Syllogism

What you'll learn

• Interpret the four standard types of statements in syllogism
• Draw Venn diagrams to represent each statement type
• Derive valid conclusions from two premises
• Apply immediate inferences (conversion) to check conclusions

Key concepts

The Four Standard Statement Types

Memory trick: The vowels A, E, I, O come from the Latin words Affirmo (I affirm) and nEgo (I deny). A and I are positive; E and O are negative.

Venn Diagram Representations

All A are B — Circle A is completely inside circle B.

  [ B [ A ] ]

No A is B — Circles A and B do not overlap at all.

  [A]    [B]

Some A are B — Circles A and B partially overlap.

  [A ∩ B]

Some A are not B — Part of A is outside B.

  [ A [A∩B] B ]  — part of A is outside B

Drawing Valid Conclusions — The Standard Rules

When given two premises, apply these tested rules:

Golden rule: A conclusion can only be as strong as the weaker premise. If either premise is "Some," the conclusion cannot be "All."

Worked Example 1

Statements:

1. All birds are animals.
2. All animals are living beings.

Conclusion I: All birds are living beings. Conclusion II: Some living beings are birds.

Using Venn diagram: Birds ⊂ Animals ⊂ Living Beings → Birds ⊂ Living Beings.

• Conclusion I: All birds are living beings → True (All A are B, All B are C → All A are C).
• Conclusion II: Some living beings are birds → True (since all birds are living beings, some living beings must be birds — conversion of Conclusion I).

Both conclusions follow.

Worked Example 2

Statements:

1. Some pens are books.
2. No book is a pencil.

Conclusion I: Some pens are not pencils. Conclusion II: No pen is a pencil.

• Premise 1 is "Some A are B" (Particular Positive).
• Premise 2 is "No B is C" (Universal Negative).
• Rule: Some A are B + No B is C → Some A are not C.
• Conclusion I: Some pens are not pencils → Follows.
• Conclusion II: No pen is a pencil → Too strong (we only know SOME pens are books; other pens may or may not be pencils) → Does not follow.

Immediate Inferences (Conversion)

Sometimes the conclusion is just a converted form of one of the given statements.

Example: "All cats are mammals" → "Some mammals are cats" is a valid immediate inference.

Either-Or Conclusions

When neither Conclusion I nor Conclusion II independently follows, check if together they form a complementary pair:

• Conclusion I: Some A are B
• Conclusion II: No A is B

These are complementary (one must be true). In such cases, answer: "Either I or II follows."

Quick check

1. Statements: All flowers are trees. No tree is water. Conclusion: No flower is water. Does it follow?
2. Statements: Some dogs are cats. All cats are birds. Conclusion: Some dogs are birds. Does it follow?
3. Convert the statement "All teachers are learners" using immediate inference.
4. Can the conclusion "All A are C" follow if one of the premises is "Some A are B"? Why or why not?
5. Statements: No river is a mountain. Some mountains are trees. What valid conclusion follows?

Open the Practice tab for graded questions on Syllogism.