Transcript Quantificational Logic

Logic with quantifiers
aka
First-Order Logic
Predicate Logic
Quantificational Logic
7/12/2016
Harry Lewis/CS20/CSCI E-120/with thanks
to Albert R. Meyer
1
Predicates
•
•
•
•
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
7/12/2016
Harry Lewis/CS20/CSCI E-120/with thanks
to Albert R. Meyer
2
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
7/12/2016
Harry Lewis/CS20/CSCI E-120/with thanks
to Albert R. Meyer
3
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 True
P(6,-6)⋀¬P(1,2) is
7/12/2016
Harry Lewis/CS20/CSCI E-120/with thanks
to Albert R. Meyer
4
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 True
P(6,-6)⋀¬P(1,2) is True
7/12/2016
Harry Lewis/CS20/CSCI E-120/with thanks
to Albert R. Meyer
5
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
• Let R(x,y) := “x≥0 ⋀ x+y=0”
– Then ∀y∃x (R(x,y) ⋁ R(y,x)) is …?
• ∀y ∃x ((x≥0 ⋀ x+y=0) ⋁ (y≥0 ⋀ y+x=0)): True!
7/12/2016
Harry Lewis/CS20/CSCI E-120/with thanks
to Albert R. Meyer
6
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 …
7/12/2016
Harry Lewis/CS20/CSCI E-120/with thanks
to Albert R. Meyer
7
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)) ?
7/12/2016
Harry Lewis/CS20/CSCI E-120/with thanks
to Albert R. Meyer
8
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))
⤳ ∃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”
7/12/2016
Harry Lewis/CS20/CSCI E-120/with thanks
to Albert R. Meyer
9