Episode 8 — Aptitude and Reasoning / 8.22 — Syllogism

8.22.a Concepts and Formulas -- Syllogism

1. What is a Syllogism?

A syllogism is a form of deductive reasoning consisting of:

Premises (given statements assumed to be TRUE)
Conclusions (statements to be tested for validity)

Key Rule: Premises are ALWAYS considered TRUE, even if they contradict real-world knowledge.

Example:

  Premise 1: All cats are dogs.         (Accept as TRUE)
  Premise 2: All dogs are birds.        (Accept as TRUE)
  Conclusion: All cats are birds.       (Test this for validity)

2. Four Types of Categorical Statements

Every syllogism statement falls into one of four categories:

Type A: Universal Affirmative -- "All A are B"

  +-----------+
  |     B     |
  |  +-----+  |
  |  |  A  |  |
  |  +-----+  |
  +-----------+

  A is completely inside B.
  Every member of A is also a member of B.

Properties of "All A are B":

Every A is definitely a B.
But NOT every B is necessarily an A.
"Some A are B" is also TRUE (implied).
"Some B are A" is also TRUE (implied).

Type E: Universal Negative -- "No A is B"

  +-----+    +-----+
  |  A  |    |  B  |
  +-----+    +-----+

  A and B are completely separate (disjoint).
  No member of A is a member of B, and vice versa.

Properties of "No A is B":

No A is B, AND No B is A (symmetric).
"Some A are not B" is TRUE (implied).
"Some B are not A" is TRUE (implied).
"Some A are B" is FALSE.
"All A are B" is FALSE.

Type I: Particular Affirmative -- "Some A are B"

  +-----+---+-----+
  |  A  | X |  B  |
  +-----+---+-----+

  X is the overlapping region.
  At least one member of A is also a member of B.

Properties of "Some A are B":

At least one A is a B.
"Some B are A" is also TRUE (converse is valid).
We CANNOT conclude "All A are B."
We CANNOT conclude "No A is B."

Type O: Particular Negative -- "Some A are not B"

  +----------+
  |  A       |
  |    +-----+-----+
  |    |  X  |  B  |
  +----+-----+     |
       +-----------+

  Part of A is outside B.
  At least one member of A is NOT a member of B.

Properties of "Some A are not B":

At least one A is not a B.
We CANNOT conclude "Some B are not A" (this is a common trap!).
We CANNOT conclude "No A is B."
We CANNOT conclude "All A are B."

3. Summary Table of Statement Types

Type	Statement	Symbol	Quantity	Quality
A	All A are B	A(A,B)	Universal	Affirmative
E	No A is B	E(A,B)	Universal	Negative
I	Some A are B	I(A,B)	Particular	Affirmative
O	Some A are not B	O(A,B)	Particular	Negative

4. Venn Diagram Representations

"All A are B" (Three Possible Diagrams)

  Case 1: A is a proper subset of B     Case 2: A = B (identical sets)
  +-----------+                          +-------+
  |     B     |                          | A = B |
  |  +-----+  |                          +-------+
  |  |  A  |  |
  |  +-----+  |
  +-----------+

  Case 3: (Not possible -- A cannot extend outside B)

"No A is B"

  Only one possibility:
  +-----+    +-----+
  |  A  |    |  B  |
  +-----+    +-----+

"Some A are B" (Multiple Possible Diagrams)

  Case 1: Partial overlap     Case 2: A inside B        Case 3: B inside A
  +---+---+---+              +-----------+              +-----------+
  | A | X | B |              |     B     |              |     A     |
  +---+---+---+              |  +-----+  |              |  +-----+  |
                             |  |  A  |  |              |  |  B  |  |
                             |  +-----+  |              |  +-----+  |
                             +-----------+              +-----------+

  Case 4: A = B
  +-------+
  | A = B |
  +-------+

"Some A are not B" (Multiple Possible Diagrams)

  Case 1: Partial overlap        Case 2: Completely separate
  +------+----+-----+           +-----+    +-----+
  | A    | X  |  B  |           |  A  |    |  B  |
  +------+----+-----+           +-----+    +-----+

  Case 3: B is a proper subset of A
  +-----------+
  |     A     |
  |  +-----+  |
  |  |  B  |  |
  |  +-----+  |
  +-----------+

5. Valid Immediate Inferences (Conversions)

Conversion Rules

Original Statement	Valid Conversion
All A are B (Type A)	Some B are A (Type I)
No A is B (Type E)	No B is A (Type E)
Some A are B (Type I)	Some B are A (Type I)
Some A are not B (Type O)	NO valid conversion

Key Point:

"All A are B" does NOT mean "All B are A." "Some A are not B" does NOT mean "Some B are not A."

