Session 19

Translate the following English sentences to First order logic

Every student is an academic. 
Everybody who teaches an academic is an academic. 
Every student has at least one class.
Anyone taking a class is a student.
Jeff teaches Fred.
Fred is taking a class.

Use Modus Ponens or inference to demonstrate that "Jeff is an academic" Justify your work

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 the 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 in the 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)

Fun with sentences

An Exercise: Logic as Representation

Represent the following sentences in logic.

One more outburst like that and you’ll be in contempt of court!
A new ER is on TV tonight, if you’re interested.
Either the Yankees win the pennant, or I’m out $10.00.
Maybe I’ll come to your party, and maybe I won’t.
Well, I like Bob and I don’t like Bob.

Solutions?

1. One more outburst like that and you’ll be in contempt of court!

Straightforward translation:

haveOutburst(You) and inContempt(You)

Consequence:

inContempt(You) must be true!

Intended meaning:

haveOutburst(You) implies inContempt(You)

2. ER is on TV tonight, if you’re interested.

Straightforward translation:

interested( You, ER ) implies onTubeTonight( ER )

Consequence:

What if I am not interested?

Intended meaning:

onTubeTonight( ER ) and ( interested( You, ER )

implies canWatchTonight( You,ER ) )

3. Either the Yankees win the pennant, or I’m out $10.00.

Straightforward translation:

win( Yankees, pennant-of( AL, 2004 ) ) or lose(JBS, $10)

Consequence: What if the Yankees win? Where is the causality?

Intended meaning:

prevent( win(Yankees, pennant-of( AL, 2004 ) ),

lose(JBS, $10) )

4. Maybe I’ll come to your party, and maybe I won’t.

Straightforward translation:

( maybe( comeTo( JBS, YourParty ) ) ) or

( maybe( not comeTo( JBS, YourParty ) ) )

Consequence: The translation is a tautology!

Intended meaning:

undecided comeTo( JBS, YourParty )

Consequence: We need an operator that deals with possibility...

5. Well, I like Bob and I don’t like Bob.

Straightforward translation:

like( JBS, Bob ) and not like(JBS, Bob )

Consequence: The translation is a contradiction!

Intended meaning:

way1, way2

like( JBS, Bob, way1 ) and

not like(JBS, Bob, way2 ) )

Consequence: We need to represent states explicitly.