Syllogism

Syllogism — Study Notes for HSSC CET

Overview

Syllogism is a core logical reasoning topic that tests your ability to draw valid conclusions from given statements.

The word "syllogism" comes from Greek, meaning "reasoning together." You're given two or more statements (premises) assumed to be true, followed by conclusions. Your job is to determine which conclusions logically follow — not based on real-world truth, but purely on the logical relationship between the statements. This is where many students slip: they let common sense override logical rules.

Mastering syllogism requires understanding Venn diagrams and the four categorical statement types. Once you internalize these, even complex problems become mechanical. This topic also builds the foundation for statement-assumption questions, where you identify hidden premises that make arguments work.

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Key Concepts

• Accept statements as absolutely true: Never question whether "All cats are dogs" makes real-world sense. If the statement says it, believe it for that problem.
• Four standard statement types (A, E, I, O):
  • A-type (Universal Affirmative): "All X are Y"
  • E-type (Universal Negative): "No X is Y"
  • I-type (Particular Affirmative): "Some X are Y"
  • O-type (Particular Negative): "Some X are not Y"
• "Some" means "at least one": It could mean one, many, or even all. "Some X are Y" doesn't exclude the possibility that all X are Y.
• "Some X are Y" implies "Some Y are X": This is called converse. Particular affirmatives are convertible.
• "No X is Y" implies "No Y is X": Universal negatives are also fully convertible.
• "All X are Y" does NOT mean "All Y are X": Universal affirmatives convert only to particular — "All X are Y" gives "Some Y are X."
• Use Venn diagrams for complex cases: Draw circles representing each category. Shade or mark regions to visualize what must be true versus what might be true.
• "Either-or" conclusions: If neither individual conclusion follows alone, check if together they cover all possibilities — then "either (i) or (ii)" follows.

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Key Facts and Rules

Critical Rules to Remember:

1. All + All = All (if middle term connects): All A are B + All B are C → All A are C
2. All + No = No: All A are B + No B is C → No A is C
3. Some + All = Some: Some A are B + All B are C → Some A are C
4. Some + Some = No conclusion: Two particular statements yield no definite conclusion
5. No + No = No conclusion: Two negative statements yield no definite conclusion
6. Particular + Negative = Usually no conclusion

Complementary Pair Rule: "Some X are Y" and "Some X are not Y" form a complementary pair — at least one must always be true.

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Worked Examples

Example 1: Basic Two-Statement Problem

Statements:

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

Conclusions: (a) All mangoes are sweet. (b) Some sweet are mangoes.

Solution:

• Draw three circles: Mango inside Fruit, Fruit inside Sweet
• From All + All rule: All mangoes are sweet ✓
• Converting "All mangoes are sweet" → Some sweet are mangoes ✓
• Answer: Both (a) and (b) follow

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Example 2: Universal Negative

Statements:

1. All cats are animals.
2. No animal is a bird.

Conclusions: (a) No cat is a bird. (b) Some animals are cats.

Solution:

• All cats are animals + No animal is bird → No cat is bird (All + No = No) ✓
• "All cats are animals" converts to "Some animals are cats" ✓
• Answer: Both (a) and (b) follow

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Example 3: Particular Statements

Statements:

1. Some doctors are teachers.
2. All teachers are graduates.

Conclusions: (a) Some doctors are graduates. (b) All graduates are teachers.

Solution:

• Some doctors are teachers + All teachers are graduates → Some doctors are graduates (Some + All = Some) ✓
• "All teachers are graduates" converts only to "Some graduates are teachers," NOT "All graduates are teachers" ✗
• Answer: Only (a) follows

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Example 4: Either-Or Case

Statements:

1. All pens are books.
2. All books are papers.

Conclusions: (a) All papers are pens. (b) Some papers are not pens.

Solution:

• Neither follows definitively from the statements alone
• But these form a complementary pair — either all papers are pens, or some papers are not pens
• Answer: Either (a) or (b) follows

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Common Mistakes

• Assuming "All X are Y" means "All Y are X" → Wrong! "All cats are animals" doesn't mean "All animals are cats." Always convert universal affirmative to particular only.
• Applying real-world logic → "All dogs are cats" feels false, so students ignore it. Treat every statement as hypothetically true for that problem.
• Forgetting "Some" includes "All" → If "All X are Y," then "Some X are Y" is automatically true. Don't mark it as "doesn't follow."
• Missing the complementary pair → When asked about "Some X are Y" vs "Some X are not Y" and neither follows alone, check if they're complementary — one must be true.
• Ignoring possibility diagrams → For "Some" statements, draw multiple Venn diagram possibilities. A conclusion follows only if it's true in ALL possible diagrams.

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Quick Reference

• A = All X are Y | E = No X is Y | I = Some X are Y | O = Some X are not Y
• Convertible fully: E and I | Convertible to particular: A | Not convertible: O
• Two particulars = No conclusion | Two negatives = No conclusion
• "Some" includes the possibility of "All"
• Complementary pair: "Some are" + "Some are not" — one must always be true
• When stuck, draw all possible Venn diagrams — conclusion must hold in every case