33  Syllogisms and Deductive Logic

33.1 Introduction

A syllogism is a logical argument that applies deductive reasoning to arrive at a conclusion based on two or more premises.
These questions test your ability to reason logically and evaluate whether conclusions necessarily follow from given statements.

33.2 1) Structure of Syllogisms

A typical syllogism has:
- Major premise: A general statement.
- Minor premise: A specific statement related to the major premise.
- Conclusion: Derived logically from both.

Example:
- Premise 1: All humans are mortal.
- Premise 2: Socrates is a human.
- Conclusion: Socrates is mortal.

33.3 2) Standard Forms of Categorical Propositions

Syllogisms are usually expressed using four types of categorical statements:

  1. Universal Affirmative (A): “All A are B”
  2. Universal Negative (E): “No A are B”
  3. Particular Affirmative (I): “Some A are B”
  4. Particular Negative (O): “Some A are not B”

33.4 3) Rules of Valid Syllogism

  • At least three terms: major, minor, and middle.
  • The middle term must be distributed at least once.
  • A term cannot be distributed in the conclusion if not distributed in premises.
  • Two negative premises → no valid conclusion.
  • Two particular premises → no valid conclusion.

33.5 4) Common Patterns

33.5.1 Pattern 1: All–All

  • Premise 1: All A are B.
  • Premise 2: All B are C.
  • Conclusion: All A are C. ✅ Valid.

33.5.2 Pattern 2: All–Some

  • Premise 1: All A are B.
  • Premise 2: Some B are C.
  • Conclusion: Some A are C. ❌ Not necessarily valid (depends on overlap).

33.5.3 Pattern 3: No–Some

  • Premise 1: No A are B.
  • Premise 2: Some B are C.
  • Conclusion: Some A are not C. ❌ Invalid.

33.6 5) Venn Diagram Approach

The safest method is to draw Venn diagrams.
Steps:
1. Draw three overlapping circles (A, B, C).
2. Shade/exclude regions as per premises.
3. Check whether the conclusion always holds.

Tip: If the conclusion holds in all possible diagrams, it is valid.

33.7 6) Deductive Logic (Beyond Syllogisms)

Syllogisms are part of broader deductive reasoning:
- Modus Ponens (If–Then Rule):
- If P → Q, and P is true, then Q is true.
- Modus Tollens:
- If P → Q, and Q is false, then P is false.
- Hypothetical Syllogism:
- If P → Q, and Q → R, then P → R.
- Disjunctive Syllogism:
- P or Q; Not P → therefore Q.

33.8 7) Examples

33.8.1 Example 1

Premises:
1. All cats are animals.
2. All animals are living beings.
Conclusion: All cats are living beings. ✅ Valid.

33.8.2 Example 2

Premises:
1. All engineers are graduates.
2. Some graduates are MBAs.
Conclusion: Some engineers are MBAs. ❌ Not necessarily valid.

33.8.3 Example 3

Premises:
1. No politicians are honest.
2. Some leaders are politicians.
Conclusion: Some leaders are not honest. ✅ Valid.

33.8.4 Example 4 (Deductive Rule)

  • If it rains, the ground will be wet.
  • It rains.
  • Therefore, the ground is wet. ✅ Valid (Modus Ponens).

33.9 8) Practice Questions

  1. All doctors are professionals. Some professionals are teachers. → Conclusion: Some doctors are teachers. (Valid/Invalid?)
  2. All fruits are healthy. All apples are fruits. → Conclusion: All apples are healthy.
  3. Some books are novels. All novels are interesting. → Conclusion: Some books are interesting.
  4. No athletes are lazy. Some students are athletes. → Conclusion: Some students are not lazy.
  5. If a person studies, they will pass. Ramesh studies. → Conclusion: Ramesh will pass.

33.10 9) Suggested Answers

  1. ❌ Invalid (teachers may not overlap with doctors).
  2. ✅ Valid.
  3. ❌ Invalid (cannot say some books are interesting unless specified).
  4. ✅ Valid.
  5. ✅ Valid (Modus Ponens).

33.11 Summary

  • Syllogisms test logical necessity between premises and conclusions.
  • Four types: All, No, Some, Some not.
  • Venn diagrams are the most reliable method.
  • Deductive logic extends syllogisms to conditionals (If–Then) and disjunctions.
  • In exams: check whether conclusion follows in all cases, not just some.