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boris
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1. Just the following propositions hold true:
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1. male(Ahmed) male(Patel) male(Scott) tall(Ahmed) tall(Patel) short(Khan) short(Scott)
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Evaluate the truth of the following formula
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| x | tall(x) | male(x) | short(x) | $\lnot$short(x) | male(x) $\land \lnot$short(x) | tall(x) $\iff$ male(x) $\land \lnot$short(x) | $\forall x \bullet$tall(x) $\iff$ male(x) $\land \lnot$short(x) |
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| --- | ------- | ------- | -------- | --------------- | ----------------------------- | -------------------------------------------- | --------------------------------------------------------------- |
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| Ah | T | T | F | T | T | T | |
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| Kh | F | F | T | F | F | T | |
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| Pa | T | T | F | T | T | T | |
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| Sc | F | T | T | F | F | T | |
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| | | | | | | | T |
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2. Using appropriate binary predicates, express each of the following sentences in predicate logic
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a) Salford stores only supply stores outside of Salford.
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- $\forall x \bullet \forall y \bullet \textsf{in(x, Salford)} \land \textsf{supples}$
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b) No store supplies itself
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c) There are no stores in Eccles but there are some in Trafford.
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d) Stores do not supply stores that are supplied by stores which they supply
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e) Stores which supply each other are always in the same place
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@@ -0,0 +1,171 @@
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**Slide 1: Learning Objectives**
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- Understand when the order of quantifiers is important.
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- Understand how ∀ and ∃ are connected.
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- Use and remember scoping rules.
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- Identify bound and free variables in formulae.
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- Establish the truth of formulae in predicate logic.
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- Understand why predicate logic is described as undecidable.
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- Understand the difference between zero-order, first-order, and higher-order predicate logics.
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**Slide 2: Objectives & Recap**
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- Quantifiers and their alternative view.
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- Distributive laws of quantifiers.
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- De Morgan's laws with quantifiers.
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- Scope of quantifiers.
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- Bound and free variables.
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**Slide 3: The Universal Quantifier (∀)**
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- Pronounced as "for all".
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- Example: ∀x (human(x) ⇒ mortal(x)) = All humans are mortal.
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- In terms of conjunction: ∀x p(x) ≡ p(x₁) ∧ p(x₂) ∧... ∧ p(xₙ)
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**Slide 4: The Existential Quantifier (∃)**
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- Pronounced as "there exists" or "for some".
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- Example: ∃x (human(x) ∧ happy(x)) = Some humans are happy.
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- In terms of disjunction: ∃x p(x) ≡ p(x₁) ∨ p(x₂) ∨... ∨ p(xₙ)
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**Slide 5: Distributive Laws of Quantifiers**
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- ∀x (p(x) ∧ q(x)) ≡ (∀x p(x)) ∧ (∀x q(x))
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- ∃x (p(x) ∨ q(x)) ≡ (∃x p(x)) ∨ (∃x q(x))
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**Slide 6: The Order of Quantification**
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- Example 1 (Everyone loves someone): ∀x ∃y loves(x, y)
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- Example 2 (There is someone loved by everyone): ∃y ∀x loves(x, y)
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**Slide 7: De Morgan's Laws and Quantifiers**
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- ¬∃x p(x) ≡ ∀x ¬p(x)
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- ¬∀x p(x) ≡ ∃x ¬p(x)
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**Slide 8: Scope of Quantifiers**
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- In absence of brackets, scope extends to the end of the formula.
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- Brackets can enforce different scoping patterns.
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**Slide 9: Bound and Free Variables**
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- Bound variable: occurrence introduced by a quantifier within its scope.
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- Example:
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```
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Formula Variable status
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child(x) Only one occurrence of x, free.
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∀x child(x) ∧ clever(x) Both occurrences of x are bound by the same quantifier.
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((∀x child(x)) ∧ clever(x)) The first occurrence of x is bound, the second is free.
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```
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- Free variable: occurrence not within any quantifier's scope.
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**Slide 10: Meaning of Bound Variables**
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- The meaning of a bound variable does not depend on its name.
