Session One

Session 18 - Monday, October 1st

First Order Logic

A key to predicate logic we have been ignoring is the inclusion of quantifiers

You should recall from discrete that you can also write statements such as

" x fast(x) implies horse(x)

$ y fast(y) implies valuable(y)

Represent the following sentences in predicate logic with quantifiers

Schafer is (was) a Hoosier
Hoosiers like basketball.
Children of basketball fans are basketball fans.
Basketball fans like the month of March.
Margaret is Schafer's daughter

Last time we looked at representing the following sentences in predicate logic

Schafer is a Hoosier - hoosier(Schafer)
Hoosiers like basketball - "x hoosier(x) -> likes (x,Basketball)
Children of basketball fans are basketball fans. "x,y childOf(x,y) AND likes (y, Basketball) -> likes (x,Basketball)
Basketball fans like the month of March. "y likes(y,Basketball) -> likes(y,March)
Margaret is Schafer's daughter daughterOf(Margaret,Schafer)

Rules of Inference

Two common rules for reasoning over predicate logic representations rely on a simple rule from propositional logic involving implication, extended for reasoning with variables:

Modus Ponens (p ^ (p->q) -> q )

man(Marcus) p

"x man(x) implies person(x) p -> q

person(Marcus) q

Modus Tolens ( q' ^ (p->q) -> p' )

not( person(Lassie) ) q'

"x man(x) implies person(x) p -> q

not( man(Lassie) ) not p

Remember:

Not all statements we make about the world are true.
The "name" of a symbol does not matter in reasoning with the symbol. Only the semantics attached to it matter.
" is the Universal quantifier. It means "for all"
- "x P is true in a model m iff P is true with x being each possible object in the world.
- Roughly speaking, is the equivalent of th conjunction of instantiations of P
  - P(a) ^ P(b) ^ P(c) ....
- Typically, -> is the main connective with "
  - "x At(x,UNI) -> Smart(x) [Everyone at UNI is smart]
- Common mistake is to use ^ as the connective with "
  - "x At(x,UNI) ^ Smart(x) [Everyone is at UNI and everyone is smart]
$ is the Existential quantifier. It means "there exists"
- $ x P is true in a model m iff P is true with x being one possible object in the world.
- Roughly speaking, is equivalent of the disjunction of instantiations of P
  - P(a) V P(b) V P(c) ....
- Typically, ^ is the main connective with $
  - $x At(x,Iowa) ^ Smart(x) [Someone at Iowa is smart]
- Common mistake is to use -> as the connective with $
  - $x At(x,Iowa) -> Smart(x) [This is true if there is anyone not at Iowa]
" x " y is the same as " y " x
$ x $ y is the same as $ y $ x
$ x " y is not the same as " y $ x
- $ x " y Loves (x,y) ["There is a person who loves everyone in the world"]
- " y $ x Loves (x,y) ["Everyone int eh world is loved by at least one person"
Quantifier duality: each can be expressed using the other
- " x Likes(x,IceCream) ¬$ x ¬Likes(x,IceCream)
- $ x Likes(x,Broccoli) ¬" x ¬Likes(x,Broccoli)