Discrrete mathematics for Computer Science 08QLogic
Logic with quantifiers
aka
First-Order Logic
Predicate Logic
Quantificational Logic is a proposition with variables
Predicates
- A
predicate
- For example: P(x,y) := “x+y=0”
- (For today, universe is Z = all integers)
- P(-4,3) is
Predicates
- A
predicate
is a proposition with variables
- For example: P(x,y) := “x+y=0”
- P(-4,3) is False •
P(5,-5) is is a proposition with variables
Predicates
- A
predicate
- 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
Predicates
- A
predicate
- 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 ⋀
Quantifiers
- ∀ x Q(x) := “for all x, Q(x)”
That is, Q(x) holds for each and every value of x
- – • ∃ x Q(x) := “for some x, Q(x)”
That is, Q(x) holds for at least one value of x
- – Let Q(x) := “x-7=0”
- ∃
- – ∀ x Q(x) is false but x Q(x) is true
- – ∀ ∃ ⋁ Then y x (R(x,y) R(y,x)) is …?
- ∀ ∃ ⋀ ⋁ ⋀ y x ((x≥0 x+y=0) (y≥0 y+x=0)): True!
⋀ Let R(x,y) := “x≥0 x+y=0”
∃
Quantifiers
- ∀ is AND-like and is OR-like
- If the universe is {Alice, Bob, Carol} then
- – ∀ x Q(x) is the same as Q(Alice) Q(Bob) Q(Carol) ⋀ ⋀
- – ∃ x Q(x) is the same as Q(Alice) Q(Bob) Q(Carol) ⋁ ⋁
- In general the universe is infinite …
Rhetoric 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)) ? ⇒
⤳ ∃ x ￢ (Glitters(x) Gold(x)) ⇒ ⤳ ∃ x ￢ ( ￢ Glitters(x) Gold(x)) rewriting “ ”
∨
Negation 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)) by DeMorgan and double negation “There is something that glitters and is not gold”