Discrrete mathematics for Computer Science 08QLogic

Logic with quantifiers

aka

First-Order Logic

Predicate Logic

Quantificational Logic

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}

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}

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}

_⋀

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}

_⋀

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)”}

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)) ?

⇒

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”