Episode 8 — Aptitude and Reasoning / 8.22 — Syllogism

8.22.c Solved Examples -- Syllogism

Example 1: All + All Chain

Statements:

  1. All dogs are animals.
  2. All animals are living things.

Conclusions: I. All dogs are living things. II. Some living things are dogs.

Solution:

Step 1: Identify types. P1: All dogs are animals (Type A). P2: All animals are living things (Type A).

Step 2: Draw Venn diagram.

  +-------------------------+
  |    Living Things        |
  |  +------------------+  |
  |  |    Animals        |  |
  |  |  +------------+  |  |
  |  |  |   Dogs     |  |  |
  |  |  +------------+  |  |
  |  +------------------+  |
  +-------------------------+

Step 3: Test conclusions.

  • I. All dogs are living things -> Dogs is inside Living Things. TRUE.
  • II. Some living things are dogs -> Since dogs exist inside living things, at least some living things are dogs. TRUE.

Answer: Both I and II follow.

Example 2: All + No Chain

Statements:

  1. All roses are flowers.
  2. No flower is a thorn.

Conclusions: I. No rose is a thorn. II. Some thorns are roses.

Solution:

Step 1: P1: All A are B. P2: No B is C. Chain: All+No = No A is C.

Step 2: Venn diagram.

  +----------+    +----------+
  |  Flowers |    |  Thorns  |
  | +------+ |    |          |
  | | Roses| |    |          |
  | +------+ |    |          |
  +----------+    +----------+

Step 3: Test.

  • I. No rose is a thorn -> Roses is inside Flowers, Flowers is separate from Thorns. TRUE.
  • II. Some thorns are roses -> Thorns and Roses are completely separate. FALSE.

Answer: Only I follows.

Example 3: Some + All Chain

Statements:

  1. Some cats are black.
  2. All black things are dark.

Conclusions: I. Some cats are dark. II. All cats are dark.

Solution:

Step 1: P1: Some A are B (Type I). P2: All B are C (Type A). Chain: Some+All = Some A are C.

Step 2: Venn diagram.

  +-----------------------+
  |       Dark            |
  |  +---------+          |
  |  |  Black  |          |
  +--+--+------+----------+
  | Cats|  XX  |
  +-----+------+
  
  XX = cats that are black (and therefore dark)

Step 3: Test.

  • I. Some cats are dark -> Those cats that are black are also dark. TRUE.
  • II. All cats are dark -> Some cats may NOT be black, hence not necessarily dark. FALSE.

Answer: Only I follows.

Example 4: No Definite Conclusion (Some + Some)

Statements:

  1. Some apples are red.
  2. Some red things are balls.

Conclusions: I. Some apples are balls. II. No apple is a ball.

Solution:

Step 1: P1: Some A are B. P2: Some B are C. This is the Some+Some case -> NO definite conclusion.

Step 2: Multiple Venn diagrams:

  Diagram 1 (A and C overlap):       Diagram 2 (A and C separate):
  +----+----+----+                    +----+---+---+----+
  | A  | AB |  B |                    | A  |AB | BC| C  |
  +----+--+-+--+-+                    +----+---+---+----+
          | BC |
          +--+-+----+
             | C    |
             +------+

In Diagram 1, some apples could be balls. In Diagram 2, no apple is a ball.

Step 3: Neither conclusion follows in ALL diagrams.

Step 4: Check Either-Or: "Some apples are balls" (I) and "No apple is a ball" (E) are complementary! At least one MUST be true.

Answer: Either I or II follows.

Example 5: Universal Negative Premise

Statements:

  1. No man is a machine.
  2. All machines are useful.

Conclusions: I. No man is useful. II. Some useful things are not men.

Solution:

Step 1: P1: No A is B. P2: All B are C.

Step 2: Venn diagram.

  +-----------+    +-------------------+
  |   Men     |    |     Useful        |
  |           |    |  +-----------+    |
  |           |    |  | Machines  |    |
  |           |    |  +-----------+    |
  +-----------+    +-------------------+

But wait -- Men and Useful might or might not overlap (since Useful is bigger than Machines):

  Diagram 1: Men and Useful separate    Diagram 2: Men and Useful overlap
  +-----+   +-----------+              +----------+---------+
  | Men |   |  Useful   |              | Men      | Useful  |
  +-----+   | +-------+ |              +----+-----+-+-------+
             | |Machines| |                  |Machines|
             | +-------+ |                  +--------+
             +-----------+

Step 3: Test.

  • I. No man is useful -> In Diagram 2, men can overlap with Useful. FALSE (not in all diagrams).
  • II. Some useful things are not men -> Machines are useful and not men (No man is machine). TRUE in all diagrams.

Answer: Only II follows.

