Discrrete mathematics for computer science 08QLogic

Logic 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

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Predicates

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3/22/19

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

Harry Lewis/CS20/CSCI E-120/with thanks to Albert R. Meyer

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Predicates

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3/22/19

A predicate is a proposition with variables
For example: P(x,y) := “x+y=0”
P(-4,3) is False
P(5,-5) is

Harry Lewis/CS20/CSCI E-120/with thanks to Albert R. Meyer

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Predicates

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3/22/19

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

Harry Lewis/CS20/CSCI E-120/with thanks to Albert R. Meyer

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Predicates

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•
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•

3/22/19

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

Harry Lewis/CS20/CSCI E-120/with thanks to Albert R. Meyer

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Quantifiers

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∀x Q(x) := “for all x, Q(x)”

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∃x Q(x) := “for some x, Q(x)”

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Let Q(x) := “x-7=0”

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Let R(x,y) := “x≥0 ⋀ x+y=0”

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∀y ∃x ((x≥0 ⋀ x+y=0) ⋁ (y≥0 ⋀ y+x=0)): True!

– That is, Q(x) holds for each and every value of x
– That is, Q(x) holds for at least one value of x
– ∀x Q(x) is false but ∃x Q(x) is true
– Then ∀y∃x (R(x,y) ⋁ R(y,x)) is …?

3/22/19

Harry Lewis/CS20/CSCI E-120/with thanks to Albert R. Meyer

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Quantifiers

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∀ 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)

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3/22/19

In general the universe is infinite …

Harry Lewis/CS20/CSCI E-120/with thanks to Albert R. Meyer

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Rhetoric and Quantifiers

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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)?

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∃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

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Negation and Quantifiers

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￢∀ x P(x) ≡ ∃x ￢ P(x)

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￢∀ x (Glitters(x) ⇒ Gold(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)) 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

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