Resolution Exercise Solutions

2. Consider the following axioms:

  1. Every child loves Santa.
    ∀ x (CHILD(x) → LOVES(x,Santa))

  2. Everyone who loves Santa loves any reindeer.
    ∀ x (LOVES(x,Santa) → ∀ y (REINDEER(y) → LOVES(x,y)))

  3. Rudolph is a reindeer, and Rudolph has a red nose.
    REINDEER(Rudolph) ∧ REDNOSE(Rudolph)

  4. Anything which has a red nose is weird or is a clown.
    ∀ x (REDNOSE(x) → WEIRD(x) ∨ CLOWN(x))

  5. No reindeer is a clown.
    ¬ ∃ x (REINDEER(x) ∧ CLOWN(x))

  6. Scrooge does not love anything which is weird.
    ∀ x (WEIRD(x) → ¬ 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.
    ∀ x (BUY(x) → ∃ y (OWNS(x,y) ∧ (RABBIT(y) ∨ GROCERY(y))))

  2. Every dog chases some rabbit.
    ∀ x (DOG(x) → ∃ y (RABBIT(y) ∧ CHASE(x,y)))

  3. Mary buys carrots by the bushel.
    BUY(Mary)

  4. Anyone who owns a rabbit hates anything that chases any rabbit.
    ∀ x ∀ y (OWNS(x,y) ∧ RABBIT(y) → ∀ z ∀ w (RABBIT(w) ∧ CHASE(z,w) → HATES(x,z)))

  5. John owns a dog.
    ∃ x (DOG(x) ∧ OWNS(John,x))

  6. Someone who hates something owned by another person will not date that person.
    ∀ x ∀ y ∀ z (OWNS(y,z) ∧ HATES(x,z) → ¬ DATE(x,y))

  7. (Conclusion) If Mary does not own a grocery store, she will not date John.
    (( ¬ ∃ x (GROCERY(x) ∧ OWN(Mary,x))) → ¬ DATE(Mary,John))

4. Consider the following axioms:

  1. Every Austinite who is not conservative loves some armadillo.
    ∀ x (AUSTINITE(x) ∧ ¬ CONSERVATIVE(x) → ∃ y (ARMADILLO(y) ∧ LOVES(x,y)))

  2. Anyone who wears maroon-and-white shirts is an Aggie.
    ∀ x (WEARS(x) → AGGIE(x))

  3. Every Aggie loves every dog.
    ∀ x (AGGIE(x) → ∀ y (DOG(y) → LOVES(x,y)))

  4. Nobody who loves every dog loves any armadillo.
    ¬ ∃ x ((∀ y (DOG(y) → LOVES(x,y))) ∧ ∃ z (ARMADILLO(z) ∧ LOVES(x,z)))

  5. Clem is an Austinite, and Clem wears maroon-and-white shirts.
    AUSTINITE(Clem) ∧ WEARS(Clem)

  6. (Conclusion) Is there a conservative Austinite?
    ∃ x (AUSTINITE(x) ∧ 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.
    ∀ x (LOVES(Mary,x) → STAR(x))

  2. Any student who does not pass does not play.
    ∀ x (STUDENT(x) ∧ ¬ PASS(x) → ¬ PLAY(x))

  3. John is a student.
    STUDENT(John)

  4. Any student who does not study does not pass.
    ∀ x (STUDENT(x) ∧ ¬ STUDY(x) → ¬ PASS(x))

  5. Anyone who does not play is not a football star.
    ∀ x (¬ PLAY(x) → ¬ STAR(x))

  6. (Conclusion) If John does not study, then Mary does not love John.
    ¬ STUDY(John) → ¬ LOVES(Mary,John)

6. Consider the following axioms:

  1. Every coyote chases some roadrunner.
    ∀ x (COYOTE(x) → ∃ y (RR(y) ∧ CHASE(x,y)))

  2. Every roadrunner who says ``beep-beep'' is smart.
    ∀ x (RR(x) ∧ BEEP(x) → SMART(x))

  3. No coyote catches any smart roadrunner.
    ¬ ∃ x ∃ y (COYOTE(x) ∧ RR(y) ∧ SMART(y) ∧ CATCH(x,y))

  4. Any coyote who chases some roadrunner but does not catch it is frustrated.
    ∀ x (COYOTE(x) ∧ ∃ y (RR(y) ∧ CHASE(x,y) ∧ ¬ CATCH(x,y)) → FRUSTRATED(x))

  5. (Conclusion) If all roadrunners say ``beep-beep'', then all coyotes are frustrated.
    (∀ x (RR(x) → BEEP(x)) → (∀ y (COYOTE(y) → 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.
    ∀ x ((∃ y (HARLEY(y) ∧ RIDES(x,y))) → ROUGH(x))

  2. Every biker rides [something that is] either a Harley or a BMW.
    ∀ x (BIKER(x) → ∃ y ((HARLEY(y) ∨ BMW(y)) ∧ RIDES(x,y)))

  3. Anyone who rides any BMW is a yuppie.
    ∀ x ∀ y (RIDES(x,y) ∧ BMW(y) → YUPPIE(x))

  4. Every yuppie is a lawyer.
    ∀ x (YUPPIE(x) → LAWYER(x))

  5. Any nice girl does not date anyone who is a rough character.
    ∀ x ∀ y (NICE(x) ∧ ROUGH(y) → ¬ DATE(x,y))

  6. Mary is a nice girl, and John is a biker.
    NICE(Mary) ∧ BIKER(John)

  7. (Conclusion) If John is not a lawyer, then Mary does not date John.
    ¬ LAWYER(John) → ¬ DATE(Mary,John)

8. Consider the following axioms:

  1. Every child loves anyone who gives the child any present.
    ∀ x ∀ y ∀ z (CHILD(x) ∧ PRESENT(y) ∧ GIVE(z,y,x) → LOVES(x,z)

  2. Every child will be given some present by Santa if Santa can travel on Christmas eve.
    TRAVEL(Santa,Christmas) → ∀ x (CHILD(x) → ∃ y (PRESENT(y) ∧ 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.
    ∀ x ∀ t (FOGGY(t) → ( ∃ y (LIGHT(y) ∧ HAS(x,y)) → TRAVEL(x,t)))

  5. Any reindeer with a red nose is a source of light.
    ∀ x (RNR(x) → LIGHT(x))

  6. (Conclusion) If Santa has some reindeer with a red nose, then every child loves Santa.
    ( ∃ x (RNR(x) ∧ HAS(Santa,x))) → ∀ y (CHILD(y) → LOVES(y,Santa))

9. Consider the following axioms:

  1. Every investor bought [something that is] stocks or bonds.
    ∀ x (INVESTOR(x) → ∃ y ((STOCK(y) ∨ BOND(y)) ∧ BUY(x,y)))

  2. If the Dow-Jones Average crashes, then all stocks that are not gold stocks fall.
    DJCRASH → ∀ x ((STOCK(x) ∧ ¬ GOLD(x)) → FALL(x))

  3. If the T-Bill interest rate rises, then all bonds fall.
    TBRISE → ∀ x (BOND(x) → FALL(x))

  4. Every investor who bought something that falls is not happy.
    ∀ x ∀ y (INVESTOR(x) ∧ BUY(x,y) ∧ FALL(y) &rarrm; ¬ 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 ∧ TBRISE ) → ∀ x (INVESTOR(x) ∧ HAPPY(x) → ∃ y (GOLD(y) ∧ BUY(x,y)))

10. Consider the following axioms:

  1. Every child loves every candy.
    ∀ x ∀ y (CHILD(x) ∧ CANDY(y) → LOVES(x,y))

  2. Anyone who loves some candy is not a nutrition fanatic.
    ∀ x ( (∃ y (CANDY(y) ∧ LOVES(x,y))) → ¬ FANATIC(x))

  3. Anyone who eats any pumpkin is a nutrition fanatic.
    ∀ x ((∃ y (PUMPKIN(y) ∧ EAT(x,y))) → FANATIC(x))

  4. Anyone who buys any pumpkin either carves it or eats it.
    ∀ x ∀ y (PUMPKIN(y) ∧ BUY(x,y) → CARVE(x,y) ∨ EAT(x,y))

  5. John buys a pumpkin.
    ∃ x (PUMPKIN(x) ∧ BUY(John,x))

  6. Lifesavers is a candy.
    CANDY(Lifesavers)

  7. (Conclusion) If John is a child, then John carves some pumpkin.
    CHILD(John) → ∃ x (PUMPKIN(x) ∧ CARVE(John,x))

Gordon S. Novak Jr.