Example 6: Particular Negative

Statements:

  1. All pens are books.
  2. Some books are not pencils.

Conclusions: I. Some pens are not pencils. II. Some books are pens.

Solution:

Step 1: P1: All A are B (Type A). P2: Some B are not C (Type O).

Step 2: Venn diagram:

  +------------------+
  |    Books         |
  | +------+         |     +----------+
  | | Pens |    X    |     | Pencils  |
  | +------+         |     |          |
  +---------+--------+-----+----------+
            (Some books not pencils = X region)

The "X" (books not pencils) might or might not include pens.

Step 3: Test.

  • I. Some pens are not pencils -> The "books not pencils" region might or might not include pens. Cannot be determined. Could be true, could be false. Does NOT follow.
  • II. Some books are pens -> Since all pens are books, at least some books are pens. TRUE (conversion of All A are B gives Some B are A).

Answer: Only II follows.

Example 7: Three Premises

Statements:

  1. All A are B.
  2. All B are C.
  3. No C is D.

Conclusions: I. No A is D. II. All A are C. III. Some C are B.

Solution:

Step 1: Chain: All A are B + All B are C -> All A are C. Then: All A are C + No C is D -> No A is D.

Step 2: Venn diagram.

  +----------------------------------+    +--------+
  |              C                   |    |   D    |
  |  +------------------------+     |    |        |
  |  |          B             |     |    |        |
  |  |  +---------------+    |     |    |        |
  |  |  |      A        |    |     |    +--------+
  |  |  +---------------+    |     |
  |  +------------------------+     |
  +----------------------------------+

Step 3: Test.

  • I. No A is D -> A is inside C, C is separate from D. TRUE.
  • II. All A are C -> A inside B inside C. TRUE.
  • III. Some C are B -> B is inside C, so some C are B. TRUE.

Answer: All three (I, II, III) follow.

Example 8: Either-Or with Type A and O

Statements:

  1. Some girls are students.
  2. Some students are tall.

Conclusions: I. All girls are tall. II. Some girls are not tall.

Solution:

Step 1: Some+Some -> No definite conclusion about girls and tall.

Step 2: Check Either-Or.

  • I: "All girls are tall" (Type A)
  • II: "Some girls are not tall" (Type O)
  • Same terms (girls, tall), one is Type A, other is Type O -> Complementary pair!

Step 3: At least one must be true.

Answer: Either I or II follows.

Example 9: All-All with Reverse Conclusion

Statements:

  1. All teachers are graduates.
  2. All graduates are educated.

Conclusions: I. All teachers are educated. II. All educated people are teachers. III. Some educated people are teachers.

Solution:

  +---------------------------+
  |       Educated            |
  |  +-------------------+   |
  |  |    Graduates      |   |
  |  |  +------------+   |   |
  |  |  |  Teachers  |   |   |
  |  |  +------------+   |   |
  |  +-------------------+   |
  +---------------------------+
  • I. All teachers are educated -> Teachers inside Educated. TRUE.
  • II. All educated are teachers -> Educated is bigger than Teachers. FALSE.
  • III. Some educated are teachers -> Teachers exist inside Educated. TRUE.

Answer: I and III follow.

Example 10: No + No = No Conclusion

Statements:

  1. No car is a bus.
  2. No bus is a train.

Conclusions: I. No car is a train. II. Some trains are not cars.

Solution:

Step 1: No+No = No definite conclusion. (Two negative premises.)

Step 2: Venn diagrams:

  Diagram 1: Car and Train separate    Diagram 2: Car and Train overlap
  +-----+  +-----+  +-------+         +-----+  +-----+
  | Car |  | Bus |  | Train |         | Car=Train| Bus |
  +-----+  +-----+  +-------+         +---------+ +---+
  • I. No car is a train -> True in Diagram 1, False in Diagram 2. Does NOT follow.
  • II. Some trains are not cars -> True in Diagram 1, could be False if Car=Train in Diagram 2. Does NOT follow.

Step 3: Check Either-Or between I and II -> Not a complementary pair (I is "No A is C", II is "Some C are not A" -- different types, different terms order).

Answer: Neither I nor II follows.

Example 11: Some-No Chain

Statements:

  1. Some kings are queens.
  2. No queen is a prince.

Conclusions: I. Some kings are not princes. II. No king is a prince.

Solution:

Step 1: Some A are B + No B is C -> Some A are not C.

Step 2: Venn diagram:

  +------+---+------+    +---------+
  | Kings|KQ | Queens|   | Princes |
  +------+---+------+    +---------+

Kings that are queens cannot be princes (since no queen is prince). But other kings might or might not be princes.

