures

The pictures of the four statements, the meaning of × and the idea that one statement can be true in several pictures.

Now draw each statement with circles. In the figures × means “at least one member is certainly here”. A region without × may or may not have members.

BA
A — All A are B: the circle of A inside B
AB
E — No A is B: the circles apart
AB×
I — Some A are B: the circles cut, × in the common part
AB×
O — Some A are not B: × in the part of A outside B

Important: each figure above is only one possibility. “All A are B” has another picture too — the two circles equal (A = B). “Some A are B” also allows three or four pictures. A certain conclusion is one that stays true in all pictures. The table below lists all the possible pictures of each statement — we made it by checking every picture.

StatementPossible pictures
A — All A are Bequal (X = Y) · X inside Y
E — No A are Bapart
I — Some A are Bequal (X = Y) · X inside Y · Y inside X · cutting
O — Some A are not BY inside X · cutting · apart
TrickThat is why “All A are B” does not follow from “Some A are B” (it fails in the “cutting” picture), and why “No B is A” never follows from “All A are B”. Each time ask: does some picture break it?
BasicTry it yourself first
Statement: All A are B. Which of the following definitely follows?
BasicTry it yourself first
Statement: No rivers are hills. Which of the following definitely follows?
StandardTry it yourself first
Statement: Some A are B. Which of the following definitely follows?
StandardTry it yourself first
Statement: All stars are keys. Which of the following definitely follows?

Certain conclusions from one statement (conversion)

What survives when a statement is turned around

The full table of turning one statement around and the certain conclusions that come out of it.

Turning a statement around (swapping subject and predicate) is sometimes certain and sometimes not. This is called conversion. In the table below, for each statement, are all the statements that come out true in every picture:

StatementWhat certainly follows
All A are BSome A are B; Some B are A
No A are BSome A are not B; No B are A; Some B are not A
Some A are BSome B are A
Some A are not Bnothing
  • No A is B stays certain when reversed: no B is A.
  • Some A are B stays certain when reversed: some B are A.
  • All A are B does not stay “all” when reversed; only “some B are A” stays.
  • Some A are not B has no certain reverse — reversing it is the commonest slip.
TrickTo remember: the “no” form reverses fully (E), so does “some are” (I). “All” becomes “some” when reversed. “Some are not” does not reverse at all.
Example 1 · Basic
Statement: All B are A. Which of the following definitely follows?
Solution, step by step
  1. 1Draw the picture of the statement and test every option on it.
  2. 2“No B are A” — false in the picture [A + B].
  3. 3“All A are B” — false in the picture [A] [A + B].
  4. 4“No A are B” — false in the picture [A + B].
  5. 5“Some A are B” holds in every picture — the answer.
Answer Some A are B
All A are B → some B are A (reversed only as “some”).
BasicTry it yourself first
Statement: No A are B. Which of the following definitely follows?
BasicTry it yourself first
Statement: Some B are A. Which of the following definitely follows?
StandardTry it yourself first
Statement: All sweets are vessels. Which of the following definitely follows?
StandardTry it yourself first
Statement: No vessels are fans. Which of the following definitely follows?
StandardTry it yourself first
Statement: Some rivers are clothes. Which of the following definitely follows?
StandardTry it yourself first
Statement: All tables are stars. Which of the following definitely follows?
StandardTry it yourself first
Which statement is the direct opposite of “No sweets are stars” (false when it is true and true when it is false)?
2

Two statements — the Venn method and the rules

Draw the pictures, test each conclusion

The method of drawing two statements, the rule table, “either-or” and possibility questions.

The Venn method: draw all the pictures

A conclusion is certain only if it holds in every picture

The method of combining the pictures of two statements, a solved example and “which conclusion is certain” questions.

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The rule table for two statements

Sixteen pairs — which conclusion is certain

The certain conclusions of all 16 pairs of two statements; checked by computer.

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The “either I or II” rule

When two conclusions are direct opposites

When “either-or” applies, the three pairs and the conditions under which it does not.

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Possibility conclusions

“It is possible that …” — true in at least one picture

What possibility means, when a possibility is right though not certain and when it is wrong.

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3

Three or four statements and “only a few”

Make a chain: each statement links to the next

Three statements with four groups, the chain method and “only a few” statements.

Three statements, four groups

Join them two at a time

The method of solving long chains of statements, a table of chains and questions with three conclusions.

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“Only a few” statements

Some are and some are not — both at once

The meaning of “only a few X are Y”, the certain conclusions from it and related questions.

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4

Strategy guide

The picture, the table and four short rules

The order for each question, four short checking rules and the common mistakes.

The order of solving and four short rules

Check by rule first, then make sure with a picture

The order for a two-statement question and the four rules that can be read straight off the table.

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Common mistakes

The mistakes that keep happening

The commonest reasons for losing marks in syllogism.

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5

Mixed practice

All types mixed

From easy letter questions to hard word questions, in three levels.

Basic practice

One or two statements, in letters

Conversion, opposite statements and plain two-statement questions.

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Standard practice

In words, “either-or” and possibility

Word questions, opposite conclusions and possibility.

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Advanced practice

Four groups, “only a few”, hard possibilities

The hardest kinds: long chains, “only a few” and mixed conclusions.

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6

Revision — cards

Flip each card and test yourself

Think of the answer yourself first, then flip.

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