Logic with quantifiersaka First-Order Logic Predicate Logic Quantificational Logic 3/22/19 Harry Lewis/CS20/CSCI E-120/with thanks to Albert R.
Trang 1Logic with quantifiers
aka First-Order Logic Predicate Logic Quantificational Logic
3/22/19 Harry Lewis/CS20/CSCI E-120/with thanks to Albert R Meyer 1
Trang 2• A predicate is a proposition with variables
• For example: P(x,y) := “x+y=0”
• (For today, universe is Z = all integers)
• P(-4,3) is
3/22/19 Harry Lewis/CS20/CSCI E-120/with thanks to Albert R Meyer 2
Trang 3• A predicate is a proposition with variables
• For example: P(x,y) := “x+y=0”
• P(-4,3) is False
• P(5,-5) is
3/22/19 Harry Lewis/CS20/CSCI E-120/with thanks to Albert R Meyer 3
Trang 4• A predicate is a proposition with variables
• For example: P(x,y) := “x+y=0”
• P(-4,3) is False
• P(5,-5) is True
• P(6,-6)⋀¬P(1,2) is
3/22/19 Harry Lewis/CS20/CSCI E-120/with thanks to Albert R Meyer 4
Trang 5• A predicate is a proposition with variables
• For example: P(x,y) := “x+y=0”
• P(-4,3) is False
• P(5,-5) is True
• P(6,-6)⋀¬P(1,2) is True
3/22/19 Harry Lewis/CS20/CSCI E-120/with thanks to Albert R Meyer 5
Trang 6• ∀x Q(x) := “for all x, Q(x)”
• ∃x Q(x) := “for some x, Q(x)”
• Let Q(x) := “x-7=0”
• Let R(x,y) := “x≥0 ⋀ x+y=0”
• ∀y ∃x ((x≥0 ⋀ x+y=0) ⋁ (y≥0 ⋀ y+x=0)): True!
3/22/19 Harry Lewis/CS20/CSCI E-120/with thanks to Albert R Meyer 6
Trang 7• ∀ is AND-like and ∃ is OR-like
• If the universe is {Alice, Bob, Carol} then
Q(Alice) ⋀ Q(Bob) ⋀ Q(Carol)
Q(Alice) ⋁ Q(Bob) ⋁ Q(Carol)
• In general the universe is infinite …
3/22/19 Harry Lewis/CS20/CSCI E-120/with thanks to Albert R Meyer 7
Trang 8Rhetoric and Quantifiers
• Let Loves(x,y) := “x loves y”
• “Everybody loves Oprah”: ∀x Loves(x, Oprah)
• What does “Everybody loves somebody” mean?
∀x∃y Loves(x,y)?
∃y∀x Loves(x,y)?
• “All that glitters is not gold”
∀x (Glitters(x) ⇒ ￢ Gold(x)) ?
￢∀ x (Glitters(x) ⇒ Gold(x)) ?
3/22/19 Harry Lewis/CS20/CSCI E-120/with thanks to Albert R Meyer 8
Trang 9Negation and Quantifiers
• ￢∀ x P(x) ≡ ∃x ￢ P(x)
• ￢∃ x P(x) ≡ ∀x ￢ P(x)
• So negation signs can be pushed in to the predicates but the
quantifiers flip
• ￢∀ x (Glitters(x) ⇒ Gold(x))
∃
⤳ x ￢ (Glitters(x) ⇒ Gold(x))
∃
⤳ x ￢ ( ￢ Glitters(x) ∨ Gold(x)) rewriting “⇒”
∃
⤳ x (Glitters(x) ⋀ ￢ Gold(x)) by DeMorgan and double negation
“There is something that glitters and is not gold”
3/22/19 Harry Lewis/CS20/CSCI E-120/with thanks to Albert R Meyer 9