review : quantifiers a key to predicate logic we have been ignoring is the inclusion of quantifiers...

REVIEW : Quantifiers

• 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)

Rules of Inference

• Modus Ponens p ^ (p -> q) -> q

• In other words, if you have an implication rule in the knowledge base AND the left hand side of the implication rule (the antecedent) then you can infer the right hand side of the rule (the consequent)

Rules of Inference

• Modus Tolens q’ ^ (p -> q) -> p’

• In other words, if you have an implication rule in the knowledge base AND the negation of the consequent than you can imply the negation of the antecedent

CAUTION

• Do not try to use either p’ ^ (p -> q) -> q’

(If today is not my birthday than I can infer that I won’t get presents??? Not true!)

q ^ (p -> q) -> p

(If I get presents today it must be my birthday)

Consider the following

Using rules of inference, what can you add to this knowledge base?

For each new fact, document how it was infered.

HW#4 Assigned today

• Short paper pencil activity converting English to Predicate Logic and then using Modus Ponens and inference to prove a claim (fact).

• Due on Monday.

Some things to keep in mind

• The “name” of a symbol does not matter in reasoning with the symbol.

• Only the semantics attached to the name matter.

Some things to keep in mind

is the Universal quantifier.  It means "for all"

x P(x)  is true in a model m iff P is true with x being each possible object in the world.

• Roughly speaking, it is the equivalent of the conjunction of instantiations of PP(a) ^ P(b) ^ P(c) ....

Some things to keep in mind

• 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]

Some things to keep in mind

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 PP(a) V P(b) V P(c) ....

Some things to keep in mind

• 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]

Some things to keep in mind

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"

Some things to keep in mind

• 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))

Try writing the following:

• Brothers are Siblings

• Sibling is reflexive

• One’s mother is one’s female parent

• A first cousin is a child of a parent’s sibling

Try writing the following

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

• Straightforward Answer:haveOutburst(You) and inContempt(You)

• Consequence:inContempt(You) must be true!

• Intended Meaning:haveOutburst(You) implies inContempt(You)

The Walking Dead is on TV tonight if you’re interested

• Straightforward Answer: interested( You, The Walking Dead ) implies onTubeTonight( The Walking Dead )

• Consequence:What if I am not interested?

• Intended Meaning:onTubeTonight( The Walking Dead ) ^

(interested( You, The Walking Dead ) implies canWatchTonight( You,The Walking Dead ) )

Either the Steelers win the Super Bowl, or I’m out $10.00.

• Straightforward Answer:win( Steelers, SuperBowl(2014) ) or  lose(JBS, $10)

• Consequence:What if the Steelers win?  Where is the causality?

• Intended Meaning:Prevent( win(Steelers, SuperBowl(2014) ) , lose(JBS, $10) )

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

• Straightforward Answer:

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

(maybe( not comeTo( JBS, YourParty ) ) )• Consequence:

That’s a tautology

We need an operator that deals with possibility...

• Intended Meaning:Undecided comeTo(JBS,YourParty)

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

• Straightforward Answer:

like(JBS,Bob) ^ not like(JBS,Bob)

• Consequence:That’s a contradiction

We need to represent states explicitly.

• Intended Meaning: way1, way2 like(JBS, Bob, way1) ^ like(JBS,Bob,way2)