6. Combining Two Premises -- Conclusion Rules

Rule Table for Two Premises

When combining two premises that share a middle term (the term that appears in both):

Premise 1	Premise 2	Valid 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
All A are B	Some B are C	Some A are C (only if B is distributed)
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
No A is B	All B are C	Some C are not A (not always testable)
Some A are B	Some B are C	No definite conclusion
Some A are not B	(anything)	Usually no definite conclusion

Important:

When "Some + Some" appear together with the same middle term, NO definite conclusion can be drawn.

7. "Some Not" Derivation

"Some A are not B" can be derived when:

From "No A is B": If No A is B, then automatically "Some A are not B" (if A is non-empty).
From "Some A are B" + "No B is C":
- Some A are B, and No B is C.
- Those A that are B cannot be C.
- So: Some A are not C. (Valid)
From "All A are B" + "Some B are not C":
- NOT necessarily: Some A are not C (invalid!).
- Because the "Some B that are not C" might not overlap with A.

8. Complementary Pairs and "Either-Or" Cases

What is a Complementary Pair?

Two conclusions are complementary if one of them MUST be true (but not both simultaneously in general, though at least one is always true).

The Two Complementary Pairs:

Pair	Conclusion 1	Conclusion 2
Pair 1	"Some A are B" (Type I)	"No A is B" (Type E)
Pair 2	"All A are B" (Type A)	"Some A are not B" (Type O)

The "Either-Or" Rule:

If neither conclusion of a complementary pair follows independently from the premises, but together they cover all possibilities, then "Either Conclusion 1 or Conclusion 2" follows.

When to Apply Either-Or:

Check if Conclusion 1 follows -> NO.
Check if Conclusion 2 follows -> NO.
Check if they form a complementary pair -> YES.
Then: "Either 1 or 2 follows."

Example:

  Premises: Some A are B. Some B are C.
  Conclusion I:  Some A are C.
  Conclusion II: No A is C.

  - Conclusion I does not necessarily follow.
  - Conclusion II does not necessarily follow.
  - But "Some A are C" and "No A is C" are complementary.
  - So: Either I or II follows.

9. Handling Multiple Premises (3 or More)

Step-by-step Approach:

Take two premises at a time that share a common (middle) term.
Derive any valid conclusion from those two.
Use that derived conclusion as a new premise.
Combine with the remaining premise(s).
Repeat until all premises are used.

Example:

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

  Step 1: From P1 + P2: All A are C.
  Step 2: From "All A are C" + P3: No A is D.

  Valid conclusions: All A are C, No A is D, Some B are C, etc.

10. Distribution of Terms

A term is distributed when the statement refers to ALL members of that term.

Statement	Subject Distributed?	Predicate Distributed?
All A are B	Yes (A)	No (B)
No A is B	Yes (A)	Yes (B)
Some A are B	No (A)	No (B)
Some A are not B	No (A)	Yes (B)

Rules for Valid Syllogism:

The middle term must be distributed in at least one premise.
If a term is distributed in the conclusion, it must be distributed in the premise.
At least one premise must be affirmative (two negatives yield no conclusion).
If one premise is negative, the conclusion must be negative.
If both premises are particular, no valid conclusion.

11. Common Alternate Wordings

Exam questions often use varied phrasing. Here are equivalences:

Exam Phrasing	Standard Form
"All A are B"	All A are B (Type A)
"Every A is B"	All A are B (Type A)
"Each A is B"	All A are B (Type A)
"Any A is B"	All A are B (Type A)
"No A is B"	No A is B (Type E)
"No A are B"	No A is B (Type E)
"None of the A is B"	No A is B (Type E)
"A is never B"	No A is B (Type E)
"Some A are B"	Some A are B (Type I)
"A few A are B"	Some A are B (Type I)
"Most A are B"	Some A are B (Type I) -- in logic
"Many A are B"	Some A are B (Type I)
"Some A are not B"	Some A are not B (Type O)
"Not all A are B"	Some A are not B (Type O)
"A few A are not B"	Some A are not B (Type O)
"All A are not B"	AMBIGUOUS -- could mean "No A is B" or "Not all A are B"

Important Warning:

"All A are not B" is ambiguous in English. In exams, it usually means "No A is B" (Type E). But be cautious.

12. Possibility-Based Questions

Modern exams (especially banking) include possibility questions:

Types:

"Is it possible that All A are B?"
"Can 'No A is B' be true?"

Rule:

A possibility is TRUE if there exists at least one valid Venn diagram of the premises where the given possibility holds.

A possibility is FALSE only if the given possibility contradicts the premises in every possible Venn diagram.

Example:

  Premise: Some A are B.
  
  "Is it possible that All A are B?"  -> YES (one valid diagram has A inside B)
  "Is it possible that No A is B?"   -> NO (contradicts "Some A are B")

Next: 8.22.b Tips, Tricks, and Shortcuts