Resolution Exercise Solutions

2. Consider the following axioms:

1. Every child loves Santa.
   &forall x (CHILD(x) &rarr LOVES(x,Santa))
2. Everyone who loves Santa loves any reindeer.
   &forall x (LOVES(x,Santa) &rarr &forall y (REINDEER(y) &rarr LOVES(x,y)))
3. Rudolph is a reindeer, and Rudolph has a red nose.
   REINDEER(Rudolph) &and REDNOSE(Rudolph)
4. Anything which has a red nose is weird or is a clown.
   &forall x (REDNOSE(x) &rarr WEIRD(x) &or CLOWN(x))
5. No reindeer is a clown.
   &not &exist x (REINDEER(x) &and CLOWN(x))
6. Scrooge does not love anything which is weird.
   &forall x (WEIRD(x) &rarr &not LOVES(Scrooge,x))
7. (Conclusion) Scrooge is not a child.
   &not CHILD(Scrooge)

3. Consider the following axioms:

1. Anyone who buys carrots by the bushel owns either a rabbit or a grocery store.
   &forall x (BUY(x) &rarr &exist y (OWNS(x,y) &and (RABBIT(y) &or GROCERY(y))))
2. Every dog chases some rabbit.
   &forall x (DOG(x) &rarr &exist y (RABBIT(y) &and CHASE(x,y)))
3. Mary buys carrots by the bushel.
   BUY(Mary)
4. Anyone who owns a rabbit hates anything that chases any rabbit.
   &forall x &forall y (OWNS(x,y) &and RABBIT(y) &rarr &forall z &forall w (RABBIT(w) &and CHASE(z,w) &rarr HATES(x,z)))
5. John owns a dog.
   &exist x (DOG(x) &and OWNS(John,x))
6. Someone who hates something owned by another person will not date that person.
   &forall x &forall y &forall z (OWNS(y,z) &and HATES(x,z) &rarr &not DATE(x,y))
7. (Conclusion) If Mary does not own a grocery store, she will not date John.
   (( &not &exist x (GROCERY(x) &and OWN(Mary,x))) &rarr &not DATE(Mary,John))

4. Consider the following axioms:

1. Every Austinite who is not conservative loves some armadillo.
   &forall x (AUSTINITE(x) &and &not CONSERVATIVE(x) &rarr &exist y (ARMADILLO(y) &and LOVES(x,y)))
2. Anyone who wears maroon-and-white shirts is an Aggie.
   &forall x (WEARS(x) &rarr AGGIE(x))
3. Every Aggie loves every dog.
   &forall x (AGGIE(x) &rarr &forall y (DOG(y) &rarr LOVES(x,y)))
4. Nobody who loves every dog loves any armadillo.
   &not &exist x ((&forall y (DOG(y) &rarr LOVES(x,y))) &and &exist z (ARMADILLO(z) &and LOVES(x,z)))
5. Clem is an Austinite, and Clem wears maroon-and-white shirts.
   AUSTINITE(Clem) &and WEARS(Clem)
6. (Conclusion) Is there a conservative Austinite?
   &exist x (AUSTINITE(x) &and CONSERVATIVE(x))

( ( (not (Austinite x)) (Conservative x) (Armadillo (f x)) )

  ( (not (Austinite x))  (Conservative x)  (Loves x (f x)) )
  ( (not (Wears x))  (Aggie x) )
  ( (not (Aggie x))  (not (Dog y))  (Loves x y) )
  ( (Dog (g x))  (not (Armadillo z))  (not (Loves x z)) )
  ( (not (Loves x (g x)))  (not (Armadillo z))  (not (Loves x z)) )
  ( (Austinite (Clem)) )
  ( (Wears (Clem)) )
  ( (not (Conservative x))  (not (Austinite x)) ) )

5. Consider the following axioms:

1. Anyone whom Mary loves is a football star.
   &forall x (LOVES(Mary,x) &rarr STAR(x))
2. Any student who does not pass does not play.
   &forall x (STUDENT(x) &and &not PASS(x) &rarr &not PLAY(x))
3. John is a student.
   STUDENT(John)
4. Any student who does not study does not pass.
   &forall x (STUDENT(x) &and &not STUDY(x) &rarr &not PASS(x))
5. Anyone who does not play is not a football star.
   &forall x (&not PLAY(x) &rarr &not STAR(x))
6. (Conclusion) If John does not study, then Mary does not love John.
   &not STUDY(John) &rarr &not LOVES(Mary,John)

6. Consider the following axioms:

1. Every coyote chases some roadrunner.
   &forall x (COYOTE(x) &rarr &exist y (RR(y) &and CHASE(x,y)))
2. Every roadrunner who says ``beep-beep'' is smart.
   &forall x (RR(x) &and BEEP(x) &rarr SMART(x))
3. No coyote catches any smart roadrunner.
   &not &exist x &exist y (COYOTE(x) &and RR(y) &and SMART(y) &and CATCH(x,y))
4. Any coyote who chases some roadrunner but does not catch it is frustrated.
   &forall x (COYOTE(x) &and &exist y (RR(y) &and CHASE(x,y) &and &not CATCH(x,y)) &rarr FRUSTRATED(x))
5. (Conclusion) If all roadrunners say ``beep-beep'', then all coyotes are frustrated.
   (&forall x (RR(x) &rarr BEEP(x)) &rarr (&forall y (COYOTE(y) &rarr FRUSTRATED(y)))

( ( (not (Coyote x)) (RR (f x)) )

  ( (not (Coyote x))  (Chase x (f x)) )
  ( (not (RR x))  (not (Beep x))  (Smart x) )
  ( (not (Coyote x))  (not (RR y))  (not (Smart y))  (not (Catch x y)) )
  ( (not (Coyote x))  (not (RR y))  (not (Chase x y)) (Catch x y)
    (Frustrated x) )
  ( (not (RR x))  (Beep x) )
  ( (Coyote (a)) )
  ( (not (Frustrated (a))) ) )

7. Consider the following axioms:

1. Anyone who rides any Harley is a rough character.
   &forall x ((&exist y (HARLEY(y) &and RIDES(x,y))) &rarr ROUGH(x))
2. Every biker rides [something that is] either a Harley or a BMW.
   &forall x (BIKER(x) &rarr &exist y ((HARLEY(y) &or BMW(y)) &and RIDES(x,y)))
3. Anyone who rides any BMW is a yuppie.
   &forall x &forall y (RIDES(x,y) &and BMW(y) &rarr YUPPIE(x))
4. Every yuppie is a lawyer.
   &forall x (YUPPIE(x) &rarr LAWYER(x))
5. Any nice girl does not date anyone who is a rough character.
   &forall x &forall y (NICE(x) &and ROUGH(y) &rarr &not DATE(x,y))
6. Mary is a nice girl, and John is a biker.
   NICE(Mary) &and BIKER(John)
7. (Conclusion) If John is not a lawyer, then Mary does not date John.
   &not LAWYER(John) &rarr &not DATE(Mary,John)

8. Consider the following axioms:

1. Every child loves anyone who gives the child any present.
   &forall x &forall y &forall z (CHILD(x) &and PRESENT(y) &and GIVE(z,y,x) &rarr LOVES(x,z)
2. Every child will be given some present by Santa if Santa can travel on Christmas eve.
   TRAVEL(Santa,Christmas) &rarr &forall x (CHILD(x) &rarr &exist y (PRESENT(y) &and GIVE(Santa,y,x)))
3. It is foggy on Christmas eve.
   FOGGY(Christmas)
4. Anytime it is foggy, anyone can travel if he has some source of light.
   &forall x &forall t (FOGGY(t) &rarr ( &exist y (LIGHT(y) &and HAS(x,y)) &rarr TRAVEL(x,t)))
5. Any reindeer with a red nose is a source of light.
   &forall x (RNR(x) &rarr LIGHT(x))
6. (Conclusion) If Santa has some reindeer with a red nose, then every child loves Santa.
   ( &exist x (RNR(x) &and HAS(Santa,x))) &rarr &forall y (CHILD(y) &rarr LOVES(y,Santa))

9. Consider the following axioms:

1. Every investor bought [something that is] stocks or bonds.
   &forall x (INVESTOR(x) &rarr &exist y ((STOCK(y) &or BOND(y)) &and BUY(x,y)))
2. If the Dow-Jones Average crashes, then all stocks that are not gold stocks fall.
   DJCRASH &rarr &forall x ((STOCK(x) &and &not GOLD(x)) &rarr FALL(x))
3. If the T-Bill interest rate rises, then all bonds fall.
   TBRISE &rarr &forall x (BOND(x) &rarr FALL(x))
4. Every investor who bought something that falls is not happy.
   &forall x &forall y (INVESTOR(x) &and BUY(x,y) &and FALL(y) &rarr &not HAPPY(x))
5. (Conclusion) If the Dow-Jones Average crashes and the T-Bill interest rate rises, then any investor who is happy bought some gold stock.
   ( DJCRASH &and TBRISE ) &rarr &forall x (INVESTOR(x) &and HAPPY(x) &rarr &exist y (GOLD(y) &and BUY(x,y)))

10. Consider the following axioms:

1. Every child loves every candy.
   &forall x &forall y (CHILD(x) &and CANDY(y) &rarr LOVES(x,y))
2. Anyone who loves some candy is not a nutrition fanatic.
   &forall x ( (&exist y (CANDY(y) &and LOVES(x,y))) &rarr &not FANATIC(x))
3. Anyone who eats any pumpkin is a nutrition fanatic.
   &forall x ((&exist y (PUMPKIN(y) &and EAT(x,y))) &rarr FANATIC(x))
4. Anyone who buys any pumpkin either carves it or eats it.
   &forall x &forall y (PUMPKIN(y) &and BUY(x,y) &rarr CARVE(x,y) &or EAT(x,y))
5. John buys a pumpkin.
   &exist x (PUMPKIN(x) &and BUY(John,x))
6. Lifesavers is a candy.
   CANDY(Lifesavers)
7. (Conclusion) If John is a child, then John carves some pumpkin.
   CHILD(John) &rarr &exist x (PUMPKIN(x) &and CARVE(John,x))