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AI & Data Mining/Week 22/Chapter 22 Validity and Inference Rules.md

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+1. A syllogism is an instance of a form of reasoning in which a conclusion is drawn from two given or assumed propositions; a common or middle term is present in the two premises but not in the conclusion, which may be invalid.
+2. Aristotle
+
+**Double Negation ¬ Elim ¬ ¬ p p**
+
+| Propositions | Premises     | Conclusion |
+| ------------ | ------------ | ---------- |
+| p            | $\neg\neg p$ | p          |
+| T            | T            | T          |
+| F            | F            | F          |
+
+**Hypothetical syllogism; this says that if p implies q and q implies r, then it can be logically concluded that p implies r. p ⇒ q q ⇒ r p ⇒ r**
+
+| Propositions |     |     | Premises       |                | Conclusion     |
+| ------------ | --- | --- | -------------- | -------------- | -------------- |
+| p            | q   | r   | $p \implies q$ | $q \implies r$ | $p \implies r$ |
+| T            | T   | T   | T              | T              | T              |
+| T            | T   | F   | T              | F              |                |
+| T            | F   | T   | F              | T              |                |
+| T            | F   | F   | F              | T              |                |
+| F            | T   | T   | T              | T              | T              |
+| F            | T   | F   | T              | F              |                |
+| F            | F   | T   | T              | T              | T              |
+| F            | F   | F   | T              | T              | T              |
+
+1. Involves linking implications together in a sequential manner, much like the links in a chain.
+
+**p ∨ q q Therefore, p**
+
+| Propositions |     | Premises   |     | Conclusion |
+| ------------ | --- | ---------- | --- | ---------- |
+| $p$          | $q$ | $p \lor q$ | $q$ | p          |
+| T            | T   | T          | T   | T          |
+| T            | F   | T          | F   | T          |
+| F            | T   | T          | T   | F          |
+| F            | F   | F          | F   | F          |
+
+**p ⇒ q q ⇒ p Therefore, p ∧ q**
+
+| Propositions |     | Premises       |                | Conclusion  |
+| ------------ | --- | -------------- | -------------- | ----------- |
+| p            | q   | $p \implies q$ | $q \implies p$ | $p \land q$ |
+| T            | T   | T              | T              | T           |
+| T            | F   | F              | T              | F           |
+| F            | T   | T              | F              | F           |
+| F            | F   | T              | T              | F           |
+
+$p \implies q$
+$r \implies s$
+$p \lor r$ (p disjunction (or) r)
+Conclusion: $q \lor s$
+
+The "Constructive Dilemma": If the disjunction of the antecedent of two implications holds then the disjunction of the conclusions also must hold

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

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+
+
+**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.