**Detailed Notes on Lectures 9 & 10: Validity and Inference Rules**

**Slide 1: Learning Objectives**
- Define the notion of validity in an argument.
- Establish validity using truth tables.
- Demonstrate invalidity using truth tables.
- Understand inference rules.

**Slide 2: Contents**
- Objectives
- Transformational proofs are not sufficient.
- Comparison of deduction with induction.
- Validity.
- Demonstrating validity/invalidity using truth tables.
- Problem with truth tables.
- Inference rules.
- Summary, reading, and references.

**Slide 3: Transformational Proofs do not Suffice**
- Understanding transformations of formulas is useful but insufficient.
- Logic uses rules of inference to deduce true propositions from other true propositions.
- Invalid premises cannot lead to valid conclusions, preventing proofs of contradictions or useless systems.

**Slide 4: Premises and Conclusions**
- An argument consists of premises (basis for accepting) and a conclusion.
- Example:
  - Premises: Every adult is eligible to vote; John is an adult.
  - Conclusion: Therefore, John is eligible to vote.

**Slide 5: Deduction vs. Induction**
- Deductive arguments: Conclusion is wholly justified by premises.
- Inductive arguments: More general new knowledge inferred from facts or observations.

**Slide 6: Valid vs. Invalid Arguments**
- Valid arguments: Conclusion always true when premises are true.
- Invalid arguments: At least one assignment where premises are true, but conclusion is false.

**Slide 7: Example of Valid Argument**
- If John is an adult, then he is eligible to vote (premise).
- John is an adult (premise).
- Therefore, John is eligible to vote (conclusion).

**Slide 8: Example of Valid Argument with False Conclusion**
- If I catch the 19:32 train, I'll arrive in Glasgow at 19:53 (premise).
- I catch the 19:32 train (premise).
- Therefore, I arrive in Glasgow at 19:53 (conclusion) – Factually false but valid argument.

**Slide 9: Example of Invalid Argument**
- If I win the lottery, then I am lucky (premise).
- I do not win the lottery (premise).
- Therefore, I am unlucky (conclusion) – Invalid argument with factually true premises and conclusion.

**Slide 10: Demonstrating Validity Using Truth Tables**
- View argument as implication (p ⇒ q).
- If premises entail conclusion, then argument is valid.

**Slide 12: Demonstrating Validity Using Truth Table (Example)**
- Argument: If John is an adult, then he is eligible to vote; John is an adult; Therefore, John is eligible to vote.
- Atomic Propositions: p (John is an adult), q (John is eligible to vote).

| p | q | p ⇒ q | p ∧ q |
|---|---|------|-------|
| T | T |   T  |    T   |
| F | T |   F  |    F   |

- Argument is valid because conclusion (q) is always true when premises are true.

**Slide 13: Viewing Argument as Implication**
- If premises logically imply conclusion, argument is valid.
- Example: ((p ⇒ q) ∧ p) ⇒ q

**Slide 15: Demonstrating Invalidity Using Truth Tables**
- Argument is invalid if there's at least one assignment where premises are true, but conclusion is false.

**Slide 16: Demonstrating Invalidity Using Truth Table (Example)**
- Argument: p ⇔ q; p ⇒ r; Therefore, p – Invalid argument.

| p | q | r | p ⇔ q | p ⇒ r |
|---|---|------|-------|--------|
| T | T | T   |   T   |     T  |
| F | T | F   |   F   |     F  |

- Argument is invalid because there's a row where premises are true, but conclusion (p) is false.

**Slide 17: Exercise**
- Demonstrate the invalidity of the argument: p ∨ q; ¬p; Therefore, ¬q.

**Slide 18: Solution to Exercise**
- Atomic Propositions: p, q.

| p | q | p ∨ q | ¬p |
|---|---|------|-----|
| F | T |   T  |   T  |

- Argument is invalid because there's a row where premises are true, but conclusion (¬q) is false.

**Slide 19: A Problem with Truth Tables**
- Using truth tables to establish validity becomes tedious as the number of variables increases.

**Slide 20: Deductive Proofs**
- Approach to establishing validity using a series of simpler arguments known to be valid.
- Uses laws of logic (logical equivalences) and inference rules.

**Slide 21: Inference Rules**
- Primitive valid argument forms eliminating or introducing logical connectives.
- Categories: Intro (introduces connective), Elim (eliminates connective).

**Slide 22: The Layout of an Inference Rule**
- Premises (above the line): List of formulas already in proof.
- Conclusion (below the line): What may be deduced by applying the inference rule.

**Slide 23: Conjunction (∧Intro)**
- Introduces the connective ∧.
- Example: p, q; Therefore, p ∧ q.

**Slide 24: Simplification (∧Elim)**
- Eliminates the connective ∧.
- Example: p ∧ q; Therefore, p.

**Slide 25: Addition (∨Intro)**
- Introduces the connective ∨.
- Example: p; Therefore, p ∨ q.

**Slide 26: Exercise on Disjunctive Syllogism**
- Demonstrate the validity of the inference rule using a truth table.

**Slide 27: Solution to Exercise**
- Atomic Propositions: p, q.

| p | q | ¬p |
|---|---|-----|
| F | T |   T  |

- Argument is valid because conclusion (q) is always true when premises are true.

**Slide 28: Modus Ponens (⇒Elim)**
- Eliminates the connective ⇒.
- Example: p ⇒ q; p; Therefore, q.

**Slide 29: Modus Tollens (⇒Elim)**
- Eliminates the connective ⇒.
- Example: p ⇒ q; ¬q; Therefore, ¬p.

**Slide 30: Other Inference Rules**
- Double Negation (¬Elim): ¬¬p; Therefore, p.
- Laws of Equivalence (⇔Elim): p ⇔ q; Therefore, p ⇒ q and q ⇒ p.

**Slide 31: Transitive Inference Rules**
- Transitivity of Equivalence: If p ≡ q and q ≡ r, then p ≡ r.
- Hypothetical Syllogism: If p ⇒ q and q ⇒ r, then p ⇒ r.

**Slide 32: Summary**
- Valid arguments: Conclusion always true when premises are true.
- Invalid arguments: At least one assignment where premises are true, but conclusion is false.
- Truth tables demonstrate invalidity.
- Inference rules deduce true propositions from other true propositions.

**Slide 33: Reading and References**
- Russell, Norvig (2022). Artificial Intelligence. 4th Edition.
- Nissanke (1999). Introductory Logic and Sets for Computer Scientists.
- Gray (1984). Logic, Algebra and Databases.