  • I. Some kings are not princes -> Those kings that are queens cannot be princes. TRUE.
  • II. No king is a prince -> Some kings (not queens) might be princes. Does NOT follow.

Answer: Only I follows.

Example 12: Conversion Check

Statements:

  1. All poets are dreamers.
  2. No dreamer is a realist.

Conclusions: I. No poet is a realist. II. No realist is a poet. III. Some dreamers are poets.

Solution:

  +-------------+    +------------+
  |  Dreamers   |    |  Realists  |
  | +---------+ |    |            |
  | |  Poets  | |    |            |
  | +---------+ |    |            |
  +-------------+    +------------+
  • I. No poet is a realist -> Poets inside Dreamers, Dreamers separate from Realists. TRUE.
  • II. No realist is a poet -> Same reasoning. TRUE. (No A is B implies No B is A.)
  • III. Some dreamers are poets -> Poets exist inside Dreamers. TRUE.

Answer: All three follow.

Example 13: "Most" and "Few" Translation

Statements:

  1. Most birds can fly.
  2. All things that fly have wings.

Conclusions: I. Some birds have wings. II. All birds have wings.

Solution:

Step 1: "Most birds can fly" = "Some birds can fly" (Type I in logic).

Step 2: Some A are B + All B are C -> Some A are C.

  +----------------------------+
  |       Have Wings           |
  |  +------------------+     |
  |  |    Can Fly       |     |
  +--+--+--+------------+-----+
  | Birds  |XXXX|
  +--------+----+
  • I. Some birds have wings -> Those birds that fly have wings. TRUE.
  • II. All birds have wings -> Not all birds fly (only "most"). Does NOT follow.

Answer: Only I follows.

Example 14: Possibility-Based Question

Statements:

  1. All roses are flowers.
  2. Some flowers are red.

Conclusions: I. Some roses are red. II. All roses being red is a possibility.

Solution:

Step 1: All A are B + Some B are C -> No definite conclusion about A and C.

Step 2:

  Diagram 1 (roses and red overlap):    Diagram 2 (roses and red separate):
  +-------------------+                 +-------------------+
  |    Flowers        |                 |    Flowers        |
  | +------+  +-----+ |                | +------+  +-----+ |
  | | Roses|  | Red | |                | | Roses|  | Red | |
  | |  +---+--+--+  | |                | +------+  +-----+ |
  | |  | X |  |  |  | |                +-------------------+
  | +--+---+  +--+--+ |
  +-------------------+
  • I. Some roses are red -> Not true in all diagrams. Does NOT follow (as a definite conclusion).
  • II. All roses being red is a possibility -> In some valid diagram, all roses can be inside red. TRUE (as a possibility).

Answer: Only II follows.

Example 15: Three-Circle Problem

Statements:

  1. Some A are B.
  2. All B are C.
  3. No C is D.

Conclusions: I. Some A are C. II. Some A are not D. III. No B is D.

Solution:

Step 1: From P1+P2: Some A are B, All B are C -> Some A are C. From P2+P3: All B are C, No C is D -> No B is D. From "Some A are C" + P3: Some A are C, No C is D -> Some A are not D.

  +------------------+     +--------+
  |       C          |     |   D    |
  |  +----------+    |     |        |
  |  |    B     |    |     |        |
  +--+--+--+----+----+     +--------+
  | A   |XX|
  +-----+--+
  • I. Some A are C -> TRUE.
  • II. Some A are not D -> TRUE.
  • III. No B is D -> B inside C, C separate from D. TRUE.

Answer: All three follow.

Example 16: Tricky "Some Not" Problem

Statements:

  1. All A are B.
  2. Some B are not C.

Conclusions: I. Some A are not C. II. Some B are A.

Solution:

Step 1: P1 is All A are B. P2 is Some B are not C. The "Some B not C" may or may not include any A.

  Diagram 1: A overlaps with "B not C"    Diagram 2: A is entirely within "B and C"
  +------------------+                     +------------------+
  |       B          |                     |       B          |
  | +-----+   +----+ |                    | +----+-----+     |
  | |  A  |   | C  | |                    | | C  |  A  |     |
  | | (not C) +----+ |                    | +----+--+--+ XX  |
  | +-----+          |                    |         (XX=B not C)
  +------------------+                     +------------------+
  • I. Some A are not C -> True in Diagram 1, False in Diagram 2. Does NOT follow.
  • II. Some B are A -> Since All A are B, conversion gives Some B are A. TRUE.

Answer: Only II follows.

Example 17: Complete Overlap Possibility

Statements:

  1. All dogs are pets.
  2. All cats are pets.

Conclusions: I. Some dogs are cats. II. No dog is a cat.