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- Example:
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```
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∀x (child(x) ∧ clever(x)) ⇒ ∃y loves(y, x)
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∀B (child(B) ∧ clever(B)) ⇒ ∃C loves(C, B)
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```
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**Slide 11: Meaning of Free Variables**
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- Free variables denote unknowns or unspecified objects.
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- Example:
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```
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∀x (child(x) ∧ clever(x)) ⇒ x is loved.
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∀x (child(x) ∧ clever(x)) ⇒ z is loved.
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```
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**Slide 12: Exercise**
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- Identify bound and free variables in the formula:
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```
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∃x taller(y, x) ∃x ∃y taller(x, y) ∧ taller(x, z)
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```
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- Solution:
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- In the first formula: x is bound, y is free.
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- In the second formula: y is bound by ∃y, z is free. Both occurrences of x are bound and refer to the same variable.
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**Slide 13: The Equality Symbol (=)**
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- ⊢ Richard has at least two brothers:
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```
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∃x ∃y (brother(x, richard) ∧ brother(y, richard) ∧ ¬(x = y))
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```
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- Definition of sibling using parent:
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```
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∀x ∀y sibling(x, y) ≡ (¬(x = y) ∧ ∃m ∃f ¬(m = f) ∧ parent(m, x) ∧ parent(f, x) ∧ parent(m, y) ∧ parent(f, y))
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```
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**Slide 14: Establishing the Truth Values of Formulae**
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- Example (slide 21 and 22):
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- Individuals: Ahmed, Khan, Patel, Scott.
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- Properties: male, tall, short.
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- Formula: ∀x (male(x) ⇒ tall(x) ∨ short(x))
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- Truth table shows the formula is false (Patel is male but not tall or short).
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**Slide 15: Exercise**
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- Given individuals, properties, and true propositions as in slide 23.
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- Evaluate the truth of: ∀x ¬male(x) ⇒ short(x)
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- Solution (slide 24):
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```
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x male(x) ¬male(x) short(x) ¬male(x) ⇒ short(x)
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Ahmed T F F F T
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Khan F T T T T
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Patel T F F F T
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Scott T F T F F
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```
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- The formula is false (Patel is male but not short).
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**Slide 16: Example Involving ∃**
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- Given individuals, properties, and true propositions as in slide 25.
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- Formula: ∃x (male(x) ∧ ¬tall(x) ⇒ short(x))
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- Truth table (slide 26):
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```
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x male(x) ¬tall(x) short(x) ¬tall(x) ⇒ short(x) male(x) ∧ ¬tall(x) ⇒ short(x)
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Ahmed T F F F T T
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Khan F F T F F F
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Patel T F F F F F
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Scott T T T F T F
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```
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- The formula is true (Scott is male, not tall but short).
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**Slide 17: Exercise**
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- Given individuals, properties, and true propositions as in slide 27.
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- Evaluate the truth of: ∃x (male(x) ∧ (tall(x) ∨ ¬short(x)))
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- Solution (slide 28):
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```
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x male(x) tall(x) short(x) ¬short(x) tall(x) ∨ ¬short(x) male(x) ∧ (tall(x) ∨ ¬short(x))
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Ahmed T T F T T T T
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Khan F F T F F F F
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Patel T T F F T T F
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Scott T F T F F F F
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```
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- The formula is true (Ahmed and Patel are males, tall or not short).
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**Slide 29: Predicate Logic is Undecidable**
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- Universal quantification introduces computational impossibility when testing truth values with an infinite number of possible values.
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**Slide 30: First-order Predicate Logic**
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- Quantifiers refer only to objects (constants), not predicate or function names.
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- Propositional logic is zero-order logic.
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**Slide 31: Summary**
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- Order of quantifiers matters when both ∀ and ∃ are present.
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- Quantifiers are connected through negation, obey De Morgan's laws.
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- Scope of quantifiers extends to the end of the formula without brackets.
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- Bound variables are introduced by a quantifier within its scope; free variables are not within any quantifier's scope.
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**Slide 32: Reading, References and Acknowledgements**
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- Reading from Artificial Intelligence textbook by Russell and Norvig.
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- References: Introductory Logic and Sets for Computer Scientists by Nissanke.
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