**Slide 1: Learning Objectives**

- Understand when the order of quantifiers is important.
- Understand how ∀ and ∃ are connected.
- Use and remember scoping rules.
- Identify bound and free variables in formulae.
- Establish the truth of formulae in predicate logic.
- Understand why predicate logic is described as undecidable.
- Understand the difference between zero-order, first-order, and higher-order predicate logics.

**Slide 2: Objectives & Recap**

- Quantifiers and their alternative view.
- Distributive laws of quantifiers.
- De Morgan's laws with quantifiers.
- Scope of quantifiers.
- Bound and free variables.

**Slide 3: The Universal Quantifier (∀)**

- Pronounced as "for all".
- Example: ∀x (human(x) ⇒ mortal(x)) = All humans are mortal.
- In terms of conjunction: ∀x p(x) ≡ p(x₁) ∧ p(x₂) ∧... ∧ p(xₙ)

**Slide 4: The Existential Quantifier (∃)**

- Pronounced as "there exists" or "for some".
- Example: ∃x (human(x) ∧ happy(x)) = Some humans are happy.
- In terms of disjunction: ∃x p(x) ≡ p(x₁) ∨ p(x₂) ∨... ∨ p(xₙ)

**Slide 5: Distributive Laws of Quantifiers**

- ∀x (p(x) ∧ q(x)) ≡ (∀x p(x)) ∧ (∀x q(x))
- ∃x (p(x) ∨ q(x)) ≡ (∃x p(x)) ∨ (∃x q(x))

**Slide 6: The Order of Quantification**

- Example 1 (Everyone loves someone): ∀x ∃y loves(x, y)
- Example 2 (There is someone loved by everyone): ∃y ∀x loves(x, y)

**Slide 7: De Morgan's Laws and Quantifiers**

- ¬∃x p(x) ≡ ∀x ¬p(x)
- ¬∀x p(x) ≡ ∃x ¬p(x)

**Slide 8: Scope of Quantifiers**

- In absence of brackets, scope extends to the end of the formula.
- Brackets can enforce different scoping patterns.

**Slide 9: Bound and Free Variables**

- Bound variable: occurrence introduced by a quantifier within its scope.
  - Example:
    ```
    Formula             Variable status
    child(x)            Only one occurrence of x, free.
    ∀x child(x) ∧ clever(x) Both occurrences of x are bound by the same quantifier.
    ((∀x child(x)) ∧ clever(x)) The first occurrence of x is bound, the second is free.
    ```
- Free variable: occurrence not within any quantifier's scope.

**Slide 10: Meaning of Bound Variables**

- The meaning of a bound variable does not depend on its name.
  - Example:
    ```
    ∀x (child(x) ∧ clever(x)) ⇒ ∃y loves(y, x)
    ∀B (child(B) ∧ clever(B)) ⇒ ∃C loves(C, B)
    ```

**Slide 11: Meaning of Free Variables**

- Free variables denote unknowns or unspecified objects.
  - Example:
    ```
    ∀x (child(x) ∧ clever(x)) ⇒ x is loved.
    ∀x (child(x) ∧ clever(x)) ⇒ z is loved.
    ```

**Slide 12: Exercise**

- Identify bound and free variables in the formula:
  ```
  ∃x taller(y, x) ∃x ∃y taller(x, y) ∧ taller(x, z)
  ```
- Solution:
  - In the first formula: x is bound, y is free.
  - In the second formula: y is bound by ∃y, z is free. Both occurrences of x are bound and refer to the same variable.

**Slide 13: The Equality Symbol (=)**

- ⊢ Richard has at least two brothers:
  ```
  ∃x ∃y (brother(x, richard) ∧ brother(y, richard) ∧ ¬(x = y))
  ```
- Definition of sibling using parent:
  ```
  ∀x ∀y sibling(x, y) ≡ (¬(x = y) ∧ ∃m ∃f ¬(m = f) ∧ parent(m, x) ∧ parent(f, x) ∧ parent(m, y) ∧ parent(f, y))
  ```

**Slide 14: Establishing the Truth Values of Formulae**

- Example (slide 21 and 22):
  - Individuals: Ahmed, Khan, Patel, Scott.
  - Properties: male, tall, short.
  - Formula: ∀x (male(x) ⇒ tall(x) ∨ short(x))
  - Truth table shows the formula is false (Patel is male but not tall or short).

**Slide 15: Exercise**

- Given individuals, properties, and true propositions as in slide 23.
- Evaluate the truth of: ∀x ¬male(x) ⇒ short(x)
- Solution (slide 24):
  ```
  x     male(x)    ¬male(x)   short(x)   ¬male(x) ⇒ short(x)
  Ahmed    T      F          F         F        T
  Khan     F      T          T         T        T
  Patel    T      F          F         F        T
  Scott    T      F          T         F        F
  ```
- The formula is false (Patel is male but not short).

**Slide 16: Example Involving ∃**

- Given individuals, properties, and true propositions as in slide 25.
- Formula: ∃x (male(x) ∧ ¬tall(x) ⇒ short(x))
- Truth table (slide 26):
  ```
  x     male(x)   ¬tall(x)    short(x)   ¬tall(x) ⇒ short(x)   male(x) ∧ ¬tall(x) ⇒ short(x)
  Ahmed    T      F          F         F        T            T
  Khan     F      F          T         F        F            F
  Patel    T      F          F         F        F            F
  Scott    T      T          T         F        T            F
  ```
- The formula is true (Scott is male, not tall but short).

**Slide 17: Exercise**

- Given individuals, properties, and true propositions as in slide 27.
- Evaluate the truth of: ∃x (male(x) ∧ (tall(x) ∨ ¬short(x)))
- Solution (slide 28):
  ```
  x     male(x)   tall(x)    short(x)   ¬short(x)   tall(x) ∨ ¬short(x)   male(x) ∧ (tall(x) ∨ ¬short(x))
  Ahmed    T      T          F         T            T             T            T
  Khan     F      F          T         F            F             F            F
  Patel    T      T          F         F            T             T            F
  Scott    T      F          T         F            F             F            F
  ```
- The formula is true (Ahmed and Patel are males, tall or not short).

**Slide 29: Predicate Logic is Undecidable**

- Universal quantification introduces computational impossibility when testing truth values with an infinite number of possible values.

**Slide 30: First-order Predicate Logic**

- Quantifiers refer only to objects (constants), not predicate or function names.
- Propositional logic is zero-order logic.

**Slide 31: Summary**

- Order of quantifiers matters when both ∀ and ∃ are present.
- Quantifiers are connected through negation, obey De Morgan's laws.
- Scope of quantifiers extends to the end of the formula without brackets.
- Bound variables are introduced by a quantifier within its scope; free variables are not within any quantifier's scope.

**Slide 32: Reading, References and Acknowledgements**

- Reading from Artificial Intelligence textbook by Russell and Norvig.
- References: Introductory Logic and Sets for Computer Scientists by Nissanke.