Solution:

  Diagram 1: Dogs and cats overlap      Diagram 2: Dogs and cats separate
  +------------------------+            +------------------------+
  |         Pets           |            |         Pets           |
  | +------+---+------+   |            | +------+    +------+   |
  | | Dogs | X | Cats |   |            | | Dogs |    | Cats |   |
  | +------+---+------+   |            | +------+    +------+   |
  +------------------------+            +------------------------+
  • I. Some dogs are cats -> True in D1, False in D2. Does NOT follow.
  • II. No dog is a cat -> True in D2, False in D1. Does NOT follow.

Step 3: I (Type I) and II (Type E) with same terms -> Complementary pair!

Answer: Either I or II follows.

Example 18: Reverse Direction

Statements:

  1. No pen is an eraser.
  2. All erasers are rubber.

Conclusions: I. No pen is rubber. II. Some rubber things are not pens. III. Some rubber things are erasers.

Solution:

  +---------+   +----------------+
  |  Pens   |   |    Rubber      |
  |         |   | +----------+   |
  |         |   | | Erasers  |   |
  |         |   | +----------+   |
  +---------+   +----------------+
  (Pens may or may not overlap with Rubber, but not with Erasers)
  • I. No pen is rubber -> Pens might overlap with the non-eraser part of Rubber. Does NOT follow.
  • II. Some rubber things are not pens -> Erasers are rubber and not pens. TRUE.
  • III. Some rubber things are erasers -> All erasers are rubber. TRUE.

Answer: II and III follow.

Example 19: Four Conclusions

Statements:

  1. Some stars are planets.
  2. All planets are celestial.

Conclusions: I. Some stars are celestial. II. All stars are celestial. III. Some celestial things are planets. IV. Some celestial things are stars.

Solution:

From P1+P2: Some stars are planets, All planets are celestial -> Some stars are celestial. From P2 conversion: All planets are celestial -> Some celestial are planets.

  +---------------------+
  |     Celestial       |
  |  +------------+     |
  |  |  Planets   |     |
  +--+----+--+----+-----+
  | Stars |XX|
  +-------+--+
  • I. Some stars are celestial -> TRUE.
  • II. All stars are celestial -> Not all stars are necessarily planets. Does NOT follow.
  • III. Some celestial things are planets -> TRUE (conversion of P2).
  • IV. Some celestial things are stars -> TRUE (conversion of conclusion I).

Answer: I, III, and IV follow.

Example 20: All Four Statement Types

Statements:

  1. All A are B.
  2. No B is C.
  3. Some C are D.

Conclusions: I. No A is C. II. Some D are C. III. Some D are not B.

Solution:

From P1+P2: All A are B + No B is C -> No A is C. From P2+P3: No B is C + Some C are D -> Some D are not B (via Some C are D, those C are not B, so those D that are C are not B). Actually: From P3 conversion: Some C are D -> Some D are C. From P2: No B is C -> No C is B. Combined with Some C are D: Some D (that are C) are not B.

  +--------+    +----------+---+--------+
  |   B    |    |    C     |CD |   D    |
  | +----+ |    |          +---+        |
  | |  A | |    |                       |
  | +----+ |    +-----------------------+
  +--------+
  • I. No A is C -> TRUE.
  • II. Some D are C -> Conversion of P3. TRUE.
  • III. Some D are not B -> D that are C cannot be B (since No B is C). TRUE.

Answer: All three follow.

Example 21: Edge Case -- Same Terms

Statements:

  1. All mangoes are fruits.
  2. All fruits are mangoes.

Conclusions: I. All mangoes are fruits. II. All fruits are mangoes.

Solution:

Both premises together mean Mangoes = Fruits (identical sets).

  +--------------+
  | Mangoes =    |
  | Fruits       |
  +--------------+
  • I. All mangoes are fruits -> TRUE (given directly).
  • II. All fruits are mangoes -> TRUE (given directly).

Answer: Both I and II follow.

Example 22: The "At Least" Interpretation

Statements:

  1. Some boys are tall.
  2. Some tall people are strong.
  3. All strong people are healthy.

Conclusions: I. Some boys are healthy. II. Some tall people are healthy.

Solution:

From P2+P3: Some tall are strong + All strong are healthy -> Some tall are healthy. From P1+P2: Some boys are tall + Some tall are strong -> Some+Some = No definite conclusion about boys and strong.

  +-------------------+
  |    Healthy        |
  |  +----------+    |
  |  |  Strong  |    |
  +--+---+--+---+----+
  | Tall |XX|
  +--+---+--+
  |Boys|YY|
  +----+--+
  • I. Some boys are healthy -> Cannot chain Some+Some. Does NOT follow.
  • II. Some tall people are healthy -> From P2+P3 chain. TRUE.

Answer: Only II follows.

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