G4G0-2/AI & Data Mining/Week 22/Week 22 Validity and Inference Rules.md

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.