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.
    ¬ &exist x (REINDEER(x) &and CLOWN(x))

  6. Scrooge does not love anything which is weird.
    &forall x (WEIRD(x) &rarr ¬ LOVES(Scrooge,x))

  7. (Conclusion) Scrooge is not a child.
    ¬ 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.

  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 ¬ DATE(x,y))

  7. (Conclusion) If Mary does not own a grocery store, she will not date John.
    (( ¬ &exist x (GROCERY(x) &and OWN(Mary,x))) &rarr ¬ DATE(Mary,John))

4. Consider the following axioms:

  1. Every Austinite who is not conservative loves some armadillo.
    &forall x (AUSTINITE(x) &and ¬ 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.
    ¬ &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 ¬ PASS(x) &rarr ¬ PLAY(x))

  3. John is a student.

  4. Any student who does not study does not pass.
    &forall x (STUDENT(x) &and ¬ STUDY(x) &rarr ¬ PASS(x))

  5. Anyone who does not play is not a football star.
    &forall x (¬ PLAY(x) &rarr ¬ STAR(x))

  6. (Conclusion) If John does not study, then Mary does not love John.
    ¬ STUDY(John) &rarr ¬ 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.
    ¬ &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 ¬ 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 ¬ 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.
    ¬ LAWYER(John) &rarr ¬ 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.

  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 ¬ 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 ¬ 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 ¬ 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.

  7. (Conclusion) If John is a child, then John carves some pumpkin.
    CHILD(John) &rarr &exist x (PUMPKIN(x) &and CARVE(John,x))

Gordon S. Novak Jr.