Discrrete mathematics for Computer Science 08QLogic

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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
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 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”

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