This is a local copy of this page (Gordon Novak, UT Austin). It is provided because the original has non-standard HTML for logic connectors, which do not display properly.
Resolution Exercise Solutions
2. Consider the following axioms:
- Every child loves Santa.
∀ x (CHILD(x) → LOVES(x,Santa))
- Everyone who loves Santa loves any reindeer.
∀ x (LOVES(x,Santa) → ∀ y (REINDEER(y)
→ LOVES(x,y)))
- Rudolph is a reindeer, and Rudolph has a red nose.
REINDEER(Rudolph) ∧ REDNOSE(Rudolph)
- Anything which has a red nose is weird or is a clown.
∀ x (REDNOSE(x) → WEIRD(x) ∨ CLOWN(x))
- No reindeer is a clown.
¬ ∃ x (REINDEER(x) ∧ CLOWN(x))
- Scrooge does not love anything which is weird.
∀ x (WEIRD(x) → ¬ LOVES(Scrooge,x))
- (Conclusion) Scrooge is not a child.
¬ CHILD(Scrooge)
3. Consider the following axioms:
- 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))))
- Every dog chases some rabbit.
∀ x (DOG(x) → ∃ y (RABBIT(y) ∧ CHASE(x,y)))
- Mary buys carrots by the bushel.
BUY(Mary)
- 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)))
- John owns a dog.
∃ x (DOG(x) ∧ OWNS(John,x))
- 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))
- (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:
- Every Austinite who is not conservative loves some armadillo.
∀ x (AUSTINITE(x) ∧ ¬ CONSERVATIVE(x) →
∃ y (ARMADILLO(y) ∧ LOVES(x,y)))
- Anyone who wears maroon-and-white shirts is an Aggie.
∀ x (WEARS(x) → AGGIE(x))
- Every Aggie loves every dog.
∀ x (AGGIE(x) → ∀ y (DOG(y) → LOVES(x,y)))
- Nobody who loves every dog loves any armadillo.
¬ ∃ x ((∀ y (DOG(y) → LOVES(x,y))) ∧
∃ z (ARMADILLO(z) ∧ LOVES(x,z)))
- Clem is an Austinite, and Clem wears maroon-and-white shirts.
AUSTINITE(Clem) ∧ WEARS(Clem)
- (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:
- Anyone whom Mary loves is a football star.
∀ x (LOVES(Mary,x) → STAR(x))
- Any student who does not pass does not play.
∀ x (STUDENT(x) ∧ ¬ PASS(x) → ¬ PLAY(x))
- John is a student.
STUDENT(John)
- Any student who does not study does not pass.
∀ x (STUDENT(x) ∧ ¬ STUDY(x) → ¬ PASS(x))
- Anyone who does not play is not a football star.
∀ x (¬ PLAY(x) → ¬ STAR(x))
- (Conclusion) If John does not study, then Mary does not love John.
¬ STUDY(John) → ¬ LOVES(Mary,John)
6. Consider the following axioms:
- Every coyote chases some roadrunner.
∀ x (COYOTE(x) → ∃ y (RR(y) ∧ CHASE(x,y)))
- Every roadrunner who says ``beep-beep'' is smart.
∀ x (RR(x) ∧ BEEP(x) → SMART(x))
- No coyote catches any smart roadrunner.
¬ ∃ x ∃ y (COYOTE(x) ∧ RR(y) ∧ SMART(y) ∧
CATCH(x,y))
- 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))
- (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:
- Anyone who rides any Harley is a rough character.
∀ x ((∃ y (HARLEY(y) ∧ RIDES(x,y))) → ROUGH(x))
- Every biker rides [something that is] either a Harley or a BMW.
∀ x (BIKER(x) → ∃ y ((HARLEY(y) ∨ BMW(y)) ∧
RIDES(x,y)))
- Anyone who rides any BMW is a yuppie.
∀ x ∀ y (RIDES(x,y) ∧ BMW(y) → YUPPIE(x))
- Every yuppie is a lawyer.
∀ x (YUPPIE(x) → LAWYER(x))
- Any nice girl does not date anyone who is a rough character.
∀ x ∀ y (NICE(x) ∧ ROUGH(y) → ¬ DATE(x,y))
- Mary is a nice girl, and John is a biker.
NICE(Mary) ∧ BIKER(John)
- (Conclusion) If John is not a lawyer, then Mary does not date John.
¬ LAWYER(John) → ¬ DATE(Mary,John)
8. Consider the following axioms:
- Every child loves anyone who gives the child any present.
∀ x ∀ y ∀ z (CHILD(x) ∧ PRESENT(y) ∧
GIVE(z,y,x) → LOVES(x,z)
- 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)))
- It is foggy on Christmas eve.
FOGGY(Christmas)
- 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)))
- Any reindeer with a red nose is a source of light.
∀ x (RNR(x) → LIGHT(x))
- (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:
- Every investor bought [something that is] stocks or bonds.
∀ x (INVESTOR(x) → ∃ y ((STOCK(y) ∨ BOND(y))
∧ BUY(x,y)))
- If the Dow-Jones Average crashes, then all stocks that are
not gold stocks fall.
DJCRASH → ∀ x ((STOCK(x) ∧ ¬ GOLD(x)) →
FALL(x))
- If the T-Bill interest rate rises, then all bonds fall.
TBRISE → ∀ x (BOND(x) → FALL(x))
- Every investor who bought something that falls is not happy.
∀ x ∀ y (INVESTOR(x) ∧ BUY(x,y) ∧ FALL(y)
→ ¬ HAPPY(x))
- (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:
- Every child loves every candy.
∀ x ∀ y (CHILD(x) ∧ CANDY(y) → LOVES(x,y))
- Anyone who loves some candy is not a nutrition fanatic.
∀ x ( (∃ y (CANDY(y) ∧ LOVES(x,y))) →
¬ FANATIC(x))
- Anyone who eats any pumpkin is a nutrition fanatic.
∀ x ((∃ y (PUMPKIN(y) ∧ EAT(x,y))) → FANATIC(x))
- Anyone who buys any pumpkin either carves it or eats it.
∀ x ∀ y (PUMPKIN(y) ∧ BUY(x,y) →
CARVE(x,y) ∨ EAT(x,y))
- John buys a pumpkin.
∃ x (PUMPKIN(x) ∧ BUY(John,x))
- Lifesavers is a candy.
CANDY(Lifesavers)
- (Conclusion) If John is a child, then John carves some pumpkin.
CHILD(John) → ∃ x (PUMPKIN(x) ∧ CARVE(John,x))
Gordon S. Novak